WCSD Mathematics Curriculum Guide
Grade 4 Unit 9: Fractions and Decimals
Big Conceptual Idea: Number and Operations, Fractions, Measurement and Data
Read all unit 9 lessons, unit content highlights and applicable Teacher’s Reference Manual excerpts prior to unit instruction.
Instructional notes and support:
This unit focuses on exploring fractions and decimals. Assess meeting standards to these benchmarks.
Do not assess or teach percentages in this grade level. Multiplication of fractions in this unit meets
standard. Have students explore Lesson 9.9 division of decimals to help relate the operations of
multiplication and division yet do not assess these ideas. Division of fractions should not be assessed or a
major focus at this time.
Helpful Hints and Tips about using Everyday Mathematics
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All students at all ability levels should fully participate in the math block.
At the beginning of each lesson is an overview. It is called At-a-glance. This page shows the
whole lesson divided into three parts and highlights materials, vocabulary, assessment, and
more.
o Part 1 is Teaching the Lesson. It provides main instructional activities for the lesson.
o Part 2 is Ongoing Learning and Practice. It supports previously introduced concepts and
skills; essential for maintaining skills.
o Part 3 is Differentiation Options. It includes options for supporting the needs of ALL
students; usually an extension of Part 1.
The purple boxes throughout the book provide notes on how to adjust the activity to provide support for ELL students.
Part 3 of each lesson provides differentiation options, some include additional suggestions for ELL support.
o IEP students may benefit from these adjusted activities as well.
o Teachers/support staff may find activities in Part 3 helpful when ‘pushing-into’ classrooms to scaffold and support
student learning.
Gold stars with the red check indicate daily assessment, ongoing assessment, and unit assessment.
o Ongoing assessment: Recognizing Student Achievement (RSA) is included in every lesson. These are noted in the
teacher’s manual as pink stars on the answer page of the Student Math Journal page.
o Ongoing assessment: Informing Instruction is included in many lessons to help guide instruction.
o Unit assessments provide several types of periodic assessment including online assessment tools, progress check,
written assessment, and open response.
Everyday Mathematics includes valuable games that should be used to reinforce and enhance mathematical concepts and
skills learned throughout the units. All students need to be afforded time to play the games, especially those who require
the most practice. All rules for the games can be found in the Student Reference Book in the blue games section or online
on eSuite.
o Suggestions for building games into your day:
 Include games as part of the daily morning routine.
 Devote the first or last 10 minutes of math block to playing games from the current unit.
 Designate one math block per week as a game day. Set up stations that feature the unit games. Ask
parent volunteers to assist in the rotation of students through these stations.
 Set up a Games Corner that features some of the students’ favorite games. Encourage students to visit
this corner during free time. Change the games frequently to maintain student interest.
Advanced Preparation sections can be found at the beginning of each lesson on the bottom of the At-a-glance page in the
Lesson Organizer.
Interactive whiteboard-ready ePresentations are available at www.everydaymathonline.com to help you teach the lesson
and are noted in the Teacher’s Manual throughout the lessons.
Helpful Hints and Tips for Navigating Through the Planning Units:
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Bolded standards in the left column indicate that those standards are the focus of the lesson.
Not bolded standards in the left column indicate that those standards are being reviewed/practiced in the lesson.
Planning Units to the Common Core Curriculum Resources
All inquiries should be made to dtrakas@washoeschools.net. (2014 4th u9)
WCSD Mathematics Curriculum Guide
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Bolded Mathematical Practices in the left column indicate that they are the focus Practices when referring to the Guided
Questions in the middle column.
Not bolded Mathematical Practices in the left column indicate that they may be used during the Guided Questions and the
Discussion Questions in the middle column.
Bolded questions in the middle column are referred to as the Guided Questions.
Not bolded questions in the middle column are referred to as the Discussion Questions.
Enduring Understandings:
By the end of this unit students are secure in:
 Expressing a fraction with a denominator 10 as an equivalent fraction with a denominator of 100 (4.NF.5)
 Using decimal notation for fractions with denominators of 10 or 100 (4.NF.6)
 Evaluating the magnitude of fractions/decimals and place these relative to benchmarks using diagrams such as
number line diagrams including those that may feature a measurement scale. (4.NF.1)
Students are exploring a developing understanding in:
 Using the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses
of objects, and money, including problems involving simple fractions or decimals, and problems that require
expressing measurements given in a larger unit in terms of a smaller unit (4.MD.2).
Some generalized misconceptions or partial understandings that teachers may observe during instruction include:
Students treat decimals as whole numbers when making comparison of two decimals. They
4.NF.5, 6 & 7
think the longer the number, the greater the value. For example, they think that .03 is
greater than 0.3. Furthermore, students do not understand how the places in decimal
notation have the same correspondence (places to the left are 10 times greater than the
places to their immediate right) as the places in whole numbers (Georgia Department of Ed., Unit 5, p.
6).
Essential Academic Vocabulary:
Review Vocabulary:
*Note: Use these words during instruction to begin to develop definitions.
(Vocabulary from prior grades or units)
decimal
fraction
tenths
hundredths
Extended vocabulary (that may be used in the course of instruction): terminating decimal, repeating decimal (should not be an
instructional focus yet may come up during discussions on what students know about decimals).
*Note: Use these words during instruction to begin to develop definitions.
NVACS
(Content
and
Practices)
Guiding and Possible Discussion Questions
Instructional Approaches &
Content Clarifications
Lesson 9.1: Fractions, Decimals, and Percents – Focus on Fractions and Decimals
4.NF.1
4.NF.6
MP1, MP2,
MP3, MP4,
MP6
Why is it important to find many names for
numbers?
Math Message Follow-Up
What are different ways to explain percent
situations?
This lesson support 4.NF.6, consider focusing instruction on fractions and
decimals. Throughout the lesson connect that 100 % is naming the whole.
Many students encounter percents in summative grading systems. This may be
a natural context for some students to better understand their own
achievement; in addition to, having another frame for understanding rational
numbers.
What does the 100% box represent?
Mental math and reflexes for converting fractions into equivalent fractions
Planning Units to the Common Core Curriculum Resources
All inquiries should be made to dtrakas@washoeschools.net. (2014 4th u9)
WCSD Mathematics Curriculum Guide
Finding Equivalent Names for Percents
How does the grid help you determine the
fraction and decimal name? Focus on the
fractions out of 100 as percent is not a 4th grade
standard.
and considering omitting the percents and using this as part of your formative
assessment process.
Math Message Omit
Part 1 (consider using Math Journal p 248, 249, 250) Share student ideas from
the math message. Focus is using 10-by-10 square grids to represent percents.
Part 2 Students play Fraction Match with a partner. Directions are in Student
Reference book p 243. Fraction cards are in Math Masters p 473-476.
Consider using Math Masters p 278 to as a quick assessment of students’
ability to find equivalent fractions.
Part 3 Consider using Finding 50% of a Square, yet focus on part-to-whole
instead of fraction language.
Decimal Designs
What is a decimal fraction and how can it be
represented?
Part 1:
4.NF.5
4.NF.6
MP1, MP2,
MP4, MP5,
MP6, MP7
● How many squares are shaded of x?
● How many squares total are in the figure?
● What decimal fraction represents the shaded part?
How do you know?
● How would you read the decimal fraction (or
decimal) you have written?
Part 2:
● How many squares are shaded of x?
● How many squares total are in the figure?
● What decimal fraction represents the shaded part?
How do you know?
● How would you read the decimal fraction (or
decimal) you have written?
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_4_Unit5Framework.pdf
Consider including this lesson to connect learning from Units 4 & 7, as well as,
access background knowledge before moving deeper.
Part 1
Consider changing language “1 out of 10” to “1 of 10”. The language “out of”
tends to reinforce a common misconception that the numerator and
denominator of fractions are two separate whole numbers.
Part 2
Consider changing language “1 out of 10” to “1 of 10”. The language “out of”
tends to reinforce a common misconception that the numerator and
denominator of fractions are two separate whole numbers.
Lesson 9.2: Converting “Easy” Fractions to Decimals and Percents - Focus on Fractions and Decimals
4.NF.1
4.NF.5
4.NF.6
4.MD.3
MP1, MP2,
MP4, MP6,
MP7
How can using equivalent names help you to
solve problems?
Focus on the fractions and decimals for this lesson. Allow students to practice
the language found in the fraction and decimals.
Math Message Follow-Up
How might you use the decimal to figure out the
number of problems each student missed? How
might you use the fraction? The grid?
Mental math and reflexes for converting fractions into equivalent fractions,
omit percents
Math Message (Math Journal p 252) Complete problem 1
Part 1 (consider using Math Journal p 252, 253) Share student ideas from the
math message. Focus is renaming fractions to decimals and finding fractional
amounts. As a class, discuss student strategies/solutions to problem 1.
Connect these ideas to the 10-by-10 grid as scripted in the Teacher’s Lesson
Guide p 729. Have students work with a partner to complete Math Journal p
252, consider omitting percent correlation.
Part 2 Students play Rugs and Fences with a partner. Directions are in Student
Reference book p 260 and 261. Game cards and data sheet are in Math
Masters p 498-502.
Part 3 Consider using for differentiation options.
Competing the Table of Equivalent Names for
Fractions
What patterns can you find in the equivalencies
table?
How could this pattern help you find more
equivalent decimals? Fractions?
Expanding Decimals and Money
4.NF.7
MP1, MP2,
MP4, MP5,
MP6, MP7,
MP 8
When can tenths and hundredths be used
interchangeably?
When you compare two decimals, how can
you determine which one has the greater
value?
file:///C:/Users/sroggensack/Desktop/CCGPS_Math_4_Unit5Framework.pdf
Consider including this lesson to connect fraction and decimals to money.
How do the dimes represent decimal fractions? The
pennies?
Planning Units to the Common Core Curriculum Resources
All inquiries should be made to dtrakas@washoeschools.net. (2014 4th u9)
WCSD Mathematics Curriculum Guide
How does a money model help you represent tenths
and hundredths?
What strategies did you use to add tenths and
hundredths?
Pages from omitted lessons you may want to consider using:
Lesson 9.4: Consider using Math Master p. 286 (change percents to decimals)
Math Boxes on p 259 (omit question 2)
Lesson 9.5: Consider using Math Journal p. 342-343, limit to naming equivalent
fractions
Math Boxes p. 260 (omit percents only)
Lesson 9.6: Maintain Math Journal p. 261A & 261B for preparation for Smarter
Balance
Math Boxes p. 262 (modify problem 2 to name a fraction instead of a percent;
omit question 5 & 6)
Lesson 9.7: Consider using Math Boxes p. 266 (modify problem 1 to fractions
instead of percents; omit question 5)
Lesson 9.8: Multiplication of Decimals
Consider maintaining to support 4.MD.2
What do you need to know about place value
to estimate products of decimals?
4.NBT.5
4.MD.2
MP1, MP2,
MP4, MP5,
MP6, MP7
Estimating Products of Decimals
How might you use the number model to
estimate where the decimal point belongs?
Multiplying Decimals
What strategies can you use for estimating the
product?
Why is it important to estimate before solving a
problem?
Mental math and reflexes for estimating products and writing a number model
to match
Math Message Math Journal p 268, solve math message problem
Part 1 (consider using Math Journal p 268 and 269) Share student solution
strategies from the math message. Focus is to find the product of decimals by
multiplying and estimating. As a class, practice estimating products, record
responses, and discuss the estimates. Use script section from the Teacher’s
Lesson Guide p 764. Have students work with a partner to complete the rest of
Journal p 268 and 269. Students may like to use computation grids from Math
Masters p 404 and/or 434.
Part 2 Students play Over and Up Squares with a partner. Directions are in
Student Reference book p 257. Gameboard/Record sheet can be found in
Math Masters p 494.
Part 3 Consider using for differentiation options.
Use Study Link in class to support student reasoning with magnitude of number
(MP 1, 2, 3, 6)
Modification/refocus Recommended
Consider supporting students by having them use diagrams including the
number line to support 4.MD.2.
Consider modifying problems to “simple fractions or decimals” as stated in
4.MD.2.
Focus on 4.MD.2 applications and evaluating numbers that may include a
decimal. Focus operations on whole numbers and have children use a
calculator to check the reasonableness of their whole number estimates.
Lesson 9.9: Division of Decimals
4.OA.3
4.NBT.6
4.MD.2
4.G.1
4.G.2
MP1, MP2,
How is this similar to placing the decimal when
multiplying decimals?
Math Message Follow-Up
Think of a number story that could be solved by
dividing 4.2 by 7.
Mental math and reflexes for estimating quotients and writing a number
model to match
Math Message Creating a number story that could be solved by dividing 4.2 by
7
Part 1 (consider using Math Journal p 270 and 271) Share student stories and
solutions from the math message. Focus is to use estimation to place decimal
points in the answers involving division of fractions. As a class, practice looking
Planning Units to the Common Core Curriculum Resources
All inquiries should be made to dtrakas@washoeschools.net. (2014 4th u9)
WCSD Mathematics Curriculum Guide
MP3, MP4,
MP6, MP7,
MP8
Which of these problems could you imagine
solving in real life?
Estimating Quotients of Decimals
How are you using estimation to accurately
place the decimal?
at quotients and deciding where the decimal point should go by estimating the
quotient, record responses, and discussing the reasoning. Use script section
from the Teacher’s Lesson Guide p 770. Have students work with a partner to
complete the rest of Journal p 270 and 271. Students may like to use
computation grids from Math Masters p 404.
Part 2 Students play Polygon Pair-Up with a partner. Directions are in Student
Reference book p 258. Game cards can be found in Math Masters p 496 and
497.
Part 3 Consider using for differentiation options.
Modification/refocus Recommended
Consider supporting students by having them use diagrams including the
number line to support 4.MD.2.
Consider modifying problems to “simple fractions or decimals” as stated in
4.MD.2.
Focus on 4.MD.2 applications and evaluating numbers that may include a
decimal. Focus operations on whole numbers and have children use a
calculator to check the reasonableness of their whole number estimates.
Taxi Trouble
When can tenths and hundredths be used
interchangeably?
When you compare two decimals, how can
you determine which one has the greater
4.NF.5
4.NF.6
4.NF.7
4.MD.2
Consider including this lesson before the assessment to have students applying
all the mathematics they’ve worked on in this unit. Also, consider groupings in
partners or larger groups based on student needs.
value?
MP1, MP2,
MP4, MP5,
MP6, MP7
file:///C:/Users/sroggensack/Desktop/CCGPS_Math_4_Unit5Framework.pdf
How do the dimes represent decimal fractions? The
pennies?
How does a money model help you represent tenths
and hundredths?
What strategies did you use to add tenths and
hundredths?
Lesson 9.10: Unit 9 Review and Assessment
Progress
Check
9
Add assessment data and child-watching
observations to the anecdotal data you have
collected during this unit.
Depending what lessons were taught and the concepts that were taught in
each lesson, it is recommended that the assessment be modified to match the
students’ learning.
What are you noticing about each student’s
learning profile?
eSuite 2012 offers comprehensive assessment materials. Consider using.
Unit assessment may not need modifications.
How will you record this and communicate this
with other teachers, the student’s family, or as a
mathematical trajectory of learning?
Consider adding a rich mathematical task or open response item to the
assessment or within the Unit.
Everyday Math provides several opportunities for this throughout the unit.
Consider having the grade level teacher teams look at student work samples
together and discuss the represented student understanding or
misconceptions.
All lessons available on eSuite 2012- www.everydaymathonline.com.
Connections, Resources, Support:
Lesson
9.1
Extension activities:
Popcorn math: (Consider omitting percents or using as an area of differentiation.)
Planning Units to the Common Core Curriculum Resources
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WCSD Mathematics Curriculum Guide
http://www.learnnc.org/lp/pages/3762
Fraction models:
http://illuminations.nctm.org/Activity.aspx?id=3519
Connections:
Using a meter stick for fractions and decimals:
Lesson
9.2
“The term percent is simply another name for hundredths and, as such, is a standardized ratio with a denominator of
100. Physical models provide the main link among fractions, decimals and percents. Limit the percents to familiar
fractions (halves, thirds, fourths, fifths, eighths) or easy percents (1/10, 1/100) and use numbers compatible with these
fractions. The focus of these exercises is the relationship involved” (Van de Walle, 2013, p.274-277).
Lesson
9.3
Fraction and percent tool:
http://illuminations.nctm.org/Activity.aspx?id=4147
http://illuminations.nctm.org/Activity.aspx?id=3519
Lesson
9.4
Lesson
9.8
When computing with decimals, it is important to ask the students to estimate the answers. Prior to computing ask the students
to make estimates by rounding to nice whole numbers. Estimating helps students focus on the meaning of the numbers and
operations and not on counting decimal places. Students will need a solid understanding of decimal place value prior to
computing with decimals.
When multiplying decimals, connect their understanding to fractions. So when multiplying .3 x .4, you are multiplying 3/10 x
4/10 which equals 12/100. This helps students make sense that .3 x .4 = .12. Students need to understand that when multiplying
two numbers less than one, the product may be smaller than both the factors. We can think of this type of multiplication as
finding part of, this helps them make sense that the product will be smaller. Often times students try to apply a
misunderstanding that when you multiply, the answer "gets bigger". This is true in whole numbers but does not apply to
fractions and decimals. We apply this same thinking to division of decimals. When dividing two decimals less than one (such as
0.5 ÷ 0.1) the quotient will be greater than either decimal. So the quotient will be 5. Students may be confused by this if they are
under the misconception that when you divide the quotient will be smaller.
Videos:
https://learnzillion.com/lessonsets/229-multiply-and-divide-by-decimals-to-the-hundredths
Planning Units to the Common Core Curriculum Resources
All inquiries should be made to dtrakas@washoeschools.net. (2014 4th u9)
WCSD Mathematics Curriculum Guide
Lesson
9.9
Illustrative Division Context problems – why do we use decimals:
http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/000/000/292/original/illustrative_mathematics
_292.pdf?1420666161
How can you use 45 ÷ 3 = 15 to help you with 4.5 ÷ 3?
Lesson
9.10
Student Outcomes/Evidence based practice:
See the Everyday Math Assessment Handbook Unit 5 for Recognizing Student Achievement indicators (RSA), writing and reasoning
prompts, and formative assessment processes (called Informing Instruction in EM). Consider not using the open response question
for this unit.
The CCSS defines procedural fluency as skill in carrying out procedures flexibly, accurately, efficiently and appropriately (CCSS, p. 6).
Please see Washoe County School District Curriculum & Instruction webpage for fact strategy information.
Common Core State Standards Progression Documents:
K-6 Numbers and Operations Base Ten.
K-6 Numbers and Operations Fractions (pp. 5-9)
Other resources:
Van de Walle, J., Bay-Williams, B., Lovin, L., Karp, K. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate
Instruction for Grades 3-5. Pearson.
Chapter 17: Developing Concepts of Decimals and Percents
Planning Units to the Common Core Curriculum Resources
All inquiries should be made to dtrakas@washoeschools.net. (2014 4th u9)
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Constructing Task: Decimal Designs
TASK CONTENT: Representing decimals and finding equivalent fractions
tenths and hundredths
between
STANDARDS FOR MATHEMATICAL CONTENT
MCC4.NF.5 Express a fraction with denominator 10 as an equivalent fraction
with
denominator 100, and use this technique to add two fractions with respective
denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/1001.
MCC4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as
62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
STANDARDS FOR MATHEMATICAL PRACTICE TO BE EMPHASIZED
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
BACKGROUND KNOWLEDGE
While students will have previous experiences expressing fractions with denominators of 10 or 100 as
fractions, this will be their first experiences with using decimal notation and investigation into decimal
fractions. Students’ understanding of decimal numbers develops in grades 4-5 as follows.
4th Grade – Focus on the relationship between decimal fractions and decimal numbers and investigate
the relationship between decimal fractions and decimal numbers, limit to tenths and
hundredths, order decimals to hundredths, add decimal fractions with denominators of 10
and 100 (respectively)
th
5 Grade – Compare decimals up to thousandths, use decimals in operations
ESSENTIAL QUESTIONS
● What is a decimal fraction and how can it be represented?
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 32 of 127
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
MATERIALS
●
●
●
●
●
●
“Decimal Designs: Part 1” student recording sheet
“Decimal Designs: Part 1, Table, Page 1” student recording sheet
“Decimal Designs: Part 1, Table, Page 2” student recording sheet
“Decimal Designs: Part 2” student recording sheet
“Decimal Designs: Part 2, Table” student recording sheet
Crayons or colored pencils
GROUPING
Individual/Partner Task
NUMBER TALKS
Continue utilizing the different strategies in number talks and revisiting them based on the needs of your
students. Catherine Fosnot has developed problem “strings” which may be included in number talks to further
develop mental math skills. See Mini-lessons for Operations with Fractions, Decimals, and Percents by Kara
Louise Imm, Catherine Twomey Fosnot and Willem Uittenbogaard. (Mini-lessons for Operation with Fraction,
Decimal, and Percent, 2007, Kara Louise Imm, Catherine Twomey Fosnot and Willem Uittenbogaard)
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students will work with occurrences out of 10 and 100, translating them into decimal fractions and
then decimals. Students will also explore and investigate the relationship between tenths and hundredths in a
visual model and in decimal notation. Students will also begin to rename tenths using hundredths.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 33 of 127
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Comments
This lesson could be introduced by sharing shaded 10-frames and 100 grids to represent a decimal fraction or
decimal. For example, share with students some of the designs below.
Discuss strategies students could use to count the number of shaded squares. Did they use multiplication? (e.g.
Did they count the number of shaded squares in one part and multiply that number by the number of identical
parts in the design? Did they count the number of unshaded squares and subtract from 100?) Once students have
determined the decimal fraction and fraction for their favorite design ask students to share their thinking.
Finding the number of shaded squares is one way to give students an opportunity to think about pairs that make
100. As students make their decimal designs on the 10 x 10 grid, ask them if they have more shaded or
unshaded. If they have more shaded, ask them to count the number of squares that are UNSHADED and
subtract that number from 100 (i.e. think about what number added to the number of unshaded squares would
equal 100). This is a great opportunity to review numbers that add up to 100 and for students to explain how
they know how many squares are shaded.
During the introduction or mini-lesson, students may need specific instruction on writing and reading decimal
fractions and decimals.
5 For example, the 10ths square below shows 5 out of 10 shaded boxes. As a fraction, that
would be written as , and read, “five tenths.” As a decimal, it would be written as 0.5, and read, “five tenths.”
10
The 100 grid below shows 28 shaded squares out of 100. As a fraction, that would be
28
, and read, “twenty-
100
eight hundredths.” As a decimal, it would be written as 0.28 and read, twenty-eight hundredths.”
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 34 of 127
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
5
or 0. 5
10
28
100
or 0. 28
It is important for students to recognize that it doesn’t matter where the fractional parts are placed. They can be
scattered (above left) or they can be connected (above right).
Task Directions
PART 1
First, students will follow the directions below from the “Decimal Designs: Part 1” student recording sheet.
Create tenths and hundredths designs and label them accurately.
Next, students will follow the directions below for the “Decimal Designs, Table” student recording sheet.
1. Look at the example in the table below. Read the following questions and discuss how you would
answer them with your partner.
● What do you notice about how “1 out of 10” is written in
decimal fraction form?
● What do you notice about how “1 out of 10” is written in
decimal form?
● How are they alike? How are they different?
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 35 of 127
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
2. Complete the table below. Fill in the last three rows of the table
from the “Decimals Designs” student recording sheet.
3. Look at the example in the table below. Read the following questions and discuss how you would
answer them with your partner.
● What do you notice about how “29 out of 100” is written
in decimal fraction form?
● What do you notice about how “29 out of 100” is written
in decimal form?
● How are they alike? How are they different?
4. Complete the table below. Fill in the last three rows of the table
from the “Decimals Designs” student recording sheet.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 36 of 127
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
PART 2
First, students will follow the directions below from the “Decimal Designs: Part 2” student recording sheet.
Create tenths and hundredths designs and label them accurately.
Next, students will follow the directions below for the “Decimal Designs: Part 2, Table” student recording
sheet.
1. Look at the example in the table below. Read the following questions and discuss how you would
answer them with your partner.
● What do you notice about how “2 out of 10” is written in decimal form using tenths?
● What do you notice about how “20 out of 100” is written in decimal form using hundredths?
2. How are they alike? How are they different?
3. Complete the table below. Fill in the last four rows of the table from the “Decimals Designs: Part 2”
student recording sheet.
FORMATIVE ASSESSMENT QUESTIONS
Part 1:
● How many squares are shaded out of 10 (or 100)?
● How many squares total are in the figure?
● What decimal fraction represents the shaded part? How do you know?
● What decimal represents the shaded part? How do you know?
● How would you read the decimal fraction (or decimal) you have written?
● Which students are able to accurately write decimal fractions to describe a shaded region of a design?
● Which students are able to accurately write decimals to describe a shaded region of a design?
● Which students are able to accurately read numbers written in decimal fraction or decimal form?
Part 2:
● How many squares are shaded out of 10 (or 100)?
● How many squares total are in the figure?
● What decimal fraction represents the shaded part? How do you know?
● What decimal represents the shaded part? How do you know?
● How would you read the decimal fraction (or decimal) you have written?
● How are the models of tenths related to the models of hundredths?
● What do the models of the tenths and hundredths have in common? What is different?
● How can a decimal written in tenths be written as a decimal expressed in hundredths?
● Which students are able to accurately write decimal fractions to describe a shaded region of a design?
● Which students are able to accurately write decimals to describe a shaded region of a design?
● Which students are able to accurately read numbers written in decimal fraction or decimal form?
● Which students were able to connect the representations of tenths to the equivalent representation of
hundredths?
DIFFERENTIATION
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
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Dr. John D. Barge, State School Superintendent
July 2014 Page 37 of 127
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Fourth Grade Mathematics Unit 5
Extension
● Students can be encouraged to conduct a survey of 10 people or 100 people and report the results as a
decimal fraction.
Intervention
● Some students may need to continue to represent the decimal fractions and decimals using base 10
blocks. See “Ten is the Winner” and “Rolling Around with Decimals” in this unit for more information
about using base 10 blocks to represent decimal fractions and decimals.
TECHNOLOGY
● http://www.ixl.com/math/grade-4/what-decimal-number-is-illustrated This online game is a quiz activity
for illustrating decimals as part of the base-ten system. It can be used as a mini-lesson for this task,
additional practice or for remediation purposes.
● http://www.harcourtschool.com/activity/con_math/g04c22a.html This activity has students match
fractions with denominators of 10 or 100 with their matching decimals. It can be used for additional
practices or for remediation purposes.
● http://www.prometheanplanet.com/en-us/Resources/Item/41735/reading-and-writing-decimals This
resource can be used with an ActivSlate or Smartboard to discuss reading and writing decimals in many
different ways. It can be used as a mini-lesson for this task, additional practice or for remediation
purposes.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 38 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Name
Date
Decimal Designs: Part 1
Create tenths and hundredths designs and label them
accurately.
shaded boxes out of 10
Decimal Fraction
Decimal
shaded boxes out of 100
Decimal Fraction
shaded boxes out of 10
Decimal Fraction
Decimal
shaded boxes out of 100
Decimal Fraction
shaded boxes out of 10
Decimal Fraction
Decimal
Decimal Designs: Part 1
Table
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 39 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
1. Look at the example in the table below. Read the following questions and discuss how you would answer
them with your partner.
● What do you notice about how “1 out of 10” is written in decimal fraction form?
● What do you notice about how “1 out of 10” is written in decimal form?
● How are they alike? How are they different?
2. Complete the table below. Fill in the last three rows of the table from the “Decimals Designs: Part 1”
student recording sheet.
Table, Page 2
1. Look at the example in the table below. Read the following questions and discuss how you would
answer them with your partner.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 40 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Decimal Designs: Part 1
Table, Page 2
1. Look at the example in the table below. Read the following questions and discuss how you would
answer them with your partner.
 What do you notice about how “29 out of 100” is written in decimal fraction form?
 What do you notice about how “29 out of 100” is written in decimal form?
 How are they alike? How are they different?
2. Complete the table below. Fill in the last three rows of the table from the “Decimals Designs” student
recording sheet.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 41 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Name
Date
Decimal Designs: Part 2
Create tenths and hundredths designs that represent the same amount and label
them accurately.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 42 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Name
Date
Decimal Designs: Part 2
Table
1. Look at the example in the table below. Read the following questions and
discuss
how you would answer them with your partner.
● What do you notice about how “2 out of 10” is written in decimal form using tenths?
● What do you notice about how “20 out of 100” is written in decimal form using hundredths?
2. How are they alike? How are they different?
3. Complete the table below. Fill in the last four rows of the table from the “Decimals Designs: Part 2”
student recording sheet.
Input
2 out of 10
20 out of 100
Output
Decimal Fraction
(using tenths)
Decimal
(using tenths)
2/10
0.2
20/100
0.20
8 out of 10
80 out of 100
out of 10
out of 100
out of 10
out of 100
Performance Task: Flag Fractions
MATHEMATICS GRADE 4 UNIT 5: Fractions and De cimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 43 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Scaffolding Task: Expanding Decimals with Money
TASK CONTENT: Building decimal fractions and decimals in expanded notation
STANDARDS FOR MATHEMATICAL CONTENT
MCC4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons
are valid only when the two decimals refer to the same whole. Record the results of the comparisons with the
symbols >, +, or <, and justify the conclusions, e.g. by using a visual model.
STANDARDS FOR MATHEMATICAL PRACTICE TO BE EMPHASIZED
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make sure of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Using our money system, where the dime represents tenths and the penny represents hundredths, students may
more easily see decimals as parts of a whole, with the whole being one dollar. Decimal fractions such as 45/100
can be easily modeled using dimes and pennies as 4 dimes and 5 pennies. This allows the students to easily see
45/100 as 40/10 + 5/100 as well as 4/10 + 5/100.
4 Dimes and 5 Pennies
40 Pennies and 5 Pennies
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 55 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
ESSENTIAL QUESTIONS
● When can tenths and hundredths be used interchangeably?
● When you compare two decimals, how can you determine which one has the greater value?
MATERIALS
●
●
10 dimes and 10 pennies for each pair
“Expanding Decimals with Money” Recording Sheet
GROUPING
Individual or partner grouping
NUMBER TALKS
Continue utilizing the different strategies in number talks and revisiting them based on the needs of your
students. Catherine Fosnot has developed problem “strings” which may be included in number talks to further
develop mental math skills. See Mini-lessons for Operations with Fractions, Decimals, and Percents by Kara
Louise Imm, Catherine Twomey Fosnot and Willem Uittenbogaard. (Mini-lessons for Operation with Fraction,
Decimal, and Percent, 2007, Kara Louise Imm, Catherine Twomey Fosnot and Willem Uittenbogaard)
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 56 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Comments
As students develop their decimal understanding, we need to continually emphasize the link between fraction
concepts and our base-ten place value system. Revisiting the link between decimal fractions and decimals often
and working with familiar contexts for decimal fractions will help build that bridge. Additionally, continuing to
help students see decimals as a continuation of our base-ten whole number system will help them apply the
rules of whole numbers within fraction situations. This lesson helps students see decimals and decimal fractions
in expanded form, much like they have done expanded form using whole numbers. This ability to expand tenths
and hundredths will help in later tasks as students add tenths and hundredths.
Students need to develop the ability to think flexibly about decimals in a variety of contexts. One of the contexts
of decimals they are most familiar with is that of our money system.
Task:
Review with students that pennies represent hundredths of a dollar and dimes represent tenths of a dollar. Have
students compare this model of decimals with base-ten models they have used previously.
● Which pieces of the base ten model match with the dimes? With the pennies? With the dollar?
Review expanded form notations using whole numbers. Model who to write a decimal fraction in expanded
form based on students’ previous knowledge.
45/100 = 40/100 + 5/100 = 4/10 + 5/100
Student Directions:
Pull a handful of coins from your bag of dimes and pennies. Fill in the table below with the decimal represented
by your coins. Write your decimals in expanded notation using both the dime and penny combination and how
you would represent it if you only used pennies. See the example in the table.
FORMA
Decimal Made with Pennies
TIVE
Decimal Made with Pennies
and Dimes
Decimal
ASSESS
(with Expanded Notation)
(with Expanded Notation)
MENT
QUESTI
30 pennies + 6 pennies
3 dimes + 6 pennies
0.36
ONS
30/100 + 6/100
3/10 +6/100
●
ow do the
dimes
represent
●
●
●
●
●
decimal fractions? The pennies?
How does a money model help you represent tenths and hundredths?
What strategies did you use to add tenths and hundredths?
Where students able to move easily from tenths to hundredths?
Did students see the connection between the money models and the base ten model previously used?
How did I assess for student understanding?
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 57 of 127
All Rights Reserved
H
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
●
Did students see the pattern that occurs with decimal fractions?
DIFFERENTIATION
Extension
●
Provide students all types of coins and/or bills in the bag of money and have them complete the same
activity having to change all coins into “decimal fraction” friendly coins and justify the exchanges.
Intervention
● Have students create the money amount using base ten models and place the coins on top of the
base-ten blocks they match with in order to write the decimals. Have students write the value of each
place (tenths and hundredths) directly under the model on place value mats.
TECHNOLOGY
● http://www.ixl.com/math/grade-3/put-decimal-numbers-in-order This online game is a quiz activity for
putting two digit decimals in order. It can be used for additional practice or for remediation purposes.
● http://www.prometheanplanet.com/en-us/Resources/Item/68401/the-great-decimal-race This resource
can be used with an ActivSlate or Smartboard. It can be used for a mini-lesson, additional practice or
for remediation purposes.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 58 of 127
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Expanding Decimals with Money
Pull a handful of coins from your bag of dimes and pennies. Fill in the table below with the decimal
represented by your coins. Write your decimals in expanded notation using both the dime and penny
combination and how you would represent it if you only used pennies. See the example in the table.
Decimal
Decimal Made with Pennies
(with Expanded Notation)
Decimal Made with Pennies
and Dimes
(with Expanded Notation)
0.36
30 pennies + 6 pennies
30/100 + 6/100
3 dimes + 6 pennies
3/10 +6/100
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 59 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Constructing Task: Taxi Trouble
TASK CONTENT: Add tenths and hundredths, compare decimals
STANDARDS FOR MATHEMATICAL CONTENT
MCC4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator
100, and use this technique to add two fractions with respective denominators 10 and 100. For
example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/1001.
MCC4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example,
rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
MCC4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the same whole. Record the results of
the comparisons with the symbols >, +, or <, and justify the conclusions, e.g. by using a visual
model.
STANDARDS FOR MATHEMATICAL PRACTICE TO BE EMPHASIZED
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make sure of structure.
BACKGROUND KNOWLEDGE
Students may need some background knowledge built on how taxi companies charge for their
services. Many of them have a flat fee plus an additional rate per mile or fraction of a mile
traveled. Often the flat fee is a distractor from the per mile rate. It is important that students make
predictions from their initial reading of the rate and then compare that with the actual result. This
will show them how important it is to do the math when making choices on how to spend their
money!
ESSENTIAL QUESTIONS
●
How can decimal fractions help me determine the best choices on how to spend my
money?
MATERIALS
●
Paper
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 119 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
●
●
Pencils
“Taxi Trouble” Student Sheet
GROUPING
Individual or partner
NUMBER TALKS
Continue utilizing the different strategies in number talks and revisiting them based on the needs
of your students. Catherine Fosnot has developed problem “strings” which may be included in
number talks to further develop mental math skills. See Mini-lessons for Operations with
Fractions, Decimals, and Percents by Kara Louise Imm, Catherine Twomey Fosnot and Willem
Uittenbogaard. (Mini-lessons for Operation with Fraction, Decimal, and Percent, 2007, Kara
Louise Imm, Catherine Twomey Fosnot and Willem Uittenbogaard)
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Task:
●
Introduce the problem. Make sure students understand they are to defend their choice
and use mathematics (shown in number and word form) to defend their choices.
● Have students briefly read the task and make predictions about which Taxi Company
they think will be the best deal. Have them explain their thinking for their predictions.
Students will follow the directions below from the “Taxi Trouble” recording sheet.
Sam is in downtown Atlanta and needs to take a taxi 5 miles to the convention center. There is a
sign posted with the different taxi companies and their rate.
Taxi Company A: $4.00 sitting fee and 30/100 of a dollar for every 1/10 of a mile.
Taxi Company B: Free sitting fee and 5/10 of a dollar for every 1/10 of a mile.
Taxi Company C: $10.00 sitting fee and 2/10 of a dollar for every 1/10. (Sam has a 1/10 off of
your total price coupon)
Which Taxi cab company should Sam choose to ride to the convention center?
FORMATIVE ASSESSMENT QUESTIONS
Have each pair or group share their work. Focus their discussion on:
● How are you determining the cost of the ride for each Taxi Company?
● How are you organizing your work?
● Where have you used decimal fractions and decimal to defend your thinking?
● Which company they thought was best
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 120 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
●
●
●
●
●
●
●
The mathematical justification for their thinking
○ The methods they used for determining the cost of each company
○ How they combined the tenths and hundredths
After and while groups are sharing, have them look for groups that had efficient
strategies, the similarities between the methods used, and the differences between the
methods used.
Which strategies for combining tenths and hundredths did you see today that
worked best?
Where you surprised by the results?
What did you learn about the decimal representations of the money being spent?
Were students able to find the correct price for each company using decimals and
decimal fractions?
How did students show connections between tenths and hundredths?
DIFFERENTIATION
Extension
●
Have students create their own taxi company and write its sitting fee and charge per
mile in terms of tenths of a mile. Have them compare their company’s price with the
companies listed.
Intervention
● Have students use grids, money manipulatives, and/or other concrete models to build
each amount of money for the ride. Use this concrete model as the basis for the
number representations they use to explain their thinking.
TECHNOLOGY
● http://nlvm.usu.edu/en/nav/frames_asid_334_g_2_t_1.html?from=category_g_2_t_1.html
This interactive number line useful for zooming in to show smaller and smaller unit
fractions. It can be used for additional practice.
● https://wwwk6.thinkcentral.com/content/hsp/math/hspmath/ca/common/mega_math_9780153663963
_/megamathcd6/cm/launch.html?strActivityName=g36_3_2_N&strAssignID=1 This
resource allows students to locate decimals that represent a given fraction. It can be used
for additional practice or for remediation purposes.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 121 of 127
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Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics Unit 5
Name
Date
Taxi Trouble
Sam is in downtown Atlanta and needs to take a taxi 5 miles to the convention center. There is a
sign posted with the different taxi companies and their rate.
Taxi Company A: $4.00 sitting fee and 30/100 of a dollar for every 1/10 of a mile.
Taxi Company B: Free sitting fee and 5/10 of a dollar for every 1/10 of a mile.
Taxi Company C: $10.00 sitting fee and 2/10 of a dollar for every 1/10. (Sam has a 1/10 off of
your total price coupon)
Which Taxi cab company should Sam choose to ride to the convention center? Use math words,
numbers, models, and symbols to explain and justify your choice.
MATHEMATICS GRADE 4 UNIT 5: Fractions and Decimals
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 Page 122 of 127
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