Fourth Grade Mathematics
Incorporated Throughout the Year
Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reason of others.
Model with mathematics
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Grading Period 3
Number and Operations- Fractions
Extend understanding of fraction equivalence and ordering.
1.
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts
differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
2.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a
benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of
comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3.
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation.
Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8
+ 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using
properties of operations and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual
fraction models and equations to represent the problem.
4.
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the
Columbus City Schools
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Fourth Grade Mathematics
conclusion by the equation 5/4 = 5 × (1/4).
Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual
fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the
problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of
roast beef will be needed? Between what
two whole numbers does your answer lie?
b.
Understand decimal notation for fractions, and compare decimal fractions.
5.
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective
denominators 10 and 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
6.
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a
number line diagram.
7.
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same
whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Measurement and Data
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
1.
Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of
measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know
that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1,
12), (2, 24), (3, 36), ...
2.
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems
involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent
measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
3
-Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
-Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not
a requirement at this grade.
4
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What Can I Do At Home?
Grade 4
Third Grading Period
During the second grading period, in math,
your child will be expected to:
Number and Operations-Fractions
 Use pictures and models to explain why
two fractions are equivalent.
 Generate equivalent fractions.
 Compare two fractions. The fractions can
have different numerators and
denominators.
 Add and subtract fractions with the same
denominators.
 Add and subtract mixed numbers that have
the same denominators.
 Solve word problems involving adding and
subtracting fractions with like
denominators.
 Multiply a fraction by a whole number.
 Solve word problems involving multiplying
a fraction by a whole number.
 Express fractions with a denominator of 10
or 100 as a decimal.
 Compare two decimals to hundredths. Use
the symbols <, > or =.
.
Measurement and Data
 Within a unit of measurement, express
larger units of measurement in terms of a
smaller unit (e.g. meters to centimeters).
Here’s what you can do to help your child
master these skills:
 Cut an 8 ½ x 11 sheet of paper into 4 identical
rectangles. Use one rectangle as the whole. Fold
and cut the remaining rectangles so that one
represents halves, one represents fourths, and the
other represents eighths. Use these “fraction
strips” to talk about equivalent fractions with
your child.
 Use snacks to help your child compare fractions.
For example, why is ¼ of a granola bar a smaller
piece than ½ of a granola bar? Why is ¾ of a
banana more than 1/2?
 Ask your child problems about adding or
subtracting fractional parts. For example, we
ordered two pizzas, one sausage and one
pepperoni. Each pizza was cut into sixths. If you
ate 2 slices of the sausage pizza and 2 slices of
the pepperoni pizza, how much pizza did you eat?
(2/6 + 3/6 = 5/6)
 Look for examples of mixed numbers in your
home. For example, there may be 1 ½ bagels left
after breakfast or there might be 2 ¼ gallons of
milk in the refrigerator.
 Have you child help you bake. Ask them
questions like “If we put in 1/3 cups of flour five
times, how many 1/3 cups of flour will we use?”
 Help your child practice representing tenths and
hundredths. Draw a rectangle and divide it into
tenths. Shade in parts. Ask your child to write
the fraction and the decimal. Find a 100 grid
online and do the same thing with hundredths.
 Use pennies to help your child compare decimals.
Give them two different piles of pennies. Ask
them to represent the number of pennies in each
group as a fraction of a dollar. Then ask them to
write each fraction as a decimal. Have them
compare the two decimals.
 Measure different items in your house using a
big unit (e.g. feet, meters, etc.). Help your
child convert the measurement into smaller
units within the same system.
Mathematics Model Curriculum
Grade Level: Fourth Grade
Grading Period: 3
Common Core Domain
Time Range: 15 Days
Number and Operations - Fractions
Common Core Standards
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with
attention to how the number and size of the parts differ even though the two fractions themselves are the same
size. Use this principle to recognize and generate equivalent fractions.
2. Compare two fractions with different numerators and different denominators, e.g., by creating common
denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons
are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols
>, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Developing an understanding of fraction equivalence, addition and subtraction of fractions with like
denominators, and multiplication of fractions by whole numbers
Students develop understanding of fraction equivalence and operations with fractions. They recognize that two
different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing
equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions,
composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of
fractions and the meaning of multiplication to multiply a fraction by a whole number.
Content Elaborations
This section will address the depth of the standards that are being taught.
from ODE Model Curriculum
Ohio has chosen to support shared interpretation of the standards by linking the work of multistate partnerships as
the Mathematics Content Elaborations. Further clarification of the standards can be found through these reliable
organizations and their links:
Achieve the Core Modules, Resources
Hunt Institute Video examples
Institute for Mathematics and Education Learning Progressions Narratives
Illustrative Mathematics Sample tasks
National Council of Supervisors of Mathematics (NCSM) Resources, Lessons, Items
National Council of Teacher of Mathematics (NCTM) Resources, Lessons, Items
Partnership for Assessment of Readiness for College and Careers (PARCC) Resources, Items
Expectations for Learning (Tasks and Assessments)
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
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Mathematics Model Curriculum
Expectations for Learning (in development) from ODE Model Curriculum
Ohio has selected PARCC as the contractor for the development of the Next Generation Assessments for
Mathematics. PARCC is responsible for the development of the framework, blueprints, items, rubrics, and scoring
for the assessments. Further information can be found at Partnership for Assessment of Readiness for College and
Careers (PARCC). Specific information is located at these links:
Model Content Framework
Item Specifications/Evidence Tables
Sample Items
Calculator Usage
Accommodations
Reference Sheets
Sample assessment questions are included in this document.
The following website has problem of the month problems and tasks that can be used to assess students and help
guide your lessons.
http://www.noycefdn.org/resources.php
http://illustrativemathematics.org
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx
http://nrich.maths.org
At the end of this topic period students will demonstrate their understanding by….
Some examples include:
Constructed Response
Performance Tasks
Portfolios
***A student’s nine week grade should be based on multiple academic criteria (classroom assignments and teacher made
assessments).
Instructional Strategies
Columbus Curriculum Guide strategies for this topic are included in this document.
Also utilize model curriculum, Ohiorc.org, textbooks, InfoOhio, etc. as resources to provide instructional
strategies.
Websites
http://illuminations.nctm.org
http://illustrativemathematics.org
http://insidemathematics.org
http://www.lessonresearch.net/lessonplans1.html
Instructional Strategies from ODE Model Curriculum
Instructional Strategies
Students’ initial experience with fractions began in Grade 3. They used models such as number lines to locate unit
fractions, and fraction bars or strips, area or length models, and Venn diagrams to recognize and generate
equivalent fractions and make comparisons of fractions.
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Mathematics Model Curriculum
Students extend their understanding of unit fractions to compare two fractions with different numerators and
different denominators.
Students should use models to compare two fractions with different denominators by creating common
denominators or numerators. The models should be the same (both fractions shown using fraction bars or both
fractions using circular models) so that the models represent the same whole. The models should be represented in
drawings. Students should also use benchmark fractions such as ½ to compare two fractions. The result of the
comparisons should be recorded using <, >, and = symbols.
Instructional Resources/Tools from ODE Model Curriculum
Pattern blocks
Fraction bars or strips
Number lines
Misconceptions/Challenges
The following are some common misconceptions for this topic. As you teach the lessons, identify the misconceptions/challenges that
students have with the concepts being taught.
Students think that when generating equivalent fractions they need to multiply or divide either the numerator or
denominator, such as, changing ½ to sixths. They would multiply the denominator by 3 to get 6, instead of
multiplying the numerator by 3 also. Their focus is only on the multiple of the denominator, not the whole fraction.
Students need to use a fraction in the form of one such as 3/3 so that the numerator and denominator do not contain
the original numerator or denominator.
Please read the Teacher Introductions, included in this document, for further understanding.
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Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
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Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
Number of Groups Unknown (“How
many groups?” Division
Equal
Groups
3×6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Group Size Unknown
(“How many in each group?”
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Arrays, 4
Area, 5
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
a × ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? × b = p, and p ÷ b = ?
Unknown Product
Compare
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
a×b=?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
Group Size Unknown: (a × ? = p and p ÷ a = ?)
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In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
students used math tools to solve the problem, they should include a picture to represent how the
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tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
Act It Out
Make a Table
Find a Pattern
Guess and Check
Make a List
Draw a Picture
When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
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to act it out. Occasionally whole group thinking with the teacher modeling how to record
strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Teacher Introduction
Fractions
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Number and Operations-Fractions domain is helpful when assessing
student work and student thinking. This information could be used to guide classroom
discussions, understand student misconceptions and provide differentiation opportunities. The
focus of instruction is the Common Core Mathematics Standards.
Fractions are a natural part of the constructed knowledge that students develop on their own
before they come to school. From the time they are young, students have had multiple
experiences with dividing and partitioning objects and sets of objects into equal and/or fair
shares.
“There are eight cookies and six people. How can we share them?”
“I have ten pieces of candy. How can I divide them between my friend and myself so we
both get the same amount?”
For students to become mathematically powerful in their use of fractions, we must make
connections to the personal contexts which students have developed outside of school. Students
may approach fraction problems differently and divergent thinking should be valued and
encouraged. Students are encouraged to “make sense” of the instructed fraction curriculum at
school through the constructed knowledge they bring from home.
The Common Core State Standards for third grade focus on students’ understanding of the
meaning of fractions as the partitioning of shapes into equal parts, understanding fractions as
number and placing them on a number line, reasoning about equivalent fractions, and comparing
fractions. Third graders should develop a firm understanding of unit fractions (fractions with a
numerator of 1) so that they can then build fractions from these unit fractions. (For example,
3
1
can be seen as having 3 of the ’s together.) On the number line, students must understand
5
5
that the whole is the unit interval between two whole numbers, often 0 to 1. The Common Core
State Standards encourage students to think of a number line as an infinite ruler. Through
students’ work with fractions on a number line, they will notice that some fractions can be placed
1
5
on the same point on a number line (i.e.
and
) and are therefore equal. This early work
2
10
with equivalent fractions will help to prepare them for more in depth work in fourth grade.
Finally, third graders will compare fractions with like denominators or like numerators. Students
should understand that for unit fractions, the fraction with the larger denominator is smaller since
the whole has been partitioned into more, and therefore smaller, pieces. Third graders should
recognize that comparisons of fractions are only valid when the two fractions refer to the same
whole.
Fractions are relational numbers that express the relationship between a part and a whole
quantity. Fractions exist when a whole, which could be either one whole object or a set of many
objects, has been partitioned or broken into equal or fair shares. Students need practice
partitioning, dividing, or sharing an object or a group of objects into equal parts. Two equal
parts are called halves, three equal parts are called thirds, four equal parts are called fourths, etc.
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Students need to be able to give the name of the parts into which the whole has been divided and
also know the name of just one part of the whole.
Example:
This square is divided into fourths. Each piece is called one-fourth of the whole.
This circle is divided into thirds. Each piece is called one-third of the whole.
This group of objects has been divided into two equal groups or halves. Each group is one-half
of the whole group of objects.
Fraction symbols are introduced after students have had ample opportunity to partition, identify,
and name fractions. Students need a firm understanding of the difference between the numerator
and the denominator and will correctly be able to name any fraction when that distinction is clear
in their minds.
Numerator
(Top Number of the Fraction)
The numerator refers to the parts of the whole
that are counted. It names the count and tells
how many. This is the counting number.
Denominator
(Bottom Number of the Fraction)
The denominator refers to the whole. It
represents either how many equal pieces the
one whole object has been divided into or how
many total objects make up the whole set of
objects.
Counting fractional parts helps students develop the language of numerator and denominator.
For example, students can count out a set of halves using pattern blocks. If the whole used is the
hexagon, then students first need to partition the whole block (hexagon) into two equal blocks to
create halves. What shape will divide the hexagon block into 2 parts or halves? (trapezoid)
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
1
2
2
2
3
2
1
5
(five-halves) or 2
(two and one-half)
2
2
(one-half)
(two-halves) or
(three-halves) or 1
1
2
6
(six-halves) or 3 (three)
2
1 (one)
1
7
(seven-halves) or 3
(three and one-half)
2
2
(one and one-half)
4
(four-halves) or 2 (two)
2
8
(eight-halves) or 4 (four)
2
Students start by selecting one trapezoid and naming it one-half and then adding another
trapezoid and naming it two-halves. Continue adding trapezoids one at a time until students
understand that the denominator does not change because the part of the whole is not changing.
What does change is the numerator or the number of pieces that are being counted. Giving the
total number of pieces shows students the counting sequence and solidifies the difference
between the numerator and the denominator.
Grade four expectations in this domain are limited to fractions with denominators 3, 3, 4, 5,
6, 8, 10, 12, and 100.
Using pattern blocks also gives students the opportunity to use the models to make the
connection between the improper fractions created and the corresponding mixed number. They
can see that two-halves is the same as one whole hexagon and that three-halves is the same as
one whole hexagon and one-half of another or 1
1
2
hexagons. Seven halves make 3 whole
hexagons and one-half of another, so students can see that
7
2
is the same as 3
1
2
hexagons.
Count a second time using halves and show how the improper fractions can also be named using
1
mixed numbers (e.g., 1 , 2 or 1, 3 or 1 , 4 or 2, etc.).
2
2
2
2
2
When students have various experiences counting fractional parts beyond one whole using
physical models the symbols for showing mixed numbers become easier to understand. Give
students practice counting by dividing the hexagon into other fractional parts (make thirds with
the blue rhombi or sixths with triangles). If the hexagon is partitioned into 6 equal parts using
the triangles, then when given 13 triangles students can name them thirteen-sixths or they can see
that two whole hexagons can be formed with one triangle left so
13
6
2
1
6
or two whole objects
with one-sixth of another whole object left over.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Fraction Models
Students need exposure and concrete experience with the three different types of fraction
models: region/area, set, and length/measurement.
Region/Area Models
Region/area models divide a region/area into smaller, equal parts and compare the parts with the
whole. Examples include geometric models, pattern blocks, geoboards, and paper folding.
Model
Fraction
5 parts being counted (shaded)
3 fractional parts total in the whole
Mixed Number = 1
2
3
4 parts being counted (shaded)
6 fractional parts total in the whole
5 parts being counted (shaded)
2 fractional parts total in the whole
Mixed Number = 2
1
2
Set Models
Set models use an entire set of objects to equal the whole, while the individual objects make up
the fractional parts. Set models allow students to make real world connections with fractions,
but are easily overlooked in teaching fractions. Teach students how to divide sets of objects into
groups the same way they would set up a multiplication array. The number of groups is made
vertically and then the rest of the objects are filled into the groups until the objects are equally
distributed. For example, in the first model shown below, the students would take the squares
and make four groups vertically with the first four squares. The rest of the squares are then
equally distributed into the four groups that have been created. The number of objects in each
group is found by counting horizontally across one row.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Model
1
4
of 8 = 2
1
4
1
3
of 12 = 4
Put 12 objects into 3 equal groups. Count how
many circles are in one-third (4).
4
12
of 10 = 6
3
5
1
2
Put 8 objects into 4 equal groups. Count how
many squares are in each group (2).
2
8
1
3
3
5
Fraction
Put 10 objects into 5 equal groups. Count how
many triangles are in three of the groups to
find three-fifths (6).
6
10
of 12 = 6
12 + 6 = 18 donuts in a dozen and a half.
How many donuts are there in a dozen and a
half? Students know that there are 12 in one
whole dozen so to determine how many are in
one-half dozen put 12 objects into 2 equal
groups.
Students sometimes encounter difficulty in partitioning manipulatives into sets. For example,
divide 15 counters into thirds (three sets). Each part, or set, contains five counters, but the
number of shares shows thirds. Having students put out the number of sets needed vertically
first, and then filling in to determine the amount in each set should help to eliminate some of the
confusion about what fraction is shown and how many of the whole set are found in that
fractional part.
Length/Measurement Models
Length/measurement models are similar to region/area models, but instead compare lengths.
Number lines, rods, rulers, and strip models are examples.
Whole
Halves
Thirds
Fourths
Fourths
Eighths
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
0
8
1
8
inches
2
8
1
3
8
4
8
5
8
2
6
8
3
8
8
7
8
4
9
8
5
10
8
6
mm
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Students count the spaces (not the lines) between the numbers on the ruler to determine how
many parts the whole (each inch on the customary ruler or each centimeter on the metric ruler)
has been divided into. There are four spaces between the end of the ruler and 1 (and between 1
and 2, 2 and 3, etc.) on the ruler above so each line represents one-fourth of an inch. The first
line is
1
of an
4
inch, the second line is
2
1
or of an
4
2
inch, and the third line is
3
4
of an inch. When
using a ruler students are able to view mixed numbers in a practical context. They can see that
3
1
4
is 3 whole inches and 1 out of the 4 parts that make the next inch. So 3
1
4
is more than 3 but
less than 4 and is closer to 3 inches than it is to 4 inches.
Fraction Properties
Comparing fractions may sometimes be an overwhelming task for students. Building an
understanding of fraction properties will assist students with fraction comparisons. Many of the
difficulties students experience with fractions occur with the inverse relationships between the
number of parts and the size of the parts. In all other mathematical experiences larger numbers
indicate a larger quantity, or more. Students incorrectly transfer this concept to fractions:
“Four is more than three, so fourths should be greater than thirds.”
However, the larger the number of parts, the smaller the size of the parts because the whole
has been divided into more pieces so each piece is smaller. Students must construct this
relationship through repeated experiences with fraction models. Students should work with and
understand three different fraction comparison properties.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
1. More of the Same Size Parts
(same denominator, different numerator)
Example: Compare three-fourths and onefourth.
2. Same Number of Parts But Parts Are
Different Sizes
(same numerator, different denominator)
Example: Compare two-thirds and twofourths.
3. Different Number of Parts and Parts are
Different Sizes
(different numerator, different denominator)
Example: Compare one-fifth and two-thirds
Both fractions are fourths, so compare the
counting number (or numerator). Three is
greater than one, so three-fourths is greater
than one-fourth.
Both fractions have the same counting number
(or numerator). However, thirds are larger
than fourths (the greater the number of parts,
the smaller the size of the parts), so two-thirds
is greater than two-fourths.
Several strategies are listed in the table below
to help students compare fractions that have
different numerators and different
denominators.
When students are asked to compare fractions that contain different-sized numerators and
denominators, it is important for them to have experience with different methods that can help
them make the comparison. Students can use several methods to compare fractions.
Method
Example
Number Lines
5
1
and
8
4
Students divide two lines that are the same
length into the fractional parts of the two
fractions that are being compared. They then
look to see where the fractions are on the line
and make the comparison.
5 1
1 5
or
8 4
4 8
Example: Compare
Method
1
4
0 1
2
8
8
2
4
3
8
4
8
3
4
5
8
6
8
1
7
8
1
Example
Models (fraction bars)
3
7
Example: Compare and
4
10
Students divide two rectangles that are the
same size into the fractional parts of the
fractions that are being compared. They then
shade in the number of pieces in each rectangle
to represent the fractions and then make the
3 7
7 3
comparison.
or
4 10
10 4
Grade 4 Fraction Equivalence
and Ordering
0
Page 15 of 64
3
4
7
10
Columbus City Schools 2013-2014
Benchmarks of 0,
1
,
2
1
Half of 6 is 3 so
1
,
2
and it is two-sixth away from
or 1 and then
look at the second fraction and determine if it
is closest to 0,
1
, or 1. Students then use that
2
information to compare the fractions.
5
6
1
1
or
8
8
5
6
5
6
is only
1
.
2
5
is
6
1
1
1
or 0.
is less than
2
8
2
4
1
because half of 8 is 4 so is equal to , so
8
2
1
5
is closest to 0. Since is closer to 1 and
8
6
1
5
is closer to 0, then is the greater
8
6
closer to 1 than
fraction.
Equivalent Forms
Example: Compare
1
2
is equal to .
one-sixth away from one whole because
adding one more sixth will create a whole
5
1
and
6
8
Students look at the first fraction and
Example: Compare
determine if it is closest to 0,
3
6
2
5
and
3
12
Students look at the denominator that has the
larger number and see if that number can be
divided equally into the number of groups
represented by the denominator that is the
smaller number. If so, an equivalent fraction
can be found for the fraction with the
denominator that is the smaller number and
then the fractions with the same denominators
2 5
5 2
are compared.
or
3 12
12 3
12 can be divided equally into 3 groups.
There are 4 items in each group. So to find
2
of 12 you need to count the number of
3
items in 2 out of the 3 groups made from the
2 8
12 items or 4 + 4 = 8.
Now both
3 12
fractions have a denominator of 12 and the
numerators can be compared to determine
which fraction is larger. Eight is larger than
8
5
2 5
5 so
which means
.
12 12
3 12
The understanding of the relative size of fractions helps students compare fractions more easily.
It also teaches students to rely less on paper and pencil computational skills and more on number
sense, an important lifelong skill.
When students order mixed numbers they should use the same methods as when comparing only
fractions, however, they must first compare and order the whole numbers. If the whole numbers
are different, then the comparison can be done without looking at the fractions by finding the
greatest whole number and ordering based only on the whole number. If the whole numbers are
the same, then the student’s next step is to compare and order the fractional parts using methods
such as the ones mentioned above.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Kiara and Josie are sharing a pizza. Kiara ate
2
1
of the pizza while Josie ate .
8
2
Did they eat the same amount? Use words, numbers, and/or pictures to explain
your reasoning.
Using numbers, pictures, and/or words show one equivalent fraction for
Grade 4 Fraction Equivalence
and Ordering
Page 17 of 64
3
.
4
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Kiara and Josie are sharing a pizza. Kiara ate
2
1
of the pizza while Josie ate .
8
2
Did they eat the same amount? Use words, numbers, and/or pictures to explain
your reasoning.
Answer: Josie ate more because
1
2
is equal to
4
,
8
which is greater than
2
.
8
A 2-point response states that Josie ate more pizza than Kiara and shows all supporting
work
.
A 1-point response states that Josie ate more than Kiara, but gives a weak or no
explanation.
A 0-point response shows no mathematical understanding of the problem.
Using numbers, pictures, and/or words, show one equivalent fraction for
Answer: Answers will vary, but all answers need to be an equivalent form of
3
.
4
3
.
4
6
9
12
, or
( , or
, etc.)
8
12
16
A 2-point response includes a drawing, numbers, or words to show another fraction that is
3
equal to .
4
A 1-point response shows a minor error in computation or reasoning.
A 0-point response shows no mathematical understanding of the problem.
Grade 4 Fraction Equivalence
and Ordering
Page 18 of 64
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Which number sentence is true?
 A.
5
8
1
4
 B.
1
2
3
4
 C.
1
2
2
4
 D.
5
8
10
20
Draw two models to show that
Grade 4 Fraction Equivalence
and Ordering
1
2
is equal to .
4
8
Page 19 of 64
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which number sentence is true?
 A.
5
8
1
4
 B.
1
2
3
4
 C.
1
2
2
4
 D.
5
8
10
20
Answer: C
Draw two models to show that
1
2
is equal to .
4
8
For example:
Answer: Drawings will vary.
A 2-point response draws two congruent shapes that accurately show that
as
1
is the same
4
2
.
8
A 1-point response shows a flaw in reasoning.
A 0-point response shows no mathematical understanding of the task.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Put the following fractions in order from least to greatest.
7 1 2
, ,
12 6 3
 A.
7 1 2
, ,
12 6 3
 B.
1 7 2
,
,
6 12 3
 C.
2 1 7
, ,
3 6 12
 D.
1 2
7
,
,
6 3 12
Use the bars below to illustrate the following fractions:
3
8
,
1
4
,
1
2
. On the line
below order the fractions from least to greatest.
__________________________________________________________________
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Put the following fractions in order from least to greatest.
7 1 2
, ,
12 6 3
 A.
7 1 2
, ,
12 6 3
 B.
1 7 2
,
,
6 12 3
 C.
2 1 7
, ,
3 6 12
 D.
1 2
7
,
,
6 3 12
Answer: B
3 1 1
, . On the line below order the
8 4 2
Use the bars below to illustrate the following fractions: ,
fractions from least to greatest.
1 3 1
, ,
4 8 2
A 2-point response includes correct illustrations for all three fractions and orders the
fractions from least to greatest.
A 1-point response includes correct illustrations but does not order the fractions from least
to greatest.
A 0-point response includes only one correct fraction or shows no mathematical
understanding of this task.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Lisa has
2
3
of a Double Dew candy bar, Michael has
bar, and Lei has
1
2
of a Double Dew candy
5
of a Double Dew candy bar. Put them in order from who has
6
the largest piece of candy bar to who has the smallest. Explain your answer.
Place the two fractions on the number line in the appropriate places.
7
8
0
Grade 4 Fraction Equivalence
and Ordering
and
1
4
1
2
Page 23 of 64
1
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Lisa has
2
3
1
of a Double Dew candy bar, Michael has
bar, and Lei has
2
of a Double Dew candy
5
of a Double Dew candy bar. Put them in order from who has
6
the largest piece of candy bar to who has the smallest. Explain your answer.
Answer: Lei has the largest (
5
) followed by Lisa (
6
3
2
), and then Michael ( ).
6
3
A 2-point response correctly orders the fractions from biggest to smallest and includes a
complete explanation (students can draw pictures; compare the fractions to 0, ½, and 1; or
find a common denominator of 6).
A 1-point response correctly orders the fractions but includes a weak or no explanation or
includes a complete explanation with an error in the ordering of the fraction.
A 0-point response shows no understanding of this task.
Place the two fractions on the number line in the appropriate places.
7
Answer:
and
8
0
1
4
1
4
1
2
7
8
1
A 2-point response correctly places both fractions on the number line.
A 1-point response correctly places only one of the fractions on the line.
A 0-point response indicates no understanding of this task.
Grade 4 Fraction Equivalence
and Ordering
Page 24 of 64
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Which group of fractions is ordered correctly from smallest to largest?
 A.
1 1 1
,
,
2 3 4
 B.
1 1 1
,
,
8 5 4
 C.
1 1 1
,
,
6 8 3
 D.
2 2 2
, ,
3 5 7
Compare these fractions.
1
4
1
6
Which symbol below should be placed in the box to make a true statement?
 A. <
 B. =
 C. >

D. It is impossible to tell.
Grade 4 Fraction Equivalence
and Ordering
Page 25 of 64
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which group of fractions is ordered correctly from smallest to largest?
 A.
1 1 1
,
,
2 3 4
 B.
1 1 1
,
,
8 5 4
 C.
1 1 1
,
,
6 8 3
 D.
2 2 2
, ,
3 5 7
Answer: B
Compare these fractions.
1
4
1
6
Which symbol below should be placed in the box to make a true statement?
 A. <
 B. =
 C. >

D. It is impossible to tell.
Answer: C
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Which fraction is greater?
3
5
or
8
8
Explain your reasoning using words and pictures.
Doris found the following cards that fell off the number line.
1
2
2
3
1
3
2
8
Show how she should order them on the number line.
Grade 4 Fraction Equivalence
and Ordering
Page 27 of 64
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which fraction is greater?
3
5
or
8
8
Explain your reasoning using words and pictures.
Answer:
5
is larger.
8
A 2-point response states that
5
is larger and shows supporting work, using numbers and
8
pictures.
A 1-point response has the correct answer with no explanation or has a strong explanation
with a minor computational error.
A 0-point response shows no mathematical understanding of the problem.
Doris found the following cards that fell off the number line.
1
2
2
3
1
3
2
8
2
8
1
3
1
2
2
3
Show how she should order them on the number line.
Answer:
2 1 1 2
, , ,
8 3 2 3
Grade 4 Fraction Equivalence
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Columbus City Schools 2013-2014
A 2-point
andresponse
Ordering puts the cards in correct order from least to greatest on a
number line.
TEACHING STRATEGIES/ACTIVITIES
Vocabulary: numerator, denominator, equivalent, unit fraction, equivalent, common
denominator, common numerator, benchmark fractions, visual fraction models, greater
than, less than, equal to, equal parts, order, compare, least, greatest, set, whole, equivalent,
model, number line, proper fraction, improper fraction, array, rectangular array
1. The word problems below can be presented to students in a variety of ways. Some options
include: using the questions to create a choice-board, Math-O board, one problem per class
session, one problem each evening for homework, partner work or as an assessment question.
The strength of problem solving lies in the rich discussion afterward. As the school year
progresses students should be able to justify their own thinking as well as the thinking of
others. This can be done through comparing strategies, arguing another student’s solution
strategy or summarizing another student’s sharing.
1
1
Will ate of a pizza and Grace ate
of a different pizza. Will claims he ate more
4
2
pizza than Grace. Describe how Will’s claim could be true.
1
The shape below is of a larger shape. Draw what the shape may look like.
5
1
The shape below is of a larger shape. Draw what the shape may look like.
3
1
Herman wrote a fraction that was smaller than . What fraction could he have
4
written? Use an area model to justify your answer.
1
Joellen and Carl both shaded in
of their grids. Joellen used a 10 x 10 grid and Carl
4
used an 8 x 8 grid. How many squares did each student shade? Why did they each
shade different numbers?
Joellen
Carl
On the number line below fill in a fraction that will make this inequality true:
3
___ >
4
Ashley and Mary Kate had two identical cakes for their birthday party. After the
party, one-half of Ashley’s cake was left. Five-eighths of Mary Kate’s cake was left.
Which cake had more leftovers? Explain your answer.
7
The judges at the state fair pie baking contest have eaten
of the peach pie. They
12
3
have eaten of the apple pie. Which pie have they eaten more of? Use a number
4
line model to explain your answer.
5
1
Will
round to or 1 whole? Explain how you know.
8
2
Grade 4 Fraction Equivalence
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Parris estimated that he had less than half his homework completed. He had finished
7
of the page. Use a fraction bar model to convince him that his estimate is too
12
low.
At the class party, Jermaine had two slices of sausage pizza. Helena ate one slice of
cheese pizza. Describe a situation in which both students ate the same amount of
pizza.
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 4 on the right side of the page. Open “Unit 2
Framework Student Edition.” These tasks are from the Georgia Department of Education.
The tasks build on one another. You may not use all of the tasks. There are four different
types of tasks. Scaffolding tasks build up to the constructing tasks which develop a deep
understanding of the concept. Next, there are practice tasks and finally performance tasks
which are a summative assessment for the unit. Select the tasks that best fit with your
lessons and the standards being taught at that time.
3. Read Jump, Kangaroo, Jump by Stuart J. Murphy. Each time the group is split into teams
discuss how many teams, how many are on each team, and what fraction of the total number
does each team represent? Distribute 16 color tiles. Have students divide the tiles into 2
groups. What fraction represents each group? Divide the tiles into 4 equal groups. What
fraction represents each group? What fraction represents 3 groups? If I add 4 more tiles to
my total group, what changes when you divide the total number of tiles into 2 groups and 4
groups? Have students count out 8 tiles. Tell them that the eight tiles represent the whole
set. Ask how many tiles are in one-fourth of this set. How many are in three-fourths of the
set? Have students count out 6 green tiles. If these 6 tiles represent one-half of a whole set,
how many tiles would there be in a whole set? If they represent one-fourth of a set, how
many tiles would there be in a whole set?
4. Distribute 20 color tiles to each student. Ask students to divide the pile into equal groups.
Students will divide tiles into different equal groups. Discuss the different groups students
came up with and the fractional parts of each group. Have all students divide their pile into
two equal groups, discuss how many are in each group, and write that amount as a fraction.
4 equal groups? 5 equal groups? Now divide students into pairs and have them push all the
tiles together. Ask each student to pull out an even number of color tiles other than 20.
Have students divide that into two equal groups. Discuss all the fractions that represent onehalf. Write the fractions on the board and discuss the number relationships between the
numerator and denominator (e.g.,
5 7 11
, ,
10 14 22
, etc). Continue asking students to divide the set
into four equal parts or three equal parts. This leads into a discussion of how factors of
numbers help you know how many equal sets a number can easily divide into and what
happens when a set does not divide evenly. Have students pull 1 green tile, 4 red tiles, and 3
yellow tiles. Distribute the “Fraction Parts” sheet (included in this Curriculum Guide) and
have the students place the color tiles in the first rectangle. Discuss which color is
1
1
3
represented by (green), (red), and (yellow). Students could place the different colors
2
8
8
in different places on the grid and may not keep all colors together. Discuss with students
that it does not matter if the colors are together; the fractions that represent each color are
still the same. Have students use color tiles to show fractional parts of each region and then
color each section as the tiles are removed. When discussing the fractional parts of each
model, assist students in thinking about each model that is already divided into equal parts to
think about other ways to equally divide that same model (e.g., if a model is divided into 8
Grade 4 Fraction Equivalence
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Columbus City Schools 2013-2014
equal parts and you want to shade
5.
1
red,
4
how could you divide the same model into 4 equal
parts? How many squares would be in four equal parts?).
Read The Doorbell Rang by Pat Hutchins. As you read the book, have students model
dividing the dozen cookies into halves, thirds, fourths, sixths, and twelfths by giving each
student 12 centimeter cubes (color tiles, two-color counters, or pieces of paper would also
work). Develop the concept of equivalent fractions using centimeter cubes. The story begins
with two students sharing 12 cookies. Ask students how they could show one-half with the
pieces. Model one-half on the overhead by dividing the 12 cubes into two equal piles.
Because there are 6 cubes in each pile students should determine that
1
2
6
.
12
Continue
reading the story and determining equivalent fractions of thirds, fourths, sixths and twelfths.
On the overhead show the question with eighths on a number line. Ask students how many
cubes they would need to be able to solve this problem. Give students time to solve the
problem and then ask for a volunteer to come to the overhead and share his or her solution.
Break students into pairs and have each pair of students complete one “Fractions of a Set”
worksheet (included in this Curriculum Guide). Have students share their answers with the
rest of the group when they are finished.
6. Write the fraction
4
12
on an overhead. Discuss what this fraction represents. Draw an
illustration of this fraction using a rectangle model. Which fraction with a denominator of 12
represents one-half? Which fraction with a denominator of 12 represents one? Is this
fraction closer to zero, one-half, or one? Discuss how you would justify a fraction with a
denominator of 12 is closer to one-half (e.g.,
one-half but
4
12
6
12
represents one-half and
4
12
is
2
12
away from
away from 0). Continue writing other fractions and discussing and justifying
why the fractions are closer to zero, one-half, or one. Divide the students into
pairs. Give each pair a set of “Fraction Cards Part 1”and the “Fraction Sort” sheet (included
in this Curriculum Guide). Have the students sort the fractions into three groups: those close
to zero, close to
1
,
2
and close to one. Record the fractions on the record sheet. Discuss the
fractions they sorted for each group. Select a few of the fractions that were sorted and have
students give their justification as to why the fraction was sorted in that way.
7. Divide students into pairs. Give each pair of students 36 centimeter cubes (counters, color
tiles, paperclips, etc. will also work). Direct student pairs to make a row of 12 centimeter
cubes and divide the row into thirds (three groups of four). Have them make a second row of
12 centimeter cubes below the first, but this time divide it into fourths (four groups of three).
Students make a third row of 12 below the other two and divide it into sixths (six groups of
two). Ask the students how many total centimeter cubes are in each row. They should see
that each of the rows has a total of 12 cubes. This is the size of your whole unit. Compare
sizes. Which group is smaller, the group that is one fourth of twelve, or the group that is one
sixth of twelve? Students will see that the group that is a sixth is smaller because there are
fewer centimeter cubes in that group. Then ask which group is smaller, the group that has
one fourth of twelve or a group that has one third of twelve? (fourths, because it has fewer
objects) Now represent in written form what you know,
1
6
1
4
1
.
3
Ask students if they see a
pattern? Students should determine that as the denominator gets smaller, the size of the
group gets larger so if the numerator of a group of fractions is the same they can be compared
by looking at their denominators. The larger the denominator, the smaller the pieces of the
Grade 4 Fraction Equivalence
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Columbus City Schools 2013-2014
whole or the part of the group. The smaller the denominator, the larger the pieces of the
whole or part of the group. Use centimeter cubes to compare one-half of 12 to the groups
that have already been made. If you divide the group of twelve centimeter cubes into halves
will the group be smaller or larger than the thirds? (larger) What about a group of twelfths?
Will the group be larger or smaller? (smaller) Reinforce that as the denominator gets larger,
the size of the group or the part of the whole gets smaller. Extend this activity by having
students compare fractions with numerators that are greater than one. For example, if you
had two groups of twelve centimeter cubes and you were comparing
2
3
with
1
2
then two-
thirds is two groups of 4, or 8 centimeter cubes, and one-half is 6 centimeter cubes; therefore,
2
3
>
1
.
2
8. Distribute Cuisenaire® Rods. Ask students for any connections they notice between rod
colors. (e.g., 2 light greens = 1 blue, etc.) Have students take turns placing rods on document
camera and modeling different fraction equivalence for class. Discuss with the class how
they could represent each part using fraction number sentences (e.g., 2 purple rods = one
brown rod so
1
2
+
1
= 1.
2
Distribute “Exploring Fractions with Rods I” (included in this
Curriculum Guide) Cuisenaire® Rods for students to explore equivalence for the brown and
orange rods.
9. Once students have completed the “Exploring Fractions with Cuisenaire® Rods I” worksheet
from the previous activity. Distribute “Fraction Number Line” (included in this Curriculum
Guide) and Cuisenaire® Rods. Discuss with students where the endpoints of 0 and 1 whole
should be placed on a blank number line template. It is important for students to understand
that numbers on the left of a number on a number line are decreasing and numbers to the
right of a number on the number line are increasing. Next, have students place all brown
train equivalent fractions on the number line: 0,
2 1 4 2 1 6 3 8
, , , , , , ,
8 4 8 4 2 8 4 8
,
4 2
, ,
4 2
and 1
making sure students are paying attention to proper spacing of each fraction. Give students
the following problem to solve using the number line: “Ciara was making cookies. She
needed
3
cup
4
of sugar,
1
cup
2
of butter and
2
cup
8
of flour. Did Ciara use more flour or more
sugar to make her cookies? Explain your answer.” Have students write another story problem
that can be solved using the brown equivalent fraction train and number line.
10. Distribute Cuisenaire® Rods. Have students find one rod of each color and determine the
value of each rod if the white rod is equal to 1. (White = 1, red = 2, light green = 3, purple =
4, yellow = 5, dark green= 6, black = 7, brown = 8, blue = 9, orange = 10). Discuss with
students. Distribute “Exploring with Cuisenaire® Rods II” activity sheet (included in this
Curriculum Guide). Explain to students that before, we gave the white rod the value of 1.
What if we give the purple rod the value of 1? Dark green rod the value of 1? Discuss with
students different ways to name the same fraction.
11. Distribute “Fraction Number Line” (included in this Curriculum Guide) and Cuisenaire®
Rods. Have students generate and record as many equivalencies of blue rod, black rod, etc.
and record them. If students are struggling with this concept, have them
find one set of equivalencies at a time rather than building entire train. Example: “Find a set
of rods equivalent to one red rod.”
12. Distribute “Drawing Fractions Blank Circles” or “Blank Rectangles” (included in this
Curriculum Guide) to each student. Students color the models to represent different given
fractions. Once colored, the fractions can be compared to determine which is greater or if
they are the same. Discuss the need to use same shape, same size (congruent) pieces to
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
compare fractions. Students need to write number sentences using the symbols <, >, and =.
For example students could be given the fractions
2
3
and
1
.
2
They would color in the two
circles or rectangles to represent the fractions and then use the drawings of the fractions to
compare them and write the appropriate number sentence
2
3
1
.
2
13. Give each student 2 sheets of blank white paper and a ruler. Have the students turn the
pieces of paper horizontally and tape them together. Ask them to use a ruler to draw a line
horizontally across the middle of the paper leaving about an inch on either end. Have the
students label the left end above the line 0. Have groups of three or four students share a box
of fraction circle pieces. Ask each student to find a one-fourth piece in the box. Students
line up the edge of the one-fourth piece with the 0 and then carefully roll the piece on the line
that was drawn along the circumference of the circle piece until the other edge of the piece is
on the line. Students put a mark on the line to show where the piece stopped and then label
1
4
the mark on the number line . Students move the fraction piece and put the edge on the
1
4
line and roll until the other edge is on the line and label this place on the number line 2 .
4
Continue until
3
4
and
4
4
are labeled on the line. Ask the students another name for
4
4
and
have them write the number 1 above the line. Ask each student to take out a one-third piece
and compare it to the one-fourth piece. Repeat the first activity using the one-third piece by
starting with 0 and filling in
1 2
3
, , and
3 3
3
on the line. Make comparisons on the number line
to validate the comparisons made by the students as they were holding the two pieces. Have
students continue by using halves, sixths, and eighths. Once all of the numbers are labeled
on the line make comparisons between fractions. Ask students to write number sentences
using symbols (<, >, =) to compare the fractions. If necessary, have students use the fraction
pieces to validate comparisons made on the number line.
14. Divide students into pairs. Distribute scissors for student pairs to cut out the “Fraction Line
Templates” and the “Fraction Cards Part 2” (included in this Curriculum Guide). Ask one
student in each pair to cut the Fraction Line Templates (do not cut out the different
fractional parts only the strips of each template) while the other student cuts the Fraction
Cards Part 2. When they have finished cutting, each pair will have a complete set of the
materials needed for the activity. Each pair will also need the “Fraction Number Line”
(included in this Curriculum Guide). Place the fraction cards face down on the table in a
pile. Each student draws a card. Once each student in the pair has drawn his or her card they
tell who drew the biggest fraction. The students verify the answer by using the Fraction Line
Templates to mark the fraction position on the number line. For example:
The students draw out the numbers
3
8
and
1
.
4
The students use the Fraction Line Templates
to mark where the two fractions would be on the number line by lining up the templates
under the number line. Students then write a number sentence comparing the two fractions
on the “Fraction Record Sheet” (included in this Curriculum Guide).
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
3
8
0
1
1
4
Students will see that
1
4
3
.
8
Students continue to draw fraction cards, compare the fractions, record the fractions on the
number line, and write a number sentence to compare the two fractions.
15. Divide students into pairs. Cut out the “Fraction Order” (included in this Curriculum Guide)
cards. Students shuffle the cards and deal five cards face up in the center of the table.
Students order the fractions from smallest to largest or largest to smallest (which way they
will order is established in the beginning).
16. Ask a small group of students to make a number line on a bulletin board in the classroom.
1
2
The only numbers on the number line to begin with are 0, , 1, and 2. Then cut out “Fraction
Order 2” cards (included in this Curriculum Guide) and have students order all the fraction
cards from the deck to make one large number line that contains all of the cards.
17. Divide students into groups of three or four. Give each group a set of measuring cups. Order
the cups from smallest to largest. Record the fractions in written form.
Grade 4 Fraction Equivalence
and Ordering
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RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 516, 522, 524.
Focus on using the fraction strips and number lines, not the algorithm.
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. None
Practice Master pp. None
Problem Solving Master pp. None
Reteaching Master pp. None
INTERDISCIPLINARY CONNECTIONS
1. Literature Connections
The Doorbell Rang by Pat Hutchins
Fraction Fun by David A. Adler
The Hershey’s Fraction Book by Jerry Pallotta
Jump, Kangaroo, Jump by Stuart Murphy
Piece = Part = Portion by Scott Gifford
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Problem Solving Questions
Will ate
1
4
of a pizza and Grace ate
1
2
of a different pizza. Will claims he ate more pizza
than Grace. Describe how Will’s claim could be true.
The shape below is
The shape below is
1
5
1
3
of a larger shape. Draw what the shape may look like.
of a larger shape. Draw what the shape may look like.
Herman wrote a fraction that was smaller than
1
. What fraction could he have written?
4
Use an area model to justify your answer.
1
of their grids. Joellen used a 10 x 10 grid and Carl
4
used an 8 x 8 grid. How many squares did each student shade? Why did they each
shade different numbers?
Joellen and Carl both shaded in
Joellen
Carl
Fill in a fraction that will make this inequality true and use the number line to prove
your inequality.
____ >
3
4
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Problem Solving Questions
Ashley and Mary Kate had two identical cakes for their birthday party. After the party,
one-half of Ashley’s cake was left. Five-eighths of Mary Kate’s cake was left. Which
cake had more leftovers? Explain your answer.
Lauren ate
7
3
of the peach pie. Kim ate of the apple pie. Which pie have they eaten
12
4
more of? Use the number line below to explain your answer.
Will
5
1
round to or 1 whole? Explain how you know.
8
2
Parris estimated that he had less than half his homework completed. He had finished
7
of the page. Use a fraction bar model to convince him that his estimate is too low.
12
0
1
1
2
At the class party, Jermaine had two slices of sausage pizza. Helena ate one slice of
cheese pizza. Describe a situation in which both students ate the same amount of pizza.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Problem Solving Questions Answers
Will ate
1
4
of a pizza and Grace ate
1
2
Answer Key
of a different pizza. Will claims he ate more pizza than Grace.
Describe how Will’s claim could be true.
Possible Answer: Will’s pizza was larger than Grace’s pizza as a whole. Comparing the size of
fractions works when they refer to the same whole. One-fourth of a large pizza may be bigger
than one-half of a small pizza.
The shape below is
1
of a larger
5
shape. Draw what the shape may look like.
The shape below is
1
of a larger
3
shape. Draw what the shape may look like.
Answers will vary. The first shape will consist of 5 trapezoids and the second shape will consist
of 3 trapezoids.
Herman wrote a fraction that was smaller than
1
.
4
What fraction could he have written? Draw an area
model to justify your answer.
Answers will vary. Any unit fraction with a denominator larger than 4 can be accepted.
Joellen and Carl both shaded in
1
of their
4
grids. Joellen used a 10 x 10 grid and Carl used an 8 x 8
grid. How many squares did each student shade? Why did they each shade different numbers?
Answer: Joellen will shade 25 squares and Carl will shade 16 squares. Even though they both
shaded
1
,
4
they shaded different numbers because the total number of squares in each whole
grid is different.
Fill in a fraction that will make this inequality true and use the number line to prove your inequality.
___ >
3
4
Answers will vary: May include
¼
7
,
8
½
13
,
16
15
16
¾
7/8
4/4
Example:
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Problem Solving Questions Answers
Ashley and Mary Kate had two identical cakes for their birthday party. After the party, one-half of
Ashley’s cake was left. Five-eighths of Mary Kate’s cake was left. Which cake had more leftovers?
Explain your answer.
Mary Kate
5
1
Answer: Mary Kate’s cake had more leftovers.
>
8
2
The shaded part shows how much of each
Ashley
cake was leftover.
Lauren ate
7
of the peach pie. Kim ate
12
3
of the apple pie. Which pie have they eaten more of? Use the number line
4
below to explain your answer.
Answer: The apple pie.
Peach
0
Will
1
Apple
5
1
round to
or 1 whole? Explain how you know.
8
2
Answer:
5
rounds
8
to
1 1
.
2 2
is equal to
4
5
and is
8
8
only one eighth more than
4
.
8
It is three
eighths away from 8/8, or one whole.
Parris estimated that he had less than half his homework completed. He had finished
7
12
of the page.
Use a fraction bar model to convince him that his estimate is too low.
Answer: The estimate is too low because
7
is
12
greater than
1
.
2
At the class party, Jermaine had two slices of sausage pizza. Helena ate one slice of cheese pizza.
Describe a situation in which both students ate the same amount of pizza.
Answer: If the slices were different sizes, then both students could’ve eaten the same amount of
pizza. Example:
2 1
=
4
8
1
2
Sausage
Grade 4 Fraction Equivalence
and Ordering
Cheese
Page 39 of 64
Columbus City Schools 2013-2014
Fraction Parts
Use color tiles to show fractional parts of each model. Color each section.
1
2
red,
green
3
3
1
3
1
2
blue, yellow, red, blue
8
8
4
8
1
1
1
yellow, red, green
3
2
6
Grade 4 Fraction Equivalence
and Ordering
Page 40 of 64
Columbus City Schools 2013-2014
Fractions of a Set
Name
Use centimeter cubes to determine equivalent fractions for
1 1
2
, , and .
4 2
3
2
5
and
on the number line? Explain
12
12
your answer and write in each fraction where it belongs on the number line. As
a group, be prepared to explain your answer to the class.
Which fraction is located between
0
1
2
12
A.
1
=
4
B.
1
=
2
C.
2
=
3
Grade 4 Fraction Equivalence
and Ordering
5
12
Page 41 of 64
Columbus City Schools 2013-2014
Fractions of a Set
Answer Key
Use centimeter cubes to determine equivalent fractions for
1 1
2
, , and .
4 2
3
2
5
and
on the number line? Explain your
12
12
answer and write in each fraction where it belongs on the number line. As a
group, be prepared to explain your answer to the class.
Which fraction is located between
1
0
3
12
2
12
A.
1
=
4
1
B. =
2
2
C. =
3
5
12
6
12
8
12
3
12
6
12
8
12
Since the number line is divided into twelfths, I took 12 centimeter cubes and divided the
cubes into 4 equal groups, one-fourth of the group is the same as three-twelfths. I took the
same 12 cubes and divided the cubes evenly into 2 groups, one-half of twelfths is the same
as six-twelfths. Lastly, I divided the 12 centimeters into three groups and two-thirds is the
same as eight-twelfths.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Fractions of a Set
Name
Use centimeter cubes to determine equivalent fractions for
1 1
10
, , and .
5 2
10
4
8
and
on the number line? Explain
10
10
your answer and write in each fraction where it belongs on the number line. As
a group, be prepared to explain your answer to the class.
Which fraction is located between
0
1
4
10
A.
1
=
5
B.
1
=
2
C.
10
=
10
Grade 4 Fraction Equivalence
and Ordering
Page 43 of 64
8
10
Columbus City Schools 2013-2014
Fractions of a Set
Answer Key
Use centimeter cubes to determine equivalent fractions for
1 1
10
, , and .
5 2
10
4
8
and
on the number line? Explain
10
10
your answer and write in each fraction where it belongs on the number line. As
a group, be prepared to explain your answer to the class.
Which fraction is located between
0
1
2
10
A.
1
=
5
B.
1
=
2
C.
10
=
10
4
10
5
10
8
10
2
10
5
10
1
Since the number line is divided into tenths, I took 10 centimeter cubes and divided the
cubes into 5 equal groups, one-fifth of the group is the same as 2-tenths. I took the same 10
cubes and divided the cubes evenly into 2 groups, one-half of ten is the same as five- tenths.
Lastly, I divided the 10 centimeters into ten groups and ten groups of ten is the same as a
whole.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
1
Fraction Sort
Close to Zero
Close to Half
Close to One
Grade 4 Fraction Equivalence
and Ordering
Page 45 of 64
Columbus City Schools 2013-2014
Fraction Cards Part 1
4
5
9
10
1
8
2
12
12
14
5
8
9
11
2
7
3
15
4
10
1
20
3
18
6
13
5
12
2
11
9
20
6
14
8
9
49
100
21
40
3
16
1
11
2
15
5
20
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Exploring Fractions with Cuisenaire® Rods I
Name__________________________
1. What colors can be lined up end-to-end to create the same
length as the brown rod?
a. Place a brown rod in the space below and trace the rod.
b. Find all rods that when placed end-to-end equal the brown rod.
c. Draw your fraction trains below, color them with crayons or
colored pencil and give a number sentence for each train.
Trace the brown rod below:
Number Sentence
2. Now, look at the fraction trains you drew and name as many fractional
relationships you found. For example, 4 (4 white rods) is the same as 1
8
(1 purple rod) and
2
4
(2 red rods). So
4
8
=
2
1
2
2
or . These are called
4
equivalent fractions.
Grade 4 Fraction Equivalence
and Ordering
Page 47 of 64
Columbus City Schools 2013-2014
Brown Train Equivalent Fractions:
4
8
2
8
6
8
=
1
2
or
2
4
= ________
= ________
One whole = _________ or ________ or __________
3. What colors can be lined up end-to-end to create the same length as
the orange rod?
a. Place an orange rod in the space below and trace the rod.
b. Find all rods that when placed end-to-end equal the orange rod.
c. Draw your fraction trains below, color them with crayons or
colored pencil and give a number sentence for each train.
Trace the orange rod below:
Number Sentence
4. Now, look at the fraction trains you drew and name as many fractional
relationships you found.
Grade 4 Fraction Equivalence
and Ordering
Page 48 of 64
Columbus City Schools 2013-2014
Exploring Fractions with Cuisenaire® Rods I
Answer Key
1. What colors can be lined up end-to-end to create the same
length as the brown rod?
a. Place a brown rod in the space below and trace the rod.
b. Find all rods that when placed end-to-end equal the brown rod.
c. Draw your fraction trains below, color them with crayons or
colored pencil and give a number sentence for each train.
Trace the brown rod below:
Number Sentence
Brown
1
Purple
2
1
Red
1
+ =1
+
2
1
1
+ +
1
=1
4
4
4
4
1+1+1+1+1+1+1+1
White
8
8
8
8
8
8
8
8
=1
2. Now, look at the fraction trains you drew and name as many fractional
relationships you found. For example,
(1 purple rod) and
2
4
4
(4 white rods) is the same as
8
4 1
2
2
4
1
2
(2 red rods). So = or . These are called
8
equivalent fractions.
Brown Train Equivalent Fractions:
4
8
2
8
6
8
=
=
=
1
2
1
or
2
4
4
3
4
2
4
8
2
4
8
One whole = or or
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
3. What colors can be lined up end-to-end to create the same length as
the orange rod?
a. Place an orange rod in the space below and trace the rod.
b. Find all rods that when placed end-to-end equal the orange rod.
c. Draw your fraction trains below, color them with crayons or
colored pencil and give a number sentence for each train.
Trace the orange rod below:
Number Sentence
Orange
1
Yellow
2
1
+
2
=1
1+ 1+ 1 + 1+ 1
Red
White
5
5
5
5
5
=1
1 + 1 + 1 + 1 + 1 + 1 +
10 10
10 10
1 + 1 + 1 + 1
White
10
10
10
10
10
10
=1
4. Now, look at the fraction trains you drew and name as many fractional
relationships you found.
1
2
1
5
2
5
3
5
4
5
=
=
=
=
=
5
10
2
10
4
10
6
10
8
10
Grade 4 Fraction Equivalence
and Ordering
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Grade 4 Fraction Equivalence
and Ordering
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1. Use your “Exploring fractions with Cuisenaire Rods I” activity sheet to
place all the fractions you found on the number line.
2. Next, use your number line to help you identify equivalent fractions and
solve this problem.
3. Ciara was making cookies. She needed ¾ cup of sugar, 1/2 cup of butter
and 2/8 cup of flour. Did Ciara use more flour or more sugar to make her
cookies? Explain your answer.
0
Fraction Number Line
1
Exploring Fractions with Cuisinaire Rods II
Name_____________________
1. If the purple rod has a value of one, what is the value of the :
a. white rod = __________________
b. red rod = ________________ = ________________
c. light green rod = ________________ = ________________
d. brown rod = ________________
e. orange rod + dark green rod = ________________
2. If the dark green rod has a value of one, what is the value of the:
a. white rod = _________________
b. red rod = ________________
c. light green rod = ________________
d. purple rod = ________________
e. blue rod = ___________________
f. yellow rod = _________________
g. two red rods = ________________
h. one purple rod and one white rod = ______________________
i. two light green rods = __________________________
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Exploring Fractions with Cuisinaire Rods II
Answer Key
Name_____________________
1. If the purple rod has a value of one, what is the value of the :
1
a. white rod =
4
b. red rod =
2
1
=
4
2
3
c. light green rod =
4
d. brown rod = 2
e. orange rod + dark green rod = 4
2. If the dark green rod has a value of one, what is the value of the:
1
a. white rod =
6
b. red rod =
2
1
=
6
3
3
c. light green rod =
4
d. purple rod =
1
=
2
6
2
=
6
1
3
9
3
2
6
2
e. blue rod = 1 =
f. yellow rod =
=
5
6
g. two red rods =
4
6
=
2
3
h. one purple rod and one white rod =
5
6
i. two light green rods = 1
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Drawing Fractions
Grade 4 Fraction Equivalence
and Ordering
Page 54 of 64
Columbus City Schools 2013-2014
Drawing Fractions
Grade 4 Fraction Equivalence
and Ordering
Page 55 of 64
Columbus City Schools 2013-2014
Fraction Line Templates
Cut out each strip. Do not cut apart the individual pieces of each strip.
1
6
1
3
1
2
1
6
1
6
1
3
1
6
1
2
1
6
1
3
1
6
Grade 4 Fraction Equivalence
and Ordering
Page 56 of 64
Columbus City Schools 2013-2014
Fraction Line Templates
Cut out each strip. Do not cut apart the individual pieces of each strip
1
4
1
16
1
8
1
16
1
16
1
8
1
4
1
16
1
16
1
8
1
16
1
16
1
8
1
4
1
16
1
16
1
8
1
16
1
16
1
8
1
4
1
16
1
16
1
8
1
16
1
16
1
8
Grade 4 Fraction Equivalence
and Ordering
1
16
Page 57 of 64
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
Columbus City Schools 2013-2014
Fraction Cards Part 2
Cut out the individual fraction cards.
1
2
1
4
1
8
1
16
1
6
1
12
2
8
4
8
6
8
7
8
3
4
2
16
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Fraction Cards Part 2
Cut out the individual fraction cards.
4
16
6
16
8
16
10
16
12
16
15
16
2
4
2
6
3
6
4
6
5
6
3
8
Grade 4 Fraction Equivalence
and Ordering
Page 59 of 64
Columbus City Schools 2013-2014
Fraction Cards Part 2
Cut out the individual fraction cards.
2
12
3
12
4
12
5
12
6
12
7
12
8
12
9
12
10
12
11
12
1
3
2
3
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
0
1
Fraction Number Line
Grade 4 Fraction Equivalence
and Ordering
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Fraction Record Sheet
Name
Write each fraction drawn and write an expression comparing each fraction.
Fraction Drawn Fraction Drawn Write an expression using the
or
symbols.
Select any four fractions from the fraction number line and order the fractions from
smallest to largest.
Grade 4 Fraction Equivalence
and Ordering
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Columbus City Schools 2013-2014
Fraction Order
1
6
1
8
5
6
6
10
Grade 4 Fraction Equivalence
and Ordering
1
3
1
10
7
8
2
10
Page 63 of 64
1
4
2
3
6
8
2
2
1
5
1
2
4
6
3
3
Columbus City Schools 2013-2014
Fraction Order 2
4
5
2
6
8
8
8
10
Grade 4 Fraction Equivalence
and Ordering
2 8
3
1
3 9
12
5
5
2
8
9 12
1
3
4
2
2
5
6
7
3
5
1
9
6
6
Page 64 of 64
Columbus City Schools 2013-2014
Mathematics Model Curriculum
Grade Level: Fourth Grade
Grading Period: 3
Common Core Domain
Time Range: 10 Days
Number and Operations - Fractions
Common Core Standards
Build fractions from unit fractions by applying and extending previous understandings of
operations on whole numbers.
3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same
whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way,
recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction
model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an
equivalent fraction, and/or by using properties of operations and the relationship between addition and
subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and
having like denominators, e.g., by using visual fraction models and equations to represent the problem.
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4
as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a
whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing
this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual
fraction models and equations to represent the problem. For example, if each person at a party will eat
3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will
be needed? Between what two whole numbers does your answer lie?
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Developing an understanding of fraction equivalence, addition and subtraction of fractions with like
denominators, and multiplication of fractions by whole numbers
Students develop understanding of fraction equivalence and operations with fractions. They recognize that two
different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing
equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions,
composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of
fractions and the meaning of multiplication to multiply a fraction by a whole number.
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Mathematics Model Curriculum
Content Elaborations
This section will address the depth of the standards that are being taught.
from ODE Model Curriculum
Ohio has chosen to support shared interpretation of the standards by linking the work of multistate partnerships as
the Mathematics Content Elaborations. Further clarification of the standards can be found through these reliable
organizations and their links:
Achieve the Core Modules, Resources
Hunt Institute Video examples
Institute for Mathematics and Education Learning Progressions Narratives
Illustrative Mathematics Sample tasks
National Council of Supervisors of Mathematics (NCSM) Resources, Lessons, Items
National Council of Teacher of Mathematics (NCTM) Resources, Lessons, Items
Partnership for Assessment of Readiness for College and Careers (PARCC) Resources, Items
Expectations for Learning (Tasks and Assessments)
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Expectations for Learning (in development) from ODE Model Curriculum
Ohio has selected PARCC as the contractor for the development of the Next Generation Assessments for
Mathematics. PARCC is responsible for the development of the framework, blueprints, items, rubrics, and scoring
for the assessments. Further information can be found at Partnership for Assessment of Readiness for College and
Careers (PARCC). Specific information is located at these links:
Model Content Framework
Item Specifications/Evidence Tables
Sample Items
Calculator Usage
Accommodations
Reference Sheets
Sample assessment questions are included in this document.
The following website has problem of the month problems and tasks that can be used to assess students and help
guide your lessons.
http://www.noycefdn.org/resources.php
http://illustrativemathematics.org
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx
http://nrich.maths.org
At the end of this topic period students will demonstrate their understanding by….
Some examples include:
Constructed Response
Performance Tasks
Portfolios
***A student’s nine week grade should be based on multiple academic criteria (classroom assignments and teacher made
assessments).
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Mathematics Model Curriculum
Instructional Strategies
Columbus Curriculum Guide strategies for this topic are included in this document.
Also utilize model curriculum, Ohiorc.org, textbooks, InfoOhio, etc. as resources to provide instructional
strategies.
Websites
http://illuminations.nctm.org
http://illustrativemathematics.org
http://insidemathematics.org
http://www.lessonresearch.net/lessonplans1.html
Instructional Strategies from ODE Model Curriculum
None listed at this time.
Instructional Resources/Tools
Pattern blocks
Cuisenaire rods
Fraction strips
Number lines
Misconceptions/Challenges
The following are some common misconceptions for this topic. As you teach the lessons, identify the misconceptions/challenges that
students have with the concepts being taught.
Common Misconceptions from ODE Model Curriculum
Please read the Teacher Introductions, included in this document, for further understanding.
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Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
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Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
Equal
Groups
Arrays, 4
Area, 5
Compare
3×6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
a × ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? × b = p, and p ÷ b = ?
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
Number of Groups Unknown (“How
many groups?” Division
Group Size Unknown
(“How many in each group?”
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Unknown Product
a×b=?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
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Columbus City Schools 2013-2014
Group Size Unknown: (a × ? = p and p ÷ a = ?)
In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
students used math tools to solve the problem, they should include a picture to represent how the
Grade 4 Fractions
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tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
Act It Out
Make a Table
Find a Pattern
Guess and Check
Make a List
Draw a Picture
When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
to act it out. Occasionally whole group thinking with the teacher modeling how to record
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strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Teacher Introduction
Computation with Fractions
Common Denominators
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Number and Operations - Fractions domain is helpful when assessing
student work and student thinking. This information could be used to guide classroom
discussions, understand student misconceptions and provide differentiation opportunities. The
focus of instruction is the Common Core Mathematics Standards.
When students are developing fraction number sense they must be given ample opportunity to
estimate the sums and differences of fractions before being taught algorithms for addition and
subtraction.
Example: When adding one-eighth and two-eighths, will the answer be more or less
than
1
2
?
Example: If you ate four-eighths of a pizza and your friend ate three-eighths of the
same pizza, will the amount of pizza remaining be closer to a whole pizza or
will the pizza be almost gone?
Students who have developed a good foundational sense about fractions will find adding and
subtracting fractions with common denominators a logical transition and should be able to do so
with ease. Students will remember that the top number of the fraction (numerator) is what is
being counted and the bottom number (denominator) is the number of parts in the whole. The
only number that changes when adding or subtracting fractions with common denominators is
the counting number (numerator) because the size of the pieces (denominator) does not change.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6,
8, 10. 12, 100.
Adding and subtracting fractions is very much like adding and subtracting students. For
example, if you add 5 students and 4 students you get 9 students. The number of “things”
changes but the type of “thing” does not. This also is the case with fractions. If there are 12
5
4
of the class and 4 students represent of the
12
12
9
represent of the class. The number of students in
12
students in the class, then 5 students represent
class. The two groups of students together
the group you are talking about changes, but the total number of students in the class does not.
Grade 4 Fractions
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Students can use several methods to add and subtract fractions.
Method
Example
Physical Models
counters, fraction bars, pattern blocks, color
tiles, centimeter cubes
There are 10 counters. Four-tenths are blue
and three-tenths are red. What fraction of the
total counters are either blue or red?
Students can physically move fractional parts
together or separate them and then count to
determine how many they have altogether or
how many are left. In the example:
4
3
7
10 10 10
B
B
B
B
R
R
R
Maria ate three-eighths of a pizza. Kim ate
four-eighths of the same pizza. What fraction
of the pizza
Students can look at the drawing and count
was left?
M
how many fractional pieces there are altogether
M
or how many fractional pieces are left. In the
M
K
example they can easily see that there is only 1
out of the 8 pieces left.
K
Left Over
K K
3 4 7
8 7 1
Visual Representations
draw a picture, act it out
8
8
8
8
8
8
Paper and Pencil
Students need to use their knowledge of
numerators and denominators to combine and
separate fractions with common denominators.
The denominator will not change when
combining or separating because it represents
the number of pieces into which the whole has
been divided. The number of pieces counted in
the numerator of the fractions is what needs to
be combined or separated.
Juan ate five-twelfths of a cake. Dre ate threetwelfths of the same cake. What fraction of the
cake did they eat?
5
3
8
12 12 12
The cake was divided into 12 pieces and that
did not change when the pieces eaten by each
person were combined.
In the question above, what fraction of the cake
is left?
12 8
4
12 12 12
The cake was divided into 12 pieces, so 12
pieces is equal to one whole cake. If you take
away the 8 pieces that were eaten, then there
are four pieces of cake left.
Grade 4 Fractions
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Columbus City Schools 2013-2014
Equivalent Fractions
It is important to develop a conceptual understanding of equivalent fractions before attempting to
teach students formal algorithms for finding equivalent fractions. Two fractions are equivalent if
they represent the same amount of a whole or quantity of a set. Three different models which
can be used to determine equivalent fractions are area models, length models, and set models.
Model
Example
Area Model
The student uses two congruent shapes that are
divided into different-sized pieces to determine
if two fractions are equivalent. If the area
covered by the two fractions is the same then
the fractions are equivalent.
1
4
2
8
3
6
1
2
Length Model
The student uses a number line or ruler to find
fractions that are equivalent. Lines of the same
length are divided into different fractional
parts. Students can then look to see if the
fractions are located the same distance from
the start of the line (ruler). If they are, then
they are equivalent. If they are not, then they
are not equivalent.
Grade 4 Fractions
inches
inches
Page 11 of 41
3
4
1
6
8
1
0
4
5
1
0
8
10
1
Columbus City Schools 2013-2014
Model
Example
Set Model
The student uses sets of counters (color tiles,
centimeter cubes, beans, etc.) that are two
different colors. For example, to find
equivalent fractions for 4 use four black
4 of the cubes are
12
black and 8 of the
12
cubes are white.
12
cubes and eight white cubes (or any two other
colors) for a total of 12 cubes. Line the cubes
in a 12 1 vertical array with the four black
cubes on top and the eight white cubes on the
bottom. See the example at the right.
The student makes rectangular arrays with the
four black and eight white cubes so that each
horizontal row has only one color of cubes in
it. In this case there are two horizontal rows of
black cubes and four horizontal rows of white
cubes, so two of the six cubes in each vertical
column are black and four of the six cubes in
each vertical column are white. Therefore, 2
2 of the cubes are
6
black and 4 of the
6
cubes are white.
6
is equivalent to
4 .
12
1 of the cubes are
3
black and 2 of the
3
cubes are white.
Next, arrange the four black cubes in one
horizontal row followed by two horizontal
rows of white cubes. One of the three cubes in
each vertical column is black and two of the
three cubes in each vertical column are white.
Therefore, 1 is equivalent to 4 .
3
12
The set model can be used to find equivalent
fractions with any number of total cubes. It is
important for students to realize that to make a
rectangular array the total number of counters
has to be divisible by the number of rows in
the array. Remind students that when they
create their rows each row must be made of
cubes that are all the same color.
Grade 4 Fractions
The arrays above are the only rectangular
arrays which can be made with the given cubes
where each row contains all of the same color
of cubes. Based on the arrays, the following
equivalent fractions are found using this
method.
Page 12 of 41
1
3
2
6
4 and 2
12
3
4 8
6 12
Columbus City Schools 2013-2014
Once students have explored the concept of equivalent fractions through models it is essential to
revisit comparing fractions. Students will begin to realize that renaming fractions can help them
find which fractions are greater or lesser. For example, when determining whether 3 or 5 is
4
greater, students can use pictures or models to determine that
3
4
is equivalent to
6.
8
8
With this in
mind it is easy to see that 6 out of 8 parts is more than 5 out of 8 parts so 3 5 . Models will also
4
8
help students understand that equivalent fractions are just different names for the same amount of
the whole or different arrangements for the same set of objects.
In everyday life we use fractions and decimals interchangeably. We measure trip distance using
the odometer in our cars to represent mileage as a decimal. When we speak of the distance
traveled, we use fractions. For example, the trip odometer for our drive to Cleveland may read
127.9 miles. When we read the distance we say “one hundred twenty-seven and nine-tenths
miles.” Even though fractions and decimals are related, we often segment our instruction so
students relate to fractions as parts of a region, length, or set and decimals as numbers and not
parts of a region. Fractions and decimals are interrelated concepts that are different ways of
representing the same amount. To build greater understanding, we must teach the relationship
between fractions and decimals. Students must have numerous experiences with both concepts
as different ways of showing equivalent quantities.
As students work with a variety of models and representations, the connection between fractions
and decimals is visually obvious. Students will naturally begin to relate fractions and decimals.
In the 3-4 grade band, students will be expected to translate between decimals and base ten
fractions, fractions with denominators that are multiples of 10 (10, 100, 1,000, 10,000, etc.).
Students will also identify and generate equivalent forms of fractions and decimals through
connecting physical, verbal, and symbolic representations of fractions, decimals, and whole
numbers.
1
2
=
5
10
= 0.5 =
=
= 50%
0
100%
Comparing fractions may sometimes be an overwhelming task for students. In the 3-4 grade
band, students are expected to use models and points of reference to compare commonly used
fractions. Students draw on their fraction knowledge, accessing all three different fraction
models: the area model, the set model, and the length model. Building an understanding of these
different fraction models will assist students with fraction comparisons.
Many of the difficulties students experience with fractions occur with the inverse relationships
between the number of parts and the size of the parts. In all other mathematical experiences
larger numbers indicate a larger quantity, or more. Students incorrectly transfer this concept to
fractions:
“Four is more than three, so fourths should be greater than thirds.”
However, the larger the number of parts, the smaller the size of the parts because the whole has
been divided into more pieces so each piece is smaller. Students must construct this relationship
Grade 4 Fractions
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Columbus City Schools 2013-2014
through repeated experiences with fraction models. Students should work with and understand
three different fraction comparison properties.
1. More of the Same Size Parts
(same denominator, different numerator)
Example: Compare three-fourths and onefourth.
Both fractions are fourths, so compare the
counting number (or numerator). Three is
greater than one, so three-fourths is greater
than one-fourth.
2. Same Number of Parts But Parts Are
Different Sizes
(same numerator, different denominator)
Example: Compare two-thirds and twofourths.
Both fractions have the same counting number
(or numerator). However, thirds are larger
than fourths (the greater the number of parts,
the smaller the size of the parts), so two-thirds
is greater than two-fourths.
3. Different Number of Parts and Parts are
Different Sizes
(different numerator, different
denominator)
Example: Compare one-fifth and twothirds
Several strategies are listed in the table below
to help students compare fractions that have
different numerators and different
denominators.
When students are asked to compare fractions that contain different-sized numerators and
denominators, it is important for them to have experience with different methods that can help
them make the comparison. In addition to using models, students can use several methods to
compare fractions.
Method
Example
Number Lines
5
1
and
8
4
Students divide two lines that are the same
length into the fractional parts of the two
fractions that are being compared. They then
look to see where the fractions are on the line
and make the comparison.
5 1
1 5
or
8 4
4 8
Example: Compare
Grade 4 Fractions
Page 14 of 41
0
1
4
0 1
2
8
8
2
4
3
8
4
8
3
4
5
8
6
8
1
7
8
1
Columbus City Schools 2013-2014
Method
Example
Models (fraction bars)
3
7
Example: Compare and
4
10
Students divide two rectangles that are the
same size into the fractional parts of the
fractions that are being compared. They then
shade in the number of pieces in each rectangle
to represent the fractions and then make the
3 7
7 3
comparison.
or
4 10
10 4
Benchmarks of 0, ½, 1
5
1
Example: Compare and
6
8
Students look at the first fraction and
determine if it is closest to 0, ½, or 1 and then
look at the second fraction and determine if it
is closest to 0, ½, or 1. Students then use that
information to compare the fractions.
5
6
1
1
or
8
8
5
6
Equivalent Forms
Example: Compare
2
5
and
3
12
Students look at the denominator that has the
larger number and see if that number can be
divided equally into the number of groups
represented by the denominator that is the
smaller number. If so, an equivalent fraction
can be found for the fraction with the
denominator that is the smaller number and
then the fractions with the same denominators
2 5
5 2
are compared.
or
3 12
12 3
3
4
7
10
Half of 6 is 3 so
3
1 5
is equal to .
is only one6
2 6
sixth away from one whole because adding one
more sixth will create a whole and it is two1
5
.
is closer to 1 than ½ or
2
6
1
1
4
0.
is less than because half of 8 is 4 so
8
2
8
1
1
5
is equal to so is closer to 0. Since is
2
8
6
1
5
closer to 1 and is closer to 0, then is the
8
6
sixths away from
greater fraction.
12 can be divided equally into 3 groups (see
the stars below). There are 4 items in each
2
group. So to find of 12 you need to count
3
the number of items in 2 out of the 3 groups
made from the 12 items or 4 + 4 = 8.
2 8
Now both fractions have a
3 12
denominator of 12 and the numerators can
be compared to determine which fraction is
8
5
larger. Eight is larger than 5 so
12 12
2 5
which means
.
3 12
The understanding of the relative sizes of fractions helps students compare fractions more easily.
It also teaches students to rely less on paper and pencil computational skills and more on number
sense, an important lifelong skill.
When students order mixed numbers they should use the same methods as when comparing
fractions less than one. However, they must first compare and order the whole numbers. If the
whole numbers are different, then the comparison can be done without looking at the fractions by
finding the greatest whole number and ordering based only on the whole number. If the whole
numbers are the same, then the student’s next step is to compare and order the fractional parts
using methods such as the ones mentioned above.
Grade 4 Fractions
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Miss Fedun’s class ate 4 1 pizzas at their class party. The class volunteer, Miss
3
1
Green ate pizza. Which group shows how much pizza they ate altogether?
3
 A.
 B.
 C.
Shade in the appropriate number of rectangles after the equal sign to make the
equation true. Write the correct fraction on the line below.
+
3
6
Grade 4 Fractions
=
2 _________
6
Page 16 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Miss Fedun’s class ate 4 1 pizzas at their class party. The class volunteer, Miss
3
Green ate 1 pizza. Which group shows how much pizza they ate altogether?
3
 A.
 B.
 C.
Answer: B
Shade in the appropriate number of rectangles after the equal sign to make the
equation true. Write the correct fraction on the line below.
+
3
6
2
6
=
5
6
A 2-point response correctly shades the rectangle and writes 5 on the line.
6
A 1-point response correctly shades the rectangle or writes the fraction on the line.
A 0-point response shows no mathematical understanding of this task.
Grade 4 Fractions
Page 17 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
For dinner Mr. Cooney ordered two pizzas that each had 12 pieces. After dinner
there were 3 pieces of cheese pizza and 2 pieces of pepperoni pizza remaining.
Shade in the pizza below to show what fraction of a whole pizza was remaining.
Write the fraction of the pizza that was left in the box below.
3
2
+
=
12 12
The pictures show the pies that were left after LeJuan’s birthday lunch. The
shaded areas below represent the pieces of pie that were left. What fraction of the
pies is left?
 A.
5
6
 B. 1
1
2
 C. 1
5
6
Grade 4 Fractions
Page 18 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
For dinner Mr. Cooney ordered two pizzas that each had 12 pieces. After dinner
there were 3 pieces of cheese pizza and 2 pieces of pepperoni pizza remaining.
Shade in the pizza below to show what fraction of a whole pizza was remaining.
Write the fraction of the pizza that was left in the box below.
A 2-point response shades in five sections of the
circle and writes the fraction 5 in the box.
12
A 1-point response correctly shades the circle or
writes the fraction 5 in the box.
12
A 0-point response shows no mathematical
understanding of the problem.
3
2
+
=
12 12
5
12
The pictures show the pies that were left after LeJuan’s birthday lunch. The
shaded areas below represent the pieces of pie that were left. What fraction of the
pies is left?
 A.
5
6
 B. 1
1
2
 C. 1
5
6
Grade 4 Fractions
Answer: C
Page 19 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
If you were to pour all of the liquid from these containers together, would you need
more than one empty container? Explain your answer.
1
4
1
4
3
4
1
4
Three-fourths of the circle above is shaded. Shade in the second circle so that it
has one-fourth less than three-fourths shaded.
Complete the number sentence below to represent the circles.
3
4
1
4
Grade 4 Fractions
Page 20 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
If you were to pour all of the liquid from these containers together, would you need
more than one empty container? Explain your answer.
1
4
1
4
3
4
1
4
Answer: Yes, you would need two containers because 3 1 4 which is equal to one whole
4
container. There are
2
4
4
4
more that need to go in a second container.
A 2-point response states that you would need two containers and gives a complete
explanation.
A 1-point response states that you would need two containers with a weak or no
explanation.
A 0-point response shows no mathematical understanding of the problem.
Three-fourths of the circle above is shaded. Shade in the second circle so that it
has one-fourth less than three-fourths shaded.
Complete the number sentence below to represent the circles.
3 1
2
4 4
4
A 2-point response correctly shades in one-half of the blank circle (students can shade in
any two fourths of the circle) and includes a correct answer of 2 or 1 .
4
2
A 1-point response includes a correctly shaded circle with an incorrect fraction for an
answer or a correct fractional answer with an incorrect circle.
A 0-point answer shows no mathematical understanding of this task.
Grade 4 Fractions
Page 21 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
The shaded part of the picture below shows how many pieces of cake were eaten at
dinner. If Jack and JaLynn each have another piece of cake as a snack before they
go to bed, how much of the cake has been eaten?
A. 10
12
B. 8
12
C. 6
12
As a service project, a group of fourth graders are making puppets for the
kindergarten teacher to use during story time. They are making 3 puppets and each
puppet takes
3
of a yard of material. How much material do they need to make
5
the puppets? Show your work.
Grade 4 Fractions
Page 22 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
The shaded part of the picture below shows how many pieces of cake were eaten at
dinner. If Jack and JaLynn each have another piece of cake as a snack before they
go to bed, how much of the cake has been eaten?
A. 10
12
B. 8
12
C. 6
12
Answer: B
As a service project, a group of fourth graders are making puppets for the kindergarten teacher to
3
use during story time. They are making 3 puppets and each puppet takes of a yard of material.
5
How much material do they need to make the puppets? Show your work.
4
Answer: They will need 1 yards of material.
5
3 3 9
4
3
3
3
3
9
4
1 or 3 ×
=
+
+
=
=1
1 5 5
5
5
5
5
5
5
5
3 groups of
A 2-point response includes a correct answer of 1
3
9
4
=
=1
5
5
5
4
yards of materials and shows all work.
5
4
yards but has incomplete or missing
5
work or shows all work but includes a minor computational error.
A 1-point response includes a correct answer of 1
A 0-point response shows no understanding of this task.
Grade 4 Fractions
Page 23 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Tracy has 9 bags. Each bag is
1
full of apples. How many full bags of apples
3
does Tracy have? Use multiplication to show your thinking.
Washington Elementary is having a bake sale. Mario is baking chocolate chip
cookies. The recipe calls for
1
cup of brown sugar. He is making 6 batches of the
4
same recipe. How many cups of brown sugar will he need?
Grade 4 Fractions
Page 24 of 41
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Tracy has 9 bags. Each bag is
1
full of apples.
3
How many full bags of apples does Tracy have? Use multiplication to show your
thinking.
Answer: 3 full bags of apples
9 groups of
1
=
3
1
9
= =
3
3
9×
3 full bags of apples
A 2-point response includes a correct answer of 3 full bags and shows all work.
A 1-point response includes a correct answer of 3 full bags but has incomplete or missing
work or shows all work but includes a minor computational error.
A 0-point response shows no understanding of this task.
Washington Elementary is having a bake sale. Mario is baking chocolate chip
cookies. The recipe calls for
1
cup of brown sugar. He is making 6 batches of the
4
same recipe. How many cups of brown sugar will he need? Show your work.
2
4
1
2
Answer: Joe will need 1 or 1 cups of brown sugar.
6×
1
= 6
4
1
× 1 = 6 = 1 2 or 1 1 cups of brown sugar.
4
Grade 4 Fractions
4
4
2
Page 25 of 41
Columbus City Schools 2013-2014
TEACHING STRATEGIES/ACTIVITIES
Vocabulary: numerator, denominator, part, whole, fractions, addition, subtraction, sum,
decomposition, visual fraction model, equation, mixed number, improper fraction,
equivalent, unit fraction
1. The word problems below can be presented to students in a variety of ways. Some options
include: using the questions to create a choice-board, Math-O board, one problem per class
session, one problem each evening for homework, partner work or as an assessment question.
The strength of problem solving lies in the rich discussion afterward. As the school year
progresses students should be able to justify their own thinking as well as the thinking of
others. This can be done through comparing strategies, arguing another student’s solution
strategy or summarizing another student’s sharing.
5
6
=
1
1
+
6
6
+
3
6
Write at least three other ways you can add sixths together to get
Jackie and Donna need 6
yarn and Donna has 4
2
8
3
8
5
.
6
feet of yarn to make puppets. Jackie has 2
2
8
feet of
feet of yarn. How much yarn should they expect to have
leftover?
Taylor had 3
1
8
pizzas left over from her birthday party. She passed out some of the
leftover pizza to her friends to take home and now she has 1
4
8
pizza left. How much
did Taylor pass out to her friends?
Use a fraction bar model to show the answer to 3
To make a Zombie Shake, you need
3
4
3
4
+2
cup of milk,
2
4
1
.
4
cup of root beer, and
1
cup
4
of
hot sauce. How much liquid is needed?
The pizza delivery guy lost control of his van around the curve and all his pizzas fell
into the road. He lost 12 pizzas that were cut into sixths. The neighborhood kids
found 46 of the sixths. How many of the whole pizzas were they able to recover?
How much pizza was still missing?
During a relay race, a team of 6 students each ran
1
4
of a mile. How long was the
race? Use a number line model to show how you got the answer.
During a relay race, a team of students each ran
1
3
of a mile. If the race was 4 miles
long, how many students were on the team?
Harriet made 4 batches of cookies. If each batch required
2
3
of a cup of butter, how
much butter did Harriet have to use? Use a fraction bar model to show how you got
your answer. Write a multiplication equation to represent your answer.
Pete solved a story problem and wrote the answer of
1
4
×5=
5
.
4
What might the
story problem have been?
Grade 4 Fractions
Page 26 of 41
Columbus City Schools 2013-2014
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 4 on the right side of the page. Open “Unit 3
Framework Student Edition.” These tasks are from the Georgia Department of Education.
The tasks build on one another. You may not use all of the tasks. There are four different
types of tasks. Scaffolding tasks build up to the constructing tasks which develop a deep
understanding of the concept. Next, there are practice tasks and finally performance tasks
which are a summative assessment for the unit. Select the tasks that best fit with your
lessons and the standards being taught at that time.
3. Instruct students to build the orange Cuisenaire fraction train using the orange rod, 2 yellow,
5 red and 10 white rods. Discuss the relationship between the different color rods (i.e., 2
yellow = 1 orange; 4 white = 2 reds). Ask students what fractions each of these colors might
represent. The fraction names for each color will depend on which rod is one. Tell students
that for this activity the orange rod will represent one candy bar. Distribute the “Problem
Solving with Cuisenaire Rods” activity sheet (included in this Curriculum Guide). Give
students time to explore and complete the questions. Have students share their strategies
with the whole group. Make sure the students model their answers using the Cuisenaire rods.
4. Give each student circular fraction disks or “Fraction Circles” (included in the Grids and
Graphics section of this Curriculum Guide). If you are using the paper fraction circles, have
the students cut the circles apart and put them in piles by the denominator in front of them.
Ask the students to model 5 at their desks using the fraction circles. Write 5 on the
6
6
overhead. Next, ask students to take
4
6
away from the pile on the table. Ask what operation
is used for taking away. Write a subtraction sign after the 5 on the overhead. Ask students
6
how many sixths they were asked to
take away ( 4 )
6
and write 4 after the subtraction sign
6
and then write an equal sign. Ask students how many sixths are left on the table in front of
them and write 1 after the equal sign. Model several other subtraction problems the same
6
way. Always have the class say the equation when it is complete. Ask students to explain
why the denominator does not change in the equation (because the size of the pieces does not
change).
5. Use measuring cups to explore different combinations of common fractions. For each group
of 4-6 students you will need rice, measuring cups, and a 2-cup liquid measuring cup (check
the markings on the 2-cup and make sure that it shows quarter, third, and half cups). To
minimize the mess, put the rice and cups on trays. Each student will also need a copy of the
“Fill ‘er Up” worksheet (included in this Curriculum Guide). Students fill the appropriate
measuring cups with rice and pour the amounts into the 2-cup measuring cup. For example,
if the problem is 1 cup + 1 cup =, the students would fill the 1 cup twice and pour the rice
4
4
4
into the 2-cup measure each time. They then look at the 2-cup measure to see that
cup + 1 cup is equal to 1 cup. Continue measuring and pouring for the remaining
1
4
4
2
problems.
6. Distribute “Trailhead” (included in this Curriculum Guide) to each student for additional
practice.
Grade 4 Fractions
Page 27 of 41
Columbus City Schools 2013-2014
7. Distribute color counters to each student. Students can model multiplying a whole number
and a fraction by counting out enough counters to represent the whole number. Have
students to divide the counters into as many parts as the denominator of the fraction, for
1
example, of 16 . Begin by taking sixteen counters and dividing the counters into eight
8
groups. There would be eight groups of counters with two counters in each group.
1
16 2 .
8
8. Distribute plain sheets of paper to each student. Ask students how they could use the blank
2
paper to determine 3
. Ask them if there is a way to use area models to represent this
3
problem. Students take as many pieces of paper as the whole number and fold each piece into
the number of parts that is the denominator of the fraction they are multiplying by. In this case,
each piece of paper is divided into three equal parts because the denominator of the fraction is
3
three. (If they would have been multiplying 3 by then each piece of paper would have been
4
divided into four equal pieces and three of them would be shaded). Shade two parts of each
piece of paper that has been folded into thirds (so that two-thirds of each paper is shaded). In
the three rectangles, there are six shaded thirds. Have students look at the shaded parts of the
three whole pieces of paper and see if they can make any whole rectangles out of the shaded
2
2.
parts. Altogether, six shaded thirds is the same as two whole rectangles. 3
3
6
3
2
9. Divide students into groups of four. Distribute “Recipe Cards” (included in this Curriculum
Guide) and blank index cards to each group of students. Assign each group a different
amount to increase the recipe, such as double, triple, quadruple, etc. Students rewrite the
recipe on the index cards and show the increased amounts of each ingredient.
10. Distribute one “Fraction Card” (included in this Curriculum Guide) to each student. Have
them show different ways to decompose their fraction into a sum of fractions with the same
denominator. Have students record each decomposition using an equation and a fraction
diagram.
Grade 4 Fractions
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Columbus City Schools 2013-2014
RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 564-567, 574-577
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. 125, 127
Practice Master pp. 125, 127
Problem Solving Master pp. 125, 127
Reteaching Master pp.
Grade 4 Fractions
Page 29 of 41
Columbus City Schools 2013-2014
Problem Solving Questions
5 1
1 3
= + +
6 6
6 6
Write at least three other ways you can add sixths together to get
Jackie and Donna need 6
and Donna has 4
5
.
6
3
2
feet of yarn to make puppets. Jackie has 2 feet of yarn
8
8
2
feet of yarn. How much yarn should they expect to have leftover?
8
1
pizzas left over from her birthday party. She passed out some of the
8
4
leftover pizza to her friends to take home and now she has 1 pizza left. How much
8
Taylor had 3
did Taylor pass out to her friends?
Use a fraction bar model to show the answer to 3
To make a Zombie Shake, you need
3
1
+2 .
4
4
3
2
1
cup of milk, cup of root beer, and cup of
4
4
4
hot sauce. How much liquid is needed?
Grade 4 Fractions
Page 30 of 41
Columbus City Schools 2013-2014
Problem Solving Questions
The pizza delivery guy lost control of his van around the curve and all his pizzas fell
into the road. He lost 12 pizzas that were cut into sixths. The neighborhood kids found
46 of the sixths. How many of the whole pizzas were they able to recover? How much
pizza was still missing?
During a relay race, a team of 6 students each ran
1
of a mile. How long was the race?
4
Use a number line model to show how you got the answer.
During a relay race, a team of students each ran
1
of a mile. If the race was 4 miles
3
long, how many students were on the team?
Harriet made 4 batches of cookies. If each batch required
2
of a cup of butter, how
3
much butter did Harriet have to use? Use a fraction bar model to show how you got
your answer. Write a multiplication equation to represent your answer.
Pete solved a story problem and wrote the answer of
1
5
× 5 = . What might the story
4
4
problem have been?
Grade 4 Fractions
Page 31 of 41
Columbus City Schools 2013-2014
Problem Solving Questions Answers
Answer Key
5 1
1 3
5
= + +
Write at least three other ways you can add sixths together to get .
6 6
6 6
6
Answers will vary. Possible equations:
5
1
1
1 1
1
5
1 1
1 2
= + + + +
= + + +
6
6
6
6 6
6
6
6 6
6 6
5
1 4
= +
6
6 6
Jackie and Donna need 6
4
3
2
feet of yarn to make puppets. Jackie has 2 feet of yarn and Donna has
8
8
2
feet of yarn. How much yarn should they expect to have leftover?
8
Answer: They should have
1
of a foot leftover.
8
1
pizzas left over from her birthday party. She passed out some of the leftover pizza to
8
4
her friends to take home and now she has 1
pizza left. How much did Taylor pass out to her
8
friends?
5
13
Answer: Taylor passed out 1 , or
, to her friends.
8
8
Taylor had 3
Use a fraction bar model to show the answer to 3
3
1
+2 .
4
4
Answer: 6
3
4
3
2
1
4
3
4
To make a Zombie Shake, you need
3
4
cup of milk,
2
4
=
1
4
=
4
4
=1
5+1=6
cup of root beer, and
1
cup
4
of hot sauce. How
much liquid is needed?
Answer: You need 1
Grade 4 Fractions
2
4
cups of liquid.
3
4
6
4
+
=
2
4
4
4
+
+
1
4
2
4
=
6
4
=1+
Page 32 of 41
2
4
Columbus City Schools 2013-2014
Problem Solving Questions Answers
The pizza delivery guy lost control of his van around the curve and all his pizzas fell into the road. He
lost 12 pizzas that were cut into sixths. The neighborhood kids found 46 of the sixths. How many of
the whole pizzas were they able to recover? How much pizza was still missing?
46
6
Answer: 1) They recovered 7 whole pizzas.
2) 4
2
6
=7
4
6
of the pizza was still missing. There were 72 sixths in 12 pizzas.
72/6 – 46/6 = 26/6
During a relay race, a team of 6 students each ran
1
of a mile.
4
26/6 = 4 2/6
How long was the race? Use a number
line model to show how you got the answer.
Answer: The race was 1 ½ miles long. Accept
0
¼
2/4
¾
4/4
During a relay race, a team of runners each ran
6
4
5/4
1
3
6/4
7/4
of a mile. If the race was 4 miles long, how many
runners were on the team?
Answer: There were 12 runners on the team.
?×
1
3
= 4 miles
How many thirds of a mile in 4 miles?
Harriet made 4 batches of cookies. If each batch required
2
3
of a cup of butter, how much butter did
Harriet have to use? Use a fraction bar model to show how you got your answer. Write a
multiplication equation to represent your answer.
Answer: Harriet needed 2 2/3 cups of butter.
2/3 × 4 = 8/3
8/3 = 2 2/3
Pete solved a story problem and wrote the answer of
1
×5
4
=
5
.
4
What might the story problem have
been?
Answers will vary. Answers should show an understanding of the multiplication of fractions as
repeated addition and that
Grade 4 Fractions
5
4
is a multiple of
1
4
.
Page 33 of 41
Columbus City Schools 2013-2014
Problem-Solving with Cuisenaire Rods
Name_______
__
Build the orange Cuisenaire rod train and use it to help you solve these problems. Write an
equation to match the story problem.
Amy had one whole Cuisenaire candy bar. She
gave
4
10
of it to Nikyel and
2
of it
10
Sarah has
to Emily.
How much of the candy bar does she have left?
has
3
10
2
5
of a Cuisenaire candy bar. Daniel
of a Cuisenaire candy bar. Together do
they have enough to equal a whole candy bar?
Paul gave Katherine
4
10
of a Cuisenaire candy
bar. Graham gave Katherine
5
10
2
10
of a Cuisenaire candy bar.
James had 3 times as much Cuisenaire candy
bar. How much Cuisenaire candy bar did
James have?
Grade 4 Fractions
9
10
of a Cuisenaire gave Melissa
candy bar. How much of the candy bar does
Katherine have now?
Amelia had
Olivia had
of a Cuisenaire candy bar. She
6
10
of the candy bar. How much
of the candy bar does Olivia have left?
If 5 students each had
3
10
of a Cuisenaire
candy bar, how many Cuisenaire candy bars do
they have in all?
Page 34 of 41
Columbus City Schools 2013-2014
Problem-Solving with Cuisenaire Rods
Answer Key
Name_______
__
Build the orange Cuisenaire rod train and use it to help you solve these problems. Write an
equation to match the story problem.
Amy had one whole Cuisenaire candy bar. She
gave
4
10
of it to Nikyel and
2
of it
10
Sarah has
to Emily.
How much of the candy bar does she have left?
has
3
10
2
5
of a Cuisenaire candy bar. Daniel
of a Cuisenaire candy bar. Together do
they have enough to equal a whole candy bar?
10/10-4/10=6/10
6/10-2/10=4/10
Amy has 4/10 candy bar left.
Paul gave Katherine
4
10
2/5=4/10
4/10 + 3/10= 7/10
No, they do not have enough to equal a
whole candy bar.
of a Cuisenaire candy
bar. Graham gave Katherine
5
10
4/10 + 5/10 = 9/10
Katherine has 9/10 of a Cuisenaire candy
bar.
2
10
9
10
of a Cuisenaire gave Melissa
candy bar. How much of the candy bar does
Katherine have now?
Amelia had
Olivia had
of a Cuisenaire candy bar.
James had 3 times as much Cuisenaire candy
bar. How much Cuisenaire candy bar did
James have?
of a Cuisenaire candy bar. She
6
10
of the candy bar. How much
of the candy bar does Olivia have left?
9/10 – 6/10 = 3/10
Olivia has 3/10 of a candy bar left.
If 5 students each had
3
10
of a Cuisenaire
candy bar, how many Cuisenaire candy bars do
they have in all?
3/10 + 3/10 + 3/10 + 3/10 + 3/10 = 15/10 =
10/10 + 5/10 = 1 ½ candy bars.
2/10 + 2/10 + 2/10 = 6/10 or
3 x 2/10 = 6/10
James has 6/10 of a Cuisenaire candy bar.
Grade 4 Fractions
Page 35 of 41
Columbus City Schools 2013-2014
Fill‘er Up
Name
1. 1 cup + 1 cup =
4
4
6. 1 cup + 1 cup + 1 cup =
8
8
8
2. 1 cup + 1 cup + 1 cup =
3
3
3
7. 1 cup + 1 cup =
2
2
3. 1 cup + 1 cup =
3
3
8. 3 cup + 1 cup =
4
4
4. 1 cup + 1 cup =
8
8
9. 2 cup + 1 cup =
4
4
5. 1 cup + 1 cup + 1 cup =
4
4
4
10. 1 cup + 1 cup + 1 cup + 1 cup =
2
2
2
2
11. On the back of this paper, write an addition sentence where the sum is greater
than 1. Explain how you know the sum is greater than 1.
CHALLENGE:
1 cup + 1 cup + 1 cup + 1 cup =
3
2
4
4
Grade 4 Fractions
Page 36 of 41
Columbus City Schools 2013-2014
Fill ‘er Up
Answer Key
Name
1. 1 cup + 1 cup = 2 cup or 1 cup
4
4
4
2
6. 1 cup + 1 cup + 1 cup = 3 cup
8
8
8
8
2. 1 cup + 1 cup + 1 cup = 3 cup or 1 cup 7. 1 cup + 1 cup = 2 cup or 1 cup
3
3
3
3
2
2
2
3. 1 cup + 1 cup = 2 cup
3
3
3
8. 3 cup + 1 cup = 4 cup or 1 cup
4
4
4
4. 1 cup + 1 cup = 2 cup or 1 cup
8
8
8
4
9. 2 cup + 1 cup = 3 cup
4
4
4
5. 1 cup + 1 cup + 1 cup = 3 cup 10. 1 cup + 1 cup + 1 cup + 1 cup =
4
4
4
4
2
2
2
2
4 cup or 2 cups
2
11. On the back of this paper, write an addition sentence where the sum is greater
than 1. Explain how you know the sum is greater than 1.
CHALLENGE:
1 cup + 1 cup + 1 cup + 1 cup = 1 1 cups
3
3
2
4
4
Grade 4 Fractions
Page 37 of 41
Columbus City Schools 2013-2014
Trailhead
Name_______________________
TRAILHEAD
Trailhead to picnic 2/8 mile
table
Picnic table to
3/8 mile
pond
Bear to END OF
2/8 mile
TRAIL
Trailhead to END 1 mile
OF TRAIL
POND
BEAR
END OF TRAIL
Use the above map to answer the following questions.
1. The hiker wants to walk exactly 5/8 mile from the Trailhead. Where will he end
up? ___________________________
2. How far is it from the Pond to the Bear? ________________________
3. If the hiker walks from the trailhead to the bear, then turns around and runs back
to the pond, how far did the hiker travel? ___________________
4. How far is it from the Trailhead to the bear? _______________________
5. The hiker leaves the trailhead at 11:45 a.m. If it takes the hiker ¼ hour to walk
from the Trailhead to the picnic table, ¼ hour to walk from the picnic table to the
pond, and ¼ hour to walk from the pond to the bear, what time will the hiker meet
the bear? _____________________________________
Grade 4 Fractions
Page 38 of 41
Columbus City Schools 2013-2014
Trailhead
Answer Key
Name_______________________
TRAILHEAD
POND
Trailhead to picnic 2/8 mile
table
Picnic table to
3/8 mile
pond
Bear to END OF
2/8 mile
TRAIL
Trailhead to END 1 mile
OF TRAIL
BEAR
END OF TRAIL
Use the above map to answer the following questions.
1. The hiker wants to walk exactly 5/8 mile from the Trailhead. Where will he end
up? At the pond
2. How far is it from the Pond to the Bear? 1/8 mile
3. If the hiker walks from the trailhead to the bear, then turns around and runs back
to the pond, how far did the hiker travel? 7/8 mile
4. How far is it from the Trailhead to the bear? 6/8 mile
5. The hiker leaves the trailhead at 11:45 a.m. If it takes the hiker ¼ hour to walk
from the Trailhead to the picnic table, ¼ hour to walk from the picnic table to the
pond, and ¼ hour to walk from the pond to the bear, what time will the hiker meet
the bear? 12:30 p.m.
Grade 4 Fractions
Page 39 of 41
Columbus City Schools 2013-2014
Recipe Cards
Peanut Butter Cookies
Banana Bread
cup butter
1 cup peanut butter
2
1 cup sugar
4
3 cup brown sugar
4
1 egg
1 teaspoon vanilla
2
1 1 4 cups of flour
3 teaspoon soda
4
1 teaspoon salt
8
1
cup shortening
1 cup sugar
2
2 eggs
1 3 4 cups flour
1 teaspoon soda
4
1 teaspoon salt
2
1 1 4 cups of banana
2 cup chopped walnuts
3
1
2
2
Cream together first six ingredients.
Add dry ingredients. Blend well. Shape
into one-inch balls. Bake at 350º for 8 –
10 minutes. Makes 48 cookies.
Cream together everything but the flour
and nuts. Add flour slowly. Stir in the
nuts. Pour into greased loaf pans. Bake
at 350º for 50 minutes. Makes two
loaves.
Waffles
Island Broiled Chicken
1 3 4 cups of flour
2 1 8 teaspoon baking powder
1 teaspoon salt
2
2 eggs
1 2 3 cups of milk
1 cup of vegetable oil
3
1 teaspoon of vanilla
2
cup salad oil
2 1 2 tablespoons lime juice
1 1 2 tablespoons fresh ginger
1 small garlic clove
3 teaspoon oregano
4
1 teaspoon salt
4
1 teaspoon pepper
8
2 chicken fryers, cut in half
Mix all ingredients with a beater for five
minutes until very fluffy. Makes 6
waffles.
Combine ingredients in a bowl. Add
chicken and marinate. Broil chicken
about 25 minutes. Serves 4.
Grade 4 Fractions
1
2
Page 40 of 41
Columbus City Schools 2013-2014
Fraction Cards
3
4
5
6
5
9
6
9
7
9
8
9
4
5
4
8
5
8
6
8
7
8
5
7
6
7
6
10
7
10
9
10
Grade 4 Fractions
Page 41 of 41
Columbus City Schools 2013-2014
Mathematics Model Curriculum
Grade Level: Fourth Grade
Grading Period: 3
Common Core Domain
Time Range: 10 Days
Number and Operations - Fractions
Common Core Standards
Understand decimal notation for fractions, and compare decimal fractions.
5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique
to add two fractions with respective denominators 10 and 100.4 For example, express 3/10 as 30/100, and add
3/10 + 4/100 = 34/100.
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe
a length as 0.62 meters; locate 0.62 on a number line diagram.
7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only
when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <,
and justify the conclusions, e.g., by using a visual model.
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Students develop understanding of fraction equivalence and operations with fractions. They recognize that two
different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing
equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions,
composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of
fractions and the meaning of multiplication to multiply a fraction by a whole number.
Content Elaborations
This section will address the depth of the standards that are being taught.
from ODE Model Curriculum
Ohio has chosen to support shared interpretation of the standards by linking the work of multistate partnerships as
the Mathematics Content Elaborations. Further clarification of the standards can be found through these reliable
organizations and their links:
Achieve the Core Modules, Resources
Hunt Institute Video examples
Institute for Mathematics and Education Learning Progressions Narratives
Illustrative Mathematics Sample tasks
National Council of Supervisors of Mathematics (NCSM) Resources, Lessons, Items
National Council of Teacher of Mathematics (NCTM) Resources, Lessons, Items
Partnership for Assessment of Readiness for College and Careers (PARCC) Resources, Items
Expectations for Learning (Tasks and Assessments)
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Grade 4 Decimals and Fractions
Page 1 of 51
Columbus City Schools 2013-2014
Mathematics Model Curriculum
Expectations for Learning (in development) from ODE Model Curriculum
Ohio has selected PARCC as the contractor for the development of the Next Generation Assessments for
Mathematics. PARCC is responsible for the development of the framework, blueprints, items, rubrics, and scoring
for the assessments. Further information can be found at Partnership for Assessment of Readiness for College and
Careers (PARCC). Specific information is located at these links:
Model Content Framework
Item Specifications/Evidence Tables
Sample Items
Calculator Usage
Accommodations
Reference Sheets
Sample assessment questions are included in this document.
The following website has problem of the month problems and tasks that can be used to assess students and help
guide your lessons.
http://www.noycefdn.org/resources.php
http://illustrativemathematics.org
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx
http://nrich.maths.org
At the end of this topic period students will demonstrate their understanding by….
Some examples include:
Constructed Response
Performance Tasks
Portfolios
***A student’s nine week grade should be based on multiple academic criteria (classroom assignments and teacher made
assessments).
Instructional Strategies
Columbus Curriculum Guide strategies for this topic are included in this document.
Also utilize model curriculum, Ohiorc.org, textbooks, InfoOhio, etc. as resources to provide instructional
strategies.
Websites
http://illuminations.nctm.org
http://illustrativemathematics.org
http://insidemathematics.org
http://www.lessonresearch.net/lessonplans1.html
Instructional Strategies from ODE Model Curriculum
The place value system developed for whole numbers extends to fractional parts represented as decimals.
This is a connection to the metric system. Decimals are another way to write fractions. The place-value
system developed for whole numbers extends to decimals. The concept of one whole used in fractions is
Grade 4 Decimals and Fractions
Page 2 of 51
Columbus City Schools 2013-2014
Mathematics Model Curriculum
extended to models of decimals.
Students can use base-ten blocks to represent decimals. A 10 × 10 block can be assigned the value of one
whole to allow other blocks to represent tenths and hundredths. They can show a decimal representation
from the base-ten blocks by shading on a 10 × 10 grid.
Students need to make connections between fractions and decimals. They should be able to write decimals
for fractions with denominators of 10 or 100. Have students say the fraction with denominators of 10 and
4
27
100 aloud. For example
would be “four tenths” or
would be “twenty-seven hundredths.” Also,
10
100
4
have students represent decimals in word form with digits and the decimal place value, such as
would
10
be 4 tenths.
Students should be able to express decimals to the hundredths as the sum of two decimals or fractions. This
is based on understanding of decimal place value. For example, 0.32 would be the sum of 3 tenths and 2
3
2
hundredths. Using this understanding students can write 0.32 as the sum of fractions (
+
).
10
100
Students’ understanding of decimals to hundredths is important in preparation for performing operations
with decimals to hundredths in Grade 5.
In decimal numbers, the value of each place is 10 times the value of the place to its immediate right,
Students need an understanding of decimal notations before they try to do conversions in the metric system.
Understanding of the decimal place value system is important prior to the generalization of moving the
decimal point when performing operations involving decimals.
Students extend fraction equivalence from Grade 3 with denominators of 2, 3, 4, 6 and 8 to fractions with a
denominator of 10. Provide fraction models of tenths and hundredths so that students can express a fraction
with a denominator of 10 as an equivalent fraction with a denominator of 100.
When comparing two decimals, remind students that as in comparing two fractions, the decimals need to
refer to the same whole. Allow students to use visual models to compare two decimals. They can shade in
a representation of each decimal on a 10 × 10 grid. The 10 × 10 grid is defined as one whole. The decimal
must relate to the whole.
Flexibility with converting fractions to decimals and decimals to fractions provides efficiency in solving
problems involving all four operations in later grades.
Grade 4 Decimals and Fractions
Page 3 of 51
Columbus City Schools 2013-2014
Mathematics Model Curriculum
Instructional Resources/Tools from ODE Model Curriculum
Length or area models
10 × 10 square on a grid
Decimal place-value mats
Base-ten blocks
Number lines
Misconceptions/Challenges
The following are some common misconceptions for this topic. As you teach the lessons, identify the misconceptions/challenges that
students have with the concepts being taught.
Common Misconceptions from ODE Model Curriculum
Students treat decimals as whole numbers when making comparison of two decimals. They think the longer the number, the greater the
value. For example, they think that 0.03 is greater than 0.3.
Please read the Teacher Introductions, included in this document, for further understanding.
Grade 4 Decimals and Fractions
Page 4 of 51
Columbus City Schools 2013-2014
Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
Grade 4 Decimals and Fractions
Page 5 of 51
Columbus City Schools 2013-2014
Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
Equal
Groups
Arrays, 4
Area, 5
Compare
3×6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
a × ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? × b = p, and p ÷ b = ?
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
Number of Groups Unknown (“How
many groups?” Division
Group Size Unknown
(“How many in each group?”
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Unknown Product
a×b=?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
Grade 4 Decimals and Fractions
Page 6 of 51
Columbus City Schools 2013-2014
Group Size Unknown: (a × ? = p and p ÷ a = ?)
In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
students used math tools to solve the problem, they should include a picture to represent how the
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tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
Act It Out
Make a Table
Find a Pattern
Guess and Check
Make a List
Draw a Picture
When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
to act it out. Occasionally whole group thinking with the teacher modeling how to record
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strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Teacher Introduction
The Fraction-Decimal Relationship
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Number and Operations - Fractions domain is helpful when assessing
student work and student thinking. This information could be used to guide classroom
discussions, understand student misconceptions and provide differentiation opportunities. The
focus of instruction is the Common Core Mathematics Standards.
Connecting fractions and decimals as different representations of the same quantity will help
students develop a broader view of fractions and decimals. Students in the fourth grade are
expected to identify and generate equivalent forms of fractions and decimals. For example, a
student will understand and explain that ten tenths is the same as one whole in both fraction and
decimal form.
The most commonly used decimal model is the 10 by 10 grid. On the grid, students can color in
or cover up parts to represent different fractions and decimals. On the grid, students can see that
the grid represents a whole (100/100 or 1.0). The covered or colored quantities can be expressed
as the part of the whole.
Example:
marbles
I have 60 red marbles out of 100 marbles in all. Represent the number of red
I have as a fraction and decimal.
Fraction = 60 100
Decimal = 0.60
All three of the above models represent 60 out of 100. There are also other models that could be
made. As long as 60 out of 100 squares are shaded the model represents sixty hundredths.
In addition to the 10 by 10 grid, the Hundredths Disk is an invaluable decimal-fraction model
and is actually just a round version of the familiar 10 by 10 grid. The disk is broken into 100
equal parts with lines at the multiples of 10 to represent tenths. This enables students to show
amounts less than one.
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Example: Miyesha missed 7 out of 100 problems on her test. Represent the number of
problems Miyesha missed as a fraction and a decimal.
Fraction = 7 100
Decimal = 0.07
Students could also be asked to give the fraction and decimal that represent the number of
questions that she got correct on the test.
Fraction = 93100
Decimal = 0.93
Students can use models to create equivalencies. By using a 1 by 10 grid and a 10 by 10 grid or
a circle divided into tenths and the hundredths disk, students will see that 0.4 equals 0.40
( 4 10 40100 ).
=
0.4
0.40
=
0.4
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0.40
Columbus City Schools 2013-2014
The meter stick can also be used as a length model for decimals and fractions. The metric
system relies on our base ten number system, as each unit is a multiple of 10 (0.001, 0.01, 0.1, 1,
10, 100, 1000, etc.). The meter is equal to one whole or 1.0. Units increase and/or decrease by
powers of 10. Again, units less than one meter can be expressed as fractions and decimals.
1 meter = 11 = 1.0
1 decimeter = 110 = 0.1
1 centimeter = 1100 = 0.01
1 millimeter = 11000 = 0.001
Students will be expected to translate freely between decimals and fractions. It is important that
students see the relationship between decimals and fractions so that they can express amounts
using either representation. When given a decimal, students will be expected to know the
fraction that is equivalent and vice versa.
Decimals
Fractions
Fractions
Decimals
Example: LaRhonda made 7 10 of her free throw shots at last night’s basketball game.
Represent the number of shots she made as a decimal.
Fraction = 7 10
Decimal = 0.7
Example: The barometric pressure rose 0.56 of an inch of mercury. Represent the increase in
the barometric pressure as a fraction.
Fraction = 56 100
Decimal = 0.56
Percents
Although percents are not formally assessed until the 5-7 grade band, they are a natural extension
of fraction-decimal relationships. Percents are an additional way to express the same
mathematical ideas that are represented by fractions and decimals. Another way to express a
fraction, percents can also be expressed as a part-to-whole ratio. In the case of a percent, the
whole is equal to 100.
Fraction = Part/Whole = Percent (out of 100)
Percent means per hundred. One way to remember the meaning of percent is to recognize that
there are 100 cents in one whole dollar.
100
100
= 1 whole = 1.0 = 100 % = $1.00
50 pennies = 50100 = one-half = .50 = 50% = $0.50
25 pennies = 25100 = one-fourth = .25 = 25% = $0.25
200 pennies = 200100 = 2 whole = 200% = $2.00
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Percents are an expanded idea of place value. Each one percent equals one-hundredth of the
whole.
43
= 43 hundredths = 0.43 = 43 %
100
Once students understand that percents are always out of 100 it is important to show them that
not all percents are whole numbers. To change any decimal to a percent, students can multiply
the decimal by 100.
943
1000
= 934 thousandths = 0.934 = 93.4 %
Many students rely on an algorithm to change a fraction/decimal to a percent without the
conceptual understanding. As students construct the fraction-decimal-percent relationship, they
will not need to rely on an algorithm, but can draw on their own knowledge. Developing
conceptual understanding will allow students to recognize the meaning of what they are doing
and make them more able to recognize errors.
Example: The movie theater tallied up 100 movie snack sales on Friday night. 83 people
bought popcorn, 10 people bought candy, while the remaining 7 people purchased
hot dogs. Express the sales of each snack as a fraction, decimal, and a percent.
Cameron wrote the following answers:
Popcorn: 83100 = 0.83 = 83%
Candy: 10 100 = 0.10 or 0.1 = 10%
Hot Dogs: 7 100 = 0.7 = 70%
Cameron followed the algorithm: Divide the numerator by the denominator and
move the decimal point two spaces to the right. However, after he checked his work
Cameron recognized that the sum of the fractions must equal 100/100, the sum of the
decimals must equal 1.0, and the sum of the percents must equal 100. He realized
that he made an error expressing the total number of hot dogs as a decimal and a
percent. The actual answer should be 7/100 = 0.07 = 7%. His understanding of the
concepts allowed him to fix the error that he had made
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PRACTICE ASSESSMENT ITEMS
Which decimal is equivalent to
1
?
2
 A. 0.33
 B. 0.50
 C. 0.75
 D. 0.85
Represent 4 tenths and 40 hundredths on the models below.
10ths circle
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which decimal is equivalent to
1
?
2
 A. 0.33
 B. 0.50
 C. 0.75
 D. 0.85
Answer: B
Represent 4 tenths and 40 hundredths on the models below.
10ths circle
100ths circle
Answer:
=
0.4
0.40
A 2-point response shows the correct shading of 0.4 and 0.40.
A 1-point response contains a minor error.
A 0-point response shows no mathematical understanding of the task.
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PRACTICE ASSESSMENT ITEMS
Which model is equivalent to 0.60?
 A.
 B.
 C.
 D.
Using numbers, pictures, and/or words show one equivalent decimal and one
equivalent fraction for three tenths.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubric
Which model is equivalent to 0.60?
 A.
 B.
 C.
 D.
Answer: C
Using numbers, pictures, and/or words show one equivalent decimal and one
equivalent fraction for three tenths.
Answer: Student answers will vary.
A 2-point response shows a drawing, words or numbers to illustrate three tenths as a
3
fraction ( ) and a decimal (0.3).
10
A 1-point response shows a minor error in mathematical understanding.
A 0-point response shows no mathematical understanding of this task.
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PRACTICE ASSESSMENT ITEMS
Create a number line and explain how to order the following numbers
3.5,
3
1 ,
4
0.5,
5
1
2
Draw a model that shows 0.4 < 0.6.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Create a number line and explain how to order the following numbers
3
1
5
3.5, 1 , 0.5,
4
2
Answer:
3.5
0 0.5
3
1
4
5
1
6
2
A 4-point response shows all four numbers in the correct place on the number line and has
a reasonable explanation.
A 3-point response has no more than one number in the wrong place with a strong
explanation.
A 2-point response has no more than 2 numbers in the wrong place with a weak
explanation, or has all numbers in the right place with no explanation or has all numbers in
the correct order but at incorrect places on the number line.
A 1-point response shows major errors.
A 0-point response shows no understanding of the task.
Draw a model that shows 0.4 < 0.6.
Answer:
A 2-point response draws two models that are the same size to show the area that
represents four-tenths is smaller than the area that represents six-tenths.
A 1-point response contains a minor error or draws two models that are same size and
incorrectly shades one of the models or correctly shades two models that are not
approximately the same size.
A 0-point response shows no mathematical understanding of the task.
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PRACTICE ASSESSMENT ITEMS
0.9
one-third
Write the standard fraction form for each of the above fractions. Then compare
and order the fractions from least to greatest. Explain how you determined the
1
order of these fractions using the benchmarks of 0, , and 1.
2
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
0.9
6/8
one-third
9/10
1/3
Write the standard fraction form for each of the above fractions. Then compare
and order the fractions from least to greatest. Explain how you determined the
1
order of these fractions using the benchmarks of 0, , and 1.
2
Answer: Fractions ordered from least to greatest: 1/3, 6/8, 9/10
Includes an explanation using the benchmarks of 0, ½, and 1 (e.g., the fraction 1/3
is less than ½ which makes it the smallest fraction. The other two fractions are
greater than one-half. 9/10 is one away from a whole and 6/8 is two away from a
whole, so 9/10 is greater than 6/8.
A 4-point response includes correctly writing (6/8, 9/10, and 1/3) and ordering them from
least to greatest (1/3, 6/8, 9/10) and includes a complete explanation using the benchmarks
of 0, ½ and 1. (The fraction 1/3 is less than ½ which makes it the smallest fraction. The
other two fractions are greater than 1/2 and 9/10 is only one tenth away from a whole or 1
which makes it the greatest of the three).
A 3-point response includes correctly writing (6/8, 9/10, and 1/3) and ordering them from
least to greatest (1/3, 6/8, 9/10) and includes partial understanding of the explanation given
above.
A 2-point explanation includes correctly writing (6/8, 9/10, and 1/3) and ordering them
from least to greatest (1/3, 6/8, 9/10) and an incorrect or missing explanation.
A 1-point response includes correctly writing or ordering the fractions with an incorrect or
no explanation.
A 0-point response shows no mathematical understanding of this task.
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PRACTICE ASSESSMENT ITEMS
Which decimal is represented below?
 A. 0.23
 B. 0.33
 C. 0.67
Tiffany has 10 fish. If 6 of the fish are goldfish, which picture represents the
number of fish that are not goldfish? The shaded boxes represent Tiffany’s
goldfish.
 A.
 B.
 C.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which decimal is represented below?
 A. 0.23
 B. 0.33
 C. 0.67
Answer: B
Tiffany has 10 fish. If 6 of the fish are goldfish, which picture represents the
number of fish that are not goldfish? The shaded boxes represent Tiffany’s
goldfish.
 A.
 B.
C
Answer: B
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PRACTICE ASSESSMENT ITEMS
Add.
 A. 0.81
 B. 0.70
 C. 0.80
Wendy has $1.00 to spend at the school store. She buys a ruler for $0.52. Which
picture represents the amount of money she has left?
 A.
 B.
 C.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Add.
 A. 0.81
 B. 0.70
 C. 0.80
Answer: C
Wendy has $1.00 to spend at the school store. She buys a ruler for $0.52. Which
picture represents the amount of money she has left?
 A.
 B.
 C.
Answer: B
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PRACTICE ASSESSMENT ITEMS
Which set of decimals below shows 0.4, 5.51, 0.051, 4.05 in order from greatest to
least?
 A. 0.051, 0.4, 4.05, 5.51
 B. 0.4, 5.51, 0.051, 4.05
 C. 5.51, 4.05, 0.051, 0.4
 D. 5.51, 4.05, 0.4, 0.051
Kyle ran a race in 36.8 seconds. Andrew ran the race in 50.4 seconds. Joe ran the
race in 36.6 seconds. Use the times and symbols (<, >) to write an expression that
shows the relationship among the three times.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which set of decimals below shows 0.4, 5.51, 0.051, 4.05 in order from greatest to
least?
 A. 0.051, 0.4, 4.05, 5.51
 B. 0.4, 5.51, 0.051, 4.05
 C. 5.51, 4.05, 0.051, 0.4
 D. 5.51, 4.05, 0.4, 0.051
Answer: D
Kyle ran a race in 36.8 seconds. Andrew ran the race in 50.4 seconds. Joe ran the
race in 36.6 seconds. Use the times and symbols (<, >) to write an expression that
shows the relationship among the three times.
Answer: There are two possible answers; 50.4 > 36.8 > 36.6 or 36.6 < 36.8 < 50.4
A 2-point response shows the correct answer of 50.4 > 36.8 > 36.6 or 36.6 < 36.8 < 50.4
A 1-point response contains a minor error or states that 50.4 is the largest number, 36.8 is
the middle number, and 36.6 is the smallest number, without using symbols.
A 0-point response shows no mathematical understanding of the task.
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: decimal, fraction, part, whole, tenth(s), hundredth(s), digit, numeral, equal
parts, denominator, numerator, least, greatest, place value
1. The word problems below can be presented to students in a variety of ways. Some options
include: using the questions to create a choice-board, Math-O board, one problem per class
session, one problem each evening for homework, partner work or as an assessment question.
The strength of problem solving lies in the rich discussion afterward. As the school year
progresses students should be able to justify their own thinking as well as the thinking of
others. This can be done through comparing strategies, arguing another student’s solution
strategy or summarizing another student’s sharing.
Sara got
8
10
of her math problems correct on Monday’s test. If Friday’s test has 100
questions, how many problems will she need to get correct to receive the same score
as Monday’s test? Explain how you got your answer.
Tessa claims that
9
10
is the same as
9
.
100
Use the tens and hundreds grid to prove to
Tessa that this cannot be correct.
52
100
Draw a star on the number line where
would be located. Explain how you
decided where to place the star.
0
0.1
0.2
0.3
0.4
0.5
0.6
An electrician used one wire that measured
measured
16
100
0.7
2
10
0.8
0.9
1.0
of a meter and a second wire that
of a meter. How much wire did he use altogether?
Draw a model to compare 0.07 and 0.4. Then complete the equation with <, >, or =.
0.07 ___ 0.4
During science, Divinity measured out
3
10
43
100
of a liter of hot water and Jacob measured
of a liter of oil. When they combined the two liquids how much did they have?
Solve the following equation:
The hair stylist cut
5
10
6
10
+
22
100
+
1
10
of an inch off Sharelle’s hair. Sharelle wanted it to be shorter,
so the stylist trimmed off another
30
100
of an inch. How much did she cut altogether?
Franklin solved an addition equation and got an answer of
55
.
100
Unfortunately, he
forgot which two fractions he added together to arrive at that answer. He remembers
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that one addend had a denominator of ten and the second addend had a denominator
of 100. What could Franklin’s equation have been?
A swimmer swam the backstroke in 43.35 seconds. She won the race by 0.6 of a
second. What was the second place swimmer’s time?
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 4 on the right side of the page. Open “Unit 5
Framework Student Edition.” These tasks are from the Georgia Department of Education.
The tasks build on one another. You may or may not use all of the tasks. There are four
different types of tasks. Scaffolding tasks build up to the constructing tasks which develop a
deep understanding of the concept. Next, there are practice tasks and finally performance
tasks which are a summative assessment for the unit. Select the tasks that best fit with your
lessons and the standards being taught at that time.
3. Demonstrate the similarities of the two systems of representation (fractions and decimals).
Distribute four “Fraction and Decimal Bars” or “Decimal Grids” (included in this Curriculum
Guide) that are divided into 100 equal parts. Fold the first bar into two equal parts. Shade
one half of the bar. Write a fraction and a decimal to represent each area ( 50100 , 0.50). Fold
the second bar into fourths. Shade in one-fourth of the bar. Again represent each of the areas
with the different systems of representation ( 25100 , 0.25). Continue with thirds and fifths.
4.
5.
6.
7.
Have students line all the bars vertically and have students compare the fractions or decimals
ordering from least to greatest. Practice this same concept using “Decimal Grids” (included
in this Curriculum Guide).
Divide students into pairs and distribute the “Decimal Cards” sheet (included in this
Curriculum Guide) to each pair. Have students cut the “Decimal Cards” in half on the dotted
line, giving each partner 8 cards. Each student will use the “Decimal Grids” (included in this
Curriculum Guide) to represent the numeral on their cards. Then partners will switch their
“Decimal Grids” and check to be sure that the answers are correct. Have students switch
back and order their decimals from least to greatest at the bottom of the page.
Distribute “Decimal Disk” template (included in this Curriculum Guide) and another sheet of
colored paper. Have students lay the disk on top of the colored sheet and cut out both circles
at the same time. Then have students cut along the bold line to the center of both circles.
Have students slip both circles together at the slits. Demonstrate how to use a “Decimal
Disk” to represent decimals and fractions. The disk is divided into 10 large parts (or tenths)
or 100 small parts (or hundredths). This is a circular representation of the hundredths grid.
Discuss how fractions can be represented 42 = 42 small chunks or 4 large chunks and 2
100
small chunks. Give the students fractions and decimals to represent on their disks. Then
read Alexander, Who Used to Be Rich Last Sunday by Judith Viorst. As the story is read,
the class creates a chart to represent the amount of money Alexander loses as a money
amount, fraction, and decimal. For example, Alexander spends fifteen cents on bubble gum
= 15 ¢ = 15 = .15. After the different amounts have been recorded have students model the
100
amount on their “Decimal Disk” and hold up for the teacher to check visually.
Give each student the “Decimal Squares” (included in this Curriculum Guide). Have
students cut out the grids and order the grids from least to greatest. Then have students
represent each grid as a decimal and order the decimal numbers from least to greatest.
Give each pair of students a set of “Decimal Squares” (included in this Curriculum Guide).
Students represent the value of each Decimal Square as a fraction and as a decimal. For
example:
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=
16
or .16
100
8. Copy and distribute a set of “Match Game Cards” (included in this Curriculum Guide) to
every two students. Instruct students to turn all cards face down. Player one turns over 2
cards. If the cards match player one keeps the cards and takes another turn. If the cards do
not match player 2 takes his or her turn. Play continues until all cards are matched. Pass out
“Match Game Follow Up” activity sheet (included in this Curriculum Guide) to all students
and follow the directions.
9. Distribute “Ways to Be Equal” (included in this Curriculum Guide) to each student. Have
the students color in 50 squares in the first decimal grid. Ask students what decimal they
colored in. Then have them represent the number of squares colored as a fraction and a
decimal (0.50, 50100 , 1 2 , 0.5). Continue naming different fractions and/or decimals. Have
students color in the fractions or decimals and represent each shaded grid as a fraction or a
decimal.
10. Distribute “Decimal Squares” (included in this Curriculum Guide). Call off different
decimal numbers. Have students select a different color crayon to color the portion of the
grid that corresponds to the decimal number that was called out. Have the students write the
number as a decimal and a fraction.
11. Write numbers up to a million through the hundredths on the board. Have students practice
reading and writing the word form of the numbers. Call out a specific place value and call on
students to identify the digit in that place. For additional practice distribute “What’s My
Number?” (included in this Curriculum Guide).
Grade 4 Decimals and Fractions
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Columbus City Schools 2013-2014
RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 624-627, 630-631
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp.
Practice Master pp. 137, 139
Problem Solving Master pp.
Reteaching Master pp. 137, 139
INTERDISCIPLINARY CONNECTIONS
1. Literature Connections
Alexander,Who Used to Be Rich Last Sunday by Judith Viorst
Fraction Action by Loreen Leedy
The Penny Pot by Stuart J. Murphy
“Smart” from Where the Sidewalk Ends by Shel Silverstein
Grade 4 Decimals and Fractions
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Columbus City Schools 2013-2014
Problem Solving Questions
Sara got 8/10 of her math problems correct on Monday’s test. If Friday’s test has 100
questions, how many problems will she need to get correct to receive the same score as
Monday’s test? Explain how you got your answer.
Tessa claims that 9/10 is the same as 9/100. Use the tens and hundreds grid to prove to
Tessa that this cannot be correct.
Draw a star on the number line where 52/100 would be located. Explain how you
decided where to place the star.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
An electrician used one wire that measured 2/10 of a meter and a second wire that
measured 16/100 of a meter. How much wire did he use altogether?
Draw a model to compare 0.07 and 0.4. Then complete the equation with <, >, or =.
0.07 ___ 0.4
Grade 4 Decimals and Fractions
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Columbus City Schools 2013-2014
Problem Solving Questions
During science, Divinity measured out
43
of a liter of hot water and Jacob measured
100
3
of a liter of oil. When they combined the two liquids how much did they have?
10
3
6
22
1
Solve the following equation:
+
+
10
10
100
10
5
of an inch off Sharelle’s hair. Sharelle wanted it to be shorter, so
10
30
the stylist trimmed off another
of an inch. How much did she cut altogether?
100
The hair stylist cut
Franklin solved an addition equation and got an answer of
55
. Unfortunately, he
100
forgot which two fractions he added together to arrive at that answer. He remembers
that one addend had a denominator of ten and the second addend had a denominator of
100. What could Franklin’s equation have been?
A swimmer swam the backstroke in 43.35 seconds. She won the race by 0.6 of a
second. What was the second place swimmer’s time?
Grade 4 Decimals and Fractions
Page 33 of 51
Columbus City Schools 2013-2014
Problem Solving Questions Answers
Sara got
8
10
Answer Key
of her math problems correct on Monday’s test. If Friday’s test has 100 questions, how
many problems will she need to get correct to receive the same score as Monday’s test? Explain how
you got your answer.
Answer: Sara will need to get 80/100 correct on Friday’s test.
9
10
Tessa claims that
is the same as
9
.
100
Use the tens and hundreds grid to prove to Tessa that this
cannot be correct.
.
Answer: Nine of the tens grid squares should be shaded and 90 of the hundreds grid should be
shaded to show the two numbers are equivalent.
Draw a star on the number line where
52
100
would be located. Explain how you decided where to place
the star.
0
0.1
0.2
0.3
0.4
0.5
An electrician used one wire that measured
0.6
2
10
0.7
0.8
0.9
1.0
of a meter and a second wire that measured
16
100
of a
meter. How much wire did he use altogether?
Answer: He used
36
100
of a meter.
2
20
=
10 100
20
100
+
16
100
=
36
100
Draw a model to compare 0.07 and 0.4. Then complete the equation with <, >, or =.
0.07 _<__ 0.4
Answer:
Grade 4 Decimals and Fractions
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Columbus City Schools 2013-2014
Problem Solving Questions Answers
During science, Divinity measured out
43
100
of a liter of hot water and Jacob measured
3
10
of a liter of
oil. When they combined the two liquids how much did they have?
Answer: They had
73
100
of a liter of liquid.
Solve the following equation:
Answer:
92
100
6
10
=
60
100
6
10
+
and
22
100
1
10
=
+
3
10
=
30
100
30
100
43
73
=
100 100
+
1
10
10
100
60
100
+
22
100
+
10
100
=
92
100
5
of an inch off Sharelle’s hair. Sharelle wanted it to
10
30
trimmed off another
of an inch. How much did she cut altogether?
100
80
5
50
Answer: The stylist cut
of an inch off Sharelle’s hair
=
100
100
10
The hair stylist cut
Franklin solved an addition equation and got an answer of
55
.
100
be shorter, so the stylist
50
100
+
30
100
=
80
100
Unfortunately, he forgot which two
fractions he added together to arrive at that answer. He remembers that one addend had a denominator
of ten and the second addend had a denominator of 100. What could Franklin’s equation have been?
Answers will vary. Possible equations:
25
100
+
3
55
=
100
10
;
5
100
+
5
10
=
55
15
;
100 100
+
4
55
=
100
10
A swimmer swam the backstroke in 43.35 seconds. She won the race by 0.6 of a second. What was
the second place swimmer’s time?
Answer:
The second place swimmer had a time of 43.95 seconds.
0.6 = 0.60
43.35 + .60 = 43.95
Grade 4 Decimals and Fractions
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Columbus City Schools 2013-2014
Fractions and Decimals Bars
Grade 4 Decimals and Fractions
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Columbus City Schools 2013-2014
Decimal Grids
Grade 4 Decimals and Fractions
Page 37 of 51
Columbus City Schools 2013-2014
Decimal Cards
Partner 1 Cards
.18
.72
.24
1.7
.38
.03
.69
.4
.81
.28
.44
.6
.14
1.4
.37
.07
Partner 2 Cards
Grade 4 Decimals and Fractions
Page 38 of 51
Columbus City Schools 2013-2014
Decimal Grids
Hundredths
Order decimals from least to greatest on the lines below.
_____ _____ _____ _____ _____ _____ _____ _____
least
Grade 4 Decimals and Fractions
greatest
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Columbus City Schools 2013-2014
Decimal Disk
Grade 4 Decimals and Fractions
Page 40 of 51
Columbus City Schools 2013-2014
Decimal Squares
(Hundredths)
Name___________________
__________________
__________________
__________________
__________________
__________________
___________________
_________________
__________________
___________________
___________________
Grade 4 Decimals and Fractions
___________________
Page 41 of 51
___________________
Columbus City Schools 2013-2014
Decimal Squares
(Hundredths)
Name
__________
__________
__________________
__________________
__________________
___________________
__________________
__________________
_____________________
__________________
___________________
Grade 4 Decimals and Fractions
Page 42 of 51
__________
Columbus City Schools 2013-2014
Decimal Squares
(Tenths)
Name____________________________
__________________
___________________
__________________
__________________
____________________
___________________
___________________
Grade 4 Decimals and Fractions
____________________
Page 43 of 51
___________________
Columbus City Schools 2013-2014
Match Game Cards
25
100
.25
twenty-five
hundredths
.50
fifty
hundredths
.10
ten
hundredths
thirty-two
hundredths
50
100
.75
seventyfive
hundredths
32
100
75
100
.33
thirty-three
hundredths
.72
72
100
Grade 4 Decimals and Fractions
Page 44 of 51
.32
10
100
Columbus City Schools 2013-2014
Match Game Card
90
100
.90
ninety
hundredths
.44
forty-four
hundredths
44
100
.61
sixty-one eighty-eight
hundredths hundredths
88
100
.20
twenty
hundredths
seventytwo
hundredths
33
100
Grade 4 Decimals and Fractions
Page 45 of 51
.88
61
100
20
100
Columbus City Schools 2013-2014
Match Game
Follow Up
Name
Color the 4 boxes that show the same representation the same color.
60
100
10
100
ten
hundredths
sixty
hundredths
forty-four
hundredths
44
100
Grade 4 Decimals and Fractions
.44
.10
61
100
.60
sixty-one
hundredths
.61
Page 46 of 51
Columbus City Schools 2013-2014
Match Game
Follow Up
Name
Color the 4 boxes that show the same thing the same color.
50
100
20
100
twenty
hundredths
ninety
hundredths
.20
25
100
.90
fifty
hundredths
.50
twenty-five
hundredths
90
100
Grade 4 Decimals and Fractions
.25
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Columbus City Schools 2013-2014
Ways to be Equal
Fraction ________ Decimal ________
Fraction ________ Decimal ________
Fraction ________ Decimal ________
Fraction ________ Decimal ________
Fraction ________ Decimal ________
Grade 4 Decimals and Fractions
Fraction ________ Decimal ________
Page 48 of 51
Columbus City Schools 2013-2014
Decimal Squares
(Hundredths)
Name
__________________
___________________
_________________
__________________
Grade 4 Decimals and Fractions
__________________
__________________
__________________
__________________
Page 49 of 51
__________________
___________________
___________________
__________________
Columbus City Schools 2013-2014
What’s My Number?
Name_____________________
Write the numeral that represents the number word. Compare your numbers with
the numbers at the bottom of the page.
1) one thousand, three hundred seventy-one and forty-five hundredths
2) one hundred thirty thousand, four hundred one
3) one hundred forty-four thousand, five hundred two and forty hundredths
4) fifteen thousand, four hundred forty-five and four tenths
5) thirteen thousand, four hundred one and fifty-five hundredths
6) five hundred fifty thousand, seven hundred forty-five and one tenth
7) three thousand, five hundred forty-five and four tenths
8) fourteen thousand, four hundred forty-one and forty-five hundredths
9) fourteen and four hundredths
10) one thousand, three hundred seventy-one and four tenths
14,441.45
144,502.40
14.04
1,371.45
550,745.1
3,545.4
13,401.45
130,401
15,445.4
Grade 4 Decimals and Fractions
1,371.4
Page 50 of 51
Columbus City Schools 2013-2014
What’s My Number?
Name_____________________
Answer Key
Write the numeral that represents the number word. Compare your numbers with the numbers at
the bottom of the page.
1) one thousand, three hundred seventy-one and forty-five hundredths
1,371.45
2) one hundred thirty thousand, four hundred one
130,401
3) one hundred forty-four thousand, five hundred two and forty-hundredths
144,502.40
4) fifteen thousand, four hundred forty-five and four tenths
15,445.4
5) thirteen thousand, four hundred one and fifty-five hundredths
14, 401.455
6) five hundred fifty thousand, seven hundred forty-five and one tenth
550,745.1
7) three thousand, five hundred forty-five and four tenths
3,545.4
8) fourteen thousand, four hundred forty-one and forty-five hundredths
14,441.45
9) fourteen and four hundredths
14.04 ______
10) one thousand, three hundred seventy-one and four tenths
1,371.4______
14,441.45
144,502.45
14.04
1,371.45
550,745.1
3,545.4
13,401.45
130,401
15,445.4
Grade 4 Decimals and Fractions
1,371.4
Page 51 of 51
Columbus City Schools 2013-2014
Mathematics Model Curriculum
Grade Level: Fourth Grade
Grading Period: 3
Common Core Domain
Time Range: 10 Days
Measurement and Data
Common Core Standards
Solve problems involving measurement and conversion of measurements from a larger unit
to a smaller unit.
1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml;
hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a
smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as
long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing
the number pairs (1, 12), (2, 24), (3, 36), ...
2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses
of objects, and money, including problems involving simple fractions or decimals, and problems that require
expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities
using diagrams such as number line diagrams that feature a measurement scale.
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Developing an understanding and fluency with multi-digit multiplication, and developing understanding of
dividing to find quotients involving multi-digit dividends
Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in
each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models),
place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use
efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers. Depending on the
numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate
products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain
why the procedures work based on place value and properties of operations; and use them to solve problems.
Students apply their understanding of models for division, place value, properties of operations, and the
relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable
procedures to find quotients involving multi-digit dividends. They select and accurately apply appropriate methods
to estimate and mentally calculate quotients, and interpret remainders based upon the context.
Content Elaborations
This section will address the depth of the standards that are being taught.
from ODE Model Curriculum
Ohio has chosen to support shared interpretation of the standards by linking the work of multistate partnerships as
the Mathematics Content Elaborations. Further clarification of the standards can be found through these reliable
organizations and their links:
Achieve the Core Modules, Resources
Hunt Institute Video examples
Institute for Mathematics and Education Learning Progressions Narratives
Grade 4 Measurement and Data
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Columbus City Schools 2013-2014
Mathematics Model Curriculum
Illustrative Mathematics Sample tasks
National Council of Supervisors of Mathematics (NCSM) Resources, Lessons, Items
National Council of Teacher of Mathematics (NCTM) Resources, Lessons, Items
Partnership for Assessment of Readiness for College and Careers (PARCC) Resources, Items
Expectations for Learning (Tasks and Assessments)
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Expectations for Learning (in development) from ODE Model Curriculum
Ohio has selected PARCC as the contractor for the development of the Next Generation Assessments for
Mathematics. PARCC is responsible for the development of the framework, blueprints, items, rubrics, and scoring
for the assessments. Further information can be found at Partnership for Assessment of Readiness for College and
Careers (PARCC). Specific information is located at these links:
Model Content Framework
Item Specifications/Evidence Tables
Sample Items
Calculator Usage
Accommodations
Reference Sheets
Sample assessment questions are included in this document.
The following websites have problem of the month problems, tasks and assessment questions that can be used to
assess students and help guide your lessons.
http://www.noycefdn.org/resources.php
http://illustrativemathematics.org
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx
At the end of this topic period students will demonstrate their understanding by….
Some examples include:
Constructed Response
Performance Tasks
Portfolios
***A student’s nine week grade should be based on multiple academic criteria (classroom assignments and teacher made
assessments).
Instructional Strategies
Columbus Curriculum Guide strategies for this topic are included in this document.
Also utilize model curriculum, Ohiorc.org, textbooks, InfoOhio, etc. as resources to provide instructional
strategies.
Websites
http://illuminations.nctm.org
http://www.insidemathematics.org
Grade 4 Measurement and Data
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Columbus City Schools 2013-2014
Mathematics Model Curriculum
http://illustrativemathematics.org
Instructional Strategies from ODE Model Curriculum
In order for students to have a better understanding of the relationships between units, they need to use
measuring devices in class. The number of units needs to relate to the size of the unit. They need to discover
that there are 12 inches in 1 foot and 3 feet in 1 yard. Allow students to use rulers and yardsticks to discover
these relationships among these units of measurements. Using 12-inch rulers and yardstick, students can see
that three of the 12-inch rulers, which is the same as 3 feet since each ruler is 1 foot in length, are equivalent
to one yardstick. Have students record the relationships in a two column table or t-charts. A similar strategy
can be used with rulers marked with centimeters and a meter stick to discover the relationships between
centimeters and meters.
Present word problems as a source of students’ understanding of the relationships among inches, feet and
yards.
Students are to solve word problems involving distances, intervals of time, liquid volumes, masses of
objects, and money, including problems involving simple fractions or decimals, and problems that require
expressing measurements given in a larger unit in terms of a smaller unit.
Present problems that involve multiplication of a fraction by a whole number (denominators are 2, 3, 4, 5 6,
8, 10, 12 and 100). Problems involving addition and subtraction of fractions should have the same
denominators. Allow students to use strategies learned with these concepts.
Students used models to find area and perimeter in Grade 3. They need to relate discoveries from the use of
models to develop an understanding of the area and perimeter formulas to solve real-world and
mathematical problems.
Misconceptions/Challenges
The following are some common misconceptions for this topic. As you teach the lessons, identify the misconceptions/challenges that
students have with the concepts being taught.
Common Misconceptions from ODE Model Curriculum
Students believe that larger units will give the larger measure. Students should be given multiple opportunities to
measure the same object with different measuring units. For example, have the students measure the length of a
room with one-inch tiles, with one-foot rulers, and with yard sticks. Students should notice that it takes fewer yard
sticks to measure the room than rulers or tiles.
Please read the Teacher Introductions, included in this document, for further understanding.
Grade 4 Measurement and Data
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Columbus City Schools 2013-2014
Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
Grade 4 Measurement and Data
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Columbus City Schools 2013-2014
Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
Number of Groups Unknown (“How
many groups?” Division
Equal
Groups
3×6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Group Size Unknown
(“How many in each group?”
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Arrays, 4
Area, 5
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
a × ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? × b = p, and p ÷ b = ?
Unknown Product
Compare
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
a×b=?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
Grade 4 Measurement and Data
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Columbus City Schools 2013-2014
Group Size Unknown: (a × ? = p and p ÷ a = ?)
In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
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students used math tools to solve the problem, they should include a picture to represent how the
tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
Act it Out
Draw a Picture
Find a Pattern
Guess and Check
Make a List
Make a Table
When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
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whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
to act it out. Occasionally whole group thinking with the teacher modeling how to record
strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Teacher Introduction
Measurement
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Measurement and Data domain is helpful when assessing student
work and student thinking. This information could be used to guide classroom discussions,
understand student misconceptions and provide differentiation opportunities. The focus of
instruction is the Common Core Mathematics Standards.
The ability to measure, interpret and estimate measurements, and select appropriate units plays
an important role in our everyday lives.
How can I make curtains that are long enough to cover my windows? Do I need a yard
of material or just two feet?
The weight limit for checked baggage is 50 pounds. This suitcase seems awfully heavy.
Is the suitcase under the weight limit?
1
This recipe calls for 2
cups of milk. I have less than a pint left. Will I have enough to
2
make this?
A strong understanding of measurement can help students master other concepts such as
perimeter, area, time, fractions, multiplication, division, and rounding.
Students are expected to gain familiarity with customary and metric units by understanding their
approximate size. They need to understand measurement to have the ability to select appropriate
units of measure for length, weight/mass, capacity, volume, and temperature. Students will
develop the knowledge of the relationship among units.
Once they have a clear understanding of relationships among different units they will be able to
solve real world application problems. Students will actively estimate and measure different
attributes of objects in the customary and metric systems. In addition, students will solve
conversion problems within the same measurement system while performing computations.
Hands on measurement experiences continue to be crucial to intermediate grade students. In
addition to using manipulatives, creating mental pictures of unit sizes will help students with
understanding. The referent chart that follows provides easy-to-remember images of the relative
size of standard units.
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Referent Chart
Length: Customary
Inch: the diameter of a quarter
Foot: the length of a piece of paper
Yard: the width of a doorway (little less than a meter)
Mile: 4 times around a high school track (A mile is about a kilometer and a half.)
Length: Metric
Millimeter: the thickness of a dime
Centimeter: the width of a pinky finger across the nail
Meter: the distance from the doorknob to the floor on a door
Kilometer: about 6 city blocks; a better referent for kilometer may be to choose a place
in the school neighborhood, e. g., the distance between school and McDonald’s . A
kilometer is about 2 a mile.
3
Capacity: Customary
Cup: the size of the milk carton in a school lunch
Pint: a bottle of salad dressing
Quart: a Biggie soft drink at Wendy’s
Gallon: a large jug of milk
Capacity: Metric
Milliliter: the amount of liquid in an eyedropper; about 10 raindrops
Liter: half of a 2-liter pop bottle
Kiloliter: the amount of water in a hot tub
Mass/Weight: Customary
Ounce: the weight of a piece of cheese
Pound: the weight of a loaf of bread
Ton: the weight of a Clydesdale horse
Mass/Weight: Metric
Milligram: the mass of a grain of salt
Gram: the mass of a paper clip
Kilogram: the mass of a textbook
Temperature: Celsius
0º C: the temperature at which water freezes
22º C: room temperature
37º C: normal body temperature
100º C: the temperature at which water boils
Temperature: Fahrenheit
32º F: the temperature at which water freezes
72º F: room temperature
98.6º F: normal body temperature
212º F: the temperature at which water boils
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In order for students to be successful with performing conversion problems, they must have an
understanding about the relative size of units. Memorizing rules and formulas will not help
students perform accurate conversions unless they have a conceptual foundation and examine
their answers for reasonableness.
When converting units of measure, students need to grasp one fundamental concept: the number
of units is inversely related to the size of the unit. Simply stated, the smaller the unit, the more
of that unit it will take to measure an object. The larger the unit, the less of that unit it will take
to measure an object. For example, Mikayla is measuring the capacity of her watering can. The
can holds 2 liters. If she were to measure the amount of water in milliliters, the number of
milliliters it would take to measure the capacity of the watering can would be greater than the
number of liters it takes to measure the capacity. The capacity of the can does not change, but it
takes more of a smaller unit to fill the can. To help students become proficient with conversions,
teachers can carefully build with students the process of conversion. First, have students decide
if the converted answer will be greater or less than the number of the current unit. Second, work
with students to visualize the relationship between the units. Third, have students apply their
knowledge of multiplication and division based on the size of the new answer. Conversions
begin to make sense to students when they are based on conceptual understanding.
The Metric System
The metric system is based on the place value system of powers of 10. To understand the metric
system, students must have a complete understanding of our place value (including decimals)
system. Teachers need to find an alternative to the traditional “moving decimal points”
explanation to be successful. When we teach students rules without the conceptual development,
students become confused as to how and why different rules are applied. If our teaching
acknowledges the common sense of students and encourages a sensible approach, students will
be less apprehensive about unit conversions. Reframe metric conversions by thinking of
multiplying or dividing one unit by a power of 10 to convert it to another unit. This
multiplication or division is what causes the decimal point to move.
Using the metric system for length, capacity, and mass is simple when students understand the
meaning of each metric prefix and its relative size. In addition to working with the fractional
meanings of each metric prefix, provide students with a relative size relation to the base units.
Base units include meters (length), liters (capacity), and grams (mass). Prefixes that denote units
smaller than the base unit include milli-, centi-, and deci-. Prefixes that denote units greater than
the base unit include deka-, hecto-, and kilo-. All three types of measurement (length, capacity,
and mass) use these same prefixes to differentiate between sizes. One mnemonic device that can
be used to memorize the order of the prefixes from greatest to least is King Henry Danced Until
Dawn Counting Money. The chart below shows the meaning of each prefix and the number
representation in relation to the base unit.
King
Henry
Danced
kilo- (k)
1000
hecto- (h)
100
deka- (da)
10
Until
(Unit)
L, m, or g
1
Dawn
Counting
Money
deci- (d)
0.1
centi- (c)
0.01
milli- (m)
0.001
Once students are familiar with the meaning of the prefixes and their relationship to one another,
they can use multiplication or division by a power of 10 because the metric system is based on
powers of 10. Multiplying or dividing one unit by a power of 10 changes it into another unit
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within the metric system. It is important for students to see that the product or quotient found
will have the same numerals in it as the original unit, but the decimal point will have moved to a
different location. When changing from a smaller unit to a larger unit of measurement, you must
divide (smaller unit → larger unit = divide). There will be fewer of the larger unit than the
smaller unit. When changing from a larger unit to a smaller unit of measurement, you must
multiply (larger unit → smaller unit = multiply). There will be more of the smaller unit than the
larger unit. Conversions build on conceptual knowledge of the meaning of multiplication and
division.
Example: When Giant Eagle™ had a sale on bottled water, they sold 52,300 liters in two
weeks. How many kiloliters of water does this equal?
To solve this problem, the student first conceptualizes that the number of kiloliters will be less
than the number of liters because kiloliters are larger than liters. She also knows that 1000 liters
are equal to one kiloliter. Then she realizes that to change liters, a smaller unit, to kiloliters, a
larger unit, she must divide. To divide, she realizes that 52,300 divided by 1000 is easy to do
because the decimal point must be moved three places to the left. Because 52,300 does not
currently have a decimal point, one must be placed at the end of the number before it can be
moved.
5 2 ,3 0 0. = 52.3
Solution: Giant Eagle™ sold 52.3 kiloliters of bottled water in two weeks.
After students have developed the conceptual meaning of conversions, they become easier to
perform.
Example: Mrs. Katz takes 3.8 grams of vitamins monthly. How many milligrams does this
equal?
In addition to thinking through the meaning of the units, write out the prefixes in order from
greatest to least.
k
h
da
(L, m, g)
d
c
m
Put your finger on the measurement from which you are starting (grams, or units). To reach
milligrams, move your finger to milligrams (3 spaces to the right). Since we are moving to the
right 3 spaces, we are multiplying by 1000. To change grams into milligrams, we must move the
decimal point three places to the right. Zeroes must be added as place holders, but the other
numbers do not change.
3.8 0 0
= 3,800
Solution: Mrs. Katz takes 3,800 milligrams of vitamins monthly.
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Length
Length can be defined as how long/tall/wide an object measures. Rulers, meter sticks,
yardsticks, and tape measures are common measuring tools used to measure length. Customary
units include inches, feet, yards, and miles. Students will also have to identify fractional parts of
1 1 1 1
).
2 4 8 16
an inch ( , , ,
Metric units include millimeters, centimeters, decimeters, meters,
dekameters, hectometers, and kilometers. In addition to conversions using metric units of length,
students must be able to convert between customary units of length.
12 inches = 1 foot
36 inches = 3 feet = 1 yard
5,280 feet = 1,760 yards = 1 mile
Similar to metric conversions, students must first decide if the answer will be greater or less than
the current unit. Then they must visualize the relationship between the units. Next, students
must apply their knowledge of multiplication and division to decide whether the answer needs to
be larger or smaller than the number of the current unit. However, customary conversions are
not based on powers of 10. They must be multiplied or divided by different conversion numbers.
When remainders appear, students must be aware of how to label them appropriately with the
unit of the divisor.
Example: Mrs. Traber needs 14 feet of material to sew curtains for her kitchen. How many
yards of material does Mrs. Traber need to buy?
To solve this problem, the student needs to decide whether the number of yards will be greater or
less than the number of feet. He knows that the number of yards will be less than the number of
feet because yards are greater than feet. Also, he knows that 3 feet are equal to one yard. Next,
he realizes that he will need to divide since the answer will be smaller than the current number of
units.
14 feet total ÷ 3 feet in one yard = 4 yards and 2 feet
Solution: Mrs. Traber needs 4 yards and 2 feet of material to create her curtains.
Capacity
Capacity can be defined as the amount of liquid or pourable substance (e. g., sand, beans, rice,
etc.) a container can hold. When measuring capacity, the item used to fill the container takes the
shape of the container into which it is poured. Measuring spoons, cups, and graduated cylinders
are used to measure capacity. Metric units include milliliters, centiliters, deciliters, liters,
dekaliters, hectoliters, and kiloliters. Customary units include fluid ounces, cups, pints, quarts,
half gallons, and gallons. In addition to conversions using metric units of capacity, students must
be able to convert between customary units of capacity.
8 fluid ounces = 1 cup
2 cups = 1 pint
2 pints = 1 quart
2 quarts = 1 half gallon
4 quarts = 1 gallon
2 half gallons = 1 gallon
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Similar to metric conversions, students must first decide if the answer will be greater or less than
the current unit. Then they must visualize the relationship between the units. Next, students
must apply their knowledge of multiplication and division to decide whether the answer needs to
be larger or smaller than the number of the current unit. However, customary conversions are
not based on powers of 10. They must be multiplied or divided by different conversion numbers.
When remainders appear, students must be aware of how to label them appropriately with the
unit of the divisor.
Example: Isaiah’s family drinks ten gallons of milk in one month. How many half gallons of
milk does this equal?
To solve this problem, the student needs to decide whether the number of half gallons will be
greater or less than the number of gallons. He knows that the number of half gallons will be
greater than the number of gallons because half gallons are smaller than gallons. Then he knows
that 2 half gallons are equal to one gallon. Next, he knows that he will need to multiply since the
number of half gallons will be greater than the current number of gallons.
10 gallons
2 half gallons in a gallon = 20 half gallons
Solution: Isaiah’s family drinks 20 half gallons of milk in one month.
A graphic to help students understand the conversions between the customary units of capacity is
the Gallon Guy. The body of the Gallon Guy represents a gallon. The top part of each arm and
leg represents a quart (4 quarts in one gallon). The bottom of each arm and leg represents two
pints (8 pints in one gallon). The four fingers or toes on each arm or leg represent cups (16 cups
in one gallon). Fluid ounces are not represented on the Gallon Guy, but there are 8 fluid ounces
in each cup.
cup
cup
cup
cup
pint
pint
quart
quart
pint
pint
cup
cup
cup
cup
quart
pint
pint
pint
cup
cup
cup
cup
cup
cup
cup
cup
pint
quart
Gallon
Guy
Another model of the relationships among customary units of capacity is a representation that
uses fraction bars. The doubling relationship between each progressively larger unit becomes
visually obvious. Students find it easy to draw on assessments, and it connects easily with their
fraction knowledge.
1 gallon = 2 half gallons = 4 quarts = 8 pints = 16 cups
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1 gallon
1 half gallon
1 quart
1 pint
1
cup
1
cup
1 quart
1 pint
1
cup
1 half gallon
1
cup
1 pint
1
cup
1 quart
1 pint
1
cup
1
cup
1 pint
1
cup
1
cup
1
cup
1 quart
1 pint
1
cup
1
cup
1 pint
1
cup
1
cup
1 pint
1
cup
1
cup
In addition to the Gallon Guy and fraction bars, students can use the following mnemonic device
to remember the order of the customary units of capacity from smallest to greatest:
Furry Cats Parade Quietly, Horses Gallop
(Fluid Ounces, Cups, Pints, Quarts, Half Gallons, Gallons)
A visual way to represent customary capacity conversions is the stair step model below. After
students have developed the conceptual understanding of customary conversions, they can use
the stair step model to convert. When converting smaller units to larger units, you move up the
stairs, or divide. When converting larger units to smaller units, you move down the stairs, or
multiply.
C
P
Q
H
G
F
Mass/Weight
Mass and weight are similar, but not the same. Sometimes the words are incorrectly used
interchangeably. Weight can be defined as how heavy an object is. Mass measures the amount
of matter in an object. The difference between mass and weight is that gravity influences weight,
but not mass. If you were to go to Mars you would weigh less because the gravity on Mars is
one-third of the gravity on the Earth, however, your mass would remain the same because you
are still made up of the same amount of matter. Mass/weight are measured using a variety of
scales and balances. Metric units include milligrams (mg), centigrams (cg), decigrams (dg),
grams (g), dekagrams (dag), hectograms (hg), and kilograms (kg). Customary units of
mass/weight include ounces (oz), pounds (lbs), and tons (t).
Similar to metric conversions, students must first decide if the answer will be greater or less than
the current unit. Then they must visualize the relationship between the units. Next, students must
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apply their knowledge of multiplication and division to decide whether the answer needs to be
larger or smaller than the number of the current unit. However, customary conversions are not
based on powers of 10. They must be multiplied or divided by different conversion numbers.
When remainders appear, students must be aware of how to label them appropriately with the
unit of the divisor.
16 ounces = 1 pound
2000 pounds = 1 ton
Example: Jaleel’s mom bought 53 ounces of hamburger for a family cookout. How many
pounds does this equal?
To solve this problem, the student needs to decide whether the number of pounds will be greater
or less than the number of ounces. He knows that the number of pounds will be less than the
number of ounces because pounds are greater than ounces. Then he knows that 16 ounces are
equal to one pound. Next, he knows that he will need to divide since the number of pounds will
be less than the current number of ounces.
53 ounces ÷ 16 ounces in a pound = 3 pounds and 5 ounces
Solution: Jaleel’s mom bought 3 pounds and 5 ounces of hamburger for the cookout.
Temperature
Temperature can be defined as the measure of how hot or how cold something is. It is measured
with thermometers. Customary units of temperature include degrees Fahrenheit (ºF). Metric
units of temperature include degrees Celsius (sometimes referred to as Centigrade, ºC). Other
countries around the world measure temperature using the Celsius scale, while Fahrenheit is used
more often in the United States.
100
90
F
95 96 97 98 99 100 101 102 103 104 105 106 107 108
80
70
60
50
40
30
20
30
10
0
-10
40
50
20
60
70
10
-20
-30
C
Grade 4 Measurement and Data
0
80
-10
90
F
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100
90
80
70
60
50
40
30
20
10
0
-10
-20
-30
F
Columbus City Schools 2013-2014
Example: Jabar went outside to play. He was appropriately dressed in shorts and a T-shirt. The
temperature on the weather forecast said that the average temperature for the month
was 25º. In what unit is the temperature being measured?
Room temperature is 20ºC. However, 20ºF would be below freezing (32ºF). The temperature
must be measured in degrees Celsius for Jabar to wear shorts and a t-shirt.
Solution: The temperature is being measured in degrees Celsius.
Example: The meteorologist on Channel 8 said that today’s high temperature would be 35º
Fahrenheit. What should Jasmine choose to wear on her way to school? Explain
your answer.
a) a jacket, scarf, hat, and mittens
b) a light jacket
c) a sweater
Jasmine knows that water freezes at 32ºF. 35ºF is just 3º warmer. She knows she will need to
keep very warm since the high temperature is expected to be just above freezing.
Solution: Jasmine should choose a) a jacket, scarf, hat, and mittens.
Elapsed Time
Elapsed time can be thought of as the duration of an activity. It can be expressed as the time that
has passed from the beginning of an activity to the end of that activity. This concept appears
throughout our world in our everyday life.
 The movie starts at 6:25 p.m. What time do we need to leave to see the beginning of the
movie if it takes 10 minutes to get to the theater?
th
 My grandma’s birthday is January 30 . If the Postal Service takes 5 days to deliver a card to
her, by what day do I need to mail her card?
 This meatloaf needs to bake for 45 minutes. If I put it in the oven at 5:45 p.m., will it be
finished before the company arrives at 6:15 p.m.?
 I laid my baby down for a nap at 9:15 a.m. She slept until 10:30 a.m. How long did she
sleep?
The teaching of elapsed time helps develop a student’s ability to solve problems. Students can
use a variety of strategies to solve elapsed time problems, many of which they develop on their
own. While some students may demonstrate difficulty developing their own strategies, we can
assist them by providing different approaches and breakdowns of different types of problems to
work on the conceptual knowledge they need to be able to find elapsed time. Once students have
seen the different approaches they can choose a strategy or method that works best for them.
Determining Elapsed Time Using Clocks
Determining elapsed time using clocks requires some basic mathematical skills. First students
must be able to proficiently tell time on an analog/dial and a digital clock. By the end of third
grade, students will be expected to tell time to the nearest minute. Students also need to work
with estimations of time based on the benchmark of one hour. For example, is the time elapsed
from 7:10 p.m. to 8:00 p.m. more or less than one hour? In addition, students should understand
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the importance of the number 60. Providing daily practice opportunities with these skills will
assist students in becoming more successful with elapsed time problems in a testing situation.
Students must be familiar with the number 60, as 60 is the cornerstone to understanding our
system of telling time. There are 60 seconds in a minute and 60 minutes in one hour. Just as we
regroup when we get to 10 in the base ten number system, we must regroup when we get to 60
when dealing with time. Working with numbers less than 60 will help students look at pieces of
time and recognize parts. For example, 50 minutes can also be stated as 10 minutes and 40
minutes. To solve different elapsed time problems, students can break down chunks of time into
easier numbers. For example, Henry begins football practice at 7:30 p.m. and practices for 45
minutes. What time did Henry’s practice end? Recognizing that 45 minutes can be broken into
30 minutes and 15 minutes will make it easier to determine that Henry’s practice ended at 8:15
p.m. by adding 30 minutes to 7:30 to get to the next whole hour (8:00) and then adding the
additional 15 minutes. As students become familiar with decomposing the number 60 and
numbers less than 60, working elapsed time problems becomes easier and simple computational
errors can be eliminated.
45 + 15 = 60
40 + 20 = 60
52 + 8 = 60
30 + 30 = 60
10 + 50 = 60
47 + 13 = 60
Students will be asked to solve three different types of elapsed time problems. Every elapsed
time problem will have two out of the three key pieces of information needed: the start time of
the activity, the end time of the activity, and/or the elapsed time of the activity. Teaching
students to identify the different types of problems will assist in their strategy selection to solve
each problem. In addition to choosing a strategy, the final step of checking the answer to see if it
is reasonable is important in all problems and especially important in elapsed time problems.
Students may find that they have made an initial error in reasoning after checking their answer,
e.g., a student may have a start time that is chronologically after the end time. Checking their
reasoning, strategy, and answer will build confidence and mathematical power.
Type One: The start time and the end time are provided. Students must solve for the
elapsed time.
Example: Mrs. Smith’s class begins a reading project at 1:20 p.m. The students finish their
projects at 3:05 p.m. How long did Mrs. Smith’s class work on their projects?
Start Time: 1:20 p.m.
End Time: 3:05 p.m.
Elapsed Time: ?
For Type One problems, counting on from the starting time can easily solve the problem.
1. First, estimate whether the elapsed time is greater than an hour or less than an hour.
1:20 p.m. to 3:05 p.m. is more than an hour because an hour after 1:20 p.m. is 2:20 p.m.
2. Next count the number of full hours that elapse between the start time and the end time.
1:20 p.m. + 1 hour equals 2:20 p.m. Another full hour cannot elapse because that would pass
the end time of 3:05 p.m. (2:20 p.m. + 1 hour = 3:20 p.m.)
3. Count the minutes from 2:20 p.m. to 3:05 p.m. Creating full hours will allow students to
develop the relationship between minutes and hours. Familiarity with creating the number 60
(one full hour) will prove useful when determining the number of minutes. From 2:20 p.m.
to 3:00 p.m. is 40 minutes. From 3:00 p.m. to 3:05 p.m. another five minutes elapses. When
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adding the total number of minutes together we can see that from 2:20 p.m. to 3:05 p.m. is 45
minutes.
4. Add the hours and the minutes to determine the time elapsed.
1 hour + 45 minutes = 1 hour and 45 minutes
5. Check the answer to make sure it makes sense. Go back through your computations and
check the reasonableness of the answer. It is reasonable that the time elapsed between 1:20
p.m. and 3:05 p.m. is 1 hour and 45 minutes because it had to be more than 1 hour but less
than 2 hours.
Answer: Mrs. Smith’s class worked for 1 hour and 45 minutes.
Type Two: The start time and the elapsed time are provided. Students must solve for the
end time.
Example: Kennedy is baking chocolate chip cookies. She put them in the oven at 9:35 a.m. The
cookies must bake for 45 minutes. What time will the cookies be finished?
Start Time: 9:35 a.m.
End Time: ?
Elapsed Time: 45 minutes
Similar to Type One problems, counting on from the start time can also solve the problem.
1. Break the elapsed time into parts that can be added to the start time to create full hours.
Competency with decomposing numbers will help students create parts of time. 45 minutes
can be broken into 25 minutes and 20 minutes. Adding 25 minutes to the start time will
create a full hour. 9:35 a.m. + 25 minutes = 10:00 a.m.
2. Add the remaining time.
10:00 a.m. + 20 minutes = 10:20 a.m.
3. Check the answer to make sure it makes sense. Go back through your computations and
check the reasonableness of the answer. It is reasonable that Kennedy’s cookies are finished
at 10:20 a.m. because her cookies are done 45 minutes AFTER the start time. If it took her
cookies an hour to bake, they would have been done at 10:35 a.m. Forty-five minutes is less
than an hour and 10:20 a.m. is earlier than 10:35 a.m.
Answer: The cookies will be finished at 10:20 a.m.
Students may also be assessed on a Type Two problem that provides a number of elapsed times
that must be combined and/or added on to the start time.
Example: Mr. Taylor is baking brownies. He begins at 3:15 p.m. It takes him ten minutes to
measure the ingredients, fifteen minutes to combine them, and five minutes to mix the
ingredients. The brownies must bake for a half an hour. At what time will the brownies be
finished?
Students must recognize that although this problem contains different elapsed times it is still a
Type Two problem. Again, counting on from the starting time for each step can help students
solve this problem.
Grade 4 Measurement and Data
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Columbus City Schools 2013-2014
1. Add 10 minutes to 3:15 p.m. for the time it took Mr. Taylor to measure the ingredients.
3:15 p.m. + 10 minutes = 3:25 p.m.
2. Add 15 minutes to 3:25 p.m. for the time it took Mr. Taylor to combine the ingredients.
3:25 p.m. becomes the new start time, as he measured the ingredients first.
3:25 p.m. + 15 minutes = 3:40 p.m.
3. Add 5 minutes to 3:40 p.m. for the time it took Mr. Taylor to mix the ingredients. 3:40 p.m.
becomes the new start time, as he combined the ingredients before he mixed them.
3:40 p.m. + 5 minutes = 3:45 p.m.
4. Add 30 minutes to 3:45 p.m. for the time it took the brownies to bake. 3:45 p.m. becomes
the new start time, as he mixed the ingredients before the brownies baked.
3:45 p.m. + 30 minutes = 4:15 p.m.
5. Check the answer to make sure it makes sense. Go back through your computations and
check the reasonableness of the answer. It is reasonable that Mr. Taylor’s brownies were
finished baking at 4:15 p.m. because 4:15 p.m. is AFTER the start time of 3:15 p.m. If you
combine all of the times needed to prepare and bake the brownies (10 minutes + 15 minutes
+ 5 minutes + 30 minutes) the total is 1 hour. One hour after 3:15 p.m. is 4:15 p.m. so the
answer correct.
Answer: The brownies will be finished at 4:15 p.m.
Type Three: The end time and the elapsed time are provided. Students must solve for the
start time.
Example: The movie began 2 hours and 13 minutes ago. It is now 7:27 p.m. What time did the
movie begin?
Start Time: ?
End Time: 7:27 p.m.
Elapsed Time: 2 hours and 13
minutes
Students must be able to adequately work with elapsed times that may be a combination of hours
and minutes. Counting back from the end time can easily solve Type Three problems.
1. Count backward from the end time, beginning with the hours of the elapsed time.
7:27 p.m. – 1 hour = 6:27 p.m. – 1 hour = 5:27 p.m.
2. Then use the minutes of the elapsed time and count backward from the minutes of the end
time.
5:27 p.m. – 13 minutes = 5:14 p.m.
3. Check the answer to make sure it makes sense. Go back through your computations and
check the reasonableness of the answer. It is reasonable that the movie started at 5:14 p.m.
because the start time is 2 hours and 13 minutes BEFORE the end time of 7:27 p.m. Two
hours earlier than 7:00 p.m. would be 5 p.m. and 13 minutes earlier than 27 is 14, so the
start time would be 5:14 p.m.
Ans wer: The movie started at 5:14 p.m.
Grade 4 Measurement and Data
Page 20 of 94
Columbus City Schools 2013-2014
Students may encounter difficulty when having to move backward and/or forward across the
hour. If students have had ample practice opportunities with decomposing the number 60, this
backward/forward movement will become a natural part of solving elapsed time problems.
Example: Alexis finished studying for her math test at 6:25 p. m. She studied for 45 minutes.
What time did Alexis begin studying?
Start Time: ?
End Time: 6:25 p. m.
Elapsed Time: 45 minutes
1. Estimate the time when Alexis started studying.
45 minutes is a little less than an hour. One hour before 6:25 p.m. would be 5:25 p.m., so we
know that Alexis started studying after 5:25 p.m.
2. Since Alexis studied for less than an hour, count backward from the end time using the
minutes of the elapsed time. 45 minutes is greater than the 25 minutes of the end time, we
know that the hour of the start time will fall before the 6:00 hour. 45 minutes can be broken
down into 25 minutes and 20 minutes. Counting back 25 minutes from 6:25 p.m. will put us
at 6:00 p.m. Then counting back 20 more minutes from 6:00 p.m. will put us at 5:40 p.m.,
the time Alexis started studying.
45 minutes = 25 minutes + 20 minutes
6:25 p.m. – 25 minutes = 6:00 p.m. – 20 minutes = 5:40 p.m.
3. Check the answer to make sure it makes sense. Go back through your computations and
check the reasonableness of the answer. It is reasonable that Alexis started studying at 5:40
because 5:40 p m. is 45 minutes BEFORE she finished studying.
Answer: Alexis started studying at 5:40 p.m.
Once students gain proficiency with the three different types of elapsed time problems, students
begin to develop their own strategies to solve problems. One common strategy is combining
estimation with the use of easier numbers.
Example: Mr. Newkirk started mowing his lawn at 9:13 a.m. He mows for 57 minutes. What
time did Mr. Newkirk finish mowing?
Start Time: 9:13 a.m.
End Time: ?
Elapsed Time: 57 minutes
1. Estimate the time that Mr. Newkirk finished mowing the lawn.
57 minutes is just a little less than one hour. Mr. Newkirk finished mowing the lawn around
10:13 a.m.
2. Using the simpler numbers, adjust your answer.
57 minutes is 3 minutes less than one hour. If Mr. Newkirk started at 9:13 a.m. and mowed
for one hour, he would be finished at 10:13 a.m. Since he mowed three minutes less than one
hour, Mr. Newkirk was finished at 10:10 a.m.
9:13 a.m. + 1 hour = 10:13 a.m.
10:13 a.m. – 3 minutes = 10:10 a m.
3. Check the answer to make sure it makes sense. Go back through your computations and
check the reasonableness of the answer.
Grade 4 Measurement and Data
Page 21 of 94
Columbus City Schools 2013-2014
It is reasonable that Mr. Newkirk finished mowing the lawn at 10:10 a.m. because 10:10 a.m.
is a little less than one hour after he started and 57 minutes is a little less than an hour. Also,
10:10 a.m. is 57 minutes AFTER the start time of 9:13 a.m.
Answer: Mr. Newkirk finished moving at 10:10 a.m.
Determining Elapsed Time Using a Calendar
In addition to determining elapsed time with clocks, students may be asked to determine elapsed
time using a calendar.
Example: Today is March 1st. Lucy knows that her birthday is in exactly two weeks. What day
is her birthday?
To solve this problem students need to know that one week is equal to seven days. This problem
could be solved using two different strategies.
Strategy One: If an illustration of a calendar is available find the current date (March
1st). Then count down two rows (equaling two weeks) to find the date of March 15th.
Strategy Two: Two weeks equals fourteen days. Add fourteen to the date (March 1st) to
arrive at the answer of March 15th.
Students should be familiar with both strategies, as an illustration of a calendar may not always
be available in a testing situation. Students also need to be able to move forward and backward
on the calendar.
Example: Quincy’s birthday was one week and three days ago. Today is January 16th. On what
date did Quincy’s birthday fall?
To solve this problem, students need to recognize that a backward movement on the calendar is
necessary, as Quincy’s birthday fell before January 16th. One week before January 16th is
January 9th (16 – 7 = 9). Three days prior to January 9th is January 6th (9 – 3 = 6). Quincy’s
birthday was on January 6th.
Grade 4 Measurement and Data
Page 22 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Lavon’s driveway is 42 meters in length. What is the length of the driveway in
centimeters?
 A. 42 centimeters
 B. 420 centimeters
 C. 4,200 centimeters
 D. 42,000 centimeters
Which would be the best unit to measure the distance from Columbus, Ohio to
Orlando, Florida?
 A. centimeters
 B. kilograms
 C. meters
 D. kilometers
Grade 4 Measurement and Data
Page 23 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Lavon’s driveway is 42 meters in length. What is the length of the driveway in
centimeters?
 A. 42 centimeters
 B. 420 centimeters
 C. 4,200 centimeters
 D. 42,000 centimeters
Answer: C
Which would be the best unit to measure the distance from Columbus, Ohio to
Orlando, Florida?
 A. centimeters
 B. kilograms
 C. meters
 D. kilometers
Answer: D
Grade 4 Measurement and Data
Page 24 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Jeff rode his bike 2.5 miles on Monday. Tuesday he rode 3 times as far as
Monday. He figured that he rode 6.0 miles altogether.
Jeff’s answer was incorrect. What is the correct answer?

A 10 miles

B. 5.5 miles

C. 7.5 miles

D. 6.0 miles
A store sells packages of sausage in two sizes: 8 ounces for $2.00 and 16
ounces for $3.20. How could you decide which package is the better deal?

A. You could multiply the price of each package of sausage by the number
of ounces to see how much you are paying altogether.

B. You could subtract the prices of the two packages of sausage to see how
much more you are paying for the larger package.

C. You could divide the price of each package of sausage by the number of
ounces to see how much you are paying for each ounce.

D. You could add the price of each package of sausage and the number of
ounces to see how much you are paying altogether.
Grade 4 Measurement and Data
Page 25 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Jeff rode his bike 2.5 miles on Monday. Tuesday he rode 3 times as far as
Monday. He figured that he rode 6.0 miles altogether.
Jeff’s answer was incorrect. What is the correct answer?
 A 10 miles
 B. 5.5 miles
 C. 7.5 miles
 D. 6.0 miles
Answer: A
A store sells packages of sausage in two sizes: 8 ounces for $2.00 and 16 ounces
for $3.20. How could you decide which package is the better deal?
 A. You could multiply the price of each package of sausage by the number of
ounces to see how much you are paying altogether.
 B. You could subtract the prices of the two packages of sausage to see how
much more you are paying for the larger package.
 C. You could divide the price of each package of sausage by the number of
ounces to see how much you are paying for each ounce.
 D. You could add the price of each package of sausage and the number of
ounces to see how much you are paying altogether.
Answer: C
Grade 4 Measurement and Data
Page 26 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
If you were measuring the following objects with both units, of which unit would
you have more?
Circle one answer for each item below.
Weighing your
dog
grams
kilograms
Filling the bathtub
liters
milliliters
Measuring the
length of your
classroom
yards
feet
Weighing yourself
ounces
pounds
Matt has to complete a measurement project for school. He has to choose an item
and match it with the best measurement. Help Matt with his project by writing
the letter of each item next to the best measurement.
A) Length of a baseball bat
_____
250-300 grams
B) Weight of a large orange
_____
17-19 centimeters
C) Weight of a bicycle
_____
12 kilograms
D) Length of a pencil
_____
1 meter
Grade 4 Measurement and Data
Page 27 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
If you were measuring the following objects with both units, of which unit would
you have more?
Circle one answer for each item below.
Answer:
Weighing your
dog
grams
kilograms
Filling the bathtub
liters
milliliters
Measuring the
length of your
classroom
yards
feet
Weighing yourself
ounces
pounds
A 2-point response circles all the correct units of measure.
A 1-point response circles only two correct units of measure.
A 0-point response shows no mathematical understanding of this task.
Matt has to complete a measurement project for school. He has to choose an item
and match it with the best measurement. Help Matt with his project by writing
the letter of each item next to the best measurement.
Answer:
A) Length of a baseball bat
__B
250-300 grams
B) Weight of a large orange
__D__ 17-19 centimeters
C) Weight of a bicycle
__C__ 12 kilograms
D) Length of a pencil
__A__ 1 meter
A 2-point response shows four correct answers.
A 1-point response shows at least two correct answers.
A 0-point response shows no mathematical understanding of the task.
Grade 4 Measurement and Data
Page 28 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
How many quarts are in 3 gallons?

A. 6 quarts

B. 9 quarts

C. 12 quarts

D. 15 quarts
If you need to convert a small unit of measure (inches) to a larger unit of measure
(feet), what would you do? Solve the problem below. Explain your answer and
show your work below.
36 inches = _________ feet
Grade 4 Measurement and Data
Page 29 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
How many quarts are in 3 gallons?
 A. 6 quarts
 B. 9 quarts
 C. 12 quarts
 D. 15 quarts
Answer: C
If you need to convert a small unit of measure (inches) to a larger unit of measure
(feet), what would you do? Solve the problem below. Explain your answer and
show your work below.
36 inches = _________ feet
Answer: 3 feet - Compare the units and determine how many inches are in a foot (12
inches to 1 foot). Then divide 36 inches by 12 inches to convert to feet.
A 2-point response should include an explanation using division or grouping to result in a
lower number of units (3 feet).
A 1-point response has a minor error in computation but indicates that the number would
be lower.
A 0-point response shows no mathematical understanding of the question.
Grade 4 Measurement and Data
Page 30 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
In art, Mr. Warden's class is doing a weaving project. For the project, they need
pieces of ribbon that are 1 foot in length. One spool of ribbon contains 6 yards of
ribbon. How many one-foot pieces can they make from a spool?
 A. 24
 B. 18
 C. 12
 D. 6
Dara bought a bottle that contains 895 milliliters of juice. She wants to put the
juice in a pitcher so that it is easier to pour. If her pitcher holds 1 liter of juice,
will all of the juice from the bottle fit in the pitcher? Explain your answer.
Grade 4 Measurement and Data
Page 31 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
In art, Mr. Warden's class is doing a weaving project. For the project they need
pieces of ribbon that are 1 foot in length. One spool of ribbon contains 6 yards of
ribbon. How many one-foot pieces can they make from a spool?
 A. 24
 B. 18
 C. 12
 D. 6
Answer: B
Dara bought a bottle that contains 895 milliliters of juice. She wants to put the
juice in a pitcher so that it is easier to pour. If her pitcher holds 1 liter of juice,
will all of the juice from the bottle fit in the pitcher? Explain your answer.
Answer: Yes, the juice will fit in the pitcher. 1 liter is equal to 1,000 milliliters so the
pitcher will hold 1,000 milliliters, which is more than 895 milliliters.
A 2-point response includes a correct answer of "yes" with a complete explanation.
A 1-point response includes a correct answer of "yes" with a weak or no explanation or
includes an explanation of the relationship between milliliters and liters without stating
that the juice will fit in the pitcher.
A 0-point response shows no mathematical understanding of this task.
Grade 4 Measurement and Data
Page 32 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Paula and Reginald were measuring how much material was left on a roll of
fabric. Paula measured the length of the fabric in feet and Reginald measured the
length of the fabric in yards.
Reginald found there were 12 yards of fabric on the roll. How many feet did
Paula say were left on the roll?
Are there more feet or yards of fabric? Explain your answer.
Paula and Reginald are making flags for field day. Each flag takes a length of 4
feet of fabric. They want to make 1 flag for each of the 8 classrooms in their
building. How many yards and feet of fabric do they need to complete their
project? Show your work.
Grade 4 Measurement and Data
Page 33 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Paula and Reginald were measuring how much material was left on a roll of
fabric. Paula measured the length of the fabric in feet and Reginald measured
the length of the fabric in yards.
Reginald found there were 12 yards of fabric on the roll. How many feet did
Paula say were on the roll?
36 feet
Are there more feet or yards of fabric? Explain your answer.
There are more feet than yards because one yard is the same as 3 feet. Feet
are the smaller unit of measure.
Paula and Reginald are making flags for field day. Each flag takes a length of 4
feet of fabric. They want to make 1 flag for each of the 8 classrooms in their
building. How many yards and feet of fabric do they need to complete their
project? Show your work.
4 feet 8 classrooms = 32 feet of fabric. Paula and Reginald need 32 feet
of fabric to make 8 flags. 32 feet = 10 yards 2 feet
10
3
32
3
02
A 4-point response states that Paula measured 36 feet of fabric, there would be more
feet than yards, and gives a reasonable explanation (e.g., feet are smaller, 1 yard is the
same as 3 feet), and it states that Paula and Reginald need 10 yards 2 feet of fabric in
order to make 8 flags for field day and shows their work.
A 3-point response answers all three questions appropriately but gives an incorrect or
weak explanation or does not show their work.
A 2-point response answers two of the three questions correctly but gives an incorrect,
weak or no explanation and shows their work or gives an explanation but does not show
their work.
A1-point response answers one question correctly and gives a weak explanation without
showing their work or answers one question correctly and shows their work with no
explanation.
A 0-point response shows no mathematical understanding of this task.
Grade 4 Measurement and Data
Page 34 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
At 3:00 p.m. Lori begins to make cupcakes. It takes her 10 minutes to gather all
the ingredients, 15 minutes to mix all the ingredients, and 25 minutes to bake the
cupcakes. The cupcakes cool in the pan for 15 minutes. Finally, Lori spends 10
minutes frosting the cupcakes. Draw hands on the clock to show when Lori is
finished with the cupcakes.
11
12
1
2
10
9
3
4
8
7
6
5
Alisha's mother told her to go to her room and study for two hours and then she
could go to the park to play. She has been in her room for 90 minutes studying.
Is it time for her to stop studying and go to the park? Explain your answer.
Grade 4 Measurement and Data
Page 35 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
At 3:00 p.m. Lori begins to make cupcakes. It takes her 10 minutes to gather all
the ingredients, 15 minutes to mix all the ingredients, and 25 minutes to bake the
cupcakes. The cupcakes cool in the pan for 15 minutes. Finally, Lori spends 10
minutes frosting the cupcakes. Draw hands on the clock to show when Lori is
12
finished with the cupcakes.
11
1
2
10
9
Answer: 4:15 p.m.
3
4
8
7
6
5
A 2-point response shows the correct time of 4:15 p.m.
A 1-point response shows a minor flaw in computation.
A 0-point response shows no understanding of the task.
Alisha's mother told her to go to her room and study for two hours and then she
could go to the park to play. She has been in her room for 90 minutes studying.
Is it time for her to stop studying and go to the park? Explain your answer.
Answer: No, it is not time for her to stop studying. Two hours is a total of 120 minutes.
Alisha has only studied for 90 minutes so she needs to study longer before she can go to
the park.
A 2-point response includes a correct answer with a complete explanation.
A 1-point response includes a correct answer with a weak or no explanation.
A 0-point response shows no mathematical understanding of the task.
Grade 4 Measurement and Data
Page 36 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Mr. McGowan will get home from work at noon on Saturday. He wants to
plan a family day. Listed below are several suggestions.
The State Soccer Game from 1:30 p.m. to 3:30 p.m.
Preview of a new release at Park Theaters from 1:30 p.m. to 3:45 p.m.
Country Concert at the State Fair from 3:30 p.m. to 5:00 p.m.
Jazz Concert at Grove Park from 4:00 p.m. to 5:15 p.m.
Rib Festival from 5:30 p.m. to 7:30 p.m.
Card Tournament from 6:30 p.m. to 9:00 p.m.
Ice Cream Social from 7:00 p.m. to 8:45 p.m.
Family Swim Night from 9:00 p.m. to 12:00 a.m.
Which activity would take the shortest amount of time?
Which activity would take the longest amount of time?
Create a schedule for the McGowan family showing the greatest number of
activities they can attend without arriving late to any event or leaving early from
any event.
Grade 4 Measurement and Data
Page 37 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Mr. McGowan will get home from work at noon on Saturday. He wants to plan a
family day. Listed below are several suggestions.
The State Soccer Game from 1:30 p.m. to 3:30 p.m.
Preview of a new release at Park Theaters from 1:30 p.m. to 3:45 p.m.
Country Concert at the State Fair from 3:30 p.m. to 5:00 p.m.
Jazz Concert at Grove Park from 4:00 p.m. to 5:15 p.m.
Rib Festival from 5:30 p.m. to 7:30 p.m.
Card Tournament from 6:30 p.m. to 9:00 p.m.
Ice Cream Social from 7:00 p.m. to 8:45 p.m.
Family Swim Night from 9:00 p.m. to 12:00 a.m.
Answer:
Which activity would take the shortest amount of time? Jazz Concert
Which activity would take the longest amount of time? Family Swim Night
Create a schedule for the McGowan family showing the greatest number of
activities they can attend without arriving late to any event or leaving early from
any event.
Preview of a new release at Park Theaters from 1:30 p.m. to 3:45 p.m. or State Soccer
Game from 1:30 p.m. to 3:30 p.m.
Jazz Concert at Grove Park from 4:00 p.m. to 5:30 p.m.
Ice Cream Social from 7:00 p.m. to 8:45 p.m.
Family Swim Night from 9:00 p.m. to 12:00 a.m.
A 4-point response includes an activity that takes the shortest amount of time, an activity
that takes the longest amount of time, and includes a schedule with the greatest number
of activities they can attend without arriving late to any event or leaving early from any
event.
A 3-point response includes an activity that takes the shortest amount of time, an activity
that takes the longest amount of time, and includes a schedule with a minor error or
includes a schedule but omits one of the activities that takes the shortest amount of time
or the greatest amount of time.
A 2-point response includes an activity that takes the shortest amount of time and an
activity that takes the longest amount of time but does not include a schedule or includes
a schedule but does not include an activity that takes the shortest amount of time and an
activity that takes the longest amount of time.
A 1-point response includes only one activity.
A 0-point response shows no mathematical understanding of the problem.
Grade 4 Measurement and Data
Page 38 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Bernie spent $2.89 for supplies at the school store. He paid with a five-dollar
bill. Which picture shows the amount of change he should receive?
 A.
 B.
 C.
 D.
MENU
Hamburger
Hot Dog
French Fries
Onion Rings
Milkshake
Soft Drink
$1.79
$1.49
$1.29
$1.79
$1.09
$0.99
Deion had lunch at the Snack Shack. He bought a hamburger, french fries, and a
milkshake. He paid with a ten-dollar bill. Give one combination of bills and
coins that Deion could have received as change? Use words, numbers, and/or
pictures to explain your answer.
Grade 4 Measurement and Data
Page 39 of 94
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Bernie spent $2.89 for supplies at the school store. He paid with a five-dollar
bill. Which picture shows the amount of change he should receive?

A.

B.
 C.
 D.
Answer: B
MENU
Hamburger
Hot Dog
French Fries
Onion Rings
Milkshake
Soft Drink
$1.79
$1.49
$1.29
$1.79
$1.09
$0.99
Deion had lunch at the Snack Shack. He bought a hamburger, french fries, and a
milkshake. He paid with a ten-dollar bill. Give one combination of bills and coins
that Deion could have received as change? Use words, numbers, and/or pictures to
explain your answer.
Answer: He should receive $5.83 in change.
A 2-point response includes a correct amount of change and shows a combination of bills
and/or coins that correctly illustrates $5.83.
A 1-point response includes a complete explanation with a minor computational error or a
correct amount of change with a weak or no explanation.
A 0-point response shows no mathematical understanding of the problem.
Grade 4 Measurement and Data
Page 40 of 94
Columbus City Schools 2013-2014
TEACHING STRATEGIES/ACTIVITIES
Vocabulary: measure, standard/customary units of measure (inch, feet, yard, mile, ounce,
pound, cup, quart, gallon, etc.), metric units of measure (millimeter, centimeter, decimeter,
meter, kilometer, milligram, grams, kilogram, milliliter, liter, kiloliter, etc.), length,
distance, weight, time, capacity, volume, unit, appropriate, ruler, meter stick, yard stick,
trundle wheel, balance scale, balance pan, conversion, elapsed time, minute, hour, quarter
hour, second, analog, dial, digital, quarter ‘til, quarter after, quarter past, half hour, half
past, hour hand, minute hand, day, week, month, year
These standards are a repeat of the standards taught in quarter 2. The focus this quarter is
on conversions and word problems involving fractions and decimals. Activity 1 below has
new problems and Activity 2 has a link to a unit with several activities for both standards.
The remaining activities are the same as the activities in quarter 2. Teachers may choose
activities that were not taught in quarter 2 and/or change the numbers in the activities to
reflect the focus on fractions and decimals.
1. The word problems below can be presented to students in a variety of ways. Some options
include: using the questions to create a choice-board, Math-O board, one problem per class
session, one problem each evening for homework, partner work or as an assessment question.
The strength of problem solving lies in the rich discussion afterward. As the school year
progresses students should be able to justify their own thinking as well as the thinking of
others. This can be done through comparing strategies, arguing another student’s solution
strategy or summarizing another student’s sharing.
The snakes at the zoo are given vitamins according to their length. Each snake gets one
dose per inch of length. The lengths of the zoo’s snakes are listed in the chart below.
Each snake is given one dose of vitamins per inch of length. Complete the chart with the
doses of vitamins for each snake. If the zoo had an 8-foot snake, how many doses of
vitamins would it require?
Length in Feet
Vitamin Doses
1
2
3
4
Hank ordered drain pipes for his plumbing job. Put the drain pipes in order from smallest
to largest:
63 centimeters, 600 millimeters, 215 centimeters, 1 meter
Gevon has 3 feet, 5 inches of rope. How many inches of rope does he have?
Marie walked for 1 hour and 25 minutes. Sheri walked for 1.75 hours. Michelle also
walked for some time. If the total time the three girls walked was 3 hours and 40
minutes, how long did Michelle walk?
Colin needs to fill his fish aquarium. Help him decide which metric unit of measurement
he should use and why.
Larue cut 2
1
yards
3
of fabric to make her prom dress. She then cut 1
2
yards
3
for a scarf.
How many feet of fabric did she cut?
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Bart bought a kite for $6.55 and a soda for $0.75. He paid with a $10.00 bill. How
much change did he get back? The cashier did not have bills, so she gave Bart his
change in coins. What was the fewest number of coins she could have given him?
Lulu will open a lemonade stand to raise $35 dollars for a Cedar Point ticket. She plans
to sell 8 oz. cups for $0.50 each. If she makes 2.5 gallons of lemonade, will she earn
enough to buy her ticket?
Five same-sized math textbooks in Room 32 have a total mass of 3.5 kilograms. How
many grams is one of the textbooks?
Bruce is biking to a bookstore that is 12 kilometers from his house. He bikes 2500
meters and stops to drink water. He then bikes 4800 meters before stopping to tie his
shoe. He then goes for another 1200 meters before stopping to rest. How many meters
does Bruce have to go until he reaches the bookstore? How many kilometers is that?
Use a number line to help solve the problem.
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 4 on the right side of the page. Open “Unit 7
Framework Student Edition.” These tasks are from the Georgia Department of Education.
The tasks build on one another. You may or may not use all of the tasks. There are four
different types of tasks. Scaffolding tasks build up to the constructing tasks which develop a
deep understanding of the concept. Next, there are practice tasks and finally performance
tasks which are a summative assessment for the unit. Select the tasks that best fit with your
lessons and the standards being taught at that time.
3. Distribute rulers, yardsticks, one-inch color tiles and the “Customary Length Record Sheet”
(included in this Curriculum Guide) to each group of students. Students use the color tiles,
ruler, and yardstick to determine how many inches are in a foot, how many inches are in a
yard, and how many feet are in a yard. These benchmarks are recorded on the recording
sheet. Next, on the board write five items for students to measure, such as: height of the
tallest student in your group, the width of the smallest window in the classroom, etc.
Students may use any unit of measure to take the actual measurement. Discuss what happens
when you measure something that was 20 inches and you want to figure out about how many
feet are equivalent to 20 inches. Ask how many inches equal a foot? How many inches are
left out of 20 inches? For example, 20 inches = 1ft 8in. Have students complete the rest of
their record sheets, converting when possible.
4. Use base ten rods to establish benchmarks for metric measurements of length. Distribute
adding machine tape (each piece should be a little longer than a meter), base ten rods, meter
sticks, and a “Metric Length Record Sheet” (included in this Curriculum Guide) to each
group of students. Each student will use the adding machine tape, rods, and meter sticks to
construct their own metric rulers. Begin by asking how many base ten rods are needed to
equal the length of a meter stick. Have students investigate how many rods are needed.
Students take the rods and line them along the adding machine tape that is next to the meter
stick. Students will draw a line with decimeter divisions (the length of each rod) on the
adding machine tape and label each line with the appropriate number of decimeters. Students
will determine that ten decimeters equals the length of a meter stick. How many centimeters
equal the length of a decimeter? Using the rod as a guide, mark off each individual
centimeter using a different color. There are ten centimeters in each decimeter. How many
centimeters equal a meter? (100 cm) Explain to the students that each centimeter can also be
divided into ten equal parts and each of these parts is called a millimeter. Under the first
centimeter marking have the students use another color to mark the number of millimeters in
one centimeter using a ruler. Discuss the total number of millimeters that could be drawn on
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the adding tape (e.g., under 1 cm, students would also write 10 mm, etc.). Have students
record these benchmarks. Next, distribute five various lengths of string. Tell students to
give the measurement of each string in millimeters, centimeters, decimeters, and meters and
record the measurements on the record sheet. Concept summary should include comparison
observations made by students between metric and customary units. Discussion points
should include: it is easier to convert using metric units because of the multiples of ten,
metric measurements can consist of larger numbers (centimeters vs. inches), making note of
similarities between yards/meters, cm/inches.
5. Divide students into small groups. Each group needs a balance scale and gram masses or
customary masses. Many brands of centimeter cubes also equal one gram. Students
investigate with the masses to explore equivalencies by placing smaller units in the left pan
and balancing with larger units in the right pan. Have students use the “Balancing Mass
Record Sheet” (included in this Curriculum Guide) to record the number of masses and the
weight of the masses on each side of the scale. For example,
5
1 gram
=
1
5 gram
5. Have students save and bring in various clean, empty containers from home that represent
both metric and customary liquid units of measure. At each table, distribute a collection of
empty containers/bottles and measuring cups/spoons. Graduated cylinders, beakers, and eye
droppers are good tools to use for metric units. Measuring cups, measuring spoons, water
bottles, milk cartons, juice containers, etc., work well for customary units. Using water, have
students determine and compare the capacities of containers to one another, observing
differences in shape and size. Students will measure how much water is needed to fill each
container. In addition, students will explore how many ½ cups equal 1 cup, how many ¼
cups equal 1 cup, etc. They should also explore the smaller units of measure such as
teaspoons, tablespoons, and milliliters, and how these compare to larger units; e.g., How
many teaspoons are in a tablespoon? (3) How many tablespoons are in a cup? (16) How
many milliliters are in a liter? (1000) This activity should end in a discussion including
observations of comparisons between metric and customary units and helping students find
or develop common referents for units of measure in both systems.
6. Place students into groups of four or five students. Provide each group with a bag of
M&M’s or Reese’s Pieces , several Hershey’s Miniature candy bars, a pound bag of
Twizzler’s , a Hershey’s bar, several Hershey Kisses , a balance, gram masses and ounce
weights, a ruler with inch and centimeter units, and a “Hershey’s Weights and Measures
Record Sheet”(included in this Curriculum Guide). Have students answer the various
questions on the recording page. Then read Hershey’s Weights and Measures by Jerry
Pallotta. Have each group listen for and record the measure of each piece/package of candy
from the book and compare with their measurements.
7. Divide students into groups of four. Give each group a small cup (approx. 4 oz. – school
lunch juice cups), a large cup (approx. 16 oz.-water bottle), an empty container (plastic ice
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cream buckets work well) and enough beans to fill the empty container. Students explore the
relationship between the size of the unit and how many units it takes to measure (the smaller
the unit, the more units it takes to measure an object). Students use a small cup (e.g., a juice
cup) to fill a bucket with beans (rice, blocks, etc.), counting the number of cups it takes to fill
the bucket. Then using a large cup, students again the fill the bucket counting the number of
large cups needed to fill the bucket. Students discuss with their group the relationship
between the size of the unit, in this case different sized cups, and the number of the unit
needed to fill something. Discuss the size of a milliliter and a liter. Students then work
together to determine a unit that should be used to fill containers pictured on the “Which
Container Fills Faster?” record sheet (included in this Curriculum Guide). As a concept
summary, have students share their justification for their choices. Some students may have
experiences with jars or bottles of different sizes and could explain their reasoning.
8. Distribute “Grid Paper –One Inch” (included in the Grids and Graphics section of this
Curriculum Guide), scissors, a ruler, and yardsticks to each group of students. Direct the
students to make a 1” 12” ruler from the one-inch grid paper. They will need to tape two
rows together to make a twelve-inch ruler. Each sheet of grid paper will make four rulers.
Have them number and label each inch on their grid paper ruler. On a piece of chart paper,
make three columns and label them yards, feet, and inches. Take the yardstick and ask how
many rulers or feet does it take to equal one yard? (3) Have students measure to find out.
Each yard is a group of three feet. On the chart, in the “Feet” column, write the numbers 1-3
vertically; in the “Yard” column write a 1 next to the 3 in the “Feet” column. Then have
students write the number of inches that are equal to the number of feet in each row.
Students should see that each foot is a group of 12 inches.
Yards
Feet
1
2
3
1
Inches
12
24
36
Direct students to take two yardsticks and find the number of rulers (feet) that are equivalent
to two yards (6). Ask students how they could find two groups of three without using the
rulers and yardsticks (add two groups of three or multiply 2 3). Continue this to find the
number of feet in three yards, four yards, etc. Add the numbers needed to the chart. Ask
students if they see a pattern (each time the number of yards increases by one, the number of
feet increases by three, and each time the number of feet increases by one, the number of
inches increases by 12).
For example:
Yards
1
2
3
Feet
1
2
3
4
5
6
7
8
9
Grade 4 Measurement and Data
Inches
12
24
36
48
60
72
84
96
108
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Ask students to look at the chart and decide if a yard is a big or a small unit of measure (big).
If you measured the height of the classroom door, would it measure a lot of yards? (no) Ask
if a foot is a bigger or a smaller unit of measure than a yard (smaller). If you measured the
height of the door in feet, would the number of feet be more or less than the number of yards
(more because it would take more feet since they are smaller than yards)? Discuss whether
inches are bigger or smaller than feet (smaller). If you measured the height of the door in
inches, would you have a number bigger or smaller than the number of feet (bigger, because
it would take more inches than feet to find the height of the door because inches are smaller
than feet) Students will see a pattern that the smaller the unit, the more of that unit it takes
to measure an object and the larger the unit, the less of that unit it takes to measure an object.
9. Divide students into small groups. Distribute various lengths of material (yarn, string,
ribbon, strips of paper, etc.,) longer than one yard, for students to measure. Direct each
group to measure the material in inches, feet, and yards. Then direct groups to trade their
pieces of material with another group. Direct students to use rulers or yardsticks and
measure only in inches. Once the inches have been determined, have students convert the
measurements to feet and yards. Then, have student groups meet to compare and discuss the
actual measurements of the first group with the converted measurements of the second group.
10. Distribute scissors and five different colored sheets of construction paper to every student
(red, blue, yellow, green, and orange). Begin with a red sheet. The red sheet will represent
one whole gallon. Direct students to label the sheet with the word “gallon” and set that sheet
aside. Next, take the blue sheet and fold the paper in half along the longest side so that the
shortest sides meet (hamburger fold). Using their scissors, students cut the blue paper along
the fold into two equal parts. Write “
1
2
gallon” on each half. Review the fact that when one
whole gallon is divided into two equal parts, each part represents one-half gallon, but
together they represent one whole gallon. Take the yellow paper, fold it in half along the
longest sides (hamburger fold) and then fold it in half again (another hamburger fold). Use
the scissors to cut the yellow paper along the folds into four equal parts. Label each part
“quart”. Lay the quarts on top of the gallon and half gallon parts. Discuss with students
parts that are equivalent (two half gallons equal a whole gallon, two quarts equal one half
gallon, four quarts equal one gallon). Take the green paper and fold it in half (hamburger
fold), fold it in half again, and fold it in half again. Label each section “pint”. Cut the
sections apart. Again brainstorm equal parts. Finally, take the orange paper and fold it in
half four times, making sure that the longest sides are always where the fold is. Label each
section “cup”. Cut the sections apart and brainstorm equivalent parts. As a class, create a
conversion chart for capacity. This activity could also be done with colored index cards and
students could store the pieces in envelopes for easy reference.
11. Play a math trivia game on measurement with the class. Divide the class into 5 or 6 teams.
Number each team using the numbers 1-6. Begin with Team 1 and ask equivalence questions
such as: How many seconds in a minute? How many hours in a day? How many days in a
year? How many pints in a quart? How many feet in a yard? How many centimeters in a
meter? Include questions that require some simple calculations, such as: How many minutes
in 120 seconds? How many feet in 8 yards? Teams can earn one point on simple equivalent
questions and two points on conversion questions. Each team gets one minute to respond. If
the answer is incorrect, the question goes to the next team. The team with the most points
wins the game.
12. Divide students into pairs. At each table section, distribute a set of “Equivalent Cards” and
the “Conversion Key” (included in this Curriculum Guide). Students try to obtain pairs of
equivalent measurements. Player A deals five cards to each player. Player B goes first and
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asks for an equivalent card to one of the measures on a card that he/she is holding (e.g., if
Player B is holding 8 ounces, he/she would ask Player A for the cup card or for the
1
pint
2
card). If the other player is holding the card that was requested, then Player B collects the
card and puts both cards on the table in front of her/him to be counted later and the turn goes
to Player A. If Player A did not have an equivalent card, then Player B draws a card from the
facedown pile and again the turn goes to Player A. The game continues until someone runs
out of cards or until all matches have been made. If a player has a card that can be added to a
match that is already on the table, they can put the card down in front of them and "add" it to
the match that is already shown. Players count the cards on the table in front of them and
receive one point for each card. The player with the most points wins. This would be a good
game for review/enrichment after several experiences with customary capacity, linear, and
mass measurement.
13. Read Me and the Measure of Things by Joan Sweeney. Divide students into small groups.
Give three containers labeled A, B, and C, two cups to measure with (a large one and a small
one), a container filled with enough beans to fill the largest container at each table section,
and “Me and the Measure of Things Recording Sheet” (included in this Curriculum Guide) to
each student. Have small groups fill the A, B, and C containers using the small cup first and
then the large cup. Record how many of each cup it took to fill the containers on the record
sheet. Summarize with students that the smaller the unit, the more of that unit it will take to
measure an object. The larger the unit, the fewer of that unit it will take to measure the same
object. As students grasp this relationship, conversion and/or the relationship between units
becomes much easier to understand.
14. Demonstrate for students that when converting from smaller units of measure to larger units
in any measurement system you divide. When changing from larger units to smaller units
you multiply. Distribute various measurements of different items in inches, feet, and yards
and ask students to convert the measurements. Students need to show their work. For
example:
The height of a tree is
98 inches. Convert this
height to feet.
Conversion is from smaller unit
to larger unit so you divide by
how many inches are in a foot
(12). 98 12 = 8 feet and 2
inches
15. Read Game Time by Stuart Murphy. Discuss all the ways time is measured (weeks, days,
hours, minutes, and seconds). Discuss how many days in a week, hours in a day, weeks in a
year, days in a year, minutes in an hour, etc.). Distribute the “Elapsed Time – Time Line”
(included in this Curriculum Guide) to each student. Have students cut out the timeline by
cutting along the solid black lines and taping it together along the dotted line. Demonstrate
how they can use this linear representation of time to figure elapsed time by counting by 5’s
forward or backward on the number line. Read the story again starting on page 8 and use the
Elapsed Time – Time Line to determine the times that each goal was scored during the game
between the Falcons and the Huskies (First goal 10:15 Falcons, second goal 10:20 Huskies,
third goal 10:30 Falcons, fourth goal 11:14 Huskies, and final goal 11:15 Huskies).
16. Use the Elapsed Time – Time Line from the previous activity and distribute “What Time Do
I Begin My Errands?” (included in this Curriculum Guide) to each student. Tell students that
Thomas has several errands to run after school. Have students determine when Thomas
needs to begin his errands so he will be home in time to go to his friend’s house. (Begin no
later than 3:15)
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17. Distribute a set of “Elapsed Time Match” (included in this Curriculum Guide) cards for each
group. One card (time card) shows a clock or gives an amount of time and the other card
(question card) gives an elapsed time word problem. The question cards are shuffled and
placed in a pile on the table. The time cards are placed face-up on the table. The students
take turns drawing a question card, reading the problem, and finding the correct answer card.
The student who makes the most correct matches wins.
18. Elapsed time is usually assessed one of the following three ways:
Students are given starting time and the duration of the activity and must find the end
time (Type A). The end time is missing
Students are given the duration of the activity and the end time and must find the start
time (Type B). The start time is missing.
Students are given the start time and end time and must find the duration of the
activity (Type C). The length (duration) of the activity is missing.
Display on the overhead a transparency of the “Types of Elapsed Time Questions” (included
in this Curriculum Guide). Discuss the three types of questions. Have students look at a
copy of the “Sorting Time Questions Set 1” (included in this Curriculum Guide). Have a
student read the first question. As a group determine the information given and what the
question is asking them to do. Classify that question as Type A, B, or C and record the
question number on the transparency in the appropriate row. Continue reading one question
at a time and recording its type on the transparency until all ten questions have been
categorized. Students do not need to answer these questions; the goal of the activity is for
students to practice identifying what needs to be found to answer the question. Divide the
students into groups of four. Each group will need a copy of the sorting chart and a copy of
“Sorting Time Questions Set 2” (included in this Curriculum Guide). Once student groups
have sorted this set of questions by placing the question numbers in the appropriate rows on
the chart, discuss the sort as a group. Did all of the small groups sort their questions in the
same way? If not, have the groups justify where they placed the question. Once consensus
is reached for a question, have groups make adjustments to their charts if necessary. The
table below shows the correct sorting of the questions for the two sets.
Type
Set 1
Set 2
Type A – Missing End Time
1, 4, 10
5, 6, 9
Type B – Missing Start Time
3, 6, 9
1, 8, 10
2, 5, 7, 8
2, 3, 4, 7
Type C – Missing Duration Time
19. Divide the students into small groups. Distribute “Pancake House, Joe’s Grill and Asian
Kitchen” menus and the “Order Form” (included in this Curriculum Guide) to each group.
Each group decides on one restaurant where they will place a group order. Students write
their order on the form and total the amounts. Students can use the menus to make up
problems for students to trade and solve or you can give a specific dollar amount for each
student to spend ordering where they have to order at least 3 or 4 food items.
20. Read Pigs Will Be Pigs by Amy Axelrod. Distribute the “Pigs Will Be Pigs” record sheet
(included in this Curriculum Guide) to keep a running total of how much money the pigs find
as they go on a hunt through the house looking for money. Compare the student totals with
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the totals listed at the end of the book. Then, give students a copy of the menu from the back
of the book and/or “Mama Mia’s Pizza House” menu and “Pigs Will Be Pigs Order Pad”
(included in this Curriculum Guide). Tell students they can only spend what the Pigs found
on their hunt and what Grandma Pig gave the little piglets. Have students place an order
from both menus or two different orders from the same menu.
21. Plan a class party. Give the students a budget of $25.00. Divide the students into groups of
four. Each group needs to brainstorm a list of items needed for the party within the budget.
Distribute fliers from the grocery ads or store ads to check prices. Students create a list with
prices and a total to submit for the class party.
22. Divide students into groups of three or four. Distribute “Price Cards” (included in this
Curriculum Guide). Challenge students to find items when added together that will total
exactly $5.00, $7.00, $8.00 etc,
23. Divide students into groups of four. Distribute rulers and yardsticks to each group. Students
measure the height of each student in the group, in inches, using the yardstick or the ruler.
Next, students use the rulers to measure each student’s height in feet. Finally, students use
the yardsticks to measure each person’s height in yards. Compare the equivalent
measurements for each student’s height.
24. Go to a grassy area or do this activity on the blacktop. Divide the class into four or five
groups. Mark one starting line for each group. Give each student a golf tee or a pattern
block that has a piece of masking tape on it for a marker. Have the students write their
names or initials on the tape. One at a time, students stand at the starting line and do a broad
jump. Students place their golf tee or pattern block at their landing spot. After all of the
students have jumped, instruct them to use meter sticks and/or yardsticks to measure the
lengths of all of the jumps in their group. Students can cut a piece of string equal to the
length of their jump, take it inside, and measure the string. Students should convert to show
their jumps in different units (e.g., if a student measured in inches, convert to feet and yards,
if they measured in centimeters convert to millimeters and meters, etc.) Display the strings
on a bulletin board and graph the class broad jump results.
25. Using various capacity tools (e.g., cup, pint, quart, gallon, teaspoon, tablespoon) for
customary measure, provide students with water and the worksheet “Fill Me Up and Pour Me
Out” (included in this Curriculum Guide). Have students use the tools to find equivalent
units of measure and record them on the worksheet.
26. Distribute “Which is Best?” worksheets (included in this Curriculum Guide) to each student.
Students determine which measure is most appropriate for the item described on each card
and circles that measure.
27. Distribute a set of the clocks from “Elapsed Time Match” (included in this Curriculum
Guide). Have students work in pairs and lay out all the clocks face up on their desk. Call out
two times that are on the clock cards. Students have to find the clocks that show the times
called and determine how much time has passed between the first clock card and the second
clock card.
28. Read Pigs On A Blanket by Amy Axelrod. Have students use a clock to follow along while
the teacher reads. Check to see if the students end up with the correct time.
29. Distribute Unifix® cubes to each table section. Show students the schedule for the day. Tell
them that each cube will represent 10 or 5 minutes. Ask, “How many 10 minutes intervals
are in one hour?” (6) How many 5 minute intervals are in one hour? (12) Give students
different problems using either 5 minute intervals or 10 minute intervals. Begin with 10
minute intervals and tell them that today we are going to have math from 11:00 to 12:00.
How many cubes do we need? (6) Have students build a tower with the cubes to represent
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the one hour of math. After you spend 10 minutes working on Daily Sign In ask how much
time would we have left? After you spend 20 minutes on an activity ask how much time
would we have left and how much time have we spent on math so far? Continue building
towers to represent different parts of your schedule or build a tower to represent the whole
day. Different colors of cubes could represent each subject and the number of cubes for each
subject is represented by 10 minute intervals. Students take off cubes as they go through the
day. Ask questions throughout the day like, how much time do we have left in school? How
much time do we have until we go to lunch? Library? Recess?
30. Distribute rulers and yardsticks to a small group of students. Direct students to measure the
length of each wall of the classroom. Students add the measurements of each wall to
determine the perimeter. Students will need to regroup and convert the measurements to
determine the perimeter in feet or yards. This applies the process of measurement to finding
perimeter.
31. Distribute several pieces of yarn that are different lengths. Students use measurement tools
to measure each piece, and then add all the individual lengths to get a total combined length
for all of the pieces of yarn. Students will need to convert the measurements to determine the
combined length.
32. Apply what you’ve learned in measurement to tasks you do at home. For example:
 Measure a window in the school, and determine how much material would be needed to
make curtains.
 Determine how much fencing you would need in inches, feet, yards, and meters to keep
stray dogs off the playground.
 Determine how many plants are needed to plant a border along a 30 foot walkway if
you have to plant the flowers 6 inches apart. Each flower is in a 3 inch pot.
 Figure out how many gallons of juice you need to provide 20 guests with 8 oz of juice.
33. Each student cuts a piece of string or yarn any length they choose and then estimates the
length of the string. Challenge the students to find five things in the room that are
approximately the same length. Then, ask students to measure their string, using metric
and/or customary measurement. Make mini-posters by taping the string to a piece of
construction paper and listing the items that are the same length.
34. Put a variety of classroom objects on trays. Divide the class into groups of four. Give each
group a tray and four rulers (metric and/or customary). Instruct the students to measure the
length of each object and record the name of the object and its length in their math journals.
Tell the students to compare their measurements with those of the other members of their
group. Are the measurements about the same? Do the measurements vary widely? What
should you do if the measurements are not similar? Reexamine any objects that have a wide
range of differences in measurement. Have students discuss why differences can occur,
when it would be okay for a slight difference, and when accuracy is a must.
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RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 32-33, 76-78, 82-83,
192-194, 196-197, 588-589, 592-593, 594-595, 652-653, 654-655, 656-657, 658-661
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. 10, 19, 45, 132, 134, 145,146, 147, 148
Practice Master pp. 11, 45, 46, 130, 132, 133 134, 145, 146, 147, 148
Problem Solving Master pp. 10, 19, 21, 45, 46, 130, 133, 134, 145, 146,
147, 148
Reteaching Master pp. 11, 45, 46, 130, 132, 133, 134, 145, 146, 147, 148
INTERDISCIPLINARY CONNECTIONS
1. Literature Connections
Hershey’s® Weights and Measures by Jerry Pallotta
How Big Is A Foot? by Rolf Myller
Me and the Measure of Things by Joan Sweeney
Millions to Measure by David Schwarz
2. During writing class, read How Big Is A Foot? by Rolf Myller. Stop reading the book when
the apprentice goes to jail. Discuss with the class why the King was angry and the mistake
the apprentice made. Have students write a friendly letter to the apprentice, offering him
advice. In the letter, include the date, greeting, body, closing, and signature. Have students
include references from the text to support their interpretations. Share the letters with the
class. Finish reading the book and again discuss the need for a standard measurement.
Grade 4 Measurement and Data
Page 50 of 94
Columbus City Schools 2013-2014
Problem Solving Questions
The snakes at the zoo are given vitamins according to their length.
Length in Feet
Each snake gets one dose per inch of length.
1
The lengths of the zoo’s snakes are listed in the chart.
2
Each snake is given one dose of vitamins per inch of length.
3
Complete the chart with the doses of vitamins for each snake.
4
If the zoo had an 8-foot snake, how many doses of vitamins would it require?
Vitamin Doses
Hank ordered drain pipes for his plumbing job. Put the drain pipes in order from
smallest to largest:
63 centimeters, 600 millimeters, 215 centimeters, 1 meter
Gevon has 3 feet, 5 inches of rope. How many inches of rope does he have?
Marie walked for 1 hour and 25 minutes. Sheri walked for 1.75 hours. Michelle also
walked for some time. If the total time the three girls walked was 3 hours and 40
minutes, how long did Michelle walk?
Colin needs to fill his fish aquarium. Help him decide which metric unit of
measurement he should use and why.
Grade 4 Measurement and Data
Page 51 of 94
Columbus City Schools 2013-2014
Problem Solving Questions
Larue cut 2 1 yards of fabric to make her prom dress. She then cut 1
3
2
3
yards for a
scarf. How many feet of fabric did she cut?
Bart bought a kite for $6.55 and a soda for $0.75. He paid with a $10.00 bill. How
much change did he get back?
The cashier did not have bills, so gave Bart his change in coins. What was the fewest
number of coins she could have given him?
Lulu will open a lemonade stand to raise $35 dollars for a Cedar Point ticket. She plans
to sell 8 oz. cups for $0.50 each. If she makes 2.5 gallons of lemonade, will she earn
enough to buy her ticket?
Five same-sized math textbooks in Room 32 have a total mass of 3.5 kilograms. How
many grams is one of the textbooks?
Bruce is biking to a bookstore that is 12 kilometers from his house. He bikes 2500
meters and stops to drink water. He then bikes 4800 meters before stopping to tie his
shoe. He then goes for another 1200 meters before stopping to rest. How many meters
does Bruce have to go until he reaches the bookstore? How many kilometers is that?
Use a number line to help solve the problem.
Grade 4 Measurement and Data
Page 52 of 94
Columbus City Schools 2013-2014
Problem Solving Questions Answers
The snakes at the zoo are given vitamins according to their length. Each snake gets one dose per inch
of length. The lengths of the zoo’s snakes are listed in the chart below. Each snake is given one dose
of vitamins per inch of length. Complete the chart with the doses of vitamins for each snake. If the
zoo had an 8-foot snake, how many doses of vitamins would it require?
Answer: 1ft = 12; 2ft =24; 3ft = 36; 4ft = 48
8-foot snake would take 96 doses. n x 12
Hank ordered drain pipes for his plumbing job. Put the drain pipes in order from smallest to largest:
63 centimeters, 600 millimeters, 215 centimeters, 1 meter
Answer: 600 millimeters, 63 centimeters, 1 meter, 215 centimeters
63 centimeters = 630 millimeters
1 meter = 1000 mm.
215 centimeters = 2150 mm.
Gevon has 3 feet, 5 inches of rope. How many inches of rope does he have?
Answer: He has 41 inches of rope.
3 × 12 = 36 in.
36 + 5 = 41 in.
Marie walked for 1 hour and 25 minutes. Sheri walked for 1.75 hours. Michelle also walked for some
time. If the total time the three girls walked was 3 hours and 40 minutes, how long did Michelle walk?
Answer: Michelle walked for 30 minutes.
1: 25 + 1:45 = 3 hrs. and 10 mins.
3:40 hours – 3:10 = 30 minutes.
Colin needs to fill his fish aquarium. Help him decide which metric unit of measurement he should
use and why.
Answer: Colin should use a liter measurement to fill his aquarium. If he uses a milliliter
measure, it will take too long to fill. By using a liter measure, he can save trips because the unit
is much larger than milliliters.
Grade 4 Measurement and Data
Page 53 of 94
Columbus City Schools 2013-2014
Problem Solving Questions Answers
Larue cut 2
1
3
yards of fabric to make her prom dress. She then cut 1
2
3
yards for a scarf. How
many feet of fabric did she cut?
Answer: Larue cut 12 feet of fabric.
2 1/3 yards = 7 feet
1 2/3 yards = 5 feet
7 + 5 = 12 feet
Bart bought a kite for $6.55 and a soda for $0.75. He paid with a $10.00 bill. How much change did
he get back?
The cashier did not have bills, so she gave Bart his change in coins. What was the fewest number of
coins she could have given him?
Answer: 1) Bart received $2.70 back from the cashier. 6.55 + .75 = 7.30 10.00 – 7.30 = 2.70
2) The fewest number of coins Bart would get is 12: 10 quarters and 2 dimes
Accept $0.50 or $1 coin if the student uses them.
Lulu will open a lemonade stand to raise $35 dollars for a Cedar Point ticket. She plans to sell 8 oz.
cups for $0.50 each. If she makes 2.5 gallons of lemonade, will she earn enough to buy her ticket?
Answer: Lulu won’t have enough to buy a ticket. She will only raise $20.00.
1 gallon = 128 ounces
2.5 gallons = 320 ounces
320 ÷ 8 = 40 cups
40 × $0.50 = $20.00
Five same-sized math textbooks in Room 32 have a total mass of 3.5 kilograms. How many grams is
one of the textbooks?
Answer: One textbook is 700 grams.
3.5 kg. = 3500 g.
3500 ÷ 5 = 700 g.
Bruce is biking to a bookstore that is 12 kilometers from his house. He bikes 2500 meters and stops to
drink water. He then bikes 4800 meters before stopping to tie his shoe. He then goes for another 1200
meters before stopping to rest. How many meters does Bruce have to go until he reaches the
bookstore? How many kilometers is that?
Answer: Bruce has 3500 meters to go, or 3.5 kilometers.
12 km. = 12000 m.
2500 + 4800 + 1200 = 8500 m.
12000 – 8500 = 3500 m. or 3.5 km.
Grade 4 Measurement and Data
Page 54 of 94
Columbus City Schools 2013-2014
Customary Length Record Sheet
Name
How many inches are in one foot? __________
How many feet are in one yard? __________
How many inches are in one yard? ___________
Description of item
being measured
Inches
Grade 4 Measurement and Data
Page 55 of 94
Feet
Yards
Columbus City Schools 2013-2014
Metric Length Record Sheet
Name
How many millimeters are in one centimeter? _____ one decimeter? _____
one meter? ______
How many centimeters are in one decimeter? _______ one meter? _______
How many decimeters are in one meter? _______
String
Millimeter
Centimeter Decimeter
Meter
#1
#2
#3
#4
#5
Grade 4 Measurement and Data
Page 56 of 94
Columbus City Schools 2013-2014
Balancing Mass Record Sheet
Name
5
1 gram
=
1
5 gram
=
=
=
=
=
=
=
Grade 4 Measurement and Data
Page 57 of 94
Columbus City Schools 2013-2014
Hershey’s Weights and Measures
Record Sheet
Name_______________________
The length of one M&M’s
The mass/weight of one section of a Hershey bar
The weight of a bag of Twizzler’s
The width of a Hershey miniature
How many 1 pound bags of Twizzler’s will equal a ton?
How many Twizzler’s equal 1 pound?
5 pounds?
1 gram?
How many Twizzler’s equal the length of 1 yard?
1 meter?
3 yards?
How many Hershey miniature bars equal the length of a ruler?
1 yard?
How many Hershey’s Milk Chocolate bar pieces are in one ounce? ____
1 pound?
How many Hershey Hugs equal 1 pound?
Grade 4 Measurement and Data
Page 58 of 94
5 lbs.?
Columbus City Schools 2013-2014
Which Container Fills Faster?
Name
Part One
Fill the bucket at your table using the small cup first and then the large cup. Record how many
of each cup it took to fill the bucket.
small cups
large cups
If you wanted to measure the capacity of a bathtub, would it be more appropriate to use liters or
milliliters? Explain your answer.
Part Two
Look at the pictures of the containers below. Use what you have learned from filling the bucket
to determine if you were filling them with water, whether each container should be filled using
milliliters or liters and circle the unit that should be used.
pickle jar
milliliters or kiloliters
vitamin bottle
milliliters or liters
vinegar bottle
milliliters or kiloliters
barrel
milliliters or liters
baby food jar
milliliters or liters
wading pool
milliliters or liters
Grade 4 Measurement and Data
Page 59 of 94
Columbus City Schools 2013-2014
Equivalent Cards
8 fluid
ounces
1 cup
1
pint
2
16 fluid
ounces
2 cups
1 pint
1
quart
2
32 fluid
ounces
4 cups
Page 60 of 94
Columbus City Schools 2013-2014
Grade 4 Measurement and Data
Equivalent Cards
2 pints
1 quart
64 fluid
ounces
8 cups
4 pints
2 quarts
1
128 fluid
gallon
16 cups
ounces
2
Grade 4 Measurement and Data
Page 61 of 94
Columbus City Schools 2013-2014
Equivalent Cards
8 pints
4 quarts 1 gallon
6 inches
1
foot
2
12
inches
1 foot
36
inches
3 feet
Page 62 of 94
Columbus City Schools 2013-2014
Grade 4 Measurement and Data
Equivalent Cards
36
inches
1 yard
5,280
feet
1 mile
1,760
yards
16
ounces
1 pound
2,000
pounds
1 ton
Page 63 of 94
Columbus City Schools 2013-2014
Grade 4 Measurement and Data
Grade 4 Measurement and Data
Page 64 of 94
Columbus City Schools 2013-2014
=
=
5,280 feet
16 ounces
1 pound
1 mile
3 feet
=
=
16 cups
=
=
=
8 cups
=
36 inches
=
4 cups
=
½ foot
=
2 cups
=
=
=
1 cup
=
6 inches
8 fluid
ounces
16 fluid
ounces
32 fluid
ounces
64 fluid
ounces
128 fluid
ounces
1 yard
8 pints
4 pints
2 pints
1 pint
½ pint
=
=
=
=
Conversion Key
=
12 inches
2,000
pounds
=
1,760 yards =
=
=
4 quarts
2 quarts
1 quart
½ quart
1 ton
1 mile
1 foot
1 gallon
½ gallon
Me and the Measure of Things
Record Sheet
Name________________________
Look at each container labeled A, B and C. Fill the containers using the small cup
first and then the large cup. Record how many of each cup it took to fill the
containers.
Container
Number of small cups
Number of
large cups
A
B
C
.
Use what you know about liters and milliliters to compare using small and large
cups to measure capacity with using milliliters and liters to measure capacity.
How are the different measurement units similar and different?
If you wanted to measure the amount a backyard inflatable swimming pool holds,
would you use liters or milliliters? Explain your answer.
Grade 4 Measurement and Data
Page 65 of 94
Columbus City Schools 2013-2014
Elapsed Time – Time Line
CUT ALONG SOLID BLACK LINE
PASTE ALONG DOTTED LINE
12:00
5:00
11:00
4:00
10:00
3:00
CUT ALONG SOLID BLACK LINE
6:00
9:00
2:00
8:00
1:00
7:00
6:00
12:00
Grade 4 Measurement and Data
Page 66 of 94
PASTE ALONG DOTTED LINE
Columbus City Schools 2013-2014
What Time Do I Begin My Errands?
Your friend is picking you up promptly at 5:00 p.m.
Your first errand is to drop off letters at the post office. It
takes 15 minutes to get to the post office.
From the post office you need to go to the library to return
books. It takes 10 minutes to get to the library from the post
office.
From the library you need to go to the recreation center to
drop off the raffle ticket stubs for your mother. It takes 25
minutes to get to the recreation center from the library.
After dropping off the raffle tickets you need to stop by your
grandmother’s house to pick up a recipe for your mother. It
takes fifteen minutes to get to your grandmother’s house
from the recreation center.
You should stay and visit with your grandmother for at least
30 minutes. You are 10 minutes away from your house
after you stop at your grandmother’s house.
What time will you need to begin your errands so that you will be back
home by 5:00 o’clock. __________________
Grade 4 Measurement and Data
Page 67 of 94
Columbus City Schools 2013-2014
Elapsed Time Match
11 12
1
Lasha goes to the store at 3:30 p.m. She
gets home 45 minutes later. What time
did she get home?
9
Kyle has baseball practice at 6:15 p.m.
If it takes him 20 minutes to get to the
practice field, what time should he
leave?
5:55 p.m.
It is now 9:05 a.m. What time was it 30
minutes ago?
8:35 a.m.
Kayla went to her friend’s house at 4:20
p.m. Her mother said they would eat
dinner at 5:30 p.m. How long is it
before Kayla must be home for dinner?
1 hour and
10 minutes
Sam and Joel want to see a movie that
starts at 7:25 p.m. If the movie lasts 1
hour and 55 minutes, what time will it
be done?
Steven went to a baseball game. He was
gone for 4 hours and got home at 10:35
p.m. What time did he leave for the
game?
Grade 4 Measurement and Data
Page 68 of 94
2
10
3
8
4
7
6
5
9:20 p.m.
11 12
1
2
10
9
3
8
4
7
6
5
Columbus City Schools 2013-2014
Elapsed Time Match
Teresa got on the bus at 8:10 a.m. and
arrived at school at 8:50 a.m. How long
was her bus ride?
40 minutes
Brian went to sleep at 9:00 p.m. and
woke up at 6:00 a.m. How long did he
sleep?
9 hours
Elaine and Earl want to take the COTA
bus to the mall. The bus schedule says
that the bus leaves their corner at 11:10
a.m. and arrives at the mall at 11:45 a.m.
How long will their bus ride be?
35 minutes
Yoshio is making a cake for his
mother’s birthday. The box says to
bake the cake for 35 minutes. If he puts
the cake in the oven at 2:35 p.m., what
time will the cake be done?
3:10 p.m.
William went on a hike at 7:30 a.m. He
got home at 9:10 a.m. How long did his
hike take?
1 hour and
40 minutes
The Chin family wants to take a cave
tour that lasts for 40 minutes and ends at
5:00 p.m. What time does the tour
begin?
The symphony concert started at 7:05
p.m. and ended at 9:55 p.m. How long
was the concert?
Grade 4 Measurement and Data
Page 69 of 94
11 12
1
2
10
9
3
8
4
7
6
5
2 hours and
50 minutes
Columbus City Schools 2013-2014
Elapsed Time Match
The winner of the marathon ran the race in 2
hours and 10 minutes. The last person crossed
the finish line after running for 3 hours and 45
minutes. How much faster did the winner
finish the race than the last person?
1 hour and 35 minutes
Anthony can’t wait until his mother
comes to school to have lunch with him.
It is now 10:45 a.m. and lunch is at
11:50 a.m. How much longer does
Anthony have to wait?
1 hour and
5 minutes
It is now 12:35 p.m. What time will it
be 2 hours and 10 minutes from now?
2:45 p.m.
Patricia is getting ready for school. She
wakes up at 7:30 a.m. and leaves the
house at 8:20 a.m. How long does she
have to get ready for school?
50 minutes
It is 7:40 p.m. and Carolyn has been
doing homework for 35 minutes. What
time did she start her homework?
11
12
1
2
10
9
3
8
4
7
6
5
Paul is baking brownies. He puts them
in the oven at 4:30 and figures they will
be done at 4:55. How long do the
brownies need to bake?
25 minutes
Wendy went to choir practice at 2:15
and left at 3:30. How long did choir
practice last?
1 hour and
15 minutes
Grade 4 Measurement and Data
Page 70 of 94
Columbus City Schools 2013-2014
Type A
Missing End Time
Given start time and
the duration of the
activity and must find
the end time
Type B
Missing Start Time
Given duration of the
activity and the end
time and must find
the start time
Type C
Missing How Long
Given start time and
end time and must
find the duration of
the activity
Columbus City Schools 2013-2014
Page 71 of 94
Grade 4 Measurement and Data
Sorting Time Questions
Set 1
Name
1. At 4:00 p.m. Maureen begins to make a cake. It takes her 5 minutes to get out the things she
needs, 15 minutes to mix the ingredients, and 30 minutes to bake the cake. The cake cools
for 10 minutes in the pan and another 15 minutes on the cake rack. Finally, Maureen spends
5 minutes frosting the cake.
Draw hands on the second clock to show when the cake is finally done. Write the time on
the line.
What is the time? ________________
2. Mrs. Tallchief looks at her clock which shows the following time.
She is going to see a movie that begins at 8:10 p.m.
How much time does she have before the movie starts?
 A. 1 hour 55 minutes
 B. 1 hour 45 minutes
 C. 1 hour 25 minutes
 D. 1 hour 15 minutes
Grade 4 Measurement and Data
Page 72 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Set 1
3. It takes 25 minutes for Emma and Sasha to walk to school.
What time do they leave home if they arrive at school by 8:50?
 A. 8:15
 B. 8:25
 C. 8:35
 D. 9:15
4. What time will it be in 23 minutes?
 A.
 B.
 C.
 D.
Grade 4 Measurement and Data
Page 73 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Set 1
5. Cheryl left work at 5:12 p.m. She arrived home at 6:55 p.m. How much time had elapsed
since she left work? Use words, pictures, or numbers to explain your answer.
6. George’s school starts at 9:00 a.m. His dad has to be at work an hour before George’s school
starts. Which clock shows the time that George’s dad has to be at work?
 A.

 C.
B.
 D.
7. The clocks below show how long Michael played a video game on three days. How long did
he play on each day? Use words, pictures, or numbers to explain your answers.
Day 1
Day 2
Day 3
Grade 4 Measurement and Data
Page 74 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Set 1
8. Lin’s school bus picks her up at 8:15 a.m. Her clock now says 7:40 a.m. How much time
does Lin have to get to her bus stop?
 A. 20 minutes
 B. 25 minutes
 C. 35 minutes
 D. 55 minutes
9. Marlon has been practicing playing his trumpet for 56 minutes. What time was it when he
started if this clock shows the time now?
 A.
 B.
 C.
 D.
10. The Smith family is going to the movies. They are going to see a movie that is 1 hour and 50
minutes long. The movie starts at the time shown on this clock.
What time will the movie be over?
 A. 2:55
 B. 3:55
 C. 4:55
 D. 5:05
Grade 4 Measurement and Data
Page 75 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Set 2
Name
1. Which clock shows a time that is 45 minutes earlier than 11:15?
 A.
 B.
 C.
 D.
2. The picture below shows the time you started reading.
The picture below shows the time you finished reading.
How much time did you spend reading?
 A. Just a few minutes
 B. Almost an hour
 C. Over an hour
 D. Almost 2 hours
Grade 4 Measurement and Data
Page 76 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Set 2
3. Bruno leaves school on the school bus at 3:05 p.m.
The bus drops him off in front of his house at 3:25 p.m.
How long is the bus ride for Bruno?
 A. 20 minutes
 B. 25 minutes
 C. 30 minutes
 D. 35 minutes
4. Joanie left for soccer practice at 4:25 p.m. She arrived at 5:05 p.m. How long did it take her
to get to practice?
 A. 30 minutes
 B. 40 minutes
 C. 25 minutes
 D. 35 minutes
5. Bob’s mother dropped him off at the tennis courts Monday after school at the time shown on
the clock. She picked him up 2 hours and 8 minutes later. What time did she pick him up?
 A. 5:38 p.m.
 B. 12:38 p.m.
 C. 2:08 p.m.
 D. 2:38 p.m.
Grade 4 Measurement and Data
Page 77 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Set 2
6. Sanza wants to go to the park to meet his friends, Austin and Bo. It is 11:30 a.m., and Sanza
still has some chores to do before he can play. It will take Sanza 20 minutes to clean his room,
15 minutes to walk the dog, and 5 minutes to empty the trash. What is the earliest time that
Sanza can head to the park to meet his friends?
 A. 12:05 a.m.
 B. 10:50 a.m.
 C. 12:10 p.m.
 D. 12:15p.m.
7. Noel arrived at school at 8:35 a.m. on Monday. The clock below shows the time she woke
up on Monday morning. How much time had elapsed from the time she woke up until the
time she arrived at school?
 A. 1 hour 35 minutes
 B. 1 hour 45 minutes
 C. 1 hour 50 minutes
 D. 2 hours 50 minutes
8. The clock shows the current time. Susan has to get ready in a hurry. She has to be at a
birthday party by 9:00 and it takes 8 minutes to get there. By what time does Susan need to
leave the house in order to make it to the party on time?
 A. 8:52 a.m.
 B. 9:02 a.m.
 C. 9:08 a.m.
 D. 9:52 a.m.
Grade 4 Measurement and Data
Page 78 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Set 2
9. Look at the clock below. School starts in 20 minutes.
What time does school start?
 A. 8:35
 B. 8:55
 C. 9:05
 D. 9:15
10. Mrs. Dulchay is making cupcakes for her daughter’s birthday. She needs the cupcakes to be
finished by 4:45 when the guests will begin to arrive. It will take her 15 minutes to frost the
cupcakes after they have cooled for 30 minutes. If the cupcakes take 25 minutes to bake,
what time does she need to put them in the oven so that she will be finished at 4:45? Use
words, pictures, or numbers to explain your answer.
Grade 4 Measurement and Data
Page 79 of 94
Columbus City Schools 2013-2014
Sorting Time Questions
Answer Key
Students do not need to answer any of the problems in Set 1 or Set 2, but if a question arises, the
answers are listed below.
Set 1
1. 5:20
2. A
3. B
4. C
5. 1 hour 43 minutes
6. B
7. 1 hour 20 minutes, 40 minutes, 35 minutes
8. C
9. C
10. C
Set 2
1. A
2. C
3. A
4. B
5. A
6. C
7. C
8. A
9. D
10. 3:35
Grade 4 Measurement and Data
Page 80 of 94
Columbus City Schools 2013-2014
Pancake House
Egg Combinations
Pancakes
Sunny Delight
2 eggs, hash browns, toast $2.99
Buttermilk Pancakes
3 large pancakes with butter and
maple syrup $2.99
Strawberry Pancakes
3 large pancakes with fresh strawberries and
whipping cream $3.89
Blueberry Pancakes
3 large pancakes with fresh blueberries and
whipping cream $3.89
Hearty Start
3 eggs, hash browns, choice of bacon or sausage,
toast and 2 buttermilk pancakes $6.99
Lighter Start
1 egg and toast $1.50
Fresh Start
2 eggs, bacon or sausage, toast $3.99
Little Pig Pancakes
3 large pancakes each wrapped around
breakfast sausage $5.49
Hungry Jack’s Breakfast
3 large buttermilk pancakes, 2 eggs, choice
of sausage or bacon and home fries $ 6.59
Sides
Beverages
Milk
White or chocolate
Small $0.75
Medium $1.00
Large $1.25
Toast $0.75
Bran Muffin $1.25
Pecan Roll $1.75
Cereal $1.25
Hash Browns $0.90
Bacon $1.50
Coffee
Small $1.25
Medium $1.50
Large $1.75
Juice
Orange, grapefruit, grape, pineapple
Small $.75
Medium $1.25
Large $1.50
Sausage $1.50
Grade 4 Measurement and Data
Page 81 of 94
Columbus City Schools 2013-2014
Joe’s Grill
Starters
Onion Rings
$4.99
Sliced onion, hand battered and fried with our
special sauce
Cheese Con Queso
Mild or hot with tostada chips
$3.99
Fried Cheese
$4.59
Mozzarella cheese sticks served with marinara sauce
Potato Skins
Loaded with bacon, cheese, and sour cream
$5.99
Wings
$2.99
One pound of spicy chicken wings with celery
Club Sandwich
$5.99
Turkey, ham, swiss cheese, bacon, lettuce,
tomato, on toasted white bread with chips
Chicken Combo
$6.59
Fried chicken breast with a special sauce,
lettuce, tomato on a fresh roll with fries
Cheese steak
$6.78
Shredded steak with onions, mushrooms,
and mozzarella cheese on a toasted roll with
fries
Soups and Salads
House Salad
Sandwiches
$2.99
House Salad with any sandwich
$1.49
Old Fashion Burger
$5.79
Mustard, ketchup, lettuce, pickle, tomato and
hamburger on a sesame seed bun with fries
Soups of the Day
Bowl
Cup
Cup with any sandwich
$3.29
$2.29
$1.49
Deluxe Cheeseburger
$6.99
2 cheeses, lettuce, tomato, mayo, and
hamburger on a sesame seed bun with fries
Cobb Salad
$7.29
Crispy chicken, bacon, eggs, tomato, blue cheese,
green onions, mixed greens
Grilled Chicken Salad
$7.59
Grilled chicken, cheddar cheese, cucumbers, celery,
carrots, olives, croutons
Homemade Baked Goods
Chocolate Chip Cookies
One dozen fresh from the oven
Famous Banana Cream Pie
$3.25
$2.25
Brownie ala mode
$2.50
Grade 4 Measurement and Data
Page 82 of 94
Beverages
Milk Shake
Vanilla, chocolate, or strawberry
$1.69
Fountain Drinks
$1.29
Hot or iced tea
$1.15
Lemonade
$1.20
Columbus City Schools 2013-2014
Asian Kitchen
Special Platters
(served with egg roll and fried rice)
Appetizers
General TSO’S Chicken
Battered chicken in a special sauce
Egg Roll
$3.00
2 egg rolls filled with minced pork and vegetables
Egg Drop Soup
Egg dropped in chicken broth
$1.25
Wonton Soup
Dumpling soup
$1.95
Satay Beef
Marinated slices of beef
$2.95
California Rolls
2 rice rolls with vegetables
$2.50
$4.95
Chicken in Garlic Sauce
$4.95
Sliced chicken marinated and sautéed in a spicy hot
garlic sauce
Chicken Chow Mein
$4.95
Sliced chicken breast cooked with vegetables and
served with crispy noodles
Sesame Beef
$4.95
Sliced tender beef sautéed in a special sauce with
sprouts and sesame seeds
Shrimp In Garlic Sauce
$4.95
Shrimp sautéed with peppers, mushrooms, and water
chestnuts in a spicy sauce
Moo Shu Pork
$6.95
Shredded pork with vegetables and eggs wrapped in
pancakes served with a special sauce
Sweet and Sour Pour
$5.95
Battered fried pork in a sweet and sour sauce
Szechwan String Beans
Fresh string beans in a Szechwan sauce
Vegetable Lo Mein
$3.95
Seasonal vegetables tossed with noodles in a special
sauce
Tea
(hot or iced)
Green tea
Black tea
Red tea
Vanilla tea
Raspberry tea
Grade 4 Measurement and Data
$3.95
$1.50
$1.50
$1.50
$1.50
$1.50
Fried Rice
Chicken, pork, beef or vegetable
Page 83 of 94
$4.50
Columbus City Schools 2013-2014
Order Form
Name
Quantity
Description of Item
Cost
Total
Grade 4 Measurement and Data
Page 84 of 94
Columbus City Schools 2013-2014
Pigs Will be Pigs
Name
How much money did the pigs find on their hunt through the house?
Mr. Pig
Mrs. Pig
Mrs. Pig and the Little Pigs
The Piglets
How much money did the Pigs find altogether? ________________
There was a knock at the door and Grandma Pig gave each little piglet 8
quarters, 4 dimes and 10 pennies. How much money do the Pigs have
now?
_________________
Grade 4 Measurement and Data
Page 85 of 94
Columbus City Schools 2013-2014
Mama Mia’s Pizza House
Appetizers
Garlic Bread
Salads
10 slices: $2.99
15 slices: $3.99
Side: $2.99
Lettuce, tomato, celery, carrots, olives, and
mozzarella cheese
Breadsticks & Marinara Sauce
6 sticks: $2.99
Extra sauce: $0.50
Chef: $3.99
Cheesy Sticks
Lettuce, tomato, cucumbers, olives, red
peppers, pepperoni, ham, cheddar, and
mozzarella cheese
Lightly seasoned fried cheese served with
marinara sauce: $3.69
Grilled Vegetable Salad: $5.99
Lettuce, tomato, celery, olives, green peppers,
croutons, mushrooms and onions, cheddar and
mozzarella cheese
Antipasto
Assorted Italian cheeses and breads
Serves 4: $8.99
Serves 8 $12.99
Pizzas
Crust: Thick or Thin
Sizes: 6 inch small: $2.99
14 inch Large: $9.99
12 inch Medium: $6.99
16 inch Extra Large: $11.99
Toppings:
Black Olives, Green Olives, Green Peppers, Red Peppers, Jalapeno Peppers, Mild Banana Peppers,
Mushrooms, Onions, Pepperoni, Pineapple, Tomato, Fresh Mozzarella, Mozzarella Cheese, Provolone
Cheese
Drinks
Sizes: Small: $0.99 Medium: $1.20 Large: $1.50
Coca Cola® Diet Coke® Sprite®
Minute Maid®
Dr. Pepper®
Desserts
CinnaSticks: $3.99
Butter-flavored cinnamon coated
strips served with sweet icing
Ice Cream Sundae: $2.99
Vanilla ice-cream, hot fudge,
whipped cream and nuts
Deluxe Brownie: $1.50
White and dark chocolate chips
Assorted Cookies:
Serves 2: $2.00
Serves 4: $3.50
Grade 4 Measurement and Data
Page 86 of 94
Columbus City Schools 2013-2014
Pigs Will Be Pigs
Order Pad
Name
Item
Price
Total
Item
Price
Total
Grade 4 Measurement and Data
Page 87 of 94
Columbus City Schools 2013-2014
Price Cards
Lunch Menu
Lunch Menu
Tuesday
Peanut Butter and Jelly Sandwich
Bagel with Cream Cheese
Chef Salad
Taco Salad
Hamburger and Chips
Hot Dog and Chips
$0.75
$1.25
$2.75
$1.25
$2.50
$1.25
Apple
Cookies
$0.35
$0.15
Milk
Chocolate Milk
Small Juice
Large Juice
Lemonade
Iced
SmallTea
Fries
Water
School Supply List
Pen (each)
Pens(set of five)
Small Notebook
Large Notebook
Composition Book
Ruler
Box of Crayons
Erasers
Colored Pencils
$0.50
$0.50
$0.50
$0.75
$0.75
$0.45
$0.75
$0.50
School Supply List
$0.82
$2.27
$1.36
$1.74
$0.83
$0.17
$1.72
$0.34
$0.34
Glue
Glue Stick
Scissors
Index Cards
Pencil Box
Compass
Protractor
Pencils(10 pack)
Pencil
Snack List
$0.38
$1.42
$0.58
$0.72
$1.38
$0.66
$0.63
$1.34
$0.28
Snack List
Popcorn
Nachos
Candy Bars
Soft Pretzel
Pizza Slice
Ice Cream Bar
Gum
$0.40
$1.75
$0.60
$1.45
$1.55
$0.95
$0.75
Soda
Small
Medium
Large
Water (bottled)
Coffee
Tea
Grade 4 Measurement and Data
Page 88 of 94
$0.50
$0.75
$1.00
$1.00
$0.50
$0.50
Columbus City Schools 2013-2014
Fill Me Up and Pour Me Out
Name
Use the water, measuring spoons, and measuring cups to find the following
equivalent measures.
1 fluid ounce = ____________ tablespoons
1 cup = ____________ tablespoons = ____________ fluid ounces
¼ cup = ____________ tablespoons = ____________ fluid ounces
½ cup = ____________ tablespoons = ____________ fluid ounces
2 cups = ___________ fluid ounces = ______________ pints
Use measuring cups to find equal measures.
1 pint = ______________ cups
1 quart = _____________ cups
1 gallon = _____________ cups
1 quart = _____________ pints
½ gallon = ____________ quarts
1 gallon = ____________ pints
1 gallon = ____________ quarts
Grade 4 Measurement and Data
Page 89 of 94
Columbus City Schools 2013-2014
Fill Me Up and Pour Me Out
Answer Key
Name
Use the water, measuring spoons, and measuring cups to find the following
equivalent measures.
1 fluid ounce =
2
tablespoons
1 cup =
16
tablespoons =
¼ cup =
4
tablespoons =
½ cup =
8
tablespoons =
2 cups =
32
fluid ounces =
8
fluid ounces
2
fluid ounces
4
16
fluid ounces
pints
Use measuring cups to find equal measures.
1 pint =
2
cups
1 quart =
4
cups
1 gallon =
16
cups
1 quart =
2
pints
½ gallon =
2
quarts
1 gallon =
8
pints
1 gallon =
4
quarts
Grade 4 Measurement and Data
Page 90 of 94
Columbus City Schools 2013-2014
Which is Best?
Circle the most appropriate unit of measure for each picture.
A bicycle weighs . . .
The capacity of a fish tank
would be . . .
12 grams
or
12 kilograms
20 fluid ounces
or
20 gallons
The length of a shoe is . . .
A dog weighs . . .
9 inches
or
9 feet
15 kilograms
or
15 grams
The thickness of a dime
would be . . .
The diameter of a nickel
would be . . .
1 millimeter
or
1 centimeter
2 centimeters
or
2 millimeters
The length of an
unsharpened pencil is . . .
A large pitcher of milk is
about the same as . . .
19 centimeters
or
19 meters
4 milliliters
or
4 liters
Grade 4 Measurement and Data
Page 91 of 94
Columbus City Schools 2013-2014
Which is Best?
The length of a football
The distance from
field is . . .
Columbus to Cleveland is
about . . .
91 meters
283 miles
or
or
91 millimeters
283 yards
The weight of a bowling
ball is . . .
The length of a sheet of
paper is . . .
12 ounces
or
12 pounds
28 meters
or
28 centimeters
The distance across a CD
is . . .
The amount of water in a
bathtub is . . .
12 millimeters
or
12 centimeters
45 cups
or
45 gallons
The width of a doorway is
about . . .
Ten paperclips would weigh
about . . .
1 meter
or
1 centimeter
10 grams
or
10 kilograms
Grade 4 Measurement and Data
Page 92 of 94
Columbus City Schools 2013-2014
Which is Best? Answer Key
Circle the most appropriate unit of measure for each picture.
A bicycle weighs . . .
The capacity of a fish tank
would be . . .
12 grams
or
12 kilograms
20 fluid ounces
or
20 gallons
The length of a shoe is . . .
A dog weighs . . .
9 inches
or
9 feet
15 kilograms
or
15 grams
The thickness of a dime
would be . . .
The diameter of a nickel
would be . . .
1 millimeter
or
1 centimeter
2 centimeters
or
2 millimeters
The length of an
unsharpened pencil is . . .
A large pitcher of milk is
about the same as . . .
19 centimeters
or
19 meters
4 milliliters
or
4 liters
Grade 4 Measurement and Data
Page 93 of 94
Columbus City Schools 2013-2014
Which is Best? Answer Key
The length of a football
field is . . .
91 meters
or
91 millimeters
The distance from
Columbus to Cleveland is
about . . .
283 miles
or
283 yards
The weight of a bowling
ball is . . .
The length of a sheet of
paper is . . .
12 ounces
or
12 pounds
28 meters
or
28 centimeters
The distance across a CD
is . . .
The amount of water in a
bathtub is . . .
12 millimeters
or
12 centimeters
45 cups
or
45 gallons
The width of a doorway is
about . . .
Ten paperclips would weigh
about . . .
1 meter
or
1 centimeter
10 grams
or
10 kilograms
Grade 4 Measurement and Data
Page 94 of 94
Columbus City Schools 2013-2014