Fifth 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 4
Number and Operations- Fractions
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
4.
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example,
use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) ×
(c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the
same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as
rectangular areas.
5.
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by
whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the
given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
Geometry
Classify two-dimensional figures into categories based on their properties.
3.
Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles have
four right angles and squares are rectangles, so all squares have four right angles.
4.
Classify two-dimensional figures in a hierarchy based on properties.
Graph points on the coordinate plane to solve real-world and mathematical problems.
1.
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on
Columbus City Schools
2013-2014
Fifth Grade Mathematics
each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to
travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the
names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
2.
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the
context of the situation.
Columbus City Schools
2013-2014
What Can I Do At Home?
Grade 5
Fourth Grading Period
During the fourth grading period, in math,
your child will be expected to:
Number and Operations-Fractions
♦ Multiply a fraction or whole number by a
fraction
♦ Compare the size of a product to the size one
factor on the basis of the size of the other
factor, without performing multiplication.
♦ Explain why multiplying a given number by a
fraction greater than 1 results in a product
greater than the given number and why
multiplying a given umber by a fraction less
than 1 results in a product smaller than the
given number
♦
♦
♦
♦
Geometry
Use a pair of perpendicular lines, called axes,
to define a coordinate system with the
intersection of the line (the origin) arranged to
coincide with the 0 on each line and a given
point is the lance located by an ordered pair
called coordinates.
Represent real world and mathematical
problems by graphing point in the first
quadrant of the coordinate plane, and interpret
coordinate values of points.
Understand that attributes belonging to a
category of two-dimensional figures also
belong to all subcategories of that category.
Classify two-dimensional figures.
Here’s what you can do to help your child
master these skills:
♦ Have your child help you with recipes. Ask
how much of each ingredient you would need
if you doubled or tripled the recipe. What if
you only need one-half or one-third of the
recipe?
♦ Play the game Battleship® with your child. A
similar game can also be played by drawing
shapes on grid paper and guessing where the
points are that create the shape
♦ Look on a map, have your child give the
coordinates to a selected city.
♦ Play “Find the Shape” or “Describe the Shape”
games. Have your child describe the shape of
something they see and you guess what they
are describing. Ask your child to find
something that is in the shape of a square,
rhombus, quadrilateral, etc. and tell what
attributes makes it that shape.
♦ Example of a coordinate grid:
y
1
9
8
7
6
5
4
3
2
1
0
Columbus City Schools
Cafe
Elementary
School
1 2 3 4 5 6 7 8 9 1
x
2012-2013
Mathematics Model Curriculum
Grade Level: Fifth Grade
Grading Period: 4
Common Core Domain
Time Range: 15 Days
Number and Operations - Fractions
Common Core Standards
Apply and extend previous understandings of multiplication and division to multiply and
divide fractions.
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a
fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a
sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create
a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit
fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths.
Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular
areas.
5. Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without
performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the
given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining
why multiplying a given number by a fraction less than 1 results in a product smaller than the given number;
and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
The description from the Common Core Standards Critical Area of Focus for Grade 5 says:
Developing fluency with addition and subtraction of fractions and developing understanding of the
multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole
numbers and whole numbers divided by unit fractions)
Students apply their understanding of fractions and fraction models to represent the addition and subtraction of
fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in
calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the
meaning of fractions, of multiplication and division, and the relationship between multiplication and division to
understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited
to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)
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
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Mathematics Model Curriculum
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
Connect the meaning of multiplication and division of fractions with whole-number multiplication and
division. Consider area models of multiplication and both sharing and measuring models for division.
Ask questions such as, “What does 2 × 3 mean?” and “What does 12 ÷ 3 mean?” Then, follow with
1
questions for multiplication with fractions, such as, “What does 3 × mean?” “What does 3 × 7 mean?”
4
4
3
(7 sets of 3 ) and What does 7 × 3 mean?” ( 3 of a set of 7)
4
4
4
1
1
1
1
The meaning of 4 ÷
(how many
are in 4) and ÷ 4 (how many groups of 4 are in ) also should be
2
2
2
2
illustrated with models or drawings like:
3
4
7 groups of 3
4
or
21
4
Encourage students to use models or drawings to multiply or divide with fractions. Begin with students
modeling multiplication and division with whole numbers. Have them explain how they used the model or
drawing to arrive at the solution.
Models to consider when multiplying or dividing fractions include, but are not limited to: area models using
rectangles or squares, fraction strips/bars and sets of counters.
Use calculators or models to explain what happens to the result of multiplying a whole number by a fraction
1
1
1
1
1
1
(3 ×
,4×
, 5 × …and 4 × , 4 × , 4 × ,…) and when multiplying a fraction by a number
2
2
2
2
3
4
greater than 1.
Use calculators or models to explain what happens to the result when dividing a unit fraction by a non-zero
1
1
1
whole number ( ÷ 4, ÷ 8, ÷ 16,…) and what happens to the result when dividing a whole number by
8
8
8
1
1
1
a unit fraction (4 ÷
, 8 ÷ , 12 ÷
,…).
4
4
4
Present problem situations and have students use models and equations to solve the problem. It is important for
students to develop understanding of multiplication and division of fractions through contextual situations.
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Mathematics Model Curriculum
Instructional Resources/Tools from ODE Model Curriculum
The National Library of Virtual Manipulatives: contains Java applets and activities for K-12 mathematics.
Fractions - Rectangle Multiplication – students can visualize and practice multiplying fractions using an area
representation.
Number Line Bars– Fractions: students can divide fractions using number line bars.
ORC # 5812 Divide and Conquer - Students can better understand the algorithm for dividing fractions if they
analyze division through a sequence of problems starting with division of whole numbers, followed by division of a
whole number by a unit fraction, division of a whole number by a non-unit fraction, and finally division of a
fraction by a fraction (addressed in Grade 6).
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 may believe that multiplication always results in a larger number. Using models when multiplying
with fractions will enable students to see that the results will be smaller.
Additionally, students may believe that division always results in a smaller number. Using models when
dividing with fractions will enable students to see that the results will be larger.
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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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
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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
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.
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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
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
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.
A problem solving approach to computation with fractions will engage students in using the
process skills and building fraction sense. Students must continue to build fraction concepts with
models. When computation with fractions (e.g., multiplication and division) is a new concept for
students, it is important that they have the opportunity to conceptually develop these skills using
hands-on activities and manipulatives. Discussion of real-life applications for these concepts is
also vital as students build this knowledge. A variety of manipulatives can be utilized for
fraction concepts, including fraction circles, fraction squares (or rectangles), and fraction strips.
In fifth grade, students justify why fractions need common denominators to be added or
subtracted. The need to use common denominators can be simply stated: one can only subtract
apples from apples, not apples from oranges. In order to solve fraction problems with unlike
denominators, attention must be focused on changing the way the problem is stated without
changing the problem. An understanding of equivalent fractions helps students restate problems.
Students can use prime factorization or a visual model (common-sized piece) to find a common
denominator. Using visual models will not always produce the least common denominator, but it
does give a common denominator so that the computation can be done and then the sum or
difference can be simplified.
Students can apply their knowledge of fractions to estimating and computing with mixed
numbers. Students can use what they already know by combining and/or separating the
fractional part of a mixed number and work with the whole numbers. However, students
encounter difficulty when working with mixed numbers and fractions that need to be traded.
Errors in converting fractions to whole numbers and whole numbers to fractions can be attributed
to a reliance on the base ten number system. Continued use of varied fraction models will help
students continue to develop proficiency with fraction concepts. Students can use a variety of
estimation strategies to determine the reasonableness of their answers.
Multiplication and division with fractions can be represented concretely using fraction squares.
Start by reviewing with students what it means to multiply whole numbers. Students need to be
reminded that multiplication can be represented by repeated addition, finding the total of a
number of same size groups, or an area model. To represent multiplication with fractions we can
2
use all of these models. Start with a simple problem, for example 4
, and represent it using
3
2
fraction squares or rectangles. To represent 4
, we could say that we are adding four groups
3
2
2
of together. This can be shown by drawing
four times and adding the pieces together to
3
3
8
get the product. Students can count the number of thirds to get the product, . It is important
3
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that as students work with fraction pieces, that they are also drawing a model of the pieces and
recording in number form what they are doing.
2
4 groups of
3
2 8
4
3 3
Writing the product as a mixed number is not critical at this point and could make it more
difficult for students to see the pattern of the whole number being multiplied by the numerator.
2
A next type of problem would be
times 4. This can be represented differently if we think of
3
2
the problem as
of a group of 4. To show this, draw a group of four squares (or rectangles)
3
and then section them into thirds, shading two out of every three thirds. Total the number of
8
thirds shaded to find the product of .
3
2 of a group of 4
3
2
4
3
8
3
1 3
× . This can be represented as one-half of a group of
2 5
three-fifths. To draw this, draw a square or rectangle with three-fifths shaded. Then draw a line
to cut the square into two equal pieces to find half of the three-fifths. Students can see that half
of three-fifths is equivalent to three-tenths.
3
1
of a group of
5
2
1 3 3
2 5 10
The next type of problem would be
This problem could also be represented by using two overlapping squares. Start by drawing 12
3
on a square. Then draw another square with drawn on it going in a perpendicular direction to
5
the way the first fraction was drawn. Overlap the two squares (or draw them both on the same
square), where the two shaded sections overlap is the product.
3
1
of a group of
5
2
1 3 3
2 5 10
This modeling can also be used with mixed numbers, although it will be easier to model a
problem with mixed numbers if only the second number (the group size) is a mixed number. For
1 1
1 can be modeled easily if we think of it as finding one-third of a group of 1 1 .
example,
3 4
4
1
Draw 1 using two squares, then cut them into three equal sections. It should then be easy to see
4
5
if the whole was sectioned into fourths before cutting the entire group into
12
Grade 5 Fractions
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Columbus City Schools
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that the product is
thirds. This will model for students that converting the mixed number into an improper fraction
will make the multiplication easier.
1
of a group of 1 1
3
4
1 1
1
3 4
1
of a group of 5
3
4
5
12
1 5
3 4
5
12
As students work with several of these problems they should realize that to find the product of a
problem with fractions involved they multiply the numerator times the numerator and the
denominator times the denominator. If students “discover” this algorithm on their own through
using these models, they will have a much better understanding and ability to remember it.
Division can also be represented using concrete models. Start by reviewing with students what it
means to divide with whole numbers. Dividing six by three means how many groups of three are
in six or how many groups of three can be made with a total of six. These models can make
2
5,
representing division with fractions possible. Begin with a simple problem like
3
2
representing it as “how many are in each group if
is divided into 5 groups?” This can be
3
modeled by drawing a square with two-thirds of the square shaded. Then divide the square (with
lines perpendicular to the lines used to section it into thirds) into five equally sized sections. The
2
square should make it possible to see that the quotient is
. Students may notice that this
15
2 1
picture looks the same as one that can be used to represent
. If they are able to make this
3 5
connection they are very close to discovering the algorithm for dividing fractions. Continue
working problems where a fraction is divided by a whole number until students are able to make
the connection that the reciprocal of the whole number is multiplied by the fraction to get the
quotient. Once students have made this connection and “discovered” the algorithm, students
should not need to rely on the models as much.
2
if it is divided into 5 groups?
3
2
2
5
3
15
How much is in each group of
As students work with manipulatives and models of problems, be sure they are also recording in
number form on a piece of paper what they are doing with the model or manipulative. This will
help students make the connection from the concrete to the more abstract representation of this
concept.
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Columbus City Schools
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PRACTICE ASSESSMENT ITEMS
2 in different ways. Their work is shown
On a test four students answered the problem 6
10
5
below. Which student(s) solved the problem correctly? Which student(s) did not solve the
problem correctly? Explain their mistakes.
Katie’s work: 6
10
2 = 8
5 15
Lucy’s work: 6
10
2 = 6
5 10
5 = 30
2
20
Jerry’s work: 6
10
2 = 6
5 10
2 = 12
25
5
Susan’s work: 6
10
2 = 30 = 3
5
20
2
Tyjuan and Desira went to the store to buy ten gallons of juice. Unfortunately, the store only
sold juice in half-gallons. Tyjuan said they should buy five half-gallons since 10 1 = 5.
2
Desira said they should buy 20 half-gallons because 10 1 = 20. Which person is correct?
2
Explain your answer.
Grade 5 Fractions
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Columbus City Schools
2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
On a test four students answered the problem 6
10
2
5
in different ways. Their work is shown
below. Which student(s) solved the problem correctly? Which student(s) did not solve the
problem correctly? Explain their mistakes.
Katie’s work: 6
2
5
= 8
Lucy’s work: 6
2
5
= 6
5
2
= 30
Jerry’s work: 6
2
5
= 6
2
5
= 12
10
10
10
Susan’s work: 6
2
5
10
15
10
10
20
25
= 30 = 3
20
2
Answer: Lucy and Susan are correct because they both divided using a mathematically
2 = 30 = 3 . Jerry is incorrect because he multiplied instead of
correct method: 6
10
5
20
2
dividing. Katie is incorrect because she added the numerators and denominators; she did
not divide.
A 2-point response states that Lucy and Susan are correct, that Jerry and Katie are
incorrect, and includes a complete explanation.
A 1-point response states that Lucy and Susan are correct and that Jerry and Katie are
incorrect with a weak explanation or includes a complete explanation of how to divide
8
2 without stating who was correct or incorrect.
10
5
A 0-point response shows no mathematical understanding of the task.
Tyjuan and Desira went to the store to buy ten gallons of juice. Unfortunately, the store only
sold juice in half-gallons. Tyjuan said they should buy five gallons since 10 1 = 5. Desira
2
said they should buy 20 gallons because 10 1 = 20. Which person is correct? Explain your
2
answer.
Answer: Desira is correct because there are 20 halves in 10, not 5.
A 2-point response includes a correct answer with a reasonable explanation.
A 1-point response includes a correct answer with a weak explanation.
A 0-point response shows no mathematical understanding of the task.
Grade 5 Fractions
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Columbus City Schools
2013-2014
PRACTICE ASSESSMENT ITEMS
Four friends ordered three large pizzas. The pizzas were not cut into slices, so they decided to
divide them into fourths. How many slices will they have?
Which problem could be used to answer this question?
A. 3 + 4
B. 3
1
4
C. 3
4
D. 3
1
4
In the problem below, will the quotient be a larger or smaller number than 8? Explain your
answer.
8
Grade 5 Fractions
1
4
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Columbus City Schools
2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Four friends ordered three large pizzas. The pizzas were not cut into slices, so they decided to
divide them into fourths. How many slices will they have?
Which problem could be used to answer this question?
A. 3 + 4
B. 3
1
4
C. 3
4
D. 3
1
4
Answer: D
In the problem below, will the quotient be a larger or smaller number than 8? Explain your
answer.
8
1
4
Answer: The quotient will be a larger number than 8. This problem asks how many onefourths are in 8. Each 1 whole contains 4 one-fourths so 8 wholes would contain 32onefourths; 8 1 = 32.
4
A 2-point response includes a complete explanation.
A 1-point response includes a weak explanation but demonstrates some understanding of
the concept.
A 0-point response shows no mathematical understanding of this concept.
Grade 5 Fractions
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Columbus City Schools
2013-2014
PRACTICE ASSESSMENT ITEMS
A quarterback completed 1 of his passes thrown on Friday night. If the
3
quarterback completed a total of 12 passes, how many passes were attempted?
A. 12
B. 24
C. 36
D. 38
In an elementary school there are 150 fifth grade students. 2 of the fifth grade
3
students are boys. How many girls are in the fifth grade?
A. 25
B. 50
C. 100
D. 200
Grade 5 Fractions
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Columbus City Schools
2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
A quarterback completed 1 of his passes thrown on Friday night. If the
3
quarterback completed a total of 12 passes, how many passes were attempted?
A. 12
B. 24
C. 36
D. 38
Answer: C
In an elementary school there are 150 fifth grade students. 2 of the fifth grade
3
students are boys. How many girls are in the fifth grade?
A. 25
B. 50
C. 100
D. 200
Answer: B
Grade 5 Fractions
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Columbus City Schools
2013-2014
PRACTICE ASSESSMENT ITEMS
Mary saves part of her weekly fifteen dollar allowance because she wants to
2
purchase a video game. If she saves of her weekly allowance, how many weeks
3
will it take her to save enough money to purchase a game that costs $49.50?
Explain your answer using words, numbers, or pictures.
3
2
of an apple pie on Monday. She ate of the leftover pie on Tuesday.
4
8
How much of a whole apple pie did she eat on Tuesday?
Lola ate
A.
6
of one pie
32
B. 3 of one pie
4
C.
18
9
or
of one pie
16
32
D.
8
of one pie
8
Grade 5 Fractions
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Columbus City Schools
2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Mary saves part of her weekly fifteen dollar allowance because she wants to purchase a video
2
game. If she saves
of her weekly allowance, how many weeks will it take her to save enough
3
money to purchase a game that costs $49.50? Explain your answer using words, numbers, or
pictures.
Answer will vary.
2 $30
$15.00 ×
=
= $10.00; $49.50 ÷ $10 = $4.95 so it would take 5 weeks
3
3
A 2-point response includes a correct answer and complete explanation.
A 1-point response includes a weak explanation but demonstrates some understanding of
the concept.
A 0-point response shows no mathematical understanding of this concept.
2
3
of an apple pie on Monday. She ate of the leftover pie on Tuesday. How much of
8
4
a whole apple pie did she eat on Tuesday?
Lola ate
A.
6
of one pie
32
B. 3 of one pie
4
C.
18
9
or
of one pie
16
32
D.
8
of one pie
8
Answer: C
Grade 5 Fractions
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PRACTICE ASSESSMENT ITEMS
An item that costs $220 is on sale for
3
off the original price.
4
How much money will you save on the sale?
What is the sale price of the item?
What fraction of the original cost does the person pay after the item goes on sale?
Another item that originally costs $60 was also discounted
3
off the original price.
4
Between these two items, for which would you save the most money based on its
original price?
Grade 5 Fractions
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Columbus City Schools
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
An item that costs $220 is on sale for 3 off the original price.
4
How much money will you save on the sale?
You will save $165.00.
What is the sale price of the item?
The item will cost $55.00.
What fraction of the original cost does the person pay after the item goes on sale?
1
of the original price is $55.00.
4
Another item that originally costs $60 was also discounted 3 off the original
4
price. Between these two items, for which would you save the most money based
on its original price?
The fraction off is the same but the more money you spend, the more money you save.
The $60.00 item costs $15.00 on sale at 3 off, for a savings of $45.00. The $220.00 item
4
costs $55.00 on sale at 3 off, for a savings of $165.00.
4
A 4-point response includes a correct answer similar to those listed above for each of the
four questions.
A 3-point response includes three correct answers.
A 2-point response includes two correct answers.
A 1-point response includes one correct answer.
A 0-point response shows no understanding of the question.
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: fraction, mixed number, denominator, product, factor, multiplication,
quotient, dividend, divisor, division, remainder, appropriate, reasonable, compatible,
numbers, area model, justify, resizing, evaluate, area, scaling
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
At the Columbus Zoo, of the animals are reptiles.
of the reptiles are lizards. What
8
4
fraction of animals are lizards?
2
1
At Raiderville Middle School, of the students play a sport. of the students that play a
3
2
sport are on baseball teams. What fraction of the students are on baseball teams?
2
1
Janine spends
of her paycheck and saves of the remaining amount. What fraction of
5
6
her paycheck is saved?
3
Every day Bryon ran of a mile. How far did he run after 7 days?
5
1
Mr. Templeman’s driveway is 3 meters long and of a meter wide? What is the area of
4
Mr. Templeman’s driveway?
1
Ms. Gaver told her class she has a book with a cover 5 cm wide and twice as long. She
2
instructed her students to calculate the area of the cover of the book. What is the area of
Ms. Gaver’s book?
1
The swimming pool at the new waterpark is 16 meters by 8 meters. What is the area of
4
the pool?
The label on a can of paint states that it contains enough paint to cover 20 m². I need to
1
paint a wall that measures 2 meters by 8
meters. Will I have enough paint?
2
Without multiplying, identify which of the expressions has the greater product.
4
1
× 20
or
× 20
Explain how you know.
3
3
1
3
Andrew ate
of of a pizza. Explain without finding the exact product. Did Andrew
2
4
1
3
eat more or less than
of the pizza? Did he eat more or less than
of the pizza?
2
4
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx Go to this web page,
open grade 5 on the right side of the page. Open “Unit 4 Framework Student Edition.”
These tasks are from the Georgia Department of Education. The tasks build on one another.
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3.
4.
5.
6.
Although, 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.
Envelope Game: Tape a word problem from, “Word Problems for Envelope Game and Pass
the Folder” (included in this Curriculum Guide) to the outside of an envelope. Make enough
envelopes so that each group will have a different question. Give each group an envelope
and a set of index cards. Have the group read the question on the envelope. They will work
together to solve the problem and write their answer on the index card. When the students
have finished answering their question they are to put the index card in the envelope. When
the teacher calls time, they will pass their envelope to a different group. Once they receive an
envelope from a different group they work together to answer that question on another index
card. Students are not allowed to look in the envelope at another groups answer. Each group
should answer at least 3 questions. Once they have answered their third question, have them
pass the envelope to another group. This should be a group that has not had the envelope
before. Once the group gets the last envelope, have them take all of the index cards out. The
students will work in their groups to decide which card best answers the question. Once they
have made their decision, have the groups present their question and the answer that they
think is best to the whole class.
Pass the Folder: This is very similar to the Envelope Game (see activity 3). The difference is
that the word problem is placed on the front of the folder and the “Pass the Folder Group
Directions” (included in this Curriculum Guide) are placed on the back of the folder.
Arrange students into pairs. Distribute “Window Time” activity sheet (included in this
Curriculum Guide) and “One Inch Grid” paper (included in the Grids and Graphics section of
the Curriculum Guide). Students work together to solve and justify using numbers, words
and pictures.
The following activities are repeats from quarter one.
Distribute plain sheets of paper to each student. Ask students how they could use the blank
paper to determine 2 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 three. If they would have been multiplying 3 by 3 then each piece of paper would
4
have been 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 parts. Altogether, six shaded thirds is the same as two whole rectangles
( 2 3 = 2)
3
Step 1:
Step 3:
6=2
3
Step 2:
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Distribute colored pencils and a copy of the “Folding Fractions” record sheet (included in
this Curriculum Guide) to each student. Direct students to fold the paper in half horizontally,
then use a green pencil (students can use any two different colors of pencil to do this activity)
to lightly color half of the paper. Ask the students what is 1 of 1 ? Generate responses
2
2
from the students. Demonstrate the answer by directing students to fold the whole paper in
half again but vertically. Now shade 1 of the paper purple. Where the two colors overlap is
2
the answer to
1
2
of
1
2
. What relationship does the section that contains both colors have to
the whole piece of paper ( 1 )? Point out to the students that 1 of 1 is equal to 1 of the
4
2
2
4
paper. Ask the students to take another sheet of blank paper and fold it into thirds. To find
1 of 1 students should color 1 of the paper green. Then ask them to fold the paper in half
2
3
3
the other direction. Direct them to color 1 of the paper purple. Where the green and the
2
purple overlap is the answer to
1
2
of
1.
3
They should see that the paper is now divided into
six equal pieces and one out of the six is colored green and purple, so 1 of 1 = 1 . Students
can continue this strategy with other problems. For example, to find
2
6
3
2 of 2 students
3
3
would
divide their paper into thirds and color in two sections green. They would then turn the
paper, fold it into thirds again, and color two sections purple. Where the colors overlap is the
answer to the multiplication problem. They first need to count the total number of sections
that the paper has been divided into, which is nine, and then count the number of sections
that are both green and purple (4), so 2 of 2 = 4 .
3
3
9
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. Then direct
students to divide the counters into as many parts as the denominator of the fraction, for
example, 1 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
Distribute a copy of the “Counter Fractions” (included in this Curriculum Guide) to each
student. Students work with a partner to model each problem using their counters and then
recording their work on the record sheet. Discuss as a class how students worked through
each problem. What conclusions can they make about multiplying with fractions?
8. Distribute “Centimeter Grid” paper (included in the Grids and Graphics section of this
Curriculum Guide), a copy of the “Grid Fractions” sheet (included in this Curriculum Guide),
and colored pencils to each student. Present 1 1 for students to model using the grid paper.
2
5
Since the denominators are 2 and 5, they will need to draw two 2 by 5 grids to model each
fraction. Caution students to shade one fraction using rows and the other fraction using
columns (as seen in the example below).
Grade 5 Fractions
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Columbus City Schools
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1 of a 2 by 5 grid
of a 2 by 5 grid
2
Ask students to determine how many square units are commonly shaded on both grids. They
should see that the only square that is shaded in both rectangles is the one in the lower lefthand corner.
1
5
Therefore, 1 1
5
2
1
10
. Continue with other problems and direct students to draw the area
models to represent each fraction. When students have had some practice, they can start
drawing both area models on the same rectangle and use two different colors of pencil to
show the area of each fraction. Where the pencil colors overlap is the answer to the
multiplication problem. Have students work through the problems on the Grid Fractions
sheet with a partner. Allow students time to share with each other and the class
generalizations and strategies that they develop for multiplying fractions.
9. Distribute “Centimeter Grid” paper (included in the Grids and Graphics section of this
Curriculum Guide), a copy of the “Grid Fractions II” sheet (included in this Curriculum
Guide), and colored pencils to model multiplying mixed numbers. Remind students that one
model of multiplication is to think of the problem as groups of something (e.g., 6 3 means
6 groups of 3; 2 5 means 2 of a group of 5). Present 1 2 2 to the class. First, direct
3
2
3
students to use a red pencil to model
22
3
3
on their centimeter grid paper. Students will need
to begin with a 2 by 3 model (the 2 from the denominator of the first fraction and the 3 from
the denominator of the mixed number). They color in two whole rectangles that are 2 by 3 to
represent the 2 wholes. They then color in 2 of the third rectangle. Students find 1 of their
3
2
model and count the number of sixths that are shaded. There are 8 sixths shaded so the
answer is 8 1 2 1 1 or 8 4 1 1 .
6
6
3
6
3
3
Step 1:
22
Step 2:
1 of 2 2
2
3
Step 3:
Redistribute to show that 8
3
1 of 8
2
3
8 or 1 2
6
6
6
11
3
12
6
11 .
3
Each circled section is one-half of the original rectangles. Ask students to compare these
problems with ones that do not involve mixed numbers. How are they different? How are
they similar? Can they use the same strategy to solve both kinds of problems? Ask students
to explain what they are doing numerically as they model the problem (e.g., drawing a
Grade 5 Fractions
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Columbus City Schools
2013-2014
picture is like converting the mixed number of 2 2 into the improper fraction 8 , finding half
3
of
8
3
3
is like dividing it by 2 to find that each half is the same as
8
6
).
10. Area models can also be used to model dividing with fractions. Ask students to think about
this question: can the answer (quotient) for a division problem be larger than the number
being divided (dividend)? Allow students a few minutes to think about this question and
share their thoughts with a partner. Talk with the students about the problem 5 1 . What
3
do they think this problem means? Explain that this problem is really asking them how many
one-thirds are in five. They can start with five rectangles. Each rectangle is then divided
into three equal parts (thirds) for a total of 15 sections in the five rectangles. Since each
section is the size of one-third, there are 15 one-thirds in 5 (5 1 = 15).
3
Ask students to look at the problem and answer and determine what generalizations they can
make about dividing by a fraction. Have students do another problem to see if they can
confirm their generalizations. What is 7 3 ? Students can begin by drawing 7 rectangles.
4
To find how many
3
4
there are in 7, divide each rectangle into fourths and then see how
many groups of 3 can be made.
4
There are 9 whole groups of 3 in 7. There is 1 left over, but since the question was asking
4
4
how many 3 are in 7, students need to determine what part of 3 they have with 1 . Since it
4
4
takes 3 pieces that are each 1 to make 3 , the 1 that is left over represents 1 of
4
Therefore, it can be said that 7
4
3 =91
4
3
4
3
4
3
4
.
because there are 9 1 groups of 3 in 7. Have
4
3
students describe how they found this answer in their own words (e.g., we divided each of
the 7 wholes by 4, then grouped the pieces into groups of 3, and found how many groups
were made). Modeling a fraction divided by a whole number can also be done with
rectangles. For example, in the problem 3 3 students would start with a whole rectangle.
4
The students are trying to determine what their share of 3 would be if 3 is divided into
4
4
three equal parts. First, the whole rectangle is divided into 4 equal parts and three of them
are shaded.
Grade 5 Fractions
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Columbus City Schools
2013-2014
The shaded rectangle is then divided into three parts going the other direction. The answer is
the share obtained in each of those three parts (how many are in each group).
} one share or 3
} one share or
} one share or
12
3
12
3
12
3
4
3= 3
12
Once students have an answer, they can simplify for a final answer of 1 . Provide students
4
with a copy of the “Fraction Divide” sheet (included in this Curriculum Guide). Students
work with a partner to model each problem on their sheet. Have students discuss with a
partner to develop, in their own words, a strategy for dividing fractions. What do problems
that have a fraction divided by a whole number have in common with problems that have a
whole number divided by a fraction? Is there any kind of generalization they can make that
is true for both of these kinds of problems? If the students do not see the connection, show
them that dividing by 1 gives the same answer as multiplying by 3 and that dividing by 3
3
gives the same answer as multiplying by 1 . Explain that 1 and 3 are special numbers called
3
3
reciprocals. When reciprocals are multiplied together the answer is 1. The answer to any
division problem can be found by changing it into a multiplication problem. For example, 6
3
4
4
3
could be rewritten as 6
for a final answer of 24 = 8 (i.e., there are 8 three-fourths
3
in the number 6). Show students that they could also solve 3
4
reciprocal of 3
(3
4
1
3
=
3
12
3 by multiplying 3 by the
4
).
11. Scaling Problem Solving: Arrange students into pairs. Distribute “Scaling Problem Solving”
activity sheet (included in this Curriculum Guide). Students work together to solve and
justify using numbers, words and pictures.
Grade 5 Fractions
Page 28 of 54
Columbus City Schools
2013-2014
RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 490-509
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. 110-114
Practice Master pp. 110-114
Problem Solving Master pp. 110-114
Reteaching Master pp. 110-114
Grade 5 Fractions
Page 29 of 54
Columbus City Schools
2013-2014
Problem Solving Questions
At the Columbus Zoo,
1
1
of the animals are reptiles. of the reptiles are lizards. What
8
4
fraction of animals are lizards?
At Raiderville Middle School,
2
1
of the students play a sport. of the students that play
3
2
a sport are on baseball teams. What fraction of the students are on baseball teams?
Janine spends
2
1
of her paycheck and saves of the remaining amount. What fraction
5
6
of her paycheck is saved?
Every day Bryon ran
3
of a mile. How far did he run after 7 days?
5
Mr. Templeman’s driveway is 3 meters long and
1
of a meter wide? What is the area of
4
Mr. Templeman’s driveway?
Grade 5 Fractions
Page 30 of 54
Columbus City Schools
2013-2014
Problem Solving Questions
1
2
Ms. Gaver told her class she has a book with a cover 5 cm wide and twice as long. She
instructed her students to calculate the area of the cover of the book. What is the area of
Ms. Gaver’s book?
1
4
The swimming pool at the new waterpark is 16 meters by 8 meters. What is the area of
the pool?
Problem Solving Questions
The label on a can of paint states that it contains enough paint to cover 20 m². I need to
1
2
paint a wall that measures 2 meters by 8 meters. Will I have enough paint?
Without multiplying, identify which of the expressions has the greater product.
4
× 20
3
or
1
× 20
3
Explain how you know.
1
3
of of a pizza. Explain without finding the exact product. Did Andrew
2
4
1
3
eat more or less than of the pizza? Did he eat more or less than of the pizza?
2
4
Andrew ate
Grade 5 Fractions
Page 31 of 54
Columbus City Schools
2013-2014
Problem Solving Questions Answers
At the Columbus Zoo, ⅛ of the animals are reptiles. ¼ of the reptiles are lizards. What fraction of
animals are lizards?
Answer: The shaded vertical column represents
⅛. The dark area represents ¼ of the ⅛.
The dark area represents the fraction of lizards
in the zoo.
So ¼ of 1/8 equals 1/32.
At Raiderville Middle School, ⅔ of the students play a sport. ½ of the students that play a sport are on
baseball teams. What fraction of the students are on baseball teams?
Answer:
Fraction circle could be used to model student thinking. Shade the circle to show
the ⅔ and then overlay with ½ of that.
0
⅓
⅔
1
Janine spends 2/5 of her paycheck and saves 1/6 of the remaining amount. What fraction of her
paycheck is saved?
Answer:
The rectangle represents the entire
The rectangle represents the entire
paycheck. The five
rows represent
the paycheck
split
paycheck.
The five rows represent
the paycheck
split into
and theportion
shaded portion
shows
2/5. The
The columns show the
into fifths and thefifths
shaded
shows
2/5.
split insplit
sixths.inThe
dark shaded
show 1/6 of the
columns show thepaycheck
paycheck
sixths.
Thearea
dark
remaining amount.
shaded area show3/30
1/6=of
the remaining amount.
1/10
3/30 = 1/10
2/5
2
5
1/6
1
6
Every day Bryon ran
3
of a mile. How far did he run after 7 days?
5
Answer:
3
3
3
3
3 3
3
3
21
The students should be able to interpret this as 7 × ( ).
+ + + + + + =
5
5
5
5
5 5
5
5
5
5
=1
5
so
10
=2
5
15
=3
5
and
20
1
=4
5
5
Mr. Templeman’s flower box is 3 meters long and
1
of a meter wide? What is the area of Mr.
4
Templeman’s driveway?
Answer:
1
3
m=
meter2
4
4
Grade 5 Fractions
3m ×
Page 32 of 54
Columbus City Schools
2013-2014
Problem Solving Questions Answers
1
cm wide and twice as long. She instructed her
2
students to calculate the area of the cover of the book. What is the area of Ms. Gaver’s book?
Answer:
1
1
121
1
2 × 5 cm = 11 cm long; 5 cm × 11 cm =
cm2 = 60 cm2
2
2
2
2
Ms. Gaver told her class she has a book with a cover 5
The swimming pool at the new waterpark is 16 meters by 8
1
meters. What is the area of the pool?
4
Answer:
16 meters × 8
1
meters = 132 meters2
4
Problem Solving Questions
The label on a can of paint states that it contains enough paint to cover 20 m². I need to paint a wall
1
that measures 2 meters by 8 meters. Will I have enough paint?
2
Answer:
1
Yes; 2 meters × 8 meters = 17 meters2
2
Without multiplying, identify which of the expressions has the greater product.
4
1
× 20
or
× 20
3
3
Explain how you know.
Answer:
4
1
4
1
× 20 > × 20 because
> and they are both multiplied times the same number which
3
3
3
3
is greater than one.
1
3
of of a pizza. Explain without finding the exact product. Did Andrew eat more or
2
4
1
3
less than of the pizza? Did he eat more or less than of the pizza?
2
4
Answer:
1
3
He ate less than , because there was less than a whole pizza. He also ate less than , because
2
4
1
3
less than
is less than .
2
4
Andrew ate
Grade 5 Fractions
Page 33 of 54
Columbus City Schools
2013-2014
Word Problems for Envelope Game and Pass the Folder
A small city park consists of a rectangular lawn that is 20½ meters long and 10 meters wide.
What is the area of the lawn?
A rectangular swimming pool measures 18 meters by 9¼ meters. What is the area of the
swimming pool?
The rectangular top of a table is four times as long as it is wide. Its width is 1 ¼ meters. Find the
area of the table top.
A painting in the Art Gallery measures 3 meters by 5¼ meters. What is the area of the painting?
A windowpane measures 50 cm by 10½ cm. What is the area of the windowpane?
I measured a bulletin board and found that it was 2⅓ meters wide and 3 meters tall. Find the area
of the bulletin board.
The label on a tin of paint states that it contains enough paint to cover 10m². I need to paint a
wall that measures 4 meters by 2½ meters. Will I have enough paint?
A gardener digs a flower bed that is 10 meters long and half a meter wide. What is the area of the
flower bed?
A small city park consists of a rectangular lawn that is 30½ meters long and 20 meters wide. What
is the area of the lawn?
A rectangular swimming pool measures 16 meters by 8¼ meters. What is the area of the
swimming pool?
Grade 5 Fractions
Page 34 of 54
Columbus City Schools
2013-2014
Word Problems for Envelope Game and Pass the Folder
Answer Key
A small city park consists of a rectangular lawn that is 20½ meters long and 10 meters wide.
What is the area of the lawn?
Answer: 205 square meters
A rectangular swimming pool measures 18 meters by 9¼ meters. What is the area of the
swimming pool?
Answer: 166 square meters
The rectangular top of a table is four times as long as it is wide. Its width is 1 ¼ meters. Find the
area of the table top.
Answer: 6 ¼ square meters
A painting in the Art Gallery measures 3 meters by 5¼ meters. What is the area of the painting?
Answer: 15 ¾ meters
A windowpane measures 50 cm by 10½ cm. What is the area of the windowpane?
Answer: 525 square centimeters
I measured a bulletin board and found that it was 2⅓ meters wide and 3 meters tall. Find the area
of the bulletin board.
Answer: 7 square meters
The label on a tin of paint states that it contains enough paint to cover 10m². I need to paint a
wall that measures 4 meters by 2½ meters. Will I have enough paint?
Answer: Yes, because the area of the wall is 10 square meters.
A gardener digs a flower bed that is 10 meters long and half a meter wide. What is the area of the
flower bed?
Answer: 5 square meters.
Grade 5 Fractions
Page 35 of 54
Columbus City Schools
2013-2014
Pass the Folder
Group Directions
Group Directions
In your group, read the problem on the folder and discuss how the problem can be solved.
Work together in your group to solve the problem and justify your answer.
Ways to justify your answers may include:
□ NUMBERS
□ WORDS
□ PICTURES
When the teacher calls time, put your answer in the folder, on the back check of the way you
chose to answer the problem and pass the folder to the next team.
You will then receive a new folder with a different problem. DO NOT LOOK INSIDE THE
FOLDER. Select one of the ways to represent the answer that has not been checked off.
The folders will continue to be passed until the teacher says “Review Time.”
REVIEW TIME:
Take all of the answers out of the folder.
As a group, talk about each answer.
Answer the following questions on another sheet of paper. Be prepared to share your answers
with the class.
 Do all of the answers make sense?
 Did each group solve the problem in a different way?
 How do the different ways to solve the problem compare?
 Which strategy does your group prefer? Why?
 What questions do you have about each group’s answer?
Grade 5 Fractions
Page 36 of 54
Columbus City Schools
2013-2014
Window Time
Your neighbor, Mrs. Wagner, has asked for your help with a project. She wants to put a new
1
window in her kitchen. Her new window will be 2 feet
2
1
wide and 3
feet tall. The hardware store sells glass for $5 per square foot.
4
Use 1 inch grid paper to model how many square feet of glass she should purchase. Let 1 in. =
1 foot. Explain why the area model would be the same as multiplying the side lengths of the
kitchen window. Determine how much the glass for the window will cost.
Grade 5 Fractions
Page 37 of 54
Columbus City Schools
2013-2014
Window Time
Answer Key
3
1
2
1
ft.
4
1
1
1
ft.
2
1
1
2
1
1
2
1
4
1
1
4
1
2
1
4
1
1
1
The area model shows (1 ft2 × 6) + ( 2 ft2 × 3) + ( 4 ft2 × 2) + ( 8 ft2 × 1).
1
When you count the squares the total area is 8 8 ft2.
1
5
The cost would be $5 × 8 8 ft2 = $40 8 or $40.63.
Grade 5 Fractions
Page 38 of 54
Columbus City Schools
2013-2014
Folding Fractions
Name
Draw a picture to represent what you did with your folded paper to solve each problem. The first
one is done for you as an example.
1. What is 1 of 1 ?
2
2
of
=
1
4
2. What is 1 of 1 ?
2
3
3. What is 2 of 2 ?
3
3
4. What is 3 of 1 ?
4
3
5. What is 2 of 3 ?
5
4
6. What is 4 of 4 ?
5
5
7. What connection can be made relating the two factors and the product of each problem?
Grade 5 Fractions
Page 39 of 54
Columbus City Schools
2013-2014
Folding Fractions
Answer Key
Draw a picture to represent what you did with your folded paper to solve each problem. The first
one is done for you as an example.
1. What is 1 of 1 ?
2
2
of
=
1
4
=
1
6
=
4
9
=
3
12
=
6
20
2. What is 1 of 1 ?
2
3
of
3. What is 2 of 2 ?
3
3
of
4. What is 3 of 1 ?
4
3
of
5. What is 2 of 3 ?
5
4
of
6. What is 4 of 4 ?
5
5
of
=
16
25
7. What connection can be made relating the two factors and the product of each problem?
Student answers will vary. Answers should include some mention that the numerator
of the product is the product of the numerators of the factors and the denominator of
the product is the product of the denominators of the factors.
Grade 5 Fractions
Page 40 of 54
Columbus City Schools
2013-2014
Counter Fractions
Name
Part I
Use your counters to help you solve each problem. Draw a sketch of your work and record your
answer.
1.
2
3
of 9 =
2.
1
4
of 12 =
3.
3
5
of 15 =
4.
2
7
of 21 =
Grade 5 Fractions
Page 41 of 54
Columbus City Schools
2013-2014
Part II
Before solving each of these problems, predict whether the answer will be more or less than half
of the whole number. Use words, pictures, and/or numbers to describe how you solved each
problem.
5.
2
3
12 =
Will this answer be more or less than half of 12?
6.
4
5
10 =
Will this answer be more or less than half of 10?
7.
1
2
14 =
Will this answer be more or less than half of 14?
8.
1
4
24 =
Will this answer be more or less than half of 24?
9.
3
4
32 =
Will this answer be more or less than half of 32?
Grade 5 Fractions
Page 42 of 54
Columbus City Schools
2013-2014
Counter Fractions
Answer Key
Part I
Use your counters to help you solve each problem. Draw a sketch of your work and record your
answer.
Student drawings may vary. Examples are given.
2 of 9 = 6
1.
3
2.
1
4
of 12 = 3
3.
3
5
of 15 = 9
4.
2
7
of 21 = 6
Part II
Before solving each of these problems, predict whether the answer will be more or less than half
of the whole number. Use words, pictures, and/or numbers to describe how you solved each
problem.
Student predictions and descriptions may vary. Examples are shown.
2
5.
12 =
Will this answer be more or less than half of 12?
3
This answer will be more than half of 12 because 2 is more than 1 .
3
2
3
2
12 = 8
To find 2 of 12, you can divide 12 into three equal groups and then find the amount in two
3
of those groups (12
Grade 5 Fractions
3 = 4, 4
2 = 8).
Page 43 of 54
Columbus City Schools
2013-2014
6.
4
5
10 =
Will this answer be more or less than half of 10?
This answer will be more than half of 10 because 4 is more than 1 .
5
4
5
2
10 = 8
To find 4 of 10, you can divide 10 into five equal groups and then find the amount in four
5
of those groups (10
7.
1
2
5 = 2, 2
14 =
4 = 8).
Will this answer be more or less than half of 14?
This answer will be exactly half of 14 because 1 is the same as 1 .
2
1
2
2
14 = 7
To find 1 of 14, you can divide 14 into two equal groups and then find the amount in one
2
of those groups (14
8.
1
4
2 = 7).
24 =
Will this answer be more or less than half of 24?
This answer will be less than half of 24 because 1 is less than 1 .
4
1
4
2
24 = 6
To find 1 of 24, you can divide 24 into four equal groups and then find the amount in one
4
of those groups (24
9.
3
4
32 =
4 = 6).
Will this answer be more or less than half of 32?
This answer will be more than half of 32 because 3 is more than 1 .
4
3
4
2
32 = 24
To find 3 of 32, you can divide 32 into four equal groups and then find the amount in three
4
of those groups (32
Grade 5 Fractions
4 = 8, 8
3 = 24).
Page 44 of 54
Columbus City Schools
2013-2014
Grid Fractions
Name
Use your grid paper to help you solve these problems. Record your work and answers on this
sheet.
1.
2
3
1
4
=
2.
3
4
5
6
=
3.
1
3
4
5
=
4.
3
5
2
3
=
5.
1
4
2
5
=
6.
6
7
1
2
=
7.
1
2
1
4
=
8.
2
3
3
8
=
9. What relationship do you see between the factors and the product?
Grade 5 Fractions
Page 45 of 54
Columbus City Schools
2013-2014
Grid Fractions
Answer Key
Use your grid paper to help you solve these problems. Record your work and answers on this
sheet. Examples of grids are given.
1.
2
3
1
4
= 2 =1
2.
3
4
5
6
= 15 = 5
3.
1
3
4
5
= 4
4.
3
5
2
3
= 6 =2
5.
1
4
2
5
= 2 = 1
6.
6
7
1
2
= 6 =3
7.
1
2
1
4
= 1
8.
2
3
3
8
= 6 =1
12
24
=
6
=
8
=
15
15
20
14
=
5
=
10
=
7
=
8
24
=
4
9. What relationship do you see between the factors and the product?
Student answers will vary. Answers should include some mention that the numerator
of the product is the product of the numerators of the factors and the denominator of
the product is the product of the denominators of the factors.
Grade 5 Fractions
Page 46 of 54
Columbus City Schools
2013-2014
Grid Fractions II
Name
Use your grid paper to help you solve these problems. Record your work and answers on this
sheet.
1.
3
4
12 =
2.
1
7
41 =
3.
1
2
33 =
4.
3
5
11 =
3
4
5
9
5. What can you conclude about multiplying mixed numbers and fractions?
Based on your conclusions, predict what you think the answers will be to these last two
questions. Check your predictions using your grid paper to work out the problems.
6.
41
3
1
2
=
7.
11
2
6=
8. Were your predictions correct? Why or why not?
Grade 5 Fractions
Page 47 of 54
Columbus City Schools
2013-2014
Grid Fractions II
Answer Key
Use your grid paper to help you solve these problems. Record your work and answers on this
sheet.
1.
3
4
1 2 = 15 = 1 3
2.
1
7
4 1 = 21 = 3
3.
1
2
3 3 = 15 = 1 7
4.
3
5
1 1 = 30 = 2
3
12
4
8
12
8
5
9
35
5
45
3
5. What can you conclude about multiplying mixed numbers and fractions?
Answers will vary. Students should indicate some process for multiplying mixed
numbers and fractions.
Based on your conclusions, predict what you think the answers will be to these last two
questions. Check your predictions using your grid paper to work out the problems.
6.
41
3
1
2
=21
6
7.
11
2
6 = 18 = 9
2
8. Were your predictions correct? Why or why not?
Answers will vary.
Grade 5 Fractions
Page 48 of 54
Columbus City Schools
2013-2014
Fraction Divide
Name
Show your work for each problem using pictures and numbers.
1.
6
1
2
2.
8
3
4
3.
10
2
5
4.
12
5
6
5.
3
5
3
6.
3
4
8
Grade 5 Fractions
Page 49 of 54
Columbus City Schools
2013-2014
7.
3
8
4
8.
5
6
9
9. Does dividing by 1 result in the same answer as multiplying by 2? Why or why not?
2
10. What is the reciprocal of 4? Does dividing by 4 result in the same answer as multiplying by
the reciprocal of 4? Why or why not?
11. What do problems where a fraction is divided by a whole number have in common with
problems where a whole number is divided by a fraction?
Grade 5 Fractions
Page 50 of 54
Columbus City Schools
2013-2014
Fraction Divide
Answer Key
Show your work for each problem using pictures and numbers.
1.
6
1
2
= 12 groups of 1 in 6
2.
8
3
4
= 10 2 groups of 3 in 8
3.
10
2
5
= 25 groups of 2 in 10
4.
12
5
6
= 14 2 groups of 5 in 12
5.
3
5
2
3
4
5
5
6
3= 3
15
one share out of 3 = 3
15
6.
3
4
8= 3
32
one share out of 8 = 3
32
Grade 5 Fractions
Page 51 of 54
Columbus City Schools
2013-2014
7.
3
8
4= 3
32
one share out of 4 = 3
32
8.
5
6
9= 5
54
one share out of 9 = 5
54
9. Does dividing by 1 result in the same answer as multiplying by 2? Why or why not?
2
Yes, it has the same result. Justifications will vary. Dividing by 1 can be thought of as
2
1
2
how many groups of are in something. There will be two 1 ’s
2
multiplying by 2 would have the same answer as dividing by 1 .
2
in each whole, so
10. What is the reciprocal of 4? Does dividing by 4 result in the same answer as multiplying by
the reciprocal of 4? Why or why not?
The reciprocal of 4 is 1 . Dividing by 4 results in the same answer as multiplying by 1 .
4
There are 4
dividing by
1 ’s
4
1.
4
4
in every whole, so multiplying by 4 would have the same answer as
11. What do problems where a fraction is divided by a whole number have in common with
problems where a whole number is divided by a fraction?
Answers will vary. Students may mention that they both can be solved by multiplying
by the reciprocal of the divisor.
Grade 5 Fractions
Page 52 of 54
Columbus City Schools
2013-2014
Scaling Problem Solving
1. How Much Flour?
Five days a week a baker uses 1 1 sacks of flour when baking cakes.
4
A. Will the baker use more than or less than 5 sacks of flour from Monday through
Friday? Explain your answer without solving the problem.
B) Prove your answer to Part A by calculating the number of sacks of flour the
baker will use from Monday through Friday. Show your work.
2. London to Paris
A high speed train travels at 300km per hour from London to Paris. If the train
traveled at 5 times that speed, would the amount of time the trip takes be shorter or
8
longer?
A) Will the amount of time the trip takes be longer or shorter? Explain your
answer without solving the problem.
B) Prove your answer to Part A by calculating the km per hour the train would
travel.
Grade 5 Fractions
Page 53 of 54
Columbus City Schools
2013-2014
Scaling Problem Solving
Answer Sheet
1. How Much Flour?
Five days a week a baker uses 1 1 sacks of flour when baking cakes.
4
A) Will the baker use more than or less than 5 sacks of flour from Monday through
Friday? Explain your answer without solving the problem.
Answer: The baker will use more than 5 bags. I know because he is using
more than one sack of flour a day and there are 5 days.
B) Prove your answer to Part A by calculating the number of sacks of flour the
baker will use from Monday through Friday. Show your work.
Answer: The number of sacks that the baker will use is 6 14 . Students can
prove their answer by using repeated addition, multiplication or an area
model.
Ex:
5 × 1 14 = 5 ×
5
4
= 61
4
2. London to Paris
A high speed train travels at 300km per hour from London to Paris. If the train
traveled at 7 times that speed, would the amount of time the trip takes be shorter or
8
longer?
A) Will the amount of time the trip takes be longer or shorter? Explain your
answer without solving the problem.
Answer: The trip will take longer because the train is traveling at a slower
speed because 7 is less than one.
8
B) Prove your answer to Part A by calculating the km per hour the train would
travel.
Answer: The train would be traveling at a speed of 185 km per hour.
Students can prove their answer by using multiplication.
300 × 7 = 262 1 mph
8
Grade 5 Fractions
2
Page 54 of 54
Columbus City Schools
2013-2014
Mathematics Model Curriculum
Grade Level: Fifth Grade
Grading Period: 4
Common Core Domain
Time Range: 15 Days
Geometry
Common Core Standards
Classify two-dimensional figures into categories based on their properties.
3. Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of
that category.
For example, all rectangles have four right angles and squares are rectangles, so all squares have four right
angles.
4. Classify two-dimensional figures in a hierarchy based on properties.
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Developing understanding of volume
Students recognize volume as an attribute of three-dimensional space. They understand that volume can be
measured by finding the total number of same-size units of volume required to fill the space without gaps or
overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They
select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume.
They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as
decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine
volumes to solve real world and mathematical problems.
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 5 Geometry
2 - Dimensional figures
Page 1 of 47
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
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Mathematics Model Curriculum
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
This cluster builds from Grade 3 when students described, analyzed and compared properties of twodimensional shapes. They compared and classified shapes by their sides and angles, and connected these
with definitions of shapes. In Grade 4 students built, drew and analyzed two-dimensional shapes to deepen
their understanding of the properties of two-dimensional shapes. They looked at the presence or absence of
parallel and perpendicular lines or the presence or absence of angles of a specified size to classify twodimensional shapes. Now, students classify two-dimensional shapes in a hierarchy based on properties.
Details learned in earlier grades need to be used in the descriptions of the attributes of shapes. The more
ways that students can classify and discriminate shapes, the better they can understand them. The shapes are
not limited to quadrilaterals.
Students can use graphic organizers such as flow charts or T-charts to compare and contrast the attributes of
geometric figures. Have students create a T-chart with a shape on each side. Have them list attributes of the
shapes, such as number of side, number of angles, types of lines, etc. they need to determine what’s alike or
different about the two shapes to get a larger classification for the shapes.
Pose questions such as, “Why is a square always a rectangle?” and “Why is a rectangle not always a
square?”
Instructional Resources/Tools from ODE Model Curriculum
Anglegs (from the grades 3-5 manipulatives kit)
Rectangles and Parallelograms: Students use dynamic software to examine the properties of rectangles and
parallelograms, and identify what distinguishes a rectangle from a more general parallelogram. Using
spatial relationships, they will examine the properties of two-and three-dimensional shapes.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L270: In this lesson, students classify polygons
according to more than one property at a time. In the context of a game, students move from a simple
description of shapes to an analysis of how properties are related.
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 think that when describing geometric shapes and placing them in subcategories, the last category is the
only classification that can be used.
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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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
Geometry
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Geometry 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.
A polygon is a closed curve formed only by line segments that meet at their endpoints (called
vertices). A regular polygon is a polygon that is both equilateral and equiangular. A convex
polygon has interior angles that are all less than 180º. If you draw a straight line through a
convex polygon, it will cross no more than two sides. A concave polygon has at least one
interior angle that is greater than 180º. At least one straight line drawn through a concave
polygon will cross more than two sides.
Some special polygons are triangles and quadrilaterals. Triangles are classified by their sides
and their angles. By sides, the triangles are equilateral triangles, where all three sides are the
same, isosceles triangles, where at least two sides are congruent, (some texts list equilateral
triangles as a special case of isosceles triangles, whereas others try to separate the two types of
figures), and scalene triangles, where no two sides are congruent. By angles, the triangles are
acute triangles, where all three angles are acute angles (all triangles have at least two acute
angles), right triangles, where there is one right angle (90º), and obtuse triangles, where there
is one obtuse angle. Triangles can be classified using both the angles and the sides. When
classifying triangles using both angles and sides the terms are used in conjunction, for example;
obtuse scalene, right isosceles, etc. Be careful that students realize that certain combinations of
side and angle classifications are impossible (e.g., right equilateral is impossible because in an
equilateral triangle all three angles have to be 60 which means that one of the angles could not
be 90 ). The altitude of a triangle is the segment drawn from a vertex that is perpendicular to
the opposite side (or to the line containing the opposite side). The altitude is the height of the
triangle.
Sometimes when students aren’t sure what a geometric term means they can break the word
apart and look at the prefix or suffix to see if they know what any of the parts mean, like triangle,
tri = 3, angle = angle. A triangle has 3 angles (and 3 sides). Equilateral, equi = equal, lateral =
side, equilateral means equal sides. If a polygon is equilateral, all the sides are equal. Similarly,
if students know the prefixes penta, hexa, hepta, . . . etc., they can figure out how many sides a
polygon must have by its name. Following are some polygon names:
Sides
n
3
4
5
6
7
8
9
10
12
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Name
n-gon
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon
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Quadrilaterals are also classified by their sides and angles. The following is a graphic organizer
that can be used perhaps as a bulletin board to show the relationships of the quadrilaterals.
Quadrilateral Poster
Quadrilateral
Parallelogram
Rectangle
Rhombus
Square
Square
Trapezoid
Quadrilateral – Quad = 4. Lateral = side, like a lateral in football is a pitch out to the side.
A “forward lateral” is illegal because it’s not to the side. Quad + lateral = 4 sides. A
quadrilateral is a polygon with 4 sides.
Parallelogram – A quadrilateral with opposite sides parallel. The word parallel
when printed has 3 “l’s” that are parallel to each other.
Rectangle – Rect = right (like correct or rectify). Rect + angle = right angle. A
rectangle is a quadrilateral with four right angles (sides do not need to be
mentioned).
Rhombus – A parallelogram with 4 equal sides (angles do not need to be
mentioned).
Square – A square is a quadrilateral with 4 equal sides and 4 equal (right) angles. A
square is a rectangle, a rhombus, a parallelogram, and a quadrilateral.
Trapezoid – A quadrilateral with one pair of parallel sides.
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PRACTICE ASSESSMENT ITEMS
Compare and contrast the equilateral triangle and square using their geometric
characteristics (sides, angles, vertices).
A parallelogram has four sides with both sets of opposite sides parallel. Name three
other quadrilaterals that are also parallelograms?
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Compare and contrast the equilateral triangle and square using their geometric characteristics
(sides, angles, vertices).
Answer: Answers may vary, examples of characteristics are given.
Equilateral Triangle
Square
3 vertices
4 vertices
3 equal sides
4 equal sides
3 acute angles
4 right angles
sum of interior angles = 180
sum of interior angles = 360
each angle = 60
each angle = 90
no parallel or perpendicular sides opposite sides are parallel, adjacent sides
are perpendicular
A 4-point response accurately describes characteristics of each figure including sides,
angles, and vertices.
A 3-point response accurately describes characteristics of each figure using only two of the
categories (sides, angles, vertices).
A 2-point response accurately describes characteristics of each figure using only one of the
categories (sides, angles, vertices).
A 1-point response lists characteristics from each of the three categories for only one of the
figures.
A 0-point response shows no mathematical understanding of the task.
A parallelogram has four sides with both sets of opposite sides parallel. Name three other
quadrilaterals that are also parallelograms?
Answer:
Square, rectangle, rhombus
A 2-point response lists all three shapes.
A 1-point response lists one or two other shapes.
A 0-point response shows no mathematical understanding of the task.
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PRACTICE ASSESSMENT ITEMS
A trapezoid has two sides that are parallel so it must be a parallelogram. True or
False? Explain.
Create a hierarchy diagram using the following terms: rectangles, polygons,
rhombi, quadrilaterals, parallelograms, and squares.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
A trapezoid has two sides that are parallel so it must be a parallelogram. True or False? Explain.
Answer:
False. A trapezoid has one set of parallel sides. A parallelogram has two sets of parallel
sides.
A 2-point response states that the statement is false and gives an appropriate explanation.
A 1-point response either states that the statement is false or gives an appropriate definition
of one or the other shape.
A 0-point response shows no mathematical understanding of the task.
Create a hierarchy diagram using the following terms: rectangles, polygons, rhombi,
quadrilaterals, parallelograms, and squares.
Answers will vary:
Polygon
Quadrilateral
Parallelogram
Rectangle
Rhombi
Square
A 2-point response correctly shows the hierarchy of the terms.
A 1-point response shows a hierarchy with a minor error.
A 0-point response shows no mathematical understanding of the task.
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PRACTICE ASSESSMENT ITEMS
Bonnie was comparing and contrasting the three shapes below.
She made the following chart.
Same
Different
Quadrilaterals
Side lengths
Only obtuse angles Not the same number of
Parallel lines
sets of Parallel lines
Not the same number of
angles
Determine if her chart is complete and correct. If not, create a new chart with
correct information about the shapes.
Terra and Cynthia are playing Guess the Shape. Terra draws a shape with four
congruent sides and four 90 degree angles. Cynthia draws a shape with four 90
degree angles and opposite sides congruent. What shapes did each girl draw?
A. Square and rhombus
B. Rectangle and rhombus
C. Square and rectangle
D. Trapezoid and rectangle
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Bonnie was comparing and contrasting the three shapes below.
She made the following chart.
Same
Different
Side lengths
Quadrilaterals
Only obtuse angles Not the same number of sets of
Parallel lines
Parallel lines
Not the same number of angles
Determine if her chart is complete and correct. If not, create a new chart with correct information about the shapes.
Answers will vary:
Possible answers:
Different
Same
Quadrilaterals
At least one obtuse
angle
At least one set of
parallel sides
Side lengths
Not the same number of sets
of Parallel lines
Not the same number of
angles
Terra and Cynthia are playing Guess the Shape. Terra draws a shape with four congruent sides
and four 90 degree angles. Cynthia draws a shape with four 90 degree angles and opposite sides
congruent. What shapes did each girl draw?
A. Square and rhombus
B. Rectangle and rhombus
C. Square and rectangle
D. Trapezoid and rectangle
Answer: C
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: attributes, category, subcategory, hierarchy, features, identify, classify,
describe, justify, properties, length, width, two-dimensional, line, line segment,
dimensions, vertex, vertices, exterior angle, interior angle, acute, right angle, obtuse angle,
perpendicular, parallel, intersecting, triangle, equilateral triangle, scalene triangle,
isosceles triangle, diagonal, intersecting, square, rectangle, rhombus, parallelogram,
quadrilateral, trapezoid, polygon, pentagon, decagon, hexagon, octagon
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.
All rectangles are parallelograms but not all parallelograms are rectangles. Explain why
this statement is true.
Draw two different quadrilaterals and identify them. Explain how you know these two
shapes are quadrilaterals.
What is the difference between a regular and irregular polygon? Use a ruler to draw and
label one regular polygon and one irregular polygon.
Identify the figure below. Is it a regular or irregular polygon? Explain your reasoning.
On the geoboard below, make a trapezoid. What attributes make it a trapezoid?
Identify the figure below? Describe the attributes that identify and define the figure.
What three properties do all triangles have in common?
A parallelogram is a quadrilateral with two pairs of parallel and congruent sides. Name
three other quadrilaterals that are also parallelograms. What makes each of these figures
different?
What is the relationship between the sums of the angles in these two figures?
Complete the blanks in the charts below.
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 5 on the right side of the page. Open “Unit 6
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. Have students use manipulatives (anglegs, geoboards, attribute/pattern blocks, pictures, etc.)
to model, draw, label, and explain as many geometric terms as possible. As the teacher
circulates to each student (group) take note of the geometric terminology students are using
in their labels and explanations. Have students who are using the correct geometric
terminology share whole class. Record the geometric terms and student definitions. If there
are additional terms not introduced by the students then the teacher should introduce them
and allow students to generate definitions and examples. Post the record sheet and student
work for future reference during the unit.
4. Give each student or group of students a set of anglegs. Begin by asking students to make a
polygon. As the students work in table groups ask, “What is a polygon? What are the
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attributes of a polygon?” Student answers should include closed figure (all sides connected
end to end), at least three sides, number of sides = the number of vertices = the number of
angles, sides only touch at vertex. As teacher circulates have students share their different
polygons and name them highlighting the number of sides, ex: quadrilaterals, pentagon,
hexagon, octagon, etc. If students do not make the different sided figures then ask them to
make them and discuss how the figures are named.
5. Using the materials from activity 2 above, the lesson can be extended specifically to focus on
triangles and quadrilaterals. Instruct students to make a triangle with a 90o angle. Ask
students what they notice (at least one side must be longer for a right angle). Continue asking
them to create different triangles given specific parameters. One must have 3 equal sides, one
must have a right angle, and one must have three different lengths for sides. Using the
different attributes, identify each triangle as isosceles, scalene, or equilateral. Have students
add a side to one of their triangles. Ask students the name of the new shape created? Discuss
the number of sides and get students to say quadrilateral. Students should discuss the
similarities and differences of the quadrilateral they created with the quadrilaterals of other
students at their table. To prompt students to explore different shapes ask the following
questions:
Can you create a quadrilateral with equal sides? What is it called? (square)
Could it be called anything else? (Yes, a rectangle) Why or why not? (A rectangle has 2
pairs of parallel sides and right angles)
Are all squares rectangles? (Yes, meets the definition of a rectangle. A rectangle has 2
pairs of parallel sides and right angles)
Are all rectangles squares? (No, because a square has 4 congruent sides)
What shape do you create if you lean the square so that there are no right angles?
(rhombus) What is the definition of a rhombus? (4 congruent sides)
Describe the angles. (2 acute and 2 obtuse)
Is a rhombus a rectangle? (No, it does not have right angles).
Does a rhombus have opposite sides parallel? (yes)
Create a square again. Turn the square so that there is one vertex on the top and one on
the bottom. Is it a rhombus? Why or why not? (Yes, it is a rhombus because it has 4
congruent sides)
Are all squares rhombi? (Yes, because they have four congruent sides).
Are all rhombi squares? (No, because rhombi do not have to have right angles).
What is another name for a rhombus? (parallelogram)
What other shapes are parallelograms? (squares, rectangles)
Have students push in one vertex of their quadrilateral. Tell them it is caved in and the
special term called concave polygons. Have the students push the concaved part back out
and ask them what it should be called now. (non-concave or convex polygon)
Show students a kite. As them if a kite is a square on its side (No)
What attributes does a kite have? (4 sides, two congruent adjacent sides, closed figure,
quadrilateral, etc…)
What group of shapes would you put a kite in? Why?
The students should record the differences and try to classify them according to their
attributes. Students could classify them using a table, chart, Venn diagram, etc.) Have groups
share out whole class one of the ideas that they had for classifying shapes by attributes.
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6. Copy and laminate the triangles on the “Triangle Hierarchy” worksheets (included in this
Curriculum Guide). Group the students into pairs or groups of fours. Have students cut out
and use the triangles to create a Triangle Hierarchy Diagram. You can differentiate the
activity by giving the students the terms (polygons, triangles, equilateral triangle, scalene
triangle, isosceles triangle, acute triangle, obtuse triangle, right triangle) to classify the
polygon or by allowing them to create their own attributes to classify the polygons. Provide
the students with protractors and rulers. Students could measure and label the length of the
sides and/or angles of each triangle. Have students use chart paper to create a diagram with
headings based on the attributes used to classify the triangles. Instruct the student groups to
explain their reasoning for placement of the triangles on their diagram. Use this opportunity
to introduce and reinforce mathematical language and vocabulary, (e.g., sides, parallel,
opposite, perpendicular, scalene, isosceles, angles- 90o, right, obtuse, acute, vertices,
congruent, etc.).
7. Copy and laminate the quadrilaterals on the “Quadrilateral Hierarchy” worksheets (included
in this Curriculum Guide). Group the students into pairs or groups of fours. Have students
cut out the quadrilaterals to use to create a Quadrilateral Hierarchy Diagram. You can
differentiate the activity by giving the students the terms (polygons, quadrilaterals,
parallelogram, non-parallelogram, rectangle, square, rhombus, trapezoid, kite, other) to
classify the polygon or by allowing them to create their own attributes to classify the
polygons. Provide the students with protractors and rulers. Students could measure and label
the length of the sides and or angles of each triangle. Have students use chart paper to create
a diagram with headings based on the attributes used to classify the quadrilaterals. Instruct
the student groups to explain their reasoning for placement of the quadrilaterals on their
diagram. Use this opportunity to introduce and reinforce mathematical language and
vocabulary, (e.g., sides, parallel, opposite, perpendicular, angles-- 90o, right, obtuse, acute,
vertices, congruent, symmetry, etc.)
8. Make copies of the “Design a Shape” worksheet (included in this Curriculum Guide).
1
Laminate the sets and cut pages in half to have one design per page. Pairs sit back to back
2
or across from one another with a partition. One student is the communicator and the other
student is the designer. The teacher hands the communicator one design (using an appropriate
level of difficulty) and the other student a blank sheet of paper. Each student will use the
appropriate tool to measure the lengths, angles, and circumferences. The communicator will
describe the measurements he/she has taken on the original design while the designer
constructs a duplicate without seeing the original design based on the information given by
the communicator. Once complete, the students exchange roles and the teacher assigns the
next design.
9. Ask students to brainstorm all the shapes they know that are quadrilaterals (square, rectangle,
parallelogram, rhombus, trapezoid, etc.). Draw a rectangle on the overhead projector or on
the board. Ask students to brainstorm a list of characteristics that describe the rectangle,
such as parallel lines, side lengths, angles, sum of all the angles is 360 . Reinforce and
demonstrate the words parallel, congruent, and right angles. Discuss with students that a
rectangle does not necessarily have two short sides and two long sides. If students list this as
a characteristic, you may want to draw a square on the board and remind them that a square
is just a special name for a rectangle that has four equal sides. Divide students into pairs and
ask them to draw as many different four-sided figures as they can. Distribute the
“Quadrilateral Chart” (included in this Curriculum Guide). Students complete the chart
using the figures that they drew. Have different students draw a figure on the board and
explain the attributes that describe their figure. Did anyone else have the same attributes
checked, but for a different figure? Discuss the different figures and their common attributes.
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Have student pairs use the chart to write the characteristics of common quadrilaterals. Ask
students to share the attributes that they have determined for the common quadrilaterals.
10. Give groups of students an assortment of two and three-dimensional objects. Have students
sort the objects in as many ways as possible and record their sorts. Then have students share
their sorts with the class and explain their reasoning.
11. Divide the class into teams of four or five students. Provide each group with a set of
“Characteristics and Shapes” cards (included in this Curriculum Guide). Separate the cards
into two piles. Place the shape cards face down and spread the characteristic cards out on the
table so everyone can see them. One person selects a card from the shape pile and doesn’t
show it to anyone else. Using the characteristic cards, the other players take turns asking yes
or no questions and try to guess the shape. For example: Does it have four equal sides? Are
all the angles equal? Are there any right angles? Are opposite sides parallel? Once the
shape has been guessed, the player that guessed correctly keeps that card and gets to pick
another card from the shape pile for everyone to guess. The player with the most shapes
wins the game. The object of the game is to ask as few questions as possible. After students
have played several times, ask them to share their strategies by discussing which questions
were most helpful in finding the secret shape.
12. Distribute a copy of the “Classy Triangles” (included in this Curriculum Guide) to each small
group of students. Instruct students to sort the triangles into three groups using the length of
their sides. Students will need to develop a strategy for determining when sides are the same
length without using a ruler (e.g., they could use the side of a sheet of notebook paper and
mark the lengths to compare with the other sides of the triangle). Allow students to share
their strategies with the rest of the class. What is the name for each of these three groups
(scalene, isosceles, and equilateral)? Have students take the three groups and sort the
triangles in each group by their angles (acute, obtuse, and right). Students will need to
develop a strategy for checking the angles that does not involve using a protractor. Allow
students to share their strategy with the rest of the class (e.g., using the corner of a piece of
paper to check if an angle is greater than, less than, or equal to 90 ). What are the names for
the groups of triangles they now have? Have students identify the multiple classifications for
each group (acute scalene, obtuse scalene, right scalene, acute isosceles, obtuse isosceles, and
right isosceles). Ask students to think about the angles of the equilateral triangles. Can this
group be sorted into any smaller groups by their angles (no)? Discuss the definition of
equilateral to help students be able to justify why equilateral triangles can only have acute
angles (equilateral requires that a triangle also be equiangular, since all angles are the same
and a triangle has a total of 180 , each angle must be acute 60 angles).
13. Provide students with a copy of “Triangle Investigations” (included in this Curriculum
Guide). Distribute protractors and rulers. Have students measure all the angles for the
triangles and the lengths of the sides. When this is completed, distribute “Triangle
Investigations Too” (included in this Curriculum Guide). Have students complete the page
and then discuss the properties of the triangles.
14. Have students use the characteristics to write a definition for a square and a rhombus. Repeat
the same procedure for a rectangle and parallelogram. Compare and contrast the
characteristics of a rectangle and of a parallelogram noting similarities and differences using
mathematical vocabulary.
15. Distribute “Geometric Designs” (included in this Curriculum Guide). Students create
geometric designs that include the characteristics that are listed on the worksheet. Display
the final pictures and have students look at the drawings and find the geometric shapes, lines,
and line segments that they contain.
16. Use the “Characteristics and Shapes” pages (included in this Curriculum Guide) and read
characteristics out loud. Have students use geoboards to create the shape. Challenge
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students to find at least two other characteristics for the shape they made. Have students
work with a partner to develop a short list of characteristics (properties) that would define
that figure. What is the shortest list they can create that would define the figure?
17. Give students the following vocabulary: quadrilateral, rhombus, square, rectangle,
parallelogram, and trapezoid. Instruct students to create a poster which illustrates which of
these types of figures are subsets of the others (see example in the Teacher Introduction).
Students also write a statement about each shape and justify its placement on the graphic
organizer.
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RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 340-346
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. 76-78
Practice Master pp. 76-78
Problem Solving Master pp. 76-78
Reteaching Master pp. 76-78
INTERDISCIPLINARY CONNECTIONS
1. Literature Connections:
The Most Important Thing Is…, by Myrna Bertheau
Three Pigs, One Wolf, and Seven Magic Shapes, by Grace Maccarone
Pablo’s Tree, by Pat Mora
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Problem Solving Questions
All rectangles are parallelograms but not all parallelograms are rectangles. Explain why
this statement is true.
Draw two different quadrilaterals and identify them. Explain how you know these two
shapes are quadrilaterals.
What is the difference between a regular and irregular polygon? Use a ruler to draw and
label one regular polygon and one irregular polygon.
Identify the figure below. Is it a regular or irregular polygon? Explain your reasoning.
On the geoboard below, make a trapezoid. What attributes make it a trapezoid?
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Problem Solving Questions
Identify the figure below? Describe the attributes that identify and define the figure.
What three properties do all triangles have in common?
A parallelogram is a quadrilateral with two pairs of parallel and congruent sides. Name
three other quadrilaterals that are also parallelograms. What makes each of these figures
different?
What is the relationship between the sums of the angles in these two figures?
Complete the blanks in the charts below.
Triangles
Right,
Polygons
A) ______________
Isosceles,
Trapezoid, B) _______________________,
C) ___________________
D) _______________________, Rhombus
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Problem Solving Questions Answers
All rectangles are parallelograms but not all parallelograms are rectangles. Explain why this statement
is true.
Answer: All rectangles are parallelograms because they have parallel and congruent sides. Not
all parallelograms are rectangles because rectangles must have four right angles.
Draw two different quadrilaterals and identify them. Explain how you know these two shapes are
quadrilaterals.
Answers may vary. The two figures must have four sides.
square
rectangle
parallelogram
trapezoid
rhombus
pentagon
What is the difference between a regular and irregular polygon? Use a ruler to draw and label one
regular polygon and one irregular polygon.
Answers may vary: A regular polygon has all of its sides and angles congruent.
A square is regular
This parallelogram is irregular.
Identify the figure below. Is it a regular or irregular polygon? Explain your reasoning.
Answer: This figure is a scalene triangle. It is irregular because not all of the sides and angles
are congruent.
On the geoboard below, make a trapezoid. What attributes make it a trapezoid?
Answers may vary: The figure must have only
one set of parallel sides.
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Problem Solving Questions Answers
Identify the figure below? Describe the attributes that identify and define the figure.
Answer: the figure is a parallelogram. The attributes that make
it a parallelogram are it has four sides, it has two sets of
parallel sides, and the opposite angles are congruent.
What three properties do all triangles have in common?
Answer:
1) All triangles are closed figures.
1) All triangles have three sides.
2) All triangles have an angle sum of 180°.
A parallelogram is a quadrilateral with two pairs of parallel and congruent sides. Name three other
quadrilaterals that are also parallelograms. What makes each of these figures different?
Answer: A rectangle, rhombus, and square are all parallelograms also.
A rectangle has 4 right angles and the opposite sides are congruent. A square has 4
right angles and all sides are congruent. A rhombus has opposite sides that are
congruent and no right angles.
What is the relationship between the sums of the angles in these two figures?
Answer: The angle sum of the quadrilateral is always double the angle sum of a triangle.
The angle sum of a triangle is 180° and the angle sum of a quadrilateral is 360°.
Complete the blanks in the charts below.
Answer: A) = quadrilateral
C) = equilateral or scalene
B) and D) = rectangle, square, parallelogram
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Triangle Hierarchy
Grade 5 Geometry
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Quadrilateral Hierarchy
Grade 5 Geometry
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Design a Shape
Grade 5 Geometry
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Design a Shape
Grade 5 Geometry
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Design a Shape
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Grade 5 Geometry
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Draw the
figure
1 pair of
parallel
sides
2 pairs of
parallel
sides
2 right
angles
4 right
angles
1 pair of
congruent
sides
Quadrilateral Chart
2 pairs of
congruent
sides
4 congruent
sides
Possible
names for
this figure
Characteristics and Shapes
a polygon with
three sides
triangle
a polygon with four
sides
quadrilateral
a quadrilateral with
opposite sides that
are equal in length
and parallel
parallelogram
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Characteristics and Shapes
a parallelogram
with four equal
sides and four equal
angles
square
a parallelogram
with four right
angles and opposite
sides of equal
lengths
rectangle
a parallelogram
with four equal
sides
rhombus
Grade 5 Geometry
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Characteristics and Shapes
a quadrilateral with
only one pair of
parallel sides
trapezoid
a polygon with five
sides
pentagon
a polygon with six
sides
hexagon
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Characteristics and Shapes
lines or line
segments that
intersect to form
right angles
perpendicular lines
lines or line
segments that meet
or cross each other
intersecting lines
lines or line
segments that are in
the same plane but
never intersect
parallel lines
Grade 5 Geometry
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Characteristics and Shapes
a set of points in a
plane that are the
same distance from
a given point called
the center
part of a line that
has two endpoints
and includes all
points between
those endpoints
a never ending
straight path that
continues in one
direction from an
endpoint
Grade 5 Geometry
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circle
line segment
ray
Columbus City Schools
2013-2014
Characteristics and Shapes
57
a triangle with all
acute angles
67
56
acute triangle
33
a triangle with one
obtuse angle
45
102
obtuse triangle
28
a triangle with one
right angle
90
62
right triangle
Grade 5 Geometry
2 - Dimensional figures
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Characteristics and Shapes
a triangle with no
congruent sides
scalene triangle
a triangle with two
congruent sides
isosceles triangle
a triangle with all
sides congruent
equilateral triangle
Grade 5 Geometry
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Classy Triangles
A
B
C
D
E
F
G
H
I
J
K
L
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M
N
O
P
Q
R
S
T
U
V
W
X
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Classy Triangles
Answer Key
The following triangles are acute scalene: D, O, and T.
The following triangles are obtuse scalene: G, P, and X.
The following triangles are right scalene: A, I, and R.
The following triangles are acute isosceles: B, L, and V.
The following triangles are obtuse isosceles: C, K, and U.
The following triangles are right isosceles: E, N, and S.
The following triangles are (acute) equilateral: F, H, J, M, Q, and W.
Note: Equilateral triangles will always be acute due to the fact that equilateral triangles by
definition must also be equiangular. Since all angles are the same, they must all be 60
because the total must be 180 .
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Triangle Investigations
Name
Measure the angles and side lengths for each triangle and record the measurements
on the triangles. Use centimeters for the side lengths.
Set A
Set B
Set C
Set D
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Triangle Investigations Too
Name
The triangles in the four sets have several things in common. Describe the
triangles in each set in regards to their sides and angles.
Set A
Set B
Set C
Set D
1. Which triangles have right angles? Triangles that have one 90 angle are called
triangles.
2. Which triangles have equal sides and equal angles? Triangles with equal sides
and angles are called
triangles.
3. Which triangles have no equal sides or angles? Triangles with no equal sides
and three different angles are called
triangles.
4. Which triangles have two sides with the same length and two identical angle
measures? Triangles with two identical sides and two identical angles are
called
Grade 5 Geometry
2 - Dimensional figures
triangles.
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Triangle Investigations
Answer Key
Measure the angles and side lengths for each triangle and record the measurements on the
triangles. Use centimeters for the side lengths.
Set A
Set B
36
60
2.3 cm
60
60
3.5 cm
60
14 2.9 cm
2.9 cm
2.3 cm
3.5 cm
60
2.3 cm
72 72
1.7 cm
60
36
60
3.5 cm
6.3 cm
6 cm
4.8 cm
6 cm
60
2.7 cm
45
115
4.8 cm
72
83 83
1.6 cm
60
6 cm
Set C
6.3 cm
72
2.9 cm
Set D
3.8 cm
90
60
5.3 cm
6.9 cm
2.1 cm 119
37
20
3.1 cm
6.3 cm
7.4 cm 39
24
4.5 cm
5 cm
3.9 cm
3.1 cm
102
45
30
90
3.9 cm
5.5 cm
33
5.5 cm
13
90
5.6 cm
Grade 5 Geometry
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51
3.1 cm
77
Columbus City Schools
2013-2014
1.2 cm
Triangle Investigations Too
Answer Key
The triangles in the four sets have several things in common. Describe the triangles in each set
in regards to their sides and angles.
Set A
Equilateral
1. All sides congruent.
2. All angles congruent.
Set B
Isosceles
1. Two congruent angles.
2. Two congruent sides.
Set C
Scalene
1. No sides congruent.
2. No angles congruent.
Set D
Right
1. One 90 angle.
1. Which triangles have right angles? Triangles that have one 90 angle are
right
triangles.
Set D
2. Which triangles have equal sides and equal angles? Triangles with equal sides and angles are
called
equilateral
triangles.
Set A
3. Which triangles have no equal sides or angles? Triangles with no equal sides and three
different angles are called
scalene
triangles.
Set C
4. Which triangles have two sides with the same length and two identical angle measures?
Triangles with two identical sides and two identical angles are called
isosceles
triangles.
Set B
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Geometric Designs
Make a geometric design or picture that includes at least one of all of the following
items. First plan your design before you try to draw it on your white piece of
paper. Be sure to spend most of your time on the final drawing. Be creative.
1.
An octagon
2.
Four or more obtuse angles
3.
A set of parallel lines
4.
Five squares
5.
Two or more acute angles
6.
An equilateral triangle
7.
A set of perpendicular lines
8.
An isosceles triangle
9.
A hexagon
10. A rhombus
11. Two or more circles
12. A scalene triangle
13. Three rectangles
14. A pentagon
15. A trapezoid
16. A circle with a chord
17. A rectangle with a diagonal
18. An oval
19. A rectangular prism
20. A cube
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Mathematics Model Curriculum
Grade Level: Fifth Grade
Grading Period: 4
Common Core Domain
Time Range: 15 Days
Geometry
Common Core Standards
Graph points on the coordinate plane to solve real-world and mathematical problems.
1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the
lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an
ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from
the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the
second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and
x-coordinate, y-axis and y-coordinate).
2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane,
and interpret coordinate values of points in the context of the situation.
The description from the Common Core Standards Critical Area of Focus for Grade 5 says:
STANDARDS AND CLUSTERS BEYOND THE CRITICAL AREAS OF FOCUS
Modeling numerical relationships with the coordinate plane
Based on previous work with measurement and number lines, students develop understanding of the coordinate
plane as a tool to model numerical relationships. These initial understandings provide the foundation for work with
negative numbers, and ratios and proportional relationships in Grade Six and functional relationships in further
grades.
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
Grade 5 Coordinate System
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Mathematics Model Curriculum
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).
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
Students need to understand the underlying structure of the coordinate system and see how axes make it
Grade 5 Coordinate System
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Mathematics Model Curriculum
possible to locate points anywhere on a coordinate plane. This is the first time students are working with
coordinate planes, and only in the first quadrant. It is important that students create the coordinate grid
themselves. This can be related to two number lines and reliance on previous experiences with moving
along a number line.
Multiple experiences with plotting points are needed. Provide points plotted on a grid and have students
name and write the ordered pair. Have students describe how to get to the location. Encourage students to
articulate directions as they plot points.
Present real-world and mathematical problems and have students graph points in the first quadrant of the
coordinate plane. Gathering and graphing data is a valuable experience for students. It helps them to
develop an understanding of coordinates and what the overall graph represents. Students also need to
analyze the graph by interpreting the coordinate values in the context of the situation.
Instructional Resources/Tools from ODE Model Curriculum
Grid/graph paper
From the National Council of Teachers of Mathematics, Illuminations: Finding Your Way Around Students explore two-dimensional space via an activity in which they navigate the coordinate plane.
From the National Council of Teachers of Mathematics, Illuminations: Describe the Way – In this lesson,
students will review plotting points and labeling axes. Students generate a set of random points all located
in the first quadrant.
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
When playing games with coordinates or looking at maps, students may think the order in plotting a coordinate
point is not important. Have students plot points so that the position of the coordinates is switched. For example,
have students plot (3, 4) and (4, 3) and discuss the order used to plot the points. Have students create directions for
others to follow so that they become aware of the importance of direction and distance.
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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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
Coordinate Grids
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Geometry 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.
An understanding of coordinate grids plays an important role in our ability to navigate. When
reading maps, we rely on a coordinate grid to help us locate unfamiliar points. In history, city
planners have relied upon a grid system to plan urban areas. In 1812 city planner Joel Wright
designed the first major roads in downtown Columbus, High Street and Broad Street, to run
perpendicular to one another. The rest of the downtown area was planned so that all the streets
run parallel and/or perpendicular to High and Broad. We also use the intersection of points to
plot points and create graphs in mathematics. Recreationally, the popular childhood game of
Battleship™ uses coordinate grids. Children place plastic ships at points while their adversaries
try to guess the locations: “You sunk my battleship.” Building upon these connections, students
will be expected to find and name locations in a coordinate grid.
A coordinate grid is formed by two number lines, or axes, that intersect at a common point, or an
origin. The zero points of each of the number lines coincide. The x-axis and the y-axis, are
perpendicular to one another and exist in a two-dimensional plane. The x-axis runs horizontally,
and the y-axis runs vertically. In fifth grade students will be expected to only identify first
quadrant, or positive points, on a coordinate grid.
y
10
9
8
7
6
5
4
3
2
1
0
x
1 2 3 4 5 6 7 8 9 10
An ordered pair can be defined as two points that name a location on a coordinate grid. The first
coordinate, or the x-coordinate, specifies the location along the x-axis, while the second
coordinate, or the y-coordinate, specifies the location along the y-axis. When reading a
coordinate point, the order is important. The first number (x-coordinate) tells you how many
spaces to move along the x-axis (horizontally). The second number (y-coordinate) tells you how
many spaces to move along the y-axis (vertically). As students are working only with first
quadrant points, the x-coordinate will indicate movement only to the right along the x-axis, while
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the y-coordinate will indicate movement only up along the y-axis. When reading an ordered pair
(also called coordinate pair), say the word “point” first, and then read the numbers. Students
should also write the ordered pair inside a set of parentheses with the two numbers separated by
a comma.
Example: On the grid below McDonald Hospital is located at point (2, 8). What is located at
point (5, 3)? y
10
9
8
7
6
5
4
3
2
1
0
Hospital
Library
Candy Store
x
1 2 3 4 5 6 7 8 9 10
Solution: The candy store is located at point (5, 3). However, if a mistake was made and the
order of the coordinates were reversed, students could mistakenly think the coordinate pair
names the location of the library.
Students will be expected to have two different skills in using coordinate grids. First, the ordered
pair may be provided, and students are expected to find what is located at that point. Second, a
location may be provided, and students are expected to identify the ordered pair.
y
Example: What is located at point (6, 4)?
1
09
8
7
6
5
4
3
2
1
Sal’s Cafe
Rice
Elementary
x
1 2 3 4 5 6 7 8 9 1
Solution: Rice Elementary is located at point (6, 4). Again, order is important in reading
0 a
0
coordinate pair.
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Example: Mr. Mild and his wife are going to the movies. When Mr. Mild looks at a map, he
sees that the Bargain Movie Theater is not far away from their home. Which point gives the
location of the movie theater?
y
10
9
8
7
6
5
4
3
2
1
0
Theater
Home
x
1 2 3 4 5 6 7 8 9 10
Solution: The movie theater is located at point (9, 7).
Students may be asked to look at polygons on a grid and give the ordered pairs that are at the
vertices.
Example: What are the ordered pairs at the vertices of the parallelogram?
y
10
9
8
7
6
5
4
3
2
1
0
x
1 2 3 4 5 6 7 8 9 10
Solution: The vertices of the parallelogram are at (3, 2), (3, 7), (7, 6), (7, 1). Students could list
the vertices in a different order as long as the ordered pairs within each set of parentheses are in
the correct order.
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PRACTICE ASSESSMENT ITEMS
Which selection below correctly states the ordered pair that represents point A on
y
the grid?
 A. (4, 9)
 B. (4, 2)
 C. (6, 7)
 D. (2, 4)
10
9
8
7
6
5
4
3
2
1
0
C
B
D
A
E
x
1 2 3 4 5 6 7 8 9 10
Draw a polygon on the coordinate grid. Give the name of your polygon and list
the ordered pairs that are at the vertices.
y
5
4
3
2
1
x
0
0
Grade 5 Coordinate System
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1
2
3
4
5
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which selection below correctly states the ordered pair that represents point A on
the grid?
y
 A. (4, 9)
 B. (4, 2)
 C. (6, 7)
 D. (2, 4)
10
9
8
7
6
5
4
3
2
1
0
C
B
D
A
E
x
1 2 3 4 5 6 7 8 9 10
Answer: D
Draw a polygon on the coordinate grid. Give the name of your polygon and list
y
the ordered pairs that are at the vertices.
Answers will vary depending upon the polygon
that is drawn. One example is given. This is a
rectangle with vertices at (1, 0), (3, 0), (3, 4) and (1, 4)
5
4
3
A 4-point response correctly draws and names
a polygon and gives the ordered pairs for the vertices.
A 3-point response contains a minor error (e.g, does
not give the name for the polygon or gives an incorrect
ordered pair for one of the vertices).
2
1
x
0
0
1
2
3
4
5
A 2-point response includes several errors (e.g., gives an incorrect name or does not name
the polygon and gives an incorrect ordered pair for one or two of the vertices).
A 1-point response includes a correctly drawn and named polygon with no correct
ordered pairs for the vertices or correctly plotted points that do not form a polygon.
A 0-point response shows no mathematical understanding of this task.
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PRACTICE ASSESSMENT ITEMS
y
5
y
4 y
y
3
y
2
1
0
x
0
1
2
3
4
5
Using this coordinate grid, draw all of the squares that can be made with sides on
the grid lines that have a vertex at (0, 1). One by one, list the ordered pairs for the
four vertices of each square you create and give the dimensions of each square.
The first one has been done for you.
(0, 1), (1, 1), (1, 2), (0, 2) = 1 by 1 square
Denzel wants to plot a right triangle with an area of 4 1 square units on this grid.
2
The first side of the triangle has been drawn. Draw the other two sides to complete
the right triangle. Give the ordered pairs that are at the endpoints of each line
segment. For example, the line segment that is drawn connects (1, 1) with (4, 4).
y
5
4
3
2
1
x
0
0
Grade 5 Coordinate System
1
2
3
4
5
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Using this coordinate grid, draw all of the squares that can be made with sides on
the grid lines that have a vertex at (0, 1). One by one, list the ordered pairs for the
four vertices of each square you create and give the dimensions of each square.
The first one has been done for you.
y
(0, 1), (1, 1), (1, 2), (0, 2) = 1 by 1 square
5
4
Answer: There are 3 other squares that can be made.
(0, 1), (2, 1), (2, 3), (0, 3) = 2 by 2 square
(0, 1), (3, 1), (3, 4), (0, 4) = 3 by 3 square
(0, 1), (4, 1), (4, 5), (0, 5) = 4 by 4 square
3
2
1
x
0
0
1
2
3
4
5
A 4-point response includes the three additional squares that can be made on the grid and
gives the correct ordered pairs for the vertices of each square (within the parentheses the
order of the numbers does matter, but the listing of the ordered pairs does not have to be in
order as long as all four of them are included).
A 3-point response includes the correct ordered pairs and dimensions for two additional
squares or includes the correct ordered pairs for three squares with no dimensions or the
correct dimensions for three squares with minor errors in the ordered pairs.
A 2-point response includes the dimensions for all three additional squares with several
errors in the ordered pairs or includes the correct ordered pairs and dimensions for one
additional square.
A 1-point response contains major flaws in reasoning.
A 0-point response indicates no understanding of this task.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Denzel wants to plot a right triangle with an area of 4 1 square units on this grid.
2
The first side of the triangle has been drawn. Draw the other two sides to complete
the right triangle. Give the ordered pairs that are at the endpoints of each line
segment. For example, the line segment that is drawn connects (1, 1) with (4, 4).
y
y
5
y
4 Sa
m
3 pl
e
2
T
ar
1
ge
x
0 t
As
0
1
2
3
4
5
se
Answer: There are actually two correct
right triangles that can be drawn. The third
ss
vertex can either be at (4, 1) or atm(1, 4).
en
Ordered pairs for vertex at (4,1): t
(1, 1) connects withQ(4, 1)
(4, 1) connects withue
(4, 4)
(4, 4) connects withsti
(1, 1)
Ordered pairs for vertex at (1, 4):on
(1, 1) connects with s(1, 4)
(1, 4) connects with (4, 4)
(4, 4) connects with (1, 1)
A 2-point response includes a correctly drawn right triangle with corresponding ordered
pairs for the endpoints of the three line segments.
A 1-point response includes a correctly drawn right triangle with a minor error in the
ordered pairs or correct ordered pairs with a mistake in the drawing of the triangle.
A 0-point response indicates no understanding of this task.
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PRACTICE ASSESSMENT ITEMS
y
What shape is located at (7, 4)?
A. triangle
B. square
C. heart
D. star
10
9
8
7
6
5
4
3
2
1
0
x
1 2 3 4 5 6 7 8 9 10
A house and school are plotted on the coordinate grid below. Plot the following
two points on the grid below and label them library (3, 4) and theater (8, 2).
y
10
9
8
7
6
5
4
3
2
1
0
Grade 5 Coordinate System
house
school
x
1 2 3 4 5 6 7 8 9 10
Page 17 of 44
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
What shape is located at (7, 4)?
A. triangle
B. square
C. heart
D. star
y
10
9
8
7
6
5
4
3
2
1
0
x
1 2 3 4 5 6 7 8 9 10
Answer: D
A house and school are plotted on the coordinate grid below. Plot the following
two points on the grid below and label them library (3, 4) and theater (8, 2).
y
10
9
A 2-point response correctly plots and
labels the two points.
8
7
A 1-point response correctly plots and
labels one point or correctly plots two
6
points without labeling the points.
5
10
A 0-point response shows no mathematical 4
understanding of this task.
3
2
1
0
Grade 5 Coordinate System
house
school
library
theater
x
1 2 3 4 5 6 7 8 9 10
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: coordinate grid, coordinates, plot, coordinate plane, coordinate system, yaxis, x-axis, ordered pair, quadrant, point, vertex (vertices), horizontal, vertical, origin
point, line, x-coordinate (ordinate), y-coordinate (ordinate), plot
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.
What is the ordered pair for the heart? Explain how you know.
On the coordinate grid, plot these points: Point A: (2,2) Point B: (2,7) Point C: (5,2)
Point D: (5,7) Connect A-B, B-C, C-D, and D-A. What shape is made?
What is the coordinate pair for each of the points shown? A) (__,__) B) (__,__)
C) (__,__)
Meghan draws a line segment from (1,3) to (4,4) and another line segment from (1,4) to
(4,5). If she wants to draw another line segment that is parallel and congruent, what
points could she use?
What are the coordinate pairs for each of the points of the figure shown on the grid?
What is the perimeter of the figure?
The coordinate grid shows Kayla’s neighborhood. If Kayla was going from the school to
Molly’s home, what would be the shortest possible distance?
Draw a polygon on the coordinate grid. Give the name of your polygon and the ordered
pairs that are at the vertices.
Identify the ordered pair for each of the points on the coordinate grid. A) = (__,__)
B) = (__,__) C) = (__,__) D) = (__,__) E) = (__,__) What is the area of figure
AEDB?
Julian was asked to create a drawing of a house by plotting the following points on the
coordinate grid below. As each point was plotted it was connected to the previous point
with a line segment to create the figure. The figure shows what Julian created. (8, 6), (5,
9), (2, 6), (2, 1), (8, 1), (8, 6), (2, 6) Julian plotted one of the pairs of points incorrectly.
Identify the incorrect point and explain Julian’s error.
Aricelle wants to make another rectangle on the coordinate grid that is congruent to the
one shown. If she puts one vertex at (6,1), where would the other three vertices be
placed?
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 5 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. Have students plot coordinates on the activity sheet, "Mr. Simpson's Busy Day" (included in
this curriculum guide). They will use their completed grid to answer the questions on the
following page. As an extension, have students use a “Blank Coordinate Grid” (included in
this Curriculum Guide) or any of the coordinate grids included with other activities to write
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Columbus City Schools 2013-2014
4.
5.
6.
7.
their own ordered pairs for a new snowstorm. Once they've written the ordered pairs, have
them trade papers with a partner, who will plot the coordinates on the blank grid. Each
person can then take turns asking questions about distances between points on the grid.
Make a transparency of the “Blank Coordinate Grid” (included in this Curriculum Guide) or
any of the coordinate grids included with other activities. Do not label the axes. Ask students
to give you two numbers. (Do not make any parameters, such as less than 10, etc.) The first
number that the student says will be the x-coordinate and the second number is the ycoordinate. For example, a student gives the numbers 3 and 12. Beginning at the origin (0,
0) silently count over 3 and then try to count up 12. Tell the class that that this won’t work
(don’t offer an explanation – let the students figure it out). If the first set of numbers can be
plotted on the grid, place a point at the appropriate spot. Allow another student to give two
numbers. Repeat as before. Continue until the class catches on that you will plot points that
fit on the grid, such as (2, 3) and (5, 5). Point out that every coordinate grid has a point
called an origin located at (0, 0) and label that point. Show students how to put the rest of the
numbers on the x-axis and the y-axis and then put in the x and y to label the axes. Let them
know that the numbers represent the lines not the spaces between them. When plotting
points, you always begin at the origin and go across first and then up. Plot the points for
several ordered pairs with students to practice correctly plotting points.
Using masking tape, create a 6 foot × 6 foot coordinate grid on the floor. With students
circling the grid, model how to label the x axis and y axis using an interval of 2. Invite one
student to stand on the number 2 on the x-axis, and ask another student to stand on the 4 on
the y-axis. Direct students to walk on the tape until they meet. At their meeting place mark
it with a marker (a circle about the size of masking tape works best.). Explain how this point
is identified as (2, 4). Continue with several other examples until students demonstrate an
understanding of ordered pairs.
Pass out “Jingles, the Class Hamster” activity sheet. Read the story together. Students work
together to complete the worksheet table and coordinate grid. When completed, discuss the
points on the coordinate grid that reflect different weights of the hamster at different weeks.
Next, pass out “My Own Story” worksheet. Each student will pull 12 playing cards to
generate ordered pairs randomly. Be sure that no ordered pairs are repeated. If so, pull
another card to generate new coordinates. List the ordered pairs on the recording sheet.
Students will plot their points on their coordinate grid and connect them with line segments
from left to right. Have the students observe their graph and determine labels for each axis as
well as a title for their graph. Students will then create their own short story to depict what is
occurring in their graph. Encourage students to write with enough detail that another student
reading their story would be able to recreate the graph. Having students write a rough draft
on a separate sheet of paper first is suggested. When students are finished they may share
their stories with the class and see if the students can recreate their graph on graph paper or
students may switch papers and recreate each other’s graphs.
Make a transparency of the “Blank Coordinate Grid” (included in this Curriculum Guide) or
any of the coordinate grids included with other activities and place the grid on the overhead.
Put students into two groups. Tell them that they are going to play a game in which the
winner is the team that gets four points in a row either across, up and down, or diagonally
(you could start with three in a row until students become more familiar with the game).
Have each team determine the ordered pair where they want to place their point. As teams
call out the numbers, place a point on the grid for that team. You need to have a different
color or type of mark for each team. If a team gives an ordered pair that will not go on the
grid, say, “That pair of numbers cannot be marked on this grid.” and go on to the next team.
Students will quickly figure out that the first number they say goes with the x-axis and the
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second number goes with the y-axis. Teams need to strategize and give ordered pairs to
block the other team from winning while still trying to get four in a row for their own team.
After playing a few games, discuss the terms: ordered pairs, x- axis, and y-axis. Show
students how to write the ordered pairs using parentheses and a comma between the
coordinates. Give groups of four students a coordinate grid and have them play the game.
Each pair of students is a team. One team says an ordered pair and the other team marks the
grid. Play alternates until one team has four in a row.
8. Make a transparency of “Plotting Polygons” (included in this Curriculum Guide) and also
distribute a copy to each student. Practice plotting ordered pairs and connecting the pairs
with lines to create polygons. The last ordered pair needs to be connected to the first with a
line segment to complete the polygon. Some polygons that could be created are the
following: (1, 11), (3, 11), (3, 13), (1, 13) creates a square; (5, 15), (6, 15), (7, 14), (6, 13),
(5, 13), (4, 14) creates a hexagon; (7, 11), (11, 11), (9, 14) creates a triangle; (2, 8), (4, 8),
(4, 9), (3, 10), (2, 9) creates a pentagon; (6, 5), (7, 5), (7, 10), (6, 10) creates a rectangle;
(1, 7), (2, 5), (4, 5), (5, 7) creates a trapezoid; and (8, 6), (8, 9), (11, 6) creates a right
triangle. Ask students if they see any relationship between the number of ordered pairs and
the polygon created (there are the same number of sides as there are ordered pairs). Have
students complete the page on their own by plotting the ordered pairs at the bottom of the
page, and then compare their grid with a partner. Have students create a set of ordered pairs
on a piece of paper that will create a polygon. Divide students into pairs and give each
student a blank “Blank Coordinate Grid” (included in this Curriculum Guide) or any of the
coordinate grids included with other activities and the set of ordered pairs from their partner.
Have each student plot the points they were given and create the polygon. Pairs then share
the polygons they have created to make sure that the ordered pairs were plotted correctly.
9. Distribute "Star Points" (included in this Curriculum Guide). As a group, have students
determine the ordered pairs for point A and point B. Ask students to write the ordered pairs
for the rest of the points on the star. Give students a “Blank Coordinate Grid” (included in
this Curriculum Guide) or any of the coordinate grids included with other activities and have
them draw a shape on the grid and label at least eight points with letters. Students then
determine and record the ordered pairs for the points given on the grid (shapes should be
drawn with straight lines).
10. Distribute "Guess the Shape" (included in this Curriculum Guide). Make sure that students
understand that they need to connect points with line segments as they put them on the
coordinate grid so they are connecting the correct points with each other. Once students have
completed the shape challenge them to create their own shape on a grid. Have them make a
list of ordered pairs that someone else could use to recreate their shape. Ask students to write
their ordered pairs and their name (this way when the activity is done students can check
their shape with the creator of the ordered pairs) on a piece of paper. Collect the ordered
pairs and then distribute blank grids and a set of ordered pairs to each student. Ask the
students to follow the ordered pairs to create the shape on their grid. Have students check
their completed grids with the students who created the ordered pairs for the design.
11. Distribute “Connect Plots” (included in this Curriculum Guide) and three different colors of
markers to groups of three students. Students use different colored markers to place a point
on the grid. Students take turns. On each turn, students record the ordered pair under their
name and then plot the point on the grid. The goal is to connect five points in a row.
Students can strategically block their opponents by putting a point on their line. Lines can be
vertical, horizontal, or diagonal.
12. Read The Fly on the Ceiling by Dr. Julie Glass. Distribute a copy of the “Create It”
worksheet (included in this Curriculum Guide) to each student. Have students plot the items
they would put in their toy/game room. Ask students to write the ordered pairs for each item.
13. Divide the students into pairs. Distribute a “Four-by-Four Coordinate Grid” (included in this
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Curriculum Guide) to each student. Students sit across from each other with a piece of paper
that has been folded and placed between them to shield the grids. Student A marks
an X on a place of their choice on the grid. Student B gets four chances to guess the location
of Student A’s mark, using ordered pairs. If Student B guesses on the first guess, then four
points are earned, on the second guess three points are earned, on the third guess two points
are earned, and on the fourth guess one point is earned. If Student B does not guess the
ordered pair, then Student A gives the correct answer and the next round begins with Student
B marking an X somewhere on the grid. The student with the most points at the end of the
match wins.
14. Using geoboards as a grid, number the left side and the bottom with Post-It® notes. Have
students work with a partner to construct a polygon and identify the ordered pairs at the
vertices of the polygon. Students repeat the process by creating new polygons or switching
partners.
15. Divide students into pairs to play a game similar to Battleship™. Have students face each
other with a low barrier between them (e.g., a box, a stack of books, etc.) so that they can see
each other, but not the papers on the desk. Distribute a “Coordinate Grid” (included in the
Grids and Graphics section of this Curriculum Guide) to each student. Each player draws
two rectangles on the grid without letting the other person see where they are placed. The
rectangles need to be 1 unit by 3 units and 2 units by 4 units. Giving parameters makes it
easier for students to narrow their guesses. As students become better at the game they could
make rectangles that are other sizes or any other polygons. Players alternate guessing
ordered pairs to try to determine where the rectangles are drawn on their partner’s grid. As
each ordered pair is guessed the player looking at his or her rectangles either says hit or miss.
Hit means that the ordered pair is part of the rectangle and miss means that the ordered pair is
not part of a rectangle. Students keep track of their guesses by placing an X over the point of
the ordered pairs that have been guessed. When all of the coordinates that make up both
rectangles have been hit, then the person that guessed them all wins (students need to tell
each other when one whole rectangle has been found so that the partner knows they have
found one whole figure and now they are looking for the other). Students guess all of the
ordered pairs that are on the perimeter of the rectangles before they get credit for guessing
y
that rectangle. For example,
10
9
It would take hits at (1, 7), (2, 7),
8
(3, 7), (4, 7), (5, 7), (5, 8), (5, 9),
(4, 9), (3, 9), (2, 9), (1, 9), and (1, 8) 7
to find the 2 by 4 rectangle
6
5
It would take hits at (7, 0), (8, 0),
(8, 1), (8, 2), (8, 3), (8, 4), (7, 4),
4
(7, 3), (7, 2), and (7, 1) to find the
3
1 by 4 rectangle.
2
1
0
x
1 2 3 4 5 6 7 8 9 10
16. Show students the “Blank Coordinate Grid” (included in this Curriculum Guide) or any of
the coordinate grids included with other activities. Give ordered pairs for students to plot
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that will form polygons. Have them write the ordered pairs for several polygons on another
sheet of paper. Students trade papers and then plot the ordered pairs on another copy grid to
create the polygons.
17. Divide students into pairs and distribute the “Blank Coordinate Grid” (included in this
Curriculum Guide) or any of the coordinate grids included with other activities to each
student. Each student draws a picture on his or her grid then records the ordered pairs for the
picture on a separate sheet of paper. Give each student a new piece of grid paper. Students
then switch ordered pair record sheets and plot each other’s pictures. Compare the original
picture created to the picture created from following the ordered pairs. If they are different,
determine where the errors were made.
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RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 174-175
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. 45
Practice Master pp. 45
Problem Solving Master pp. 45
Reteaching Master pp. 45
INTERDISCIPLINARY CONNECTIONS
1. Literature Connections
The Fly on the Ceiling by Dr. Julie Glass
Let’s Fly a Kite by Stewart J. Murphy
Where's That Bone? by Lucille Recht Penner
2. Read aloud Where's That Bone? by Lucille Recht Penner. After reading the text, discuss
mapping points on a geographic graph using longitude and latitude.
3. Laminate Ohio maps (one per 2-3 students). Distribute the maps and four pieces of yarn, two
that are the length of the map and two that are the width of the map, to each group of
students. Have students locate major cities and landmarks in Ohio using the yarn pieces by
looking at the coordinates on the map. For example, ahead of time figure out the coordinates
where Columbus is located. Use the yarn pieces to plot the coordinates so they make a
square around the city of Columbus. If the coordinates for Columbus are (3, 5),
(4, 5), (3, 6), and (4, 6), then one piece of yarn would be placed across the 3 line, one across
the 4 line, one down the 5 line and one down the 6 line. This would then form a square of
yarn around Columbus. Repeat the process with other cities and landmarks in the state.
Grade 5 Coordinate System
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Problem Solving Questions
10
9
8
7
6
5
4
3
2
1
What is the ordered pair for the heart? Explain how you know.
0
1 2 3 4 5 6 7 8 9 10
10
9
8
7
6
5
4
3
2
1
On the coordinate grid, plot these points:
Point A: (2,2)
Point C: (5,2)
0
1
10
9
8
7
6
5
4
3
2
1
2
3
4
5
6
7
8
9
Connect A-B, B-C, C-D, and D-A. What shape is made?
10
What is the coordinate pair for each of the points shown?
B
A) (__,__)
A
B) (__,__)
C) (__,__)
C
0
1
2
3
4
5
6
7
8
9
10
Meghan draws a line segment from (1,3) to (4,4) and
another line segment from (1,4) to (4,5). If she
wants to draw another line segment that is parallel
and congruent, what points could she use?
10
9
8
7
6
5
4
3
2
1
0
Point B: (2,7)
Point D: (5,7)
1
2
3
4
5
6
7
8
9
10
What are the coordinate pairs for each of the points of the figure
shown on the grid? What is the perimeter of the figure?
Grade 5 Coordinate System
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Columbus City Schools 2013-2014
Problem Solving Questions
The coordinate grid shows Kayla’s neighborhood. If
Kayla was going from the school to Molly’s home,
what would be the shortest possible distance?
y
5
4
Draw a polygon on the coordinate grid. Give the name of
your polygon and the ordered pairs that are at the vertices.
3
2
1
x
0
0
1
2
3
4
5
y
10
9
8
7
6
5
4
3
2
1
Identify the ordered pair for each of the points on the coordinate
grid.
A) = (__,__)
B) = (__,__)
C) = (__,__)
D) = (__,__)
E) = (__,__)
What is the area of figure AEDB?
C
B
D
A
E
x
0
1 2 3 4 5 6 7 8 9 10
Julian was asked to create a drawing of a house by plotting the
following points on the coordinate grid below. As each point was
plotted it was connected to the previous point with a line segment to
create the figure. The figure shows what Julian created.
(8, 6), (5, 9), (2, 6), (2, 1), (8, 1), (8, 6), (2, 6)
Julian plotted one of the pairs of points incorrectly. Identify the
incorrect point and explain Julian’s error.
y
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
x
Aricelle wants to make another rectangle on the coordinate
grid that is congruent to the one shown. If she puts one vertex
at (6,1), where would the other three vertices be placed?
Grade 5 Coordinate System
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Columbus City Schools 2013-2014
Problem Solving Questions Answers
Answer Key
10
9
8
7
6
5
4
3
2
1
What is the ordered pair for the heart? Explain how you know.
Answer: (8,2) The explanation should include counting 8 to the right
and up 2
0
1 2 3 4 5 6 7 8 9 10
On the coordinate grid, plot these points:
10
9
8
7
6
5
4
3
2
1
Point A: (2,2)
Point C: (5,2)
Connect A-B, B-C, C-D, and D-A. What shape is made?
0
1
10
9
8
7
6
5
4
3
2
1
0
2
3
4
5
6
7
8
9
Answer: Shown on grid. The shape is a rectangle.
10
What is the coordinate pair for each of the points shown?
B
A) (__,__)
A
C
1
2
3
4
5
6
7
8
9
B) (__,__)
Answer: A:(2,4)
C) (__,__)
B:(4,7)
C:(9,2)
10
Meghan draws a line segment from (1,3) to (4,4) and another line
segment from (1,4) to (4,5). If she wants to draw another line
segment that is parallel and congruent, what points could she use?
10
9
8
7
6
5
4
3
2
1
0
Point B: (2,7)
Point D: (5,7)
1
2
3
4
5
6
7
8
9
10
Answers may vary: The two most likely responses are (1,5) to
(4,6) [shown] and (1,2) to (4,3) [below the two line segments]
What are the coordinate pairs for each of the vertices of the
figure shown on the grid? What is the perimeter of the figure?
Answer: (1,4) (1,6) (5,6) (5,4)
The perimeter is 12.
Grade 5 Coordinate System
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Columbus City Schools 2013-2014
Problem Solving Questions Answers
The coordinate grid shows Kayla’s neighborhood. If
Kayla was going from the school to Molly’s home, what
would be the shortest possible distance?
Answer: Kayla would need to go 5 blocks.
Draw a polygon on the coordinate grid. Give the name of your
polygon and the ordered pairs that are at the vertices.
y
5
4
Answer: Answers will vary. A correct answer should name the
polygon correctly and give the ordered pairs for each vertice.
One example is shown. A rectangle with vertices at (1,0) , (2,0) ,
(2,3) ,and (1,3).
3
2
1
x
0
0
1
2
3
4
5
y
10
9
8
7
6
5
4
3
2
1
0
Identify the ordered pair for each of the points on the coordinate grid.
A) = (__,__)
B) = (__,__)
C) = (__,__)
D) = (__,__)
E) = (__,__)
What is the area of figure AEDB?
C
B
D
A
E
x
1 2 3 4 5 6 7 8 9 10
Answer: A) = (2,4) B) = (2,7) C) = (4,9) D) = (6,7) E) = (6,4)
The area of figure AEDB is 12 sq. units.
y
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
x
Julian was asked to create a drawing of a house by plotting the following
points on the coordinate grid below. As each point was plotted it was
connected to the previous point with a line segment to create the figure. The
figure shows what Julian created.
(8, 6), (5, 9), (2, 6), (2, 1), (8, 1), (8, 6), (2, 6)
Julian plotted one of the pairs of points incorrectly. Identify the incorrect
point and explain Julian’s error.
Answer: Julian plotted point (8,1) incorrectly. He plotted it as (1,8)
Aricelle wants to make another rectangle on the coordinate grid that
is congruent to the one shown. If she puts one vertex at (6,1), where
would the other three vertices be placed?
Answer: The other vertices could be at (6,3), (10,1), and (10,3) or
at (8,1), (6,5), and (8,5) or at (2,1), (6,3), and (2,3) or at (4,1), (6,5),
and (4,5)
Grade 5 Coordinate System
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Mr. Simpson’s Busy Day
Bart, the tow truck driver, was having a busy day during the first big snowstorm of the season.
People all over the city were stranded in their cars. Bart’s GPS system usually displays all the
distress calls on a grid, but today his system isn’t working correctly and is only displaying the
coordinates. Help Bart plot the distress calls on the grid below.
Lexus (3,6)
Honda Fit (8,1)
Ford Pickup (6,2)
Jaguar (1,8)
Bart’s current location is (2,1)
Mitsubishi (1,3)
Escalade (8,9)
Nissan (5,3)
Minivan (9,4)
y
12
11
10
9
8
7
6
5
4
3
2
1
0
x
1
Grade 5 Coordinate System
2
3
4
5
Page 29 of 44
6
7
8
9
10
Columbus City Schools 2013-2014
Mr. Simpson’s Busy Day
Use your completed grid to answer the questions.
1. Which car is closest to Bart’s tow truck?
_________________________________________________________________
2. Which car is farthest from Bart’s tow truck?
_________________________________________________________________
3. Which two disabled vehicles are closest to each other?
_________________________________________________________________
4. If each unit on the grid represents 2 miles, how far from the Lexus is Bart’s truck?
_________________________________________________________________
5. Bart can only drive 60 miles before he has to refuel. Plot a route that allows him to pick
up the most cars without having to refuel.
________________________________________________________________________
________________________________________________________________________
6. Bart has just picked up the minivan and has just enough gas to travel 8 miles. If the
nearest gas station is at (7,7), will he make it before he runs out of gas? How did you
determine your answer?
________________________________________________________________________
________________________________________________________________________
7. How many miles will Bart travel if he picks up all the stranded vehicles?
_______________________________________________________________________
_______________________________________________________________________
Grade 5 Coordinate System
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Columbus City Schools 2013-2014
Mr. Simpson’s Busy Day
Answer Key
Use your completed grid to answer the questions.
1. Which car is closest to Bart’s tow truck?
__________Mitsubishi________________________________________________
2. Which car is farthest from Bart’s tow truck?
________ Escalade___________________________________________________
3. Which two disabled vehicles are closest to each other?
__________ Ford Pickup and Nissan _____________________________________
4. If each unit on the grid represents 2 miles, how far from the Lexus is Bart’s truck?
_______12 miles_______________________________________________________
5. Bart can only drive 20 miles before he has to refuel. Plot a route that allows him to pick
up the most cars without having to refuel.
___Bart could pick up the Mitsubishi, then the Nissan, then the Ford and _________
____travel 18 miles._______________________________________________________
6. Bart has just picked up the minivan and has just enough gas to travel 8 miles. If the
nearest gas station is at (7,7), will he make it before he runs out of gas? How did you
determine your answer?
______He won’t make it. The gas station is 10 miles away on the shortest route.____
7. How many miles will Bart travel if he picks up all the stranded vehicles?
__Answers will vary depending on the route chosen.________________________
Grade 5 Coordinate System
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Blank Coordinate Grid
Grade 5 Coordinate System
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Jingles, the Class Hamster
Mrs. Wagner’s class voted to get a hamster for a class pet. When the hamster left the pet
store, it weighed 10 ounces. The class decided to weigh the hamster each week to make sure it
was healthy and growing strong. The first week at school, Jingles was very anxious and scared.
He lost two ounces. Mrs. Wagner told the class to stop taking Jingles out of his cage and let him
rest. The second week, Jingles gained back all of the weight he lost. The third week, Jingles’
weight doubled. The class decided that Jingles was getting fat and they bought him a hamster
wheel. Two weeks later, Jingles had lost five ounces. The next week was Christmas break. No
one remembered to feed Jingles and he lost five ounces.
Week
Hamster
weight in
ounces
0
1
2
3
4
5
6
Represent this story on the table. Plot the points on the coordinate grid. Label each axis
and add a title for the graph.
y
12
11
10
9
8
7
6
5
4
3
2
1
0
x
1 2 3 4 5 6 7 8 9 10 11 12 13
2
Grade 5 Coordinate System
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My Own Story
Title___________________
y
12
11
10
9
8
7
6
5
4
3
2
1
0
x
1 2 3 4 5 6 7 8 9 10 11 12 13
2
Ordered Pairs:
(____ , ____) (____ , ____) (____ , ____) (____ , ____) (____ , ____) (____ , ____)
Short Story:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Grade 5 Coordinate System
Page 34 of 44
Columbus City Schools 2013-2014
Plotting Polygons
Name
y
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10 11
x
Plot the following ordered pairs on the grid above. Connect each set of points to
create a polygon. What polygon did you create?
(2, 1), (3, 4), (8, 4), (7, 1)
Grade 5 Coordinate System
Page 35 of 44
Columbus City Schools 2013-2014
Plotting Polygons
Answer Key
y
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10 11
x
Plot the following ordered pairs on the grid above. Connect each set of points to
create a polygon. What polygon did you create?
(2, 1), (3, 4), (8, 4), (7, 1)
parallelogram
All figures other than the
parallelogram are for ordered pairs given in the teaching strategies.
Grade 5 Coordinate System
Page 36 of 44
Columbus City Schools 2013-2014
Star Points
Name
y
E
19
18
17
16
15
14
F
D
C
13
G
12
11
10
9
B
H
8
7
J
6
5
4
3
A
I
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
19
Write the ordered pair for each labeled point on the star.
A (
,
)
D (
,
)
G (
,
)
B (
,
)
E (
,
)
H (
,
)
C (
,
)
F (
,
)
I (
,
)
Grade 5 Coordinate System
Page 37 of 44
J (
,
)
Columbus City Schools 2013-2014
x
Star Points
Answer Key
y
E
19
18
17
16
15
14
F
D
C
13
G
12
11
10
9
B
H
8
7
J
6
5
4
3
A
I
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
19
Write the ordered pair for each labeled point on the star.
A (5, 3)
D (8, 13)
G (18, 13)
B (7, 9)
E (10, 19)
H (13, 9)
C (2, 13)
F (12, 13)
I (15, 3)
Grade 5 Coordinate System
Page 38 of 44
J (10, 7)
Columbus City Schools 2013-2014
x
Guess the Shape
Name
y
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
19
Plot the following coordinates on the grid in the order they are listed. Draw lines
to connect the points as you plot them.
(18, 3) (16, 4) (11, 8) (9, 8) (4, 9) (2, 11) (7, 10) (3, 13) (2, 17)
(3, 16) (4, 14) (8, 12) (8, 14) (7, 17) (10, 14) (11, 11) (13, 8) (17, 5)
(18, 3)
Grade 5 Coordinate System
Page 39 of 44
Columbus City Schools 2013-2014
x
Guess the Shape
Answer Key
y
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
19
Plot the following coordinates on the grid in the order they are listed. Draw lines
to connect the points as you plot them.
(18, 3) (16, 4) (11, 8) (9, 8) (4, 9) (2, 11) (7, 10) (3, 13) (2, 17)
(3, 16) (4, 14) (8, 12) (8, 14) (7, 17) (10, 14) (11, 11) (13, 8) (17, 5)
(18, 3)
Grade 5 Coordinate System
Page 40 of 44
Columbus City Schools 2013-2014
x
Connect Plots
Players take turns. Roll a number cube to determine who goes first. The first
player writes the ordered pair they are going to plot and then plots the point by
placing a colored dot on that spot on the grid. The other players check to make
sure that the point is plotted correctly. If it is not, the player loses his or her next
turn and a black X is placed on the spot, which means that other players can select
that spot. Then the next player begins his or her turn. If the point is plotted
correctly, then the next player takes a turn. Play continues until one player has
plotted five points in a row. Strategy must be used by the players to block each
other when someone is getting close to having five in a row. Points that are plotted
by the same player should be connected with a line segment. The winner of the
game is the first player to have five points in a row connected by a line segment
either across, up and down, or diagonally.
Each pair of coordinate points should be recorded on the table below before it is
plotted on the grid.
Player 1
Name:
Grade 5 Coordinate System
Player 2
Name:
Player 3
Name:
Page 41 of 44
Columbus City Schools 2013-2014
Connect Plots
Names of Players in Group
y
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
Grade 5 Coordinate System
6
7
8
9
10 11 12 13 14 15 16 17 18
Page 42 of 44
19 20
Columbus City Schools 2013-2014
x
Create It!
Name
y
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
x
1
2
Grade 5 Coordinate System
3
4
5
6
Page 43 of 44
7
8
9
10
Columbus City Schools 2013-2014
Four-by-Four Coordinate Grid
y
4
3
2
1
0
x
1
Grade 5 Coordinate System
2
Page 44 of 44
3
4
Columbus City Schools 2013-2014