Grade 3 Mathematics, Quarter 4, Unit 4.1
Representing and Interpreting Data
Overview
Number of Instructional Days:
5
(1 day = 45–60 minutes)
Content to be Learned
Mathematical Practices to Be Integrated
•
Model with mathematics.
•
Draw a scaled picture graph and a scaled bar
graph to represent a data set with several
categories.
•
Represent data in multiple ways and evaluate
the results in the context of the situation (i.e.,
compare appropriateness of various scales for
data collected).
Solve one- and two-step “How many more?”
and “How many less?” problems using
information presented in scaled bar graphs.
Essential Questions
•
How does the scale affect the graph of a data
set?
•
What comparison problems can you create
from your data?
•
Why do intervals need to be equal increments?
•
•
How can you represent your data in a way that
makes sense?
What are the parts of a graph? Why is each part
important?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 45 Grade 3 Mathematics, Quarter 4, Unit 4.1
Representing and Interpreting Data (5 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Measurement and Data
3.MD
Represent and interpret data.
3.MD.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several
categories. Solve one- and two-step “how many more” and “how many less” problems using
information presented in scaled bar graphs. For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
Common Core Standards for Mathematical Practice
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
Clarifying the Standards
Prior Learning
In grade 2, students drew picture graphs and bar graphs with single-unit scales to represent a data set with
up to four categories. They solved simple put-together, take-apart, and compare problems.
Current Learning
In Unit 1.3, students draw scaled picture graphs and scaled bar graphs to represent data sets with several
categories and to solve one-step problems. In Unit 2.2, students measure items to the nearest 1/2 and 1/4
inch and represent the data by making a line plot with an appropriate horizontal scale.
In this unit, students create and use the data from scaled bar graphs and scaled picture graphs to solve
two-step “How many more?” and “How many less?” problems.
Future Learning
In grade 4, students will make a line plot to display a data set of measurements in fractions of a unit (1/2,
1/4, 1/8). They will solve problems involving addition and subtraction of fractions by using information
presented in line plots.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 46 Grade 3 Mathematics, Quarter 4, Unit 4.1
Representing and Interpreting Data (5 days)
Additional Findings
According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, “Addition,
subtraction, multiplication and division of whole numbers come into play as students construct and
analyze frequency tables, bar graphs, picture graphs and line plots and use them to solve problems.”
(p. 15)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 47 Grade 3 Mathematics, Quarter 4, Unit 4.1
Representing and Interpreting Data (5 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 48 Grade 3 Mathematics, Quarter 4, Unit 4.2
Time and Measurement
Overview
Number of Instructional Days:
10
(1 day = 45–60 minutes)
Content to be Learned
Mathematical Practices to Be Integrated
•
Tell and write time and measure time in
intervals.
Make sense of problems and persevere in solving
them.
•
Solve problems involving addition and
subtraction of time intervals in minutes.
•
Explain to themselves the meaning of the
problem and look for ways to solve it.
•
Measure and estimate liquid volume and
masses of objects using standard units of
grams, kilograms, and liters.
•
Use concrete objects or pictures to help
conceptualize and solve problems.
•
Solve one-step problems using four operations
involving masses and volumes given in the
same unit.
•
Solve time-interval problems on a number line
diagram.
Essential Questions
•
Why does what you measure influence how
you measure?
•
About how many liters are in this container?
How do you know?
•
How could a number line help solve an
elapsed-time problem? How could you use a
model or drawing to represent the problem?
•
About how many grams is this object? How do
you know?
•
What do you use liters to measure? Grams?
Kilograms?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 49 Grade 3 Mathematics, Quarter 4, Unit 4.2
Time and Measurement (10 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Measurement and Data
3.MD
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and
masses of objects.
3.MD.1
Tell and write time to the nearest minute and measure time intervals in minutes. Solve word
problems involving addition and subtraction of time intervals in minutes, e.g., by representing
the problem on a number line diagram.
3.MD.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g),
kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units, e.g., by using
drawings (such as a beaker with a measurement scale) to represent the problem.7
6
Excludes compound units such as cm3 and finding the geometric volume of a container.
7
Excludes multiplicative comparison problems (problems involving notions of “times as much”; see Glossary,
Table 2).
Common Core Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students
check their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
Clarifying the Standards
Prior Learning
In grade 2, students generated linear measurements by estimating and measuring objects to the nearest
whole unit. They used drawings to solve one- and two-step problems involving addition and subtraction.
They learned how to tell and write time using digital and analog clocks to the nearest five minutes using
a.m. and p.m..
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 50 Grade 3 Mathematics, Quarter 4, Unit 4.2
Time and Measurement (10 days)
Current Learning
In Unit 3.1, students tell and write time to the nearest minute.
In this unit, they measure time intervals in minutes and solve problems involving addition and subtraction
of time intervals. A number line diagram is a valuable model for this. Students reason about units of mass
and volume and measure and estimate liquid volumes and masses. They solve one-step problems with all
four operations using the same unit of measure (no conversions).
Future Learning
In grade 4, students will use the four operations to solve word problems involving distances, intervals of
time, liquid volumes, masses of objects, and money. These problems may involve simple fractions and
decimals and problems that require expressing measurements given in a larger unit in terms of a smaller
unit. Students will represent measurement quantities using diagrams such as number lines that feature a
measurement scale.
Additional Findings
“Continuous measurement quantities such as liquid volume, mass and so on are an important context for
fractional arithmetic. In grade 3, children begin to get a feel for continuous measurement quantities and
solve whole number problems involving such quantities.” (PARCC Framework, p. 16)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 51 Grade 3 Mathematics, Quarter 4, Unit 4.2
Time and Measurement (10 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 52 Grade 3 Mathematics, Quarter 4, Unit 4.3
Computational Fluency
Overview
Number of Instructional Days:
13
(1 day = 45–60 minutes)
Content to be Learned
Mathematical Practices to Be Integrated
•
Use multiplication and division within one 100
to solve word problems involving equal groups,
arrays, and measurement quantities.
Attend to precision.
•
Calculate accurately and efficiently.
•
Apply properties of operations as strategies to
multiply and divide.
•
Give carefully formulated explanations to each
other.
•
Fluently multiply and divide within 100 using
strategies such as the relationship between
multiplication and division or the properties of
operations.
•
Know from memory the products of all two
1-digit numbers.
•
Multiply one-digit whole numbers by multiples
of 10 in the range 10–90, using strategies based
on place value and properties of operations.
•
Can the order of the divisor and dividend be
reversed in a division problem without
changing the quotient? If so, is this always
true? Why or why not?
•
How can you use known facts to find unknown
facts?
•
How can the patterns of multiples of 10 help
you solve problems?
Essential Questions
•
What kinds of problems in your world might be
modeled and solved with multiplication and/or
division?
•
Can the order of factors be reversed in a
multiplication problem without changing the
product? If so, is this always true? Why or why
not?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 53 Grade 3 Mathematics, Quarter 4, Unit 4.3
Computational Fluency (13 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Operations and Algebraic Thinking
3.OA
Represent and solve problems involving multiplication and division.
3.OA.3
Use multiplication and division within 100 to solve word problems in situations involving
equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a
symbol for the unknown number to represent the problem.1
1
See Glossary, Table 2.
Understand properties of multiplication and the relationship between multiplication and division.
Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is
known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2
can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.
(Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can
find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
3.OA.5
2
Students need not use formal terms for these properties.
Multiply and divide within 100.
3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties
of operations. By the end of Grade 3, know from memory all products of two one-digit
numbers.
Number and Operations in Base Ten
3.NBT
Use place value understanding and properties of operations to perform multi-digit arithmetic.4
4
3.NBT.3
A range of algorithms may be used.
Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60)
using strategies based on place value and properties of operations.
Common Core Standards for Mathematical Practice
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 54 Grade 3 Mathematics, Quarter 4, Unit 4.3
Computational Fluency (13 days)
Clarifying the Standards
Prior Learning
In grade 2, students used addition and subtraction to solve word problems dealing with linear
measurement and money. They used drawings and equations with a symbol for the unknown. Students
wrote equations to represent an even number as the sum of equal addends. They learned the relationship
between addition and subtraction and fluently added and subtracted within 100.
Current Learning
Earlier in grade 3, students learn the relationship between multiplication and division. They represent and
solve word problems dealing with arrays, measurement quantities, and equal groups. In Unit 2.3, students
use the Commutative Property of Multiplication, and they multiply and divide within 100 using drawings,
object, and other strategies.
In this unit, students learn the Associative Property of Multiplication and solve problems involving
multiplication and division. They multiply one-digit whole numbers by multiples of 10 in the range
10–90. Students multiply and divide within 100 with fluency, and by the end of grade 3, they know from
memory all products of two 1-digit numbers.
Future Learning
In grade 4, students will solve multistep word problems using all four operations. They will multiply a
whole number of up to four digits by a one-digit whole number and multiply two 2-digit numbers, using
strategies based on place value and the properties of operations. Students will recognize that in a
multidigit whole number a digit in one place represents 10 times what it represents in the place to its right.
Additional Findings
“Word problems involving equal groups, arrays and measurement quantities can be used to build students'
understanding of and skill with multiplication and division, as well as to allow students to demonstrate
their understanding of and skill with these operations.” (PARCC Framework, p. 16)
According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, “Properties of
addition and multiplication (e.g., commutativity, associativity and the distributive property) to multiply
whole numbers and apply increasingly sophisticated strategies based on these properties to solve
multiplication and division problems involving basic facts.” (p. 15)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 55 Grade 3 Mathematics, Quarter 4, Unit 4.3
Computational Fluency (13 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 56 Grade 3 Mathematics, Quarter 4, Unit 4.4
Problem-Solving Applications
Overview
Number of Instructional Days:
12
(1 day = 45–60 minutes)
Content to be Learned
Mathematical Practices to Be Integrated
•
Solve multistep problems using addition,
subtraction, multiplication, and division.
Make sense of problems and persevere in solving
them.
•
Follow the conventional order of operations
with and without parentheses.
•
Explain the meaning of a problem.
•
Use a letter to stand for an unknown quantity.
•
Look for ways to solve a problem.
•
Apply estimation strategies to decide if an
answer makes sense.
•
Use concrete objects or pictures.
•
Try different approaches and revise as
necessary.
•
Check work by solving the problem using a
different method.
•
Listen to and make connections with the
strategies of others.
Essential Questions
•
How are all four operations related?
•
•
Why do you use a letter to represent the
unknown quantity?
How can the use of parentheses or conventional
order of operations change your answer?
•
How can estimation be useful when solving
problems?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 57 Grade 3 Mathematics, Quarter 4, Unit 4.4
Problem-Solving Applications (12 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Operations and Algebraic Thinking
3.OA
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.8
Solve two-step word problems using the four operations. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including rounding.3
3
This standard is limited to problems posed with whole numbers and having whole-number answers; students
should know how to perform operations in the conventional order when there are no parentheses to specify a
particular order (Order of Operations).
Common Core Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students
check their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
Clarifying the Standards
Prior Learning
In grade 2, students represented and solved one- and two-step problems involving addition and
subtraction within 100. They solved word problems involving dollar bills, quarters, dimes, nickels, and
pennies using $ and ¢ symbols and lengths given in the same units.
Current Learning
Earlier in grade 3, students solve one- and two-step problems with all four operations to develop and
apply mathematical understandings. They represent these problems using an unknown quantity. Students
perform operations in the conventional order both when there are and are not parentheses to specify a
particular order (multiply or divide before add or subtract).
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 58 Grade 3 Mathematics, Quarter 4, Unit 4.4
Problem-Solving Applications (12 days)
In this unit, students use a letter to stand for the unknown quantity. All problems are posed with whole
numbers and having whole number answers.
Future Learning
In grade 4, students will solve multistep word problems, including problems in which the remainder must
be interpreted.
Additional Findings
“Understanding using the associative and distributive property requires students to know two conventions
for reading an expression that has more than one operation. 1. Do the operation inside the parentheses
before doing the operation outside of the parentheses. 2. If multiplication or division is written next to an
addition or subtraction, imagine parentheses around the multiplication or division.” (Progressions, p. 27)
According to Principles and Standards for School Mathematics,
“Students should have frequent opportunity to formulate, grapple with, and solve complex problems
that require a significant amount of effort and then should be encouraged to reflect on their thinking.”
(p. 52)
“It is essential that students have time to explore problems. Giving help too soon can deprive them of
the opportunity to make mathematical discoveries.” (p. 186)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 59 Grade 3 Mathematics, Quarter 4, Unit 4.4
Problem-Solving Applications (12 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 60