Woodlynne School District
Curriculum Guide
Mathematics
Grade 5
NOTE: The following pacing guide was developed during the creation of these curriculum units. The actual implementation of each
unit may take more or less time. Time should also be dedicated to preparation for benchmark and State assessments, and analysis of
student results on the same. A separate document is included at the end of this curriculum guide with suggestions and resources related
to State Assessments. The material in this document should be integrated throughout the school year, and with an awareness of the State
Testing Schedule. It is highly recommended that teachers meet throughout the school year to coordinate their efforts in implementing
the curriculum and preparing students for benchmark and State Assessments with consideration for the district’s calendar.
1
Woodlynne School District Curriculum Guide
Content Area: Mathematics
Course Title: Grade 5 Math
Grade Level: 5
Unit 1: Place Value, Multiplication and Expressions
4 weeks
Unit 2: Divide Whole Numbers
3 weeks
Unit 3: Add and Subtract Decimals
4 weeks
Unit 4: Multiply Decimals
4 weeks
Unit 5: Divide Decimals
3 weeks
Unit 6: Adding and Subtracting Fractions
3 weeks
2
Unit 7: Multiply Fractions
3 weeks
Unit 8: Dividing Fractions
3 weeks
Unit 9: Algebra: Patterns and Graphing
3 weeks
Unit 10: Convert Units of Measure
3 weeks
Unit 11: Geometry and Volume
3 weeks
Board Approved on: 9/9/14
3
Unit 1 Overview
Content Area – Mathematics
Unit 1: Place Value, Multiplication, and Expressions 6 Lessons/ Four Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale –
Students will use operations and algebraic thinking to evaluate operations in base ten.
Interdisciplinary Integration – Language Arts: Journal writing, open-ended response questions;
Consumer Economics: real-world problem-solving.
Technology Integration – SMART board, Document Camera, virtual manipulatives, Internet sites, EnVision Math Animated Software,
calculator, i-Ready
21st Century Themes –
Global awareness, financial, economic, business and
entrepreneurial literacy
21st Century Skills –
Creativity/innovation, information literacy, critical thinking/problem solving,
communication and collaboration, life and career skills, media literacy
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.8: Look for and express regularity in repeated reasoning.
Math Domain Standards:
5. OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5. OA.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For
example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as
18932 + 921, without having to calculate the indicated sum or product
5. NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right
and 1/10 of what it represents in the place to its left.
5. NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the
placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of
10.
5. NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm.
5. NBT.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on
place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation
by using equations, rectangular arrays, and/or area models.
4
Unit Essential Questions
 How can you use place value, multiplication, and
expressions to represent and solve problems?
Goals/Objectives
Students will be able to Write and interpret
numerical expressions
Understand the place value
system
Perform operations with
multi-level whole numbers
and with decimals to
hundredths.
Unit Enduring Understandings
 Reading, writing and representing whole numbers through millions, using
properties and multiplication to solve problems and using expressions to
represent and solve problems are all used in problem solving.
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Write the expression 6 x 14 on the board.
Weekly quiz on place value, properties, powers of 10,
Have students draw an array with 6 rows
and multiplication patterns.
and 14 columns on grid paper to model this
product. Encourage students to see that they
Unit test includes all skills on the quiz and multiply by
can break apart the array into 10 columns
one and two –digit numbers, relate multiplication to
and 4 columns to make the product easier to
division, numerical expressions, and grouping symbols.
find.
How is breaking apart the array helpful?
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other
Draw a 6 by 6 grid on the board, with columns
instrument
and rows labeled as shown
x
5
6
7
8
9
5
6
42
7
8
9
Model how to name a fact, and write the
product on the table. Then have students
provide other examples. Ask students to name a
related division sentence. 42 divided by 6 = 7
Have students use counters or draw pictures to
help them understand related multiplication and
division sentences. Continue until table is
complete.
5
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number
of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be
made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or
English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
Terminology: base, distributive property, evaluate, exponent, inverse operation, numerical expression,
order of operation, period
6
Unit 2 Overview
Content Area – Mathematics
Unit 2: Dividing Whole Numbers 4 Lessons/ Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale –
Students will find whole number quotients and interpret a fraction as division of the numerator by the denominator.
Interdisciplinary Integration – Language Arts: Journal writing, open-ended response questions;
Consumer Economics: real-world problem-solving.
Technology Integration – SMART board, Document Camera, virtual manipulatives, Internet sites, i-Ready
21st Century Themes –
21st Century Skills –
Global awareness, financial, economic, business Creativity/innovation, information literacy, critical thinking/problem solving,
and entrepreneurial literacy
communication and collaboration, life and career skills, media literacy
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.4: Model with mathematics.
MP.7: Look for and make use of structure.
Math Domain Standards:
5. NBT.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on
place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation
by using equations, rectangular arrays, and/or area models.
5. NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of
whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent
the problem.
Unit Essential Questions
 How can you divide whole numbers?
Unit Enduring Understandings
 Checking work, estimating for the reasonableness of quotients and identifying key words
to know how to solve a problem are important strategies in evaluating problems.
Terminology: compatible numbers, estimate, inverse operations, remainder
7
Goals/Objectives
Students will be able to Perform operations
with multi-digit whole
numbers and with
decimals to
hundredths.
Apply and extend
previous
understandings of
multiplication and
division to multiply
and divide fractions.
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
On the board, write 75 divided by 5 and sketch
a rectangle as shown:
75 divided by 5 =
(40 + 35) divided by 5
8
+
7
5
40
35
To model 75 divided by 5, draw a rectangle.
What is the next step? (Break 75 into multiples
of 5 whose sum is 75)
Distribute the 5 to 40 divided by 5 = 8 and 35
divided by 5 = 7 add 8 +7 to get the quotient of
15
Evidence of Learning
(Formative & Summative)
Weekly quiz on placing the first digit, dividing by onedigit divisor, two-digit divisor and adding partial
quotients.
Unit test includes all skills on the quiz and estimate and
divide with two-digit divisors, interpret the remainder,
and problem-solving division
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other instrument
Using 100 square flats, 10 square sticks and
single square flats, Write 426 divided by 3.
Students separate flats and sticks equally into
three groups. There is one hundred flat left
over. Regroup the 1 hundred as 10 tens. (There
should be 142 in each group, therefore the
quotient is 142.
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of
students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made
as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English
Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred
seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
8
Unit 3 Overview
Content Area – Mathematics
Unit 3: Add and Subtract Decimals 6 Lessons/ Four Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale –
Students will demonstrate an understanding of decimal and whole number place value. Students will read, write, round, compare, add and
subtract decimals.
Interdisciplinary Integration – Language Arts: Journal writing, open-ended response questions;
Consumer Economics: real-world problem-solving.
Integration – SMART board, Document Camera, virtual manipulatives, i-Ready
21st Century Themes –
Global awareness, financial, economic, business and
entrepreneurial literacy
21st Century Skills –
Creativity/innovation, information literacy, critical thinking/problem solving,
communication and collaboration, life and career skills, media literacy
Learning Targets
Practice Standards:
MP.3: Construct viable arguments and critique the reasoning of others.
MP.8: Look for and express regularity in repeated reasoning.
Math Domain Standards:
5. NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right
and 1/10 of what it represents in the place to its left.
5. NBT.3: Read, write, and compare decimals to thousandths.
5.NBT.3.a: Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 +
4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
5. NBT.3.b: Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the
results of comparisons.
5. NBT.4: Use place value understanding to round decimals to any place.
Unit Essential Questions
 How can you add and subtract decimals?
Unit Enduring Understandings
 Understanding place value helps you add and subtract decimals.
 Various methods can be used to find decimal sums and differences.
9
Terminology: Sequence, term, thousandth, benchmark, estimate, hundredth, round, tenth
Goals/Objectives
Students will be able to Understand the place
value system
Perform operations with
multi-digit whole
numbers and decimals to
hundredths
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
On the board, write the addition problem
shown below. Have students find the sum by
recording the regroupings above the addends.
Have students use base-ten blocks to model the
addition problem. Provide an example with
subtraction of decimals. Use base-ten blocks to
model the subtraction problem.
Give each student 10 index cards and have
them write a decimal up to thousandths on each
card. Have partners combine their cards,
shuffle them and then divide them into two
equal piles. Each student takes a pile and keeps
the cards facedown. Partners turn over their top
card at the same time and compare the
decimals. The student with the greater decimal
keeps the cards. If two cards are of equal value,
then each partner turns over one more card and
compares. The card with the greater value takes
all four cards. Play continues until one player
has all the cards.
Evidence of Learning
(Formative & Summative)
Weekly quiz on place value of decimals, comparing and
ordering decimals, rounding decimals.
Unit test on all skills from quiz and estimating decimal
sums and differences, adding and subtracting decimals,
pattern with decimals, adding and subtracting money,
and problem solving – choose a method.
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number
of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be
made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or
English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision Animated
Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, & www.khanacadamy.com.
10
Unit 4 Overview
Content Area – Mathematics
Unit 4: Multiply Decimals 6 Lessons/ Four Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale –
Students will explain patterns in the number of zeros of a product and placement of the decimal point when multiplying or dividing by
powers of 10.
Students will multiply decimals to hundredths using various methods.
Interdisciplinary Integration – Language Arts: Journal writing, open-ended response questions;
Consumer Economics: real-world problem-solving.
Technology Integration – SMART board, Document Camera, virtual manipulative, Internet sites, accelerated math, calculator, study
island
21st Century Themes –
21st Century Skills –
Global awareness, financial, economic, business
Creativity/innovation, information literacy, critical thinking/problem solving,
and entrepreneurial literacy
communication and collaboration, life and career skills, media literacy
Learning Targets
Practice Standards:
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
Math Domain Standards:
5. NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the
placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers
of 10.
5.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place
value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain
the reasoning used.
11
Unit Essential Questions
 How can you solve decimal multiplication
problems?
Unit Enduring Understandings
 Multiplying with whole numbers is similar to multiplying with decimals.
 Pattern, models and drawings help you solve decimal multiplication problems.
Terminology: decimal, expanded form, hundredths ,multiplication, ones, patterns, place value, product, tenths, thousandths
Goals/Objectives
Students will be able to Understand the place
value system
Perform operations with
multi-digit whole
numbers and decimals to
hundredths
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
A 10 x 10 grid can be used to model 1.0, where
each square on the grid represents onehundredth. For example, 0.2 x 1.5. Two grids
are used to model this problem.
Use a blue crayon to shade 1.5 on the model.
Use a red crayon to shade 0.2 of the blue
hundredths. The result is 30 hundredths shaded
purple. So, 0.2 x 1.5 = 0.30
Present the problem 4.92 x 1,000 to students
along with the pattern shown below. Ask then
to underline and count the number of zeros for
each factor that is a power of 10. Explain to
students that the number of zeros they
underlined and the number of places the
decimal point moves from the original factor or
4.92 is the same. Repeat with different
examples.
12
Evidence of Learning
(Formative & Summative)
Weekly Quiz on multiplication patterns, multiplying
decimals and whole numbers, multiplying with expanded
form, and multiplying money.
Unit test on all of the skills from the quiz and zeros in the
product.
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other instrument
4.92 x 1 = 4.92
4.92 x 10 = 49.2
4.92 x 100 = 492
4.92 x 1,000 = 4,920
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of
students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made
as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English
Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred
seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
13
Unit 5 Overview
Content Area – Mathematics
Unit 5: Divide Decimals 4 Lessons/ Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale – Students will add, subtract, multiply and divide decimals to hundredths using various strategies. Students
will explain patterns in multiplying and dividing with powers of ten.
Interdisciplinary Integration – Language Arts: Journal writing, open-ended response questions;
Consumer Economics: real-world problem-solving.
Technology Integration – SMART board, Document Camera, virtual manipulative, i-Ready
21st Century Themes –
Global awareness, financial, economic, business and
entrepreneurial literacy
21st Century Skills –
Creativity/innovation, information literacy, critical thinking/problem solving,
communication and collaboration, life and career skills, media literacy
Learning Targets
Practice Standards:
MP.4: Model with mathematics.
MP.7: Look for and make use of structure.
Math Domain Standards:
5. NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the
placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of
10.
5.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place
value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain
the reasoning used.
5. NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of
whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent
the problem.
Unit Essential Questions
 How can you solve decimal division problems?
Unit Enduring Understandings
 Dividing with decimals is similar to dividing with whole numbers
 Patterns, models and drawings help you solve decimal division problems
Terminology: compatible numbers, decimal, decimal point, dividend, division, divisor, estimate, hundredth, tenth
14
Goals/Objectives
Students will be able to Understand the place
value system
Perform operations with
multi-digit whole
numbers and with
decimals to hundredths
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
Give each pair of Students 1 flat, and explain
that it represents one whole. Have students
exchange 1 flat for 10 longs and explain that
each long represents one tenth of a whole.
Write “one tenth = 0.1” on the board so
students recall how to use a decimal point
correctly.
Write these division problems on the board:
1/1 = 1
1/2 = 0.5 1/5 = 0.2 1/10 = 0.1
(use the division sign) Have students model
each problem by separating 10 longs into equal
groups. The number of longs in each group is
the quotient. Work with students to count the
longs in each group by tenths. For example,
students should count 2 longs as one-tenth,
two-tenths. Have volunteers come to the board
to write the solution for each problem.
Show students 1 whole by displaying 1 flat on a
piece of paper. Label it 1.0. Provide students
with 10 longs. Allow students time to line up
longs on top of the flat to determine how many
longs are equivalent to 1 whole.
 How many longs are needed to make 1
whole? (ten)
 What part of the whole does one long
represent? (one tenth)
Display one long and label it 0.1. Have a
volunteer model 0.5 using 5 longs. Have
students repeat this activity modeling
numbers from 0.1 through 2.0. Provide
assistance to individual students as needed
15
Evidence of Learning
(Formative & Summative)
Weekly Quiz on division patterns with decimals,
dividing decimals by whole numbers, and estimating
quotients.
Unit test on all of the skills from the quiz and dividing
decimals, writing zeros in the dividend, and problemsolving with decimal operations.
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of
students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made
as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English
Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred
seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
16
Unit 6 Overview
Content Area – Mathematics
Unit 6: Add and Subtract Fractions with Unlike Denominators 4 Lessons/ Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale – Students will be able to develop fluency with addition and subtraction of fractions.
Interdisciplinary Integration – Language Arts: journal writing; open-ended extended response questions; Consumer Economics: real
world problem solving
Technology Integration –SMART board, projector, virtual manipulatives, and i-Ready
21st Century Themes –
Global Awareness, Financial Economic,
Business and Entrepreneurial Literacy,
Environmental Literacy
21st Century Skills –
Critical Thinking/Problem Solving, Communication & Collaboration, Life & Career
Skills
Learning Targets
Practice Standards:
MP.2: Reason abstractly and quantitatively.
MP.4: Model with mathematics.
Math Domain Standards:
5. NF.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent
fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
5. NF.2: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike
denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of
fractions to estimate mentally and assess the reasonableness of answers.
Unit Essential Questions
 How can you add and subtract fractions with
unlike denominators?
 How can the ability to recognize the meanings
of operations and how they relate to one
another help solve real world mathematical
problems?
Unit Enduring Understandings
 Models are used to help to find sums and differences of fractions.
 The least common denominator is used when adding and subtracting fractions.
 Understanding of the number system helps to solve real world problems.
 Computational fluency requires efficient, accurate, and flexible methods for
computing.
Terminology: common denominator, common multiple, equivalent fractions, mixed number, simplest form
17
Goals/Objectives
Students will be able to Use equivalent fractions
as a strategy to add and
subtract fractions.
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
Think Pair Share Activity:
Number Line Activity – Use a number line to
model like fraction sums. Write 8/10 + 6/10 on
the board and have pair of students sketch a
number line from 0 to 2 by fifths. Have
students describe two different ways the
number line can be used to find 8/10 + 6/10.
Ask what operation is used to write equivalent
fractions for tenths to fractions in fifths.
(division) Ask what operation is used to write
equivalent fractions for fifths to fractions in
tenths. (multiplication) Show process of adding
unlike fractions using number line with:
http://www.visualfractions.com/AddUnlike/add
unlike.html
Think Pair Share Activity:
Fraction Strips Activity – Use fraction strips to
model like fraction sums and differences.
Distribute fraction strips for sixths to each pair
of students. Write 2/6 + 3/6 on the board. Have
students use fraction strips to follow along as
you add and name the sum 5/6. Explain how
paper and pencil are used to find the sum of
two fractions that have the same denominator.
(common denominator) Write 5/6 – 4/6 on the
board. Have the students use fraction strips to
follow along as you subtract and name the
difference. (1/6) Explain how you record the
difference by writing the numerators over the
common denominator. Then put ½ + ¼ and
6/10 – 2/5 on the board. Ask the students to
add or subtract the fractions and model it using
18
Evidence of Learning
(Formative & Summative)
Weekly quiz on adding and subtracting fractions
Chapter test on adding and subtracting fractions
Constructed Response Performance Task with grading
rubric
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other
instrument
their fraction bars. Ask the pairs to explain to
their classmates how they obtained their
answer. (obtain a common denominator)
Fraction Song – Print out lyrics to fractions
song for each student. Let them listen to the
song them replay the song and have the
students sing along. The tune is to Jingle Bells.
Song and lyrics located at:
http://www.harcourtschool.com/jingles/jingles_
all/35how_easy_is_that.html
Think Pair Share Activity:
Pattern Blocks – Use pattern blocks to model
subtraction of fractions. Model the two
fractions with common denominators to be
subtracted with pattern blocks. Name the
fraction for the difference. Have the students
summarize with a partner how to subtract
fractions with the same denominator. Be sure
to point out that when subtracting fractions that
have the same denominator, just subtract the
numerators and keep the same denominator.
Ask the children how they would use pattern
blocks to solve 2/3 – 1/6. After a few minutes,
have the children share their approaches.
Model both fractions with pattern blocks. Lay
the blocks representing the smaller fraction on
top of the blocks representing the larger
fraction. Explain that the part of the larger
fraction that remains uncovered is the
difference. Another method to model is the
take away method. Here, you need to model the
larger fraction with pattern blocks. Then take
away the blocks representing the smaller
fraction, trading for pattern blocks of the proper
19
size if necessary. The remaining pattern blocks
represent the difference. Have the students
summarize with a partner how to subtract
fractions with different denominators.
Continue to have students to work in partners
to solve fraction subtraction problems.
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning
centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the
number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may
also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504
plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing
strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
20
Unit 7 Overview
Content Area – Mathematics
Unit 7: Multiplying Fractions 4 Lessons/ Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale – Students will be able to develop fluency with multiplying fractions.
Interdisciplinary Integration – Language Arts: journal writing; open-ended extended response questions; Consumer Economics: real
world problem solving
Technology Integration – i-Ready Math, SMART board, projector, virtual manipulatives, and Internet
21st Century Themes –
Global Awareness, Financial Economic, Business and
Entrepreneurial Literacy, Environmental Literacy
21st Century Skills –
Critical Thinking/Problem Solving, Communication & Collaboration, Life &
Career Skills
Learning Targets
Practice Standards:
MP.3: Construct viable arguments and critique the reasoning of others.
MP.5: Use appropriate tools strategically.
Math Domain Standards:
5. NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5. NF.4.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.
5. NF.4.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.NF.5: Interpret multiplication as scaling (resizing), by:
5. NF.5.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.
5. NF.5.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.
5. NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or
equations to represent the problem.
21
Unit Essential Questions
 How do you multiply fractions?
 How can you use a model to show the
multiplication of fractions?
 How does the size of the product compare to the
size of one factor when multiplying fractions?
 How can the ability to recognize the meanings of
operations and how they relate to one another help
solve real world mathematical problems?
Unit Enduring Understandings
 Models can be used to represent the product of fractions.
 The magnitude of numbers affects the outcome of operations on them.
 Understanding of the number system helps to solve real world problems.
 Computational fluency requires efficient, accurate, and flexible methods for
computing.
Terminology: denominator, equivalent fractions, mixed number, numerator, product, simplest form
22
Goals/Objectives
Students will be able to -
Recommended Learning
Activities/Instructional Strategies
Apply and extend
previous understanding of
multiplication of whole
numbers to multiplying
fractions.
Instructional Strategies
Think Pair Share Activity: Model Concept
Demonstrate how to use an area model to show
the product of ¼ x ½ by folding and shading
paper. Explain each step in the process and
how you obtain a product of 1/8. Then have the
students model the same problem. Have them
wire the answer and share their responses with
the group. Repeat with other problems asking
each pair to answer: Into how many equal parts
will we fold the paper vertically? How many
parts will we shade to represent each factor? In
to how many equal parts will we fold the paper
horizontally? How many equal parts are there
in the whole paper now?
Play an “I Have, Who Has” game. I Have ...
Who Has? games can be created for virtually
any topic and used as both a whole class
practice or a center activity for small groups.
Create cards with fraction multiplication
problems on them. On the top of the card
write: “I have” then the answer to a fraction
multiplication problem. Below the problem,
write “Who has”. Underneath this, write out a
fraction multiplication problem. How to Play:
Shuffle the cards and distribute one card to
each student. If any cards are left over
distribute these to random students. The first
student begins with "I have ... Who has" while
the others listen for the question and try to find
the answer that on their card. The student with
the correct answer reads the “I have” portion
and begins the next round by reading the “Who
23
Evidence of Learning
(Formative & Summative)
Grade Level quiz on multiplying fractions
Grade Level chapter test on multiplying fractions
Constructed Response Performance Task with grading
rubric
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other
instrument
has” on the same card. Play continues until the
last card is read. As students develop
confidence with the game a stop watch can be
used to time a round. Record the time on the
board so that students try each game to beat
their current best time. When using the cards as
a math center activity one student deals out the
cards to all players. Players arrange the cards
face-up in front of them. Play continues as in
the class game. Whoever has the card that
answers the question reads that answer and then
reads the question on their card. Students turn
over the cards after reading them. The first
person to turn over all of his/her cards wins the
game. Cards can be shuffled and the game
repeated.
Recipe Activity – Provide a recipe for a healthy
trail mix that serves 10 people which involves
fractions. Ask the students what you should do
if you need to serve 20 people, 10 people, 5
people, etc. Students should respond with
multiply or divide the ingredients. Distribute
the recipe and ask the students to halve the
amount to make enough for 5 people by
multiplying each ingredient by 1/2. After
students answers are checked, have stations set
up so groups of 4 or 5 students can actually
create the trail mixture by measuring
ingredients using measuring cups and
measuring spoons for their group to enjoy.
24
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning
centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the
number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may
also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504
plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing
strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit., www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
25
Unit 8 Overview
Content Area – Mathematics
Unit 8: Dividing Fractions 4 Lessons/Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale – Students will be able to develop fluency with multiplying fractions.
Interdisciplinary Integration – Language Arts: journal writing; open-ended extended response questions; Consumer Economics: real
world problem solving
Technology Integration – Accelerated Math, SMART board, Document Camera, virtual manipulatives, and Internet, i-Ready
21st Century Themes –
Global Awareness, Financial Economic, Business and
Entrepreneurial Literacy, Environmental Literacy
21st Century Skills –
Critical Thinking/Problem Solving, Communication & Collaboration, Life &
Career Skills
Learning Targets
Practice Standards:
MP.2: Reason abstractly and quantitatively.
MP.4: Model with mathematics.
Math Domain Standards:
5. NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of
whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to
represent the problem.
5. NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit
fractions.
5. NF.7.a: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
5. NF.7.b: Interpret division of a whole number by a unit fraction, and compute such quotients.
5. NF.7.c: Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by
unit fractions, e.g., by using visual fraction models and equations to represent the problem.
Unit Essential Questions
 How do you divide fractions?
 How can you use a model to show the division of
fractions?
 How does the size of the quotient compare to the
size of the original number?
Unit Enduring Understandings
 Models can be used to represent the quotient of fractions.
 The magnitude of numbers affects the outcome of operations on them.
 Understanding of the number system helps to solve real world problems.
 Computational fluency requires efficient, accurate, and flexible methods for
computing.
26
 How can the ability to recognize the meanings of
operations and how they relate to one another help
solve real world mathematical problems?
Terminology: dividend, equation, fraction, quotient, whole number
27
Goals/Objectives
Students will be able to Apply and extend
previous understanding of
multiplication and
division of whole number
to dividing fractions.
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
Fraction Dominos - To make the game: divide
28 index cards in half by drawing a line down
the center of each card. Write fractions, mixed
numbers, and whole numbers on the cards. For
example, one card might show 1/2 on one side
and 1 1/3 on the other side. Be sure to include
matches. If you make a card that says 1/3 and
4/5, you should also make cards that say 1/3
and 6/9 or 4/5 and 8/12. You may also wish to
make doubles (1/3 and 1/3 on a card). Follow
the general directions for playing dominos with
the exception that players will divide the
numbers on the domino they place on the table.
For example, if your turn starts with a domino
that says 1/3 and 1/3 on the table, you may lay
down a domino with 1/3 and 1. However,
before you can play, you will divide 1/3 by 1. If
you get the correct answer, you may lay down
the domino.
Picture This Activity - Copy one expression
from below onto each index card. Shuffle the
cards and lay them face down. Turn over the
first card. All group members then use the grid
paper to draw a picture for the division
problem. Draw either a Geo-board picture of
the problem or try to find a unique way to draw
the quotient. When you have drawn a picture of
the problem, find each quotient. When all
group members are done, compare drawings
and answers. Discuss which drawing was the
most unique and which drawing was the easiest
to understand. Play several rounds or play until
all the cards have been used. Share with the
entire class some of your group's unique ideas.
Expressions to be included on index cards: 1/2
28
Evidence of Learning
(Formative & Summative)
Weekly quiz on dividing fractions
Unit chapter test on dividing fractions
Constructed Response Performance Task with grading
rubric
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other
instrument
÷ 1/6, 3/4 ÷ 1/12, 2 ÷ 1/3, 5/6 ÷ 1/12, 1/2
÷1/10, 1 ÷ 1/8, 2/3 ÷ 1/9, 3/8 ÷ 1/16, 5/7 ÷
1/14, 4/5 ÷ 1/15
Diverse Learners (ELL, Special Ed, Gifted & Talented)-Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number
of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be
made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or
English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
29
Unit 9 Overview
Content Area – Mathematics
Unit 9: Algebra: Patterns and Graphing 4 Lessons/Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale – Students will understand how to describe, extend, and create a wide variety of patterns and functional
relationships. Students will understand how to collect, organize, and display relevant data to answer questions.
Interdisciplinary Integration – Language Arts: journal writing; open-ended extended response questions; Consumer Economics: real
world problem solving
Technology Integration – i-Ready, SMART board, projector, virtual manipulatives, and Internet
21st Century Themes –
Global Awareness, Financial Economic, Business and
Entrepreneurial Literacy, Environmental Literacy
21st Century Skills –
Critical Thinking/Problem Solving, Communication & Collaboration, Life &
Career Skills
Learning Targets
Practice Standards:
MP.4: Model with mathematics.
MP.8: Look for and express regularity in repeated reasoning.
Math Domain Standards:
5. OA.3: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form
ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.
5. MD.2: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this
grade to solve problems involving information presented in line plots.
5. G.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).
5. G.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.
30
Unit Essential Questions
 Why is it important to look for a pattern?
 What can we learn from looking at
patterns?
 What are data sets, how are they graphed,
and what can we learn from them?
Unit Enduring Understandings
 Patterns and relationships can be used to describe and quantify physical relationships.
 Patterns and relationships can be represented graphically, numerically, symbolically or
verbally.
 Data analysis is formulating questions that can be addressed, explored, and synthesized
with relevant information.
Terminology: interval, line graph, ordered pair, origin, scale, x-axis, x-coordinate, y-axis, y-coordinate
31
Goals/Objectives
Students will be able to Analyze patterns and
relationships.
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
Treasures Activity – Give each student in a pair
a copy of a quadrant I grid. Have one student
hid their grid from their partner while he or she
colors two consecutive squares on their own
grid. Have students color two more pairs of
squares. The colored areas are “treasures.”
The other student leaves his or her sheet blank
until the game begins. Once the treasurers are
hidden, the game can begin. The student that
did not color his or her grid calls out ordered
pairs. If the ordered pair lies on the border of a
colored “treasure” square, the partner should
say, “You found a treasure.” That student then
plots that point on his or her grid. When a
player finds all the points forming the treasure,
that player gets the treasure. The winner of the
game is the first student to get all three of his or
her partner’s treasures.
Student Height Data Activity – Prior to using
this activity, measure each student’s height in
inches using mixed numbers. Compile the data
and hand it out to the students. Ask the
students to represent the data on a line plot.
(You may need to provide assistance setting up
the increments.) Ask questions along the way
such as: How can a line plot help you organize
this information? How many Xs should there
be in your line plot? For which amount does
your line plot show the greatest number of
occurrences? How would you use this
information to find the mean or average or the
data?
32
Evidence of Learning
(Formative & Summative)
Weekly quiz on line plots, ordered pairs, and graphing
data
Unit chapter test on including all skills on the quiz and
line graphs, numerical patterns, as well as analyzing
relationships
Constructed Response Performance Task with grading
rubric
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number
of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be
made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or
English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
33
Unit 10 Overview
Content Area – Mathematics
Unit 10: Convert Units of Measure 4 Lessons/Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale – Students will convert among different measurement units within a given measurement system.
Interdisciplinary Integration – Language Arts: journal writing; open-ended extended response questions; Consumer Economics: real
world problem solving
Technology Integration – i-Ready, SMART board, Document Camera, and virtual manipulatives.
21st Century Themes –
Global Awareness, Financial Economic, Business and
Entrepreneurial Literacy, Environmental Literacy
21st Century Skills –
Critical Thinking/Problem Solving, Communication & Collaboration, Life &
Career Skills
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.7: Look for and make use of structure.
Math Domain Standards:
5. MD.1: Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m),
and use these conversions in solving multi-step, real world problems.
Unit Essential Questions
 What strategies can you use to compare and covert
measurements?
Unit Enduring Understandings
 Deciding whether to multiply or divide is the first step to converting
measurements.
 Organize your solution when you are solving a multistep measurement problem.
 Converting metric measurements is different from converting customary
measurements.
Terminology: capacity, decimeter, decameter, milligram, milliliter, millimeter
34
Goals/Objectives
Students will be able to Convert like
measurement units
within a given
measurement system.
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Materials: 2 cup, 2 quart, and 2 gallon
containers, rice or beans
 Point to the cup and the quart
containers. Do you think it will take
more cups or quarts to fill the gallon?
(cups) Have volunteers pour cups and
cup of beans or rice into two different
gallon containers. Why did it take more
cups? (Cups are smaller units that
quarts.)
 Point to the cup and gallon containers.
If you convert cups to gallons, will you
need more or fewer gallons? (fewer)
Will you multiply or divide? (divide)
 Repeat the activity for converting cups
to quarts, quarts to cups, gallons to
cups, and so on.
Have students make a table of ordered pairs
like the one shown below for two units of
measure. Students line graph the
relationship between the two units and draw
a line though the points.
cups 1
2
4
6
8
10
fl.
oz
16
32
48
64
80
8
Other units that can be tabled and graphed
are cups to pints, quarts to gallons, fluid ounces
to cups, quarts to cups, gallons to pints and
fluid ounces to pints.
35
Weekly quiz on customary length, customary capacity,
weight and multistep word problems.
Unit test on all skills on the quiz and metric measures,
problem solving with customary and metric
conversions and elapsed time.
Alternative or project-based assessments will be evaluated
using a teacher-selected or created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number
of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be
made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or
English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
36
Unit 11 Overview
Content Area – Mathematics
Unit 11: Geometry and Volume 4 Lessons/Three Weeks
Target Course/Grade Level – Grade 5 Math
Unit Summary and Rationale – Students will classify polygons. Students will measure volume by counting cubes and relate volume
with operations of multiplication and addition.
Interdisciplinary Integration – Language Arts: journal writing; open-ended extended response questions; Consumer Economics: real
world problem solving
Technology Integration – i-Ready, SMART board, Document Camera, virtual manipulatives, and I-Ready
21st Century Themes –
Global Awareness, Financial Economic, Business and
Entrepreneurial Literacy, Environmental Literacy
21st Century Skills –
Critical Thinking/Problem Solving, Communication & Collaboration, Life &
Career Skills
Learning Targets
Practice Standards:
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
Math Domain Standards:
5. MD.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5. MD.3.a: A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure
volume.
5. MD.3.b: A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5. MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5. MD.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5. MD.5.a: Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the
volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.
Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
5.MD.5.b: Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and mathematical problems.
5. MD.5.c: Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by
adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
5. G.3: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
5. G.4: Classify two-dimensional figures in a hierarchy based on properties.
37
Unit Essential Questions
Unit Enduring Understandings
 How do unit cubes help you build solid figures and  Identifying, describing and classifying three dimensional figures will help you
understand the volume of a rectangular prism?
differentiate between them.
 Identifying the measure of the dimensions of a rectangular prism will help you
find its volume.
Terminology: base, congruent, lateral face, polygon, polyhedron, prism, regular polygon, unit cube, volume
38
Goals/Objectives
Students will be able to Understand concepts
of volume and relate
volume to
multiplication and to
addition.
Classify twodimensional figures
into categories based
on their properties.
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Materials: centimeter cubes
Weekly Quiz on polygons, triangles, quadrilaterals,
problem solving properties of two dimensional figures,
Review volume. Ask students to explain the
and three dimensional figures.
term and how it is measured.
Can a triangle have volume? (no it is not a solid
figure) Review the fact that a rectangular prism
Unit test on all skills from quiz and unit cubes and
is a solid figure with six rectangular faces.
solid figures, understanding volume, estimating
volume, volume of rectangular prisms, applying
Name some real-world examples of rectangular
volume formulas, and finding volume of composed
prisms. (cereal boxes, tissue box number cubes)
figures.
Give each pair of student’s 36 centimeter
cubes. Have students build rectangular prisms
each with a volume of 36 cubic units. Ask
Alternative or project-based assessments will be evaluated
students to record the length, width and height
using a teacher-selected or created rubric or other
of each prism. After pairs have built several
instrument
prisms, make a table showing the lengths,
widths and heights of the prisms.
What is the product of the length, width and
height of each prism? 36
Help students see that the product of the length,
width and height of each prism is equal to the
number of cubes needed to build the prism.
Provide students with nets for various prisms
and pyramids, without the names of the figures.
Have pairs predict the shape each net will make
based on the number of faces and the shape of
the bases.
Then have each pair assemble one net and
discuss the final shape. Ask students to
compare their predictions to the final figure and
identify the polyhedron.
39
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers
and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number
of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be
made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or
English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Envision Math Teacher Edition, EnVision Student Practice Editions, EnVison Math Enrichment Edition, EnVision
Animated Math Models, EnVision Math Center and Manipulative Kit.,www.brainpop.com, www.coolmath.com, &
www.khanacadamy.com.
40
Mathematics: Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop
in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics
education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation,
and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It
Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and
relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive
disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s
own efficacy).
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.
2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two
complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given
situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily
attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand;
considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using
different properties of operations and objects.
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3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing
arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able
to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions,
communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments
that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness
of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—
explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.
Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students
learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide
whether they make sense, and ask useful questions to clarify or improve the arguments.
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.
5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil
and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic
geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,
mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They
detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they
know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions
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with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such
as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and
deepen their understanding of concepts.
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.
7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three
and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the
shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive
property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an
existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back
for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being
composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize
that its value cannot be more than 5 for any real numbers x and y.
8 Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper
elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude
they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line
through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way
terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for
the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process,
while attending to the details. They continually evaluate the reasonableness of their intermediate results.
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Common Core Standards
Mathematics | Grade 5
In Grade 5, instructional time should focus on three critical areas: (1) 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); (2) extending division to 2-digit divisors, integrating decimal fractions
into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with
whole number and decimal operations; and (3) developing understanding of volume.
(1) 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.)
(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of
operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings
of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop
fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and
fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate
power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make
sense. They compute products and quotients of decimals to hundredths efficiently and accurately.
(3) 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 1unit 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.
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Grade 5 Overview
Operations and
Algebraic
Thinking
 Write and interpret numerical
expressions.
 Analyze patterns and relationships.

Number and
Operations in
Base Ten


Number and
Operations—
Fractions


Measurement
and Data


Geometry


Understand the place value system.
Perform operations with multi-digit
whole numbers and with decimals
to hundredths.
1.
2.
3.
Use equivalent fractions as a
strategy to add and subtract
fractions.
Apply and extend previous
understandings of multiplication
and division to multiply and divide
fractions.
4.
Convert like measurement units
within a given measurement
system.
Represent and interpret data.
Geometric measurement:
Understand concepts of volume and
relate volume to multiplication and
to addition.
8.
Graph points on the coordinate
plane to solve real world and
mathematical problems.
Classify two-dimensional figures
into categories based on their
properties.
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5.
6.
7.
Make sense of
problems and persevere
in solving them.
Reason abstractly and
quantitatively.
Construct viable
arguments and critique
the reasoning 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.
Mathematic
al Practices
Operations and Algebraic Thinking 5.OA
Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
For example, express the calculation “add 8 and 7, then multiply by 2” as 2  (8 + 7). Recognize that 3  (18932 + 921) is three
times as large as 18932 + 921, without having to calculate the indicated sum or product.
Analyze patterns and relationships.
3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form
ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For
example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate
terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other
sequence. Explain informally why this is so.
Number and Operations in Base Ten 5.NBT
Understand the place value system.
1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right
and 1/10 of what it represents in the place to its left.
2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the
placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote
powers of 10.
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Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g.,
347.392 = 3  100 + 4  10 + 7  1 + 3  (1/10) + 9  (1/100) + 2  (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the
results of comparisons.
4. Use place value understanding to round decimals to any place.
3.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluently multiply multi-digit whole numbers using the standard algorithm.
6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on
place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the
calculation by using equations, rectangular arrays, and/or area models.
7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place
value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method
and explain the reasoning used.
Number and Operations—Fractions 5.NF
Use equivalent fractions as a strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent
fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4
= 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
2.
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike
denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number
sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5
+ 1/2 = 3/7, by observing that 3/7 < 1/2.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3. Interpret a fraction as division of the numerator by the denominator (a/b = a  b). Solve word problems involving division of
whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to
represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and
that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound
sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your
answer lie?
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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.
6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or
equations to represent the problem.
7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit
fractions.1
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story
context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and
division to explain that (1/3) ÷ 4 = 1/12 because (1/12)  4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for
4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to
explain that 4 ÷ (1/5) = 20 because 20  (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by
unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate
will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
1
Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a
requirement at this grade.
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Measurement and Data 5.MD
Convert like measurement units within a given measurement system.
1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m),
and use these conversions in solving multi-step, real world problems.
Represent and interpret data.
2.
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this
grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in
identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed
equally.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure
volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving
volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the
volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of
the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
b. Apply the formulas V = l  w  h and V = b  h for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by
adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
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Geometry 5.G
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.
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.
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