Fractions
Common Core State Standards
Supplemental
Fraction Unit
for
Grade Four
based on the
Common Core Standards
SCLME 
South Carolina Leaders of Mathematics Education
2012
SCLME recommends that district mathematics curriculum leaders support teachers with the
implementation of this unit by providing the necessary content knowledge so that students gain a
strong conceptual foundation of fractions.
5/3/12
Developed by SC Leaders of Mathematics Education 2011-2012
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Fractions
Common Core State Standards
The following sources and websites were used in part or whole in the creation of this
fractions unit:
Cramer, K., Behr, M., Post T., & Lesh, R. (2009). Rational Number Project: Initial
Fraction Ideas. Retrieved from
http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-09.html
Cramer, K., Wyberg, T., & Leavitt, S. (2009). Fraction Operations and Initial
Decimal Ideas. Retrieved from
http://www.cehd.umn.edu/ci/rationalnumberproject/rnp2.html
North Carolina Department of Public Instruction. Project directed by D. Polly. 4th
Grade Fractions. Retrieved from
http://maccss.ncdpi.wikispaces.net/file/view/4thGradeUnit.pdf
K-5 Math Teaching Resources. http://www.k-5mathteachingresources.com/4thgrade-number-activities.html
Developed by SC Leaders of Mathematics Education 2011-2012
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Fractions
Common Core State Standards
Time: 9-11 days (60 minute class periods)
Background
In grade 3, students begin their focus on fractions. In the study of fractions, a variety of
concrete and pictorial experiences have students represent a fraction as a
 part of a whole in an area model,
 part of a set,
 distance designated by a point on the number line (a type of linear model).
Students begin to build understanding of the idea that a fraction is a relationship of
two numbers –
 the denominator, which names the parts on the basis of how many equal parts
are in the whole, and
 the numerator, which tells the number of equal parts being considered.
Adapted from NCTM Focus in Grade 4 Teaching with Curriculum Focal Points
Unit Overview
To continue the development of fractions, this unit will support the transition to and
implementation of the Common Core State Standards. The learning activities
provided herein should engage students in both hands-on and minds-on experiences.
Students should have multiple opportunities to communicate about their thinking and
reasoning in order to build understanding. Teachers should listen carefully to students’
ideas and encourage flexibility in their thinking.
Students develop understanding of fraction equivalence and operations with
fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3),
and they develop methods for generating and recognizing equivalent fractions.
Students extend previous understandings about how fractions are built from unit
fractions, composing fractions from unit fractions, decomposing fractions into unit
fractions, and using the meaning of fractions and the meaning of multiplication to
multiply a fraction by a whole number.
In order for students to have a deep conceptual understanding of fractions, they will
use a variety of concrete materials and pictorial representations. In using best
practices, virtual manipulatives should not take the place of concrete materials.
An anchor chart is a visual recording of students’ ideas and thinking about a certain
concept and is used to connect past and future teaching and learning. For example,
on a piece of chart paper, the teacher recorded students’ ideas about what ½
means to them. This included illustrations and labels of different representations for ½.
(See Appendix A)
Models for Fractions
Area or Region Models – Fractions are based on parts of an area or region. Examples
include: circular pie pieces, pattern blocks, regular/square tiles, folded paper strips
(any shape), drawings on grids and partitioning shapes on geoboards.
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
Linear or Length Models – With length models the whole is partitioned and lengths are
compared instead of area. Materials are compared on the basis of length. Examples
include: fraction strips, Cuisenaire rods, number lines, rulers, and folded paper strips.
Set Models – In set models, the whole is understood to be a set of objects or group of
objects, and subsets of the whole make up fractional parts. Examples; in a set of 6
marbles, ½ is 3 marbles. This concept should be taught with concrete materials so that
there would be 2 groups of 3 marbles so that each group is ½ of 6.
Content Progression for CCSS
3rd grade limited to fractions with denominators 2, 3, 4, 6, and 8;
4th grade add 5, 10, 12, and 100.
Big Ideas
 Equivalence and Ordering of Fractions
 Addition and Subtraction of Fractions with Like Denominators
 Multiplication of a Fraction by a Whole Number
 Decimal Notation and Decimal Fractions
Common Core Standards (Grade 4)
*Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4,
5, 6, 8, 10, 12, and 100.
Extend understanding of fraction equivalence and ordering.
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual
fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
4.NF.2 Compare two fractions with different numerators and different denominators,
e.g., by creating common denominators or numerators, or by comparing to a
benchmark fraction such as ½. Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the results of comparisons with symbols
>, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Build fractions from unit fractions by applying and extending previous understandings
of operations on whole numbers.
4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in
more than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 +
1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8= 8/8 + 8/8 + 1/8.
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
c. Add and subtract mixed numbers with like denominators, e.g., by replacing
each mixed number with an equivalent fraction, and/or by using properties of
operations and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to
the same whole and having like denominators, e.g., by using visual fraction
models and equations to represent the problem.
4.NF.4 Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual
fraction model to represent 5/4 as the product 5 x (1/4), recording the
conclusion by the equation 5/4 = 5 x (1/4).
b. Understand a multiple of a/b as a multiple of 1/b and use this understanding to
multiply a fraction by a whole number. For example, use a visual fraction model
to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x
(a/b) = (n x a)/b).
c. Solve word problems involving multiplication of a fraction by a whole number,
e.g., by using visual fraction models and equations to represent the problem.
For example, if each person at a party will eat 3/8 of a pound of roast beef, and
there will be 5 people at the party, how many pounds of roast beef will be
needed? Between what two whole numbers does your answer lie?
Understand decimal notation for fractions, and compare decimal fractions.
4.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
= 34/100.
*Students who can generate equivalent fractions can develop strategies for adding
fractions with unlike denominators in general. But addition and subtraction with unlike
denominators in general is not a requirement at this grade.
4.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.
4.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 comparisons with the symbols >, =, or <, and justify the
conclusions, e.g., by using a visual model.
Developed by SC Leaders of Mathematics Education 2011-2012
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Fractions
Common Core State Standards
Lesson 1: Design of Fractions
Standards:
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual
fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in
more than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 +
1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8= 8/8 + 8/8 + 1/8.
SMP 3 Construct viable arguments and critique the reasoning of others.
SMP 7 Look for and make use of structure.
SMP 8 Look for and express regularity in repeated reasoning.
NOTE: SMP indicates the Standards for Mathematical Practice throughout the
document.
Lesson Learning Goals:
 Build and compare fractions in a set.
 Explain why two fractions are equivalent even through they use different numbers.
Materials: 1-inch color tiles (red, blue, yellow, green), task cards A-H, 1-inch graph
paper, crayons/markers
Engage
(8-10 minutes)
In this lesson students use 1-inch square tiles to create designs that follow certain
criteria.
“Using the tiles at your desk, create a design that is one half blue.”
Allow students a minute or two to create their design. As they do, circulate around the
room looking for simple and creative examples to share with the class.
After students complete their designs, discuss some of the differences in the class.
 Did everyone use the same colors?
 Does everybody’s design look the same? Why not? How can that be since half
of the design had to be blue?
 Did everyone use the same number of tiles? Why or why not?
 How did you decide what you were going to do to create this pattern?
 If we created another design, would you do it differently? How?
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
You may need to repeat this activity a few times before starting the Explore section of
this lesson. Before moving on, students should see that there are many different
options for each design. Just because the problem calls for a fraction in fourths
doesn’t mean they need to use four tiles. They also need to understand that they may
only receive part of the information needed to solve the problems; they will need to fill
in the rest.
Explore
(12-15 minutes)
Students work in groups of two or three to build designs with 1-inch tiles based on the
description given on a task card.
Each student builds a representation for the card. Once all students in the group have
finished, they discuss their designs and decide on which one they will use for their
representation for the class.
Once the students agree upon the design, each student will copy it onto a sheet of 1inch graph paper.
Below the picture they will write a description and an equation of all the colors used in
their design.
“Our design for card C has 1/8 yellow, 4/8 green and 3/8 red. 1/8 + 4/8 + 3/8 = 8/8 or 1
whole.” Start with Card A and work towards Card H. Most groups will not be able to
finish all 8 cards in the time allotted for the lesson.
Explain
(12-15 minutes)
Bring all the students together and have them share the results of task cards A, B, and
C.
Suggested questions
 What did you do for your task card?
 Do you think that this group’s design fits the directions?
 How can you prove it?
 Compare two different designs. How are they similar and different?
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
Time permitting, give the students 8 tiles of any color and tell them as a class you need
to make a design that is ½ red, ¼ green, 1/8 yellow and 1/8 blue.
Ask students to describe how they know how many tiles of the region match a specific
fraction.
Elaborate
(10-12 minutes)
Have students create their own task cards. Students should use 24 total tiles and use
the denominators 2, 3, 4, 6, 8 and 12.
Students need to make sure that the fractions add up to 24/24ths or 1 whole.
As students work, check to make sure that they have completed the puzzle and have
written fractions in simplest form.
Evaluation of Students
Formative: As students are building the designs, circulate throughout the room
checking for misunderstandings. Are students using only the minimum number of tiles?
Can they use more?
How did they choose the number of tiles? Why did they choose the number of tiles of
each color?
Review the students’ description for clarity.
Summative: Have students collect their descriptions of each task card they were able
to finish, and staple them together to create a book.
Plans for Individual Differences
Intervention: Students who are struggling with this activity may need help determining
the number of tiles that will be found in their design. These students may need to start
with very basic designs, using the minimum number of tiles.
Extension: Students who are ready will create task cards for a defined number of tiles
for class use. For example, the student may use 16 tiles to create a design with ¼ blue,
1/8 green, ½ red and the rest yellow.
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
Fraction Task Cards A-H
Card A
Card E
Build a design that is… Build a design that is…
 one-fourth red
 one-half red
 one-fourth green
 one-fourth yellow
Card B
Card F
Build a design that is… Build a design that is…
 two-thirds yellow
 five-twelfths blue
 one-sixth red
 two-sixths green
Card C
Card G
Build a design that is… Build a design that is…
 one-eighth yellow
 one-fifth red
 four-eighths green
 four-tenths green
 two-fifths blue
Card D
Card H
Build a design that is… Build a design that is…
 one-third blue
 one-third yellow
 two-thirds red
 one-sixth red
 one-half green
Developed by SC Leaders of Mathematics Education 2011-2012
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Fractions
Common Core State Standards
Lesson 2: Race to One
Standards:
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual
fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b.
b. Decompose a fraction into a sum of fractions with the same denominator in
more than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 +
1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8= 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing
each mixed number with an equivalent fraction, and/or by using properties of
operations and the relationship between addition and subtraction.
SMP 3 Construct viable arguments and critique the reasoning with others.
SMP 7 Look for and make use of structure.
Lesson Learning Goals:
 Name equivalent fractions.
 Apply knowledge of equivalent fractions while playing a game.
Materials: Fractions Cards, Race to One game board, Race to One rules, counters for
each game
Lesson:
Engage
(8-10 minutes)
Give each pair of students a game board. They must fill in the fractions on each
fraction bar before playing (halves, thirds, fourths, fifths, sixths, eighths, and tenths).
Introduce the game Race to One by playing a practice game with the class. Using
just the fraction cards that are equal to or less than one, shuffle the cards and place
them face down.
Start by placing one counter on each fraction bar at a location that is less than 3/4.
(Students will start at the beginning of the fraction bar during their game.)
Select a card from the pile and discuss the possible moves available to the students.
The player can move one or more than one counter during each play, but he must
move the full amount on the chosen card. If a player is not able to move the full
amount, he will lose his turn.
Once a player moves a counter exactly to the number 1 on any fraction bar, they
collect the counter. Place a new counter at the beginning of that fraction bar so
every play has 7 counters available.
Developed by SC Leaders of Mathematics Education 2011-2012
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Fractions
Common Core State Standards
Play a few hands so students become familiar with the rules of the game.
Explore
(20-30 minutes)
Students play the game Race to One in pairs. Move throughout the room observing
how the students are playing the game.
Suggested questions:
• Which counter are you moving?
• Are there other counters that you could also move? How do you know?
• Which move will help you get more counters closest to one?
Explain
(12-15 minutes)
Bring the class back together after students have played the game for about 20 – 30
minutes.
Continue your practice game from the beginning of the class, but have students
decide which counters to move and how far. Discuss the possibilities and give reasons
for each choice.
Elaborate
Continue to play this game.
(10-20 minutes)
If students need an extension, tape two Race to One boards together, and make the
game Race to Two. In this version, all the cards can be used.
Evaluation of Students
Formative:
As students are playing the game, observe them and pose questions to check for
mathematical understanding. Suggested questions are in the Explore section.
Summative:
If teachers want a summative assessment, pose an additional follow-up task:
You have the card ¾. Name 3 possible moves that you can make on the game
board.
• One move involves 1/2
• One move that includes 1/4
• One move that includes 1/6
Plans for Individual Differences
Intervention:
For students who are struggling to find equivalent fractions, provide fraction
manipulatives (fraction bars, fraction tiles) to help them.
Extension:
Play Race to Two the entire time if students need an extension.
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
Race to One
Game Rules
1. Shuffle the fraction cards that are equal to or less than 1. Place them face
down.
2. Place seven counters on the game board, one at the beginning of each
fraction bar.
3. Player 1 draws the first card off the top of the deck of fraction cards. Move a
counter (or counters) the total amount shown on the card. You can move one
or more than one counter on every turn. You must move the full value of the
fraction on the fraction card. Example: Player 1 chooses 3/5; they can move
one counter 3/5 on the fifths line or 6/10 on the tenths line. They can also move
more than one counter the following ways: ½ and 1/10, 1/5 and 4/10, or 1/3,
1/6, and 1/10.
4. Player 2 draws the next card off the top of the deck of fraction cards and
moves his counter or counters the total found on their card. Players take turns
flipping cards and moving counters.
5. When a counter lands exactly on one, the player has won the counter. Once a
player has a won a counter, another counter is placed at the beginning of the
fraction bar so that there are always 7 counters being played at one time.
6. If you are unable to move the amount found on the fraction card, your turn is
over.
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
Fraction Cards
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Common Core State Standards
Fraction Cards
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Common Core State Standards
Lesson 3: Fraction Buckets
Standards:
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual
fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
4.NF.2 Compare two fractions with different numerators and different denominators,
e.g., by creating common denominators or numerators, or by comparing to a
benchmark fraction such as ½. Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the results of comparisons with symbols
>, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
SMP 2 Reason abstractly and quantitatively.
SMP 3 Construct viable arguments and critique the reasoning of others.
Lesson Learning Goals:
 Compare two fractions with different numerators and different denominators by
comparing them to benchmarks of ½ and 1.
 Explain their reasoning when comparing fractions.
Materials: Fraction cards, Fraction Bucket labels and “buckets” (cups, containers, etc),
math journal
Lesson:
Engage
Organize the labeled buckets in order from least to greatest.
(8 - 10 minutes)
Use the labeled buckets to discuss how the fraction cards could be positioned. How
do the buckets help the students make decisions? Discuss any misunderstandings
about the buckets. Use the symbols >, <, or = along with the words.
The teacher pre-selects 5 fraction cards, one for each bucket, to demonstrate how to
place them correctly. Using one of the fraction cards, the teacher places it in the
correct bucket while thinking aloud to the class. “I have the card 4/4, so I am thinking
that if I had four parts out of 4 total parts, I would have all the parts. So I would have a
whole. I am going to place this card in the one whole bucket. 4/4 = 1 whole.” Repeat
process with another card.
As a class, determine where the next three cards would be placed. Discuss possible
reasons why each card belongs where it is placed. Look for multiple reasons. Would it
be a better fit in a different bucket? Can students support others’ ideas? Use symbols
>, <, or = along with words more than (greater than), less than or equal to.
Explain that in a few moments students will be working with a partner to place many
different fractions in the correct bucket.
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
Explore
(18-20 minutes)
Placing the Cards
Each pair of students should receive a copy of the Fraction Bucket labels, buckets
and a set of fraction cards.
Students lay out the Fraction Bucket labels and buckets in order from least to greatest.
Shuffle the fraction cards and place them face down. Take turns flipping over a
fraction card and placing it in the correct bucket. As the card is being placed in the
bucket, the student must explain why he is choosing that particular bucket. If the
partner agrees with the explanation, another card is flipped and the students
continue. If the partner does not agree with the explanation, he explains where he
thinks it goes. Both students must agree on which bucket each card will be placed in.
If the pair cannot agree, they can place the fraction card to the side for later. Repeat
until all cards have been placed.
As students do this:
Circulate throughout the room to observe the students at work. Listen to students’
reasoning as they place a card. Ask students to re-explain why a card is placed in a
certain bucket. If there is a disagreement about card placement, then listen to both
arguments, and help students find other cards that may help them make a final
decision. Ask students to use symbols (>,<,=) when they are comparing fractions.
Checking the Cards
After all the fraction cards have been placed, students can self-check by using the
answer sheet. Shuffle the fraction cards and repeat.
Explain
Discussion about the Fraction Bucket Labels
Choose one of the buckets to discuss.
(10-12 minutes)
Suggested questions:
 What strategies did you use to place fraction cards in this bucket?
 What do all the fraction cards have in common?
 Repeat with other buckets.
Share highlights from group work.
• Were there any fraction cards that you found difficult to place?
• What was difficult regarding the placement of the fraction cards?
• What strategies did other students use to place the difficult fraction cards?
Elaborate
(12-15 minutes)
In their math journal, students explain their strategies for placing a selected fraction
card in a bucket. Have students explain the meanings of the symbols (>,<,=) and
compare fractions with the symbols and words.
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Common Core State Standards
Evaluation of Students
Formative: As students are working, the teacher is evaluating the students’ abilities to
place fractions in the correct locations. Teachers are listening for the reasons why the
students think the fractions belong where they placed them. Are there fractions that
are problematic for the class?
Possible Questions:
1. Where would you place 6/10? Why would you place it in the more than half
and less than one bucket?
2. What do all the cards in the less than (<) half bucket have in common?
3. If I had two fractions, 2/5 and 2/3, how would I know which one is bigger (>)?
Summative:
Students’ work from various sections of this lesson can be analyzed as a summative
assessment.
Plans for Individual Differences
Intervention:
Remove the cards that are not ½ or 1. As the student becomes more familiar with
these cards, introduce the less than half, followed by the more than half cards. Finally
add the more than one cards.
Extension:
Placing Blank Cards
Give students blank fraction cards and have them create 2-4 fractions for their
partners to place.
Divide the cards between two students. Place the cards face down. Each student
takes his first card and places it in the correct bucket. The student with the larger card
takes his opponent’s card. If a card is misplaced, it is automatically forfeited. If there is
a tie, a second card is drawn, and the winner takes all the cards.
Developed by SC Leaders of Mathematics Education 2011-2012
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Common Core State Standards
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Common Core State Standards
Fraction Cards
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Fraction Cards and Symbols
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<
>
=
<
>
=
<
>
=
<
>
=
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Common Core State Standards
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Common Core State Standards
Lesson 4: Fraction Chain
Standards:
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual
fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
4.NF.2 Compare two fractions with different numerators and different denominators,
e.g., by creating common denominators or numerators, or by comparing to a
benchmark fraction such as ½. Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the results of comparisons with symbols
>, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
SMP 4 Model with mathematics.
SMP 5 Use appropriate tools strategically.
SMP 7 Look for and make use of structure.
Lesson Learning Goals:
 Construct a number line to show equivalence of fractions with different
denominators.
 Compare fractions with the benchmarks 0, ½, and 1 whole.
 Explain their reasoning when comparing fractions.
Materials: 6-10 foot strings (1 per group), Fraction Cards, whole number cards 0-4 (1 set
per group), paper clips, math journals, graph paper, rulers
Lesson:
Engage
(8-10 minutes)
Prior to beginning this lesson, set up strings around the room where students are going
to work in groups of four. Have one string available for a whole group discussion.
Discuss using a number line as another way to represent a fraction. Where do we see
fractions on a number line? Examples include when measuring (distance, liquid
volume), graphs, etc. Give students a ruler to demonstrate the context of ½, ¼ and
1/8 inches.
Distribute the whole number cards (0-4) to each group of students. Hold the discussion
of how the spacing between whole numbers should be equal and ask them to
arrange the number cards in increasing order on the string.
Hand out the halves number cards next. Explain to the class that they are going to
add these cards to their number line. Once complete have class discussion.
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Explore
(18-20 minutes)
Students placed the halves number cards on their number line. They can adjust the
whole number cards if needed. If a half number card has the same value as a whole
number card, the students hang it below the whole number card with a paper clip.
Once the class has set up their number lines, stop everyone and discuss the work they
have done up to this point.
 What changes to the number line did you need to make? Why?
o At this point, make sure discussion occurs as to the importance of equal
spacing between the whole numbers on the number line.
 What did you do when you had 4/2 as a card? Why did you place it under the
2?
 Are 2 and 4/2 the same number? Why? (Discuss the = sign.)
 Is there anything else that you notice about our number line at this time?
 Do you see any patterns beginning to form?
Hand out the remaining number cards (fourths, eighths, and mixed numbers). Each
group will need to place the cards on the number line. They are allowed to change or
modify their number line at any time to make the task easier. They cannot remove any
card from the number line, only move it.
As the students are completing the number line, the teacher is moving from group to
group discussing patterns that students see, any problems they may be having,
confirming or questioning students’ conjectures, and redirecting students that need
help.
Once the number line is complete and the teacher has checked it, students can
recreate it in their math journals. They can also add any notes or findings they have
made during the activity. Have students discuss fractions and symbols (<,>,=).
Explain
(12-15 minutes)
Bring the class back together to discuss the number lines. Add the halves to the group
number line to represent the work that was done before the students were allowed to
add the other number cards.
Have a student or students demonstrate how they began the process. Did they sort
the cards into piles of like denominator or other groups? What were some of the first
cards students placed? Why did they choose these cards?
Discuss changes that needed to be made to the number line as the activity
continued. Were there any patterns the students discovered as they were placing
their cards? Complete the whole group number line.
Look at a column of cards that are paper clipped together. What is similar about
these fractions? Some students might say they are all the same number. Have
students explain how they know.
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Share the notes and conjectures students wrote in their journals.
Elaborate
More Denominators
Repeat this activity using fraction cards that have thirds, sixths, and twelfths as
denominators.
Suggested questions as students work:
Could you add any of the half and fourth cards as well?
Where would they go?
Culminating task
Give students the following task:
One group placed the fraction 1 2/3 between 1 ½ and 1 ¾ on the number line. Are
they correct? Prove your answer using both a picture and a written explanation.
Evaluation of Students
Formative:
Observe students as they work and pose questions to check for mathematical
understanding.
Summative:
Students work from the Explore phase can be used as a summative assessment.
Plans for Individual Differences
Intervention:
Start with numbers 0 and 1. Once the student has placed both cards in a correct
location, she can add the whole number 2 and the fractions in between. Some
students will need to build their number line on graph paper to have equal spacing
between each number. You may need to tape two pieces of graph paper together.
Extension:
Students create a number line using all the cards (halves, thirds, fourth, sixths, eighths
and twelfths). Choose two cards and compare them. Which one is > and how do you
know? Can you justify your answer without using a mathematical algorithm?
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2
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2
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2
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2
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2
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4
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4
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4
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4 4
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4
13
4
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14 15
4 4
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8
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8
7
8
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8
9
8
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10 11
8 8
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12 13
8 8
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14 15
8 8
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16 1 1
8
4
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1 11 3
2 4
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2 1 2 1
4
2
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2 3 3 1
4
4
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3 1 3 3
2
4
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Lesson 5: Kellen’s Candy Company
Standards:
4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in
more than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 +
1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8= 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing
each mixed number with an equivalent fraction, and/or by using properties of
operations and the relationship between addition and subtraction.
SMP 4 Model with mathematics.
SMP 7 Look for and make use of structure.
SMP 8 Look for and express regularity in repeated reasoning.
Lesson Learning Goals:
 Decompose a whole unit into an addition equation where all the fractions have
the same denominator and the sum is one whole.
Materials: connecting/pop cubes, chart paper, crayons/markers, math journals
Lesson:
Engage
(10-12 minutes)
In today’s activity students build Special Bars from different colored pop cubes. Each
color will represent a different flavor of candy. The bars come in different sizes
depending on the number of candies the buyer wants. The teacher will need to make
a bar using 8 total pop cubes prior to beginning class.
“Today we are going to pretend to visit a special candy store called Kellen’s Candy
Company. At the company they have a very unique candy bar called the Special
Bar. This bar is special because the buyer of the bar is able to pick out all the flavors
that will be in the bar. This way each bar is different and the buyer can get exactly
what he wants. As a treat, each person who visits the store receives a free 8 piece
candy bar at the end of his visit.”
To personalize this task teachers may want to use their name, example: Mr. Smith,
Smith Bar. Students could even use their names when designing a bar of their own.
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“Let’s look at the Special Bar that I made on my visit.” Share with students a bar you
created that has 8 pieces.
Take a moment to discuss the flavors that are possible. (See possible flavors below.)
Suggested Questions:
• Which flavor of candy did I choose most often?
• Which flavor of candy did I choose least often?
• How do you know which candy I chose most often?
• How much of my bar is flavored blueberry? cherry? banana? lime? …
The students’ answers should be in fraction form. You are not asking how many pieces
are certain flavors, but how much of the bar is that flavor. As students tell you the
fraction for each flavor, record the fractions on the board.
If I add up the all the fractions 3/8 + 4/8 + 1/8, I will get 8/8 which is the whole candy
bar. Create an anchor chart with 8/8 in the center to model various equations that
represent the decomposition of the whole. Later in the lesson, students can add
additional equations to model 8/8 as a decomposed representation.
Today you are going to build Special Bars of different sizes and record them in your
math journal. First you will build a Special Bar that has 8 pieces of candy. Then you will
represent the bar by drawing it in your notebook. After that you will write an equation
to show the sizes of your Special Bar. You will repeat the process with Special Bars of
different sizes. (2, 3, 4, 5, 6, 8, 10, or 12 pieces)
Explore
Building and Recording Special Bars
(18-20 minutes)
Students work on building and recording different sized Special Bars. They start with a
bar that has 8 pieces of candy. As the students are building and recording the bars,
the teacher should be asking questions of the students.
• How many (flavor) pieces do you have?
• How many more pieces would you need to complete a bar?
• Which do you have more of? less of? the same amount of?
• What does your equation look like?
• How are you determining which fractions to use in your equation?
• How does your representation compare to the candy bar on the anchor chart? If it
is a different equation, add to the anchor chart.
Make sure the representations and equations that are being recorded are correct.
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Explain
(12-15 minutes)
Students rebuild their favorite Special Bar from the day. Bring the Special Bar and the
equations for the bar to a large group meeting. Students share their drawings and
discuss the equation that represents it.
Have students determine the equation before the presenting student shares it.
Elaborate
(8-10 minutes)
Students write a story problem about their Special Bar. For example, Sandy’s Special
Bar was 4/10 Cotton Candy, 5/10 Marshmallow, and 1/10 Orange. Her dog, Ripley,
ate all of the cotton candy pieces while she was at school. How much of her Special
Bar was remaining?
Students are given part of a bar, and need to complete the rest of the bar. For
example, I have 7/12 of my bar complete with banana and chocolate. I don’t want
any more banana or chocolate, but I want two more flavors. What are some of my
options?
Evaluation of Students
Formative: As you are working with the students are they able to describe each
section of the bar in fraction form? Can they create equations that equal a whole?
Summative: I have a bar with 3 licorice, 3 cotton candy, 2 apple, and 4 orange
pieces. Draw what the bar looks like. Write an equation that represents my Special
Bar.
Plans for Individual Differences
Intervention: Limit the number of types of candy per Special Bar. Start with only two
colors, and then continue to add one at a time.
Extension: Build the Mega Special Bar which is only sold for Valentine’s Day. The Mega
Special Bar has 100 pieces of candy, and can have up to 10 different types of candy.
Have students determine the equation to represent their Mega Special Bar.
Possible Flavors for the Colored Connecting Cubes
Red – Cherry
Blue – Blueberry
Light Green – Lime
White – Marshmallow
Brown – Chocolate
Black – Licorice
Yellow – Banana
Pink – Cotton Candy
Dark Green – Apple
Orange - Orange
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Lesson 6: Who am I? Puzzles
Standards:
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual
fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
4.NF.2 Compare two fractions with different numerators and different denominators,
e.g., by creating common denominators or numerators, or by comparing to a
benchmark fraction such as ½. Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the results of comparisons with symbols
>, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
SMP 1 Make sense of problems and persevere in solving them.
SMP 3 Construct viable arguments and critique the reasoning of others.
Lesson Learning Goals:
 Apply understanding of fractions to solve “Who am I?” puzzles.
 Analyze the value of fractions while solving puzzles.
 Communicate reasoning while solving puzzles.
Materials: puzzle cards, fraction manipulatives
Lesson:
Engage
Introduce the class to puzzle 1:
Puzzle 1
1/4 1/2 3/4 4/4 5/4
(10-12 minutes)
Show the first clue to the puzzle: “I am more than (>) one half.”
Which of these fractions does this clue help us eliminate? 1/4 and 1/2
Discuss with the class why this clue helps us determine which choices to eliminate.
Show the second clue to the puzzle: “My denominator is larger than or equal to my
numerator.” How does this help us get closer to the answer? This will eliminate the
fraction 5/4, leaving us 3/4 and 4/4.
Show the last clue: “I cannot be written as a whole number.” The only fraction left that
can be written as a whole number is 4/4, which can be written as 1, so the answer has
to be 3/4.
After the class has discussed how to use the clues to solve the puzzles, explain that
they will be working with a partner to solve more puzzles.
Explore
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Students work in pairs to solve the remaining Fraction Puzzles. As the students are
working, observe how they are solving the puzzles. What strategies do students use to
get started? What clues do they not understand?
When students are finished with the remaining puzzles, they attempt to write their own
fraction puzzles in their math journals. Choose any five fractions, and write clues that
will help eliminate a fraction or two at a time, but keep the others. Remember that
fraction denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 100. Have students use
symbols and words for greater than, less than, or equal to.
Students ask classmates to solve their puzzles.
Explain
(20 minutes)
As a class, discuss how students solved the puzzles. What clues were most helpful, and
what clues were least helpful? Which clues did students need help with?
Share some of the puzzles that the students made.
If time permits, work as a class to solve a few of the puzzles that students created.
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1/4
Puzzle 1
Puzzle 2
Who am I?
Who am I?
1/2
3/4
4/4
5/4
• I am > (greater than)one half.
• My denominator is larger
than or equal to my
numerator.
• I cannot be written as a
whole number.
• I am
.
2/8
2/3
3/4 2/5
7/10
• My numerator is an even
number.
• I am > (greater than) one half.
• I am written in simplest form.
• I am
.
Puzzle 3
Puzzle 4
Who am I?
Who am I?
4/6
9/12
3/5
5/12
• I am > (greater than)1/4.
• My denominator is a multiple
of three.
• I can be simplified.
• When I am simplified, my
numerator and denominator
are less than four.
• I am
.
6/8
1/2
5/12
1/4
8/10
2/3
• I am < (less than) one half.
• I am greater than one third.
• My denominator is a multiple
of three.
• I am simplified.
• I am
.
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2/4
•
•
•
•
Puzzle 5
Puzzle 6
Who am I?
Who am I?
3/9
1/5
7/12
9/10
I am greater than 1/4.
I cannot be simplified.
I am closer to 1 than one half.
I am
.
6/10
5/4
1/5
4/6
3/8
• I am < (less than) one.
• My denominator is even.
• I can be written in a different
way.
• I am another way to say 2/3.
• I am
.
Puzzle 7
Puzzle 8
Who am I?
Who am I?
4/8
5/9
1/3
3/12
• I am greater than one fourth.
• I am not another way to write
1/2.
• I am written in lowest form.
• I am less than one half.
• I am
.
2/10
7/8
4/9
2/10
9/6
4/12
• I can be simplified to a
simpler fraction.
• I am less than one.
• My denominator is a multiple
of three.
• I am closer to one half than I
am to zero.
• I am
.
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Lesson 7: Fraction Tangrams
Note: This lesson could take more than one day depending on the needs of your
students.
Standards:
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using
visual fraction models, with attention to how the number and size of the parts
differ even though the two fractions themselves are the same size. Use this
principle to recognize and generate equivalent fractions.
4.NF.2 Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators, or by
comparing to a benchmark fraction such as ½. Recognize that comparisons are
valid only when the two fractions refer to the same whole. Record the results of
comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a
visual fraction model.
SMP 1 Make sense of problems and persevere in solving them.
SMP 6 Attend to precision.
SMP 7 Look for and make use of structure.
Lesson Learning Goals:
 Compare parts/sections of a rectangle to the whole and to other
parts/sections.
Materials: Tangrams Task Sheets, Tangrams Template, math journals
http://mathforum.org/trscavo/tangrams/construct.html
Lesson:
Engage
(8-10 minutes)
Bring students together to introduce the tangram pieces if they are a new
manipulative. These 7 pieces can be used to make different shapes and designs.
Today we are going to be using a square as our main shape. Have students work
together to create the large square using all 7 pieces of the tangrams.
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This square is going to represent one whole in today’s activity. The job of the
student is to determine the fractional size of each tangram piece, if the large
square is one.
Explore
(18-20 minutes)
Let students work to figure out the fractional size of each tangram piece. It is
assumed that the entire square (picture above) represents 1 or 1 whole. Some
students may need to place pieces on top of each other to compare them.
On the tangram task sheet, students record the size of each piece. Then students
write in their math journals a description of how they determined the size of the
pieces.
If students finish early, they can begin to solve Tangram Task Sheet 2.
Explain
(12-15 minutes)
After students have finished Tangram Task Sheet 1, regroup the entire class
together. Have students share the answers and the ways they found them.
Ask students for the different ways they started and the paths they took to solve
the problems. Discuss any difficulties that students had or that you saw during the
explore stage.
What strategies help the students find the sizes of some of the pieces?
Students return to work on finishing Tangram Task Sheet 2.
Elaborate
(20-25 minutes)
Choose one of the pieces in the tangram and give it a value of 1. What fractions
do the other pieces represent?
What fractions do the other pieces represent if the small triangle is 1/2?
Evaluation of students
Formative: How are students using the fractions represented by other pieces to
determine the fraction represented by the piece they are working with?
Are students seeing the relationship of the sizes of the pieces?
Can the students explain their thinking and the steps they used to solve the
puzzles?
Summative: Collect the Tangram Task Sheet 1. Allow students to edit and rewrite
their explanations of how they solved the problem. (They may have to resolve the
task to edit and rewrite.)
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Plans for Individual Differences
Intervention: Students who are having trouble with this activity may want to start
with an easier puzzle or just a small section of the large square.
Extension: On Tangram Task Sheet 3, students can create their own puzzles and
answer sheets. Remind students they need to be as accurate as possible when
creating their own puzzles. Share their puzzles with others.
Extension 2: Students can repeat this activity with the large square having value
other than one whole.
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Tangram Task Sheet 1
If the square is one whole, what is the value of each tangram piece in the picture below?
How did you discover the size of each piece?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
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Tangram Task Sheet 2
If the square is one whole, what is the value of each tangram piece in the picture below?
How did you discover the size of each piece?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
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Tangram Task Sheet 3
Create your own tangram puzzle for your friends to solve.
This puzzle was made by:
___________________________
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Tangram Task Sheet 3
Answers
Recreate your puzzle, and record the answers for your friends to check their answers.
This puzzle was made by:
___________________________
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Lesson 8: Fraction Relay Race
Standards:
4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating
parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in
more than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8
+ 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8= 8/8 + 8/8 + 1/8.
SMP 1 Make sense of problems and persevere in solving them.
SMP 8 Look for and express regularity in repeated reasoning.
Lesson Learning Goals:
 Decompose a fraction on a number line.
 Explain how they decomposed a fraction into smaller fractions.
Materials: Race Handout
Lesson:
Engage
(15-18 minutes)
Today we are going to get ready for a relay race. Before we run the race, we
need to make a plan. Each team will have three people on it. Each person on a
team has to run in the race, but they do not need to run the same distance. There
are certain places during the race where you can hand off the baton to the next
runner. You are going to get a chance to plan the distances of each runner on
your team before the race begins.
Let’s work together to make a plan for a team. Today’s race will have different
places that a team can hand off their baton to the next runner.
Draw a line on the board with a start and finish line. Mark 5 additional locations,
equal distance apart, where students can hand off the baton. This will break the
track into 6 separate sections. Students may have a difficult time with the concept
that there are 5 locations to hand off, but 6 sections to the race. This is a good
time to discuss the fact that the distance between the marks is what we are
considering and not the marks.
Have students talk with their teammates to determine some possibilities to setting
up the race. Remember that each person doesn’t have to run the same distance.
Share a few of the students’ ideas, and ask what fraction of the race each
student will need to run.
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S
X
X
F
In this race the first runner runs 3/6 of the race, the second runner runs 2/6, and the
final runner runs 1/6.
Write an equation for each idea. 3/6 + 2/6 + 1/6 = 6/6 or 1 whole. Look for
multiple ways to set up the race.
Explore
(15-18 minutes)
Students work on planning four different races. For each race the student teams
need to find multiple ways to set up each race. They record the distance each
runner will run, and then write an equation that will equal one whole.
Explain
(12-15 minutes)
Have students share their possibilities for each race and discuss their favorite and
the reason why they chose it. Make the connection between the races and a
number line from 0 – 1. How are these similar?
Elaborate
(12-15 minutes)
Set up a race outside using cones as hand off positions. Have the students run the
race according to their plans.
What are some possibilities if we had only 2 people on a team? 4 people?
Evaluation of Students
Formative:
While students are working, observe them and pose questions to check for
mathematical understanding.
Summative:
Students’ work from the Explore phase can be used as a summative assessment.
Plans for Individual Differences
Intervention:
If students are having difficulties, provide them with fractions manipulatives
(fraction bars, fraction tiles) to help them visualize the idea of decomposing a
whole unit.
Extension:
If students are in need of an extension, have them design a relay race that is 2
laps long so they have to decompose the number 2. You could also have them
design a race that is 2 ½ laps long.
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Race 1
S
F
This race has 6 different sections to run. What are some possibilities that your team can run?
Runner 1
Runner 2
Runner 3
Equation
Which one of your options is your favorite one? Explain why.
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Race 2
S
F
This race has 4 different sections to run. What are some possibilities that your team can run?
Runner 1
Runner 2
Runner 3
Equation
Which one of your options is your favorite one? Explain why.
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Race 3
S
F
This race has 10 different sections to run. What are some possibilities that your team can run?
Runner 1
Runner 2
Runner 3
Equation
Which one of your options is your favorite one? Explain why.
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Race 4
S
F
This race has 8 different sections to run. What are some possibilities that your team can run?
Runner 1
Runner 2
Runner 3
Equation
Which one of your options is your favorite one? Explain why.
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Lesson 9: Math Situations
Standards:
4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b.
d. Solve word problems involving addition and subtraction of fractions,
referring to the same whole and having like denominators, e.g., by using visual
fraction models and equations to represent the problem.
Lesson Learning Goals:
 Solve word problems using addition of fractions.
 Solve word problems using subtraction of fractions.
 Write equation to represent solutions involving addition and subtraction of
fractions.
Materials: blue cubes, green cubes, yellow cubes (6 of each); paper bag to hold
18 blocks; large piece of paper for drawing the model (1 piece per group); chart
markers
Lesson:
Engage
10-15 minutes
Present opportunities for students to informally share and represent strategies
involving addition and subtraction with fractions.
Students will create, add, and subtract fractions in the context of visiting animal
habitats.
Introduce the lesson by saying, “A third grade teacher named Ms. Lawson has a
problem I think we can help her solve. Her 12 students have been studying animal
habitats, and she wants them to visit some habitats and then share what they
learn with the class. She would like to divide her students into 3 groups so they can
gather information from 3 different locations. Now, here’s the problem. Ms.
Lawson wants to be fair in choosing which group each student will join. So this is
her idea. She wants us to try out her plan for deciding how the groups are set up.
She will use our opinions to help her make the final plans.”
Explore
20-25 minutes
Say, “So let’s try her idea and see what we think.”
Ask the following questions and write the answers on the board:
1. What do we know about the teacher’s problem? (12 students, 3
destinations, 3 groups)
2. What does she want us to do? (Try her plan to assign students to groups
and give her our opinion.)
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Read Ms. Lawson’s idea to the class:
a. Use green cubes to represent the pond group, yellow cubes to
represent the cave group, and blue cubes to represent the forest
group.
b. Put the cubes into a bag, and have each student draw out a cube
without looking.
c. The students will divide into 3 groups based on the color cubes they
draw. Students who draw green cubes will visit a pond, students who
draw yellow cubes will visit a cave, and students who draw blue
cubes will visit a forest.
Say, “Okay, we need 12 of you to pretend you are Ms. Lawson’s students.”
Randomly select 12 students (for example, draw popsicle sticks that have
students’ names on them).
As the teacher draws each name, the students draw a cube from the bag and
form 3 groups. When the 12 students are standing in groups, the class discusses the
outcome of the drawing, as the teacher writes the results on the board.
The class divides into small groups to complete the following:
Directions:
 Decide as a group the answers to these questions.
 Have a recorder write your answers.
 Decide who will share the group’s answers during class
Answers
discussion time.
1. What fractions can be used to represent the students who pond = ______
will visit each location?
cave = ______
forest = ______
2. On the piece of paper given to your group, draw a
(on separate
fraction model to represent the results of the group drawing.
paper)
3. What fraction represents the part of Ms. Lawson’s class
who did not go to the pond?
4. Write an equation to represent your thinking for question
#3.
5. What fraction represents the difference between the part
of the class that went to the forest and the part that went to
the cave?
6. Write an equation to represent your thinking in question
#5.
7. What fraction represents the total of all 3 groups of
students?
8. Does your group think this is a good way for Ms. Lawson to choose which group
each student will join? Explain your opinion.
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Explain
The groups share their answers and opinions to the questions.
10-15 minutes
Elaborate
10-15 minutes
Give students the following problems to solve independently. Discuss student
solutions and strategies. (adapted from www.k-5mathteachingreasources.com)
Write an equation to show your thinking for each problem situation. In each equation,
underline the answer to the question.
1. Tom and Ben ordered a pizza for lunch.
They each ate 1/3 of the pizza.
How much pizza was eaten? Equation: __________________
How much pizza was left? Equation: __________________
2. On Monday I spent 3/12 of my homework time reading and 9/12 working
on a math project.
How much more time did I spend on my math project than on reading?
Equation: _______________
3. Liam and Sam shared a chocolate bar. Liam ate 7/12 and Sam ate 5/12.
Who ate more? Answer: _________________
How much more? Equation: ___________________
4. I ate 4/12 of a box of donuts. My friend ate 1/12 more than I did.
What fraction of the box of donuts did we eat in all?
Equation: _______________
Evaluation of Students
Students complete the following performance task:
Read the following story:
Shanreese, Jackson, and Tony shared a giant cookie. Shanreese ate 1/8 of the
cookie, Jackson ate 5/8, and Tony ate the rest.
1. Using the information in the story, create and write a question that could be
solved by adding fractions with like denominators. Write an equation to represent
your thinking.
2. Using the information in the story, create and write a question that could be
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solved by subtracting fractions with like denominators. Write an equation to
represent your thinking.
Plans for Individual Differences
Intervention:
Using fraction models, work with students needing extra practice to discuss and
solve the following problems. Write an equation to show their thinking for each
question.
1. 6/8 of a set of pencils need to be sharpened.
What fraction of the pencils does not need to be sharpened?
2. Three friends ate 4/6 of a birthday cake.
After dinner dad ate 1/6 of the remaining cake.
How much of the cake was left?
3. Gracie’s mom cut a pan of brownies into 12 equal squares, and gave Gracie and each
of her party guests one square. If there were 7 guests at the party, what fraction
of the brownies was eaten? How much was left?
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Lesson 10: Multiply a Whole Number by a Unit Fraction
Standard:
4.NF.4 Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
c. Understand a fraction a/b as a multiple of 1/b.
SMP 1 Make sense of problems and persevere in solving them.
SMP 3 Construct viable arguments and critique the reasoning of others
SMP 4 Model with mathematics
SMP 8 Look for and express regularity in repeated reasoning.
Lesson Learning Goals:
 Represent a fraction a/b as the numerator times the unit fraction.
Materials: fraction circles or squares, handouts, math journals
Lesson:
Engage
(8-10 minutes)
On an anchor chart or board, ask students to answer the following question:
“What does 3 x 7 mean to you?” Student responses may include the following: 21,
3 groups of 7 objects, 3 sets of 7, 7 + 7 + 7, or draw some type of picture (see
below).
Explore
(20-25 minutes)
The class will study the fraction 3/6 and determine the unit fraction (1/6) that is
repeated to create the fraction. A unit fraction is written as a fraction where the
numerator is one and the denominator is a positive integer.
Have students draw a picture in their math journal to represent 3/6.
Lead a discussion concluding with 3/6 = 1/6 + 1/6 + 1/6 which is the same as 3
groups (sets) of 1/6 or 3 x 1/6.
Have students repeat process with 7/5. The unit fraction will be 1/5 so it is 7 groups
(sets) of 1/5 or 7 x 1/5. Students could use fraction circles, squares, etc. to show
what 7/5 looks like.
Have students complete the handout Multiplying Whole Numbers and Fractions.
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Explain
(10-15 minutes)
After students finish Multiplying Whole Numbers and Fractions, bring class together.
Have students share answers and the strategies they used to find them. Discuss
any difficulties that students had or that you saw during the explore stage.
Elaborate
(20-25 minutes)
Assign Multiplying Whole Numbers and Fractions 2. Choose 1-2 problems and
have students place answers on an anchor chart or Promethean board. Discuss
any difficulties that students had or that you saw.
Evaluation of Students
Formative: While students are working, observe them and pose questions to
check for mathematical understanding.
Summative: Collect Multiplying Whole Numbers and Fractions 2. Allow students to
edit and rewrite how they solved the problem.
Plans for Individual Differences
Intervention: If students are having difficulties, provide them with whole number
multiplication practice.
Extension: Have students create word problems that contain whole numbers and
units fractions. Then have them show how to find a solution by drawing a picture,
writing out in words, and writing a multiplication sentence.
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Multiplying Whole Numbers and Fractions
For each picture shown below, write an addition and multiplication sentence.
1.
Unit is
Words: 3 groups of ____
Addition Sentence: ____ + ____ + ____ = ____
Multiplication Sentence: ____ x ____ = ____
____________________________________________________________________________
2.
Unit is
Words: ___ groups of ____
Addition Sentence: ____ + ____ + ____ = ____
Multiplication Sentence: ____ x ____ = ____
___________________________________________________________________________
3. Unit is
Addition Sentence: ____ + ____ + ____ + ____ + ____ = _____
Words: ___ groups of ___
Multiplication Sentence: ____ x ____ = _____
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Multiplying Whole Numbers and Fractions 2
For each picture shown below, write an addition and multiplication sentence.
1.
Unit is
Words: 5 groups of ____
Addition Sentence: ____ + ____ + ____ + ____ + ____ = ____
Multiplication Sentence: ____ x ____ = ____
____________________________________________________________________________
2.
Unit is
Words: ___ groups of ____
Addition Sentence: ____ + ____ + ____ + ____ = ____
Multiplication Sentence: ____ x ____ = ____
____________________________________________________________________________
3. Unit is
Words: ___ groups of ____
Addition Sentence: ____ + ____ + ____ = ____
Multiplication Sentence: ____ x ____ = ____
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Lesson 11: Multiply a Whole Number by a Unit Fraction Using a Number Line
Standard:
4.NF.4 Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b.
SMP 1 Make sense of problems and persevere in solving them.
SMP 3 Construct viable arguments and critique the reasoning of others
SMP 4 Model with mathematics
SMP 8 Look for and express regularity in repeated reasoning.
Lesson Learning Goals:
 Use a number line as a visual fraction model to represent the multiplication of
two whole numbers.
 Use a number line as a visual fraction model to represent a fraction a/b as a
multiple of 1/b.
Materials: handouts
Lesson:
Engage
(8-10 minutes)
Using the anchor chart created in Lesson 10, ask students how they would use a
number line to represent 3 x 7.
0
7
14
21
Ask students how they would use a number line to represent 1/6 + 1/6 + 1/6 which
equals 3/6. Use an anchor chart or board to record ideas.
Explore__________________________________________
________(20-25 minutes)
As a class, study the fraction 3/6 and review the unit fraction (1/6). A unit fraction is
written as a fraction where the numerator is one and the denominator is a positive
integer. Go over several different examples and then have students complete
Practice with Number Lines.
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Explain
(10-15 minutes)
After students complete Practice with Number Lines, bring class together. Have
students share answers and the strategies they used to solve them. Discuss any
difficulties that students had or that you saw during the explore stage.
Elaborate
(20-25 minutes)
Assign Practice with Number Lines 2. Choose 1-2 problems and have students
place answers on an anchor chart or Promethean board. Discuss any difficulties
that students had or that you saw.
Evaluation of Students
Formative: While students are working, observe them and pose questions to
check for mathematical understanding.
Summative: Collect Practice with Number Lines 2. Allow students to edit and
rewrite how they solved the problem.
Plans for Individual Differences
Intervention: If students are having difficulties, provide them with whole number
multiplication practice.
Extension: Have students create word problems that contain whole numbers and
units fractions. Then have them show how to find a solution by drawing a picture,
writing out in words, using a number line and writing a multiplication sentence.
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Practice with Number Lines
1. Use the number line to represent 2 x 6.
0
2. Use the number line to represent 4/7. What is the unit fraction?______
0
3. Use the number line to represent 2/9. What is the unit fraction?______
0
4. Use the number line to represent 7/5. What is the unit fraction?______
0
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Practice with Number Lines 2
1. How do you represent 2/3 on a number line?
0
What is the unit fraction of 2/3? _________
Words: ___ groups of ___
Multiplication Sentence: ____ x ____ = _____
2. How do you represent 3/10 on a number line?
0
What is the unit fraction of 3/10? ________
Words: ___ groups of ___
Multiplication Sentence: ____ x ____ = _____
3. How do you represent 5/3 on a number line?
0
What is the unit fraction of 5/3? ________
Words: ___ groups of ___
Multiplication Sentence: ____ x ____ = _____
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Lesson 12: Multiply a Whole Number by a Fraction Using Pictures and Fraction
Circles
Note: This lesson could take more than one day depending on the needs of your
students.
Standard:
4.NF.4 Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
d. Understand a multiple of a/b as a multiple of 1/b, and use this
understanding to multiply a fraction by a whole number.
e. Solve word problems involving multiplication of a fraction by a whole
number, e.g., by using visual fraction models and equations to represent the
problem.
SMP 1 Make sense of problems and persevere in solving them.
SMP 3 Construct viable arguments and critique the reasoning of others
SMP 4 Model with mathematics
SMP 8 Look for and express regularity in repeated reasoning.
Lesson Learning Goals:
 Use pictures and fraction circles to find the product of a whole number and a
fraction.
 Explain that the expression a x b can be read as “a groups of b”.
Materials: fraction circles, 2 handouts (Multiplying Fractions and 4 Word Problems)
Lesson:
Engage
(8 – 10 minutes)
Say, “Today we are going to be using pictures and fraction circles to multiply
whole numbers and fractions. I want you to draw a picture that you can use to
solve the following word problem. Write a number sentence for the problem.”
1) Riley wants to give five cookies to each of his three friends. How many
cookies will he need?
Ask students to share their pictures and number sentences. A possible picture
might look like this.
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Students’ pictures should show three groups of five cookies. (Teachers – it is
important for students to notice that this problem can be modeled using the
multiplication sentences: 3 x 5 = 15. Also help students make connections
between the words 3 groups of 5 and the mathematical expression 3 x 5.)
2) Write 2 x 4 =
on the board. Ask students to explain what the 2 and the 4
stand for in this problem. The 2 should stand for the number of groups and the 4
should stand for the number of objects in each group. Ask students to draw a
picture to find the answer 2 x 4 =
. (Teachers – walk around as students work
and ask 2-3 students to share their pictures. Have the students show and explain
their work. Be sure to emphasize that they draw 2 groups of 4 objects.)
Explore
(20 - 25 minutes)
Let students work in groups to solve the following problems with their fraction
circles. Have them also write a multiplication sentence that would answer this
problem.
1) Seth has several pizzas he wants to share with his friends. Seth wants to give
each of his 4 friends 2/5 of a pizza. How much pizza will he give away?
Teachers – The mathematical sentence will be 4 x 2/5 = 8/5 = 1 3/5 where the 4
represents the number of groups, the 2/5 represents the amount in each group,
and 8/5 or 1 3/5 represents the total amount of pizza given away. The students
should put 4 groups of 2/5 as shown below and the sum should 8/5 or 1 3/5.
Ask: How is this problem similar to the cookie problem we did previously?
2) Mariah uses 1/3 cup of brown sugar for each batch of chocolate chip
cookies she makes. How much brown sugar will she need if she makes 5
batches?
Teachers – Ask the students to solve the problem with their fraction circles.
They will need two sets of fraction circles to be able to find the answer. Ask
what the 5 means in the sentence and what the 1/3 means (5 groups of 1/3).
The final answer is 5/3 or 1 2/3. Have students record the answer both as an
improper fraction and a mixed number.
Have students complete Multiplying Fractions.
Explain
(10 – 15 minutes)
After students complete Multiplying Fractions, bring the class together. Have
students share answers and strategies they used to find them. Discuss any
difficulties that students had or that you saw during the explore stage.
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Elaborate
(20-25 minutes)
Assign Multiplying Fractions 2. Choose 1-2 problems and have students place
answers on an anchor chart or Promethean board. Discuss any difficulties that
students had or that you saw.
Evaluation of Students
Formative: While students are working, observe them and pose questions to
check for mathematical understanding.
Summative: Collect Multiplying Fractions 2. Allow students to edit and rewrite how
they solved the problem.
Plans for Individual Differences
Intervention: If students are having difficulties, provide them with whole number
multiplication practice.
Extension: Have students create word problems that contain whole numbers and
fractions. Then have them show how to find a solution by drawing a picture,
writing out in words, and writing a multiplication sentence. They may use fraction
circles if needed.
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Multiplying Fractions 2
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Lesson 13: Multiply a Whole Number by a Fraction Using Fraction Circles, Pictures,
and Mental Images
Standard:
4.NF.4 Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
b. Understand a multiple of a/b as a multiple of 1/b, and use this
understanding to multiply a fraction by a whole number.
c. Solve word problems involving multiplication of a fraction by a whole
number, e.g., by using visual fraction models and equations to represent the
problem.
SMP 1 Make sense of problems and persevere in solving them.
SMP 3 Construct viable arguments and critique the reasoning of others
SMP 4 Model with mathematics
SMP 8 Look for and express regularity in repeated reasoning.
Lesson Learning Goals:
 Multiply a whole number and a fraction using fraction circles, pictures, and
mental images.
Materials: fraction circles, handouts (3 pages)
Lesson:
Engage
(8-10 minutes)
Write 2/3 on the board. Ask the students to picture this fraction in their minds. Ask a
few students to describe what they are picturing. Write a 5 and a multiplication
sign before the 2/3. Ask the students to picture what the following statement
would represent - 5 x 2/3.
Ask the students to draw a picture of this problem to find the product in their
journal. Walk around as students work and ask two or three students to share their
pictures. Have the students show and explain their work. Be sure to emphasize that
they draw 5 groups of 2/3. Sample student work for 5 groups of two-thirds below.
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Explore
Assign Multiplying Fractions.
(20-25 minutes)
Explain
(10-15 minutes)
After students finish Multiplying Fractions, bring class together. Have students share
answers and strategies they used to find them. Discuss any difficulties that students
had or that you saw during the explore stage.
Elaborate
(20-25 minutes)
Assign Multiplying Fractions 2. Teachers – use your discretion about which
problems to assign. After students finish, choose 1-2 problems and have students
place answers on an anchor chart or Promethean board. Discuss any difficulties
that students had or that you saw.
Evaluation of Students
Formative: While students are working, observe them and pose questions to
check for mathematical understanding.
Summative: Collect the 4 word problems Multiplying Fractions 2 handout. Allow
students to edit and rewrite how they solved the problem.
Plans for Individual Differences
Intervention: If students are having difficulties, provide them with whole number
multiplication practice.
Extension: Have students create word problems that contain whole numbers and
fractions. Then have them show how to find a solution by drawing a picture,
writing out in words, and writing a multiplication sentence. They may use fraction
circles or squares if needed.
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Multiplying Fractions 2 (page 1)
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Multiplying Fractions 2 (page 2)
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Lesson 14: Multiply a Whole Number by a Fraction Using a Number Line
Note: This lesson could take more than one day depending on the needs of your
students.
Standard:
4.NF.4 Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
b. Understand a multiple of a/b as a multiple of 1/b, and use this
understanding to multiply a fraction by a whole number.
c. Solve word problems involving multiplication of a fraction by a whole
number, e.g., by using visual fraction models and equations to represent the
problem.
SMP 1 Make sense of problems and persevere in solving them.
SMP 3 Construct viable arguments and critique the reasoning of others
SMP 4 Model with mathematics
SMP 8 Look for and express regularity in repeated reasoning.
Lesson Learning Goals:
 Use number lines to multiply a whole number by a fraction.
 Use partitioning to find a fractional part of a whole number.
Materials: handouts
Lesson:
Engage
(8-10 minutes)
Pose the following problem to students: Tasha makes $12 an hour babysitting
several children. How much does she make if she babysits these children for 3
hours? Write a multiplication sentence that describes how to find the answer.
3 x 12 = 36
number of groups
(hours)
number in each group
($/hour)
amount earned in
3 hours ($)
Teacher – The previous two lessons developed the meaning for the factors in a
multiplication sentence. The first factor is typically the number of groups. The
second factor is the number of objects in each group. Many real world situations
make it difficult to have a fractional number of groups. Show the students how
they can solve the word problem using a number line.
0
12
24
36
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Explore
(20-25 minutes)
Ask students how the number line can be used to show how much Tasha will earn
if she babysits 2 hours or 7 hours. Have the students use their pencils or fingers to
trace along a number line. Have them start tracing at 0 and make jumps for each
hour worked. The number line also demonstrates how multiplication can be
demonstrated by repeated addition.
Ask students to use number line 1 from Multiplying Fractions on Number Lines to
find the amount of money Tasha will earn if she babysits 5 hours. Encourage them
to draw in the jumps on the number line and then write a multiplication sentence
that solves the problem. The multiplication sentence should be 5 x 12 = 60. Use the
other number lines for more practice.
Teachers – Present the following problem:
Suppose Tasha works ¾ of an hour. Estimate in your head how much money Tasha
will earn. (Ask the students as a group whether this amount would be more than
$12 or less than $12. Ask a few students to explain their estimate.)Use the number
line to find the exact amount of money that Tasha will earn if she babysits ¾ of an
hour and write a multiplication sentence.
Sample student work is shown below.
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Ask several students to come to the board and show how they used the number
line to solve this problem. Be sure that they show how they divided their number
line and explain how they found their answer. Also make sure they explain the
meaning of each number in the multiplication sentence.
Sample student work is shown below:
Assign Part 2: Multiplying Fractions on Number Lines (teachers use your discretion).
Explain
(10-15 minutes)
After students finish Part 2: Multiplying Fractions on Number Lines, bring the class
together. Have students share answers and the strategies they use to find them.
Discuss any difficulties that students had or that you saw during the explore stage.
Elaborate
(20-25 minutes)
Assign Part 3: Multiplying Fractions on Number Lines. Teachers – use your discretion
about which problems to assign. After students finish, choose 1-2 problems and
have students place answers on the board. Discuss any difficulties that students
had or that you saw.
Evaluation of Students
Formative: While students are working, observe them and pose questions to
check for mathematical understanding.
Summative: Collect the 4 word problems Multiplying Fractions handout. Allow
students to edit and rewrite how they solved the problem.
Plans for Individual Differences
Intervention: Teacher may need to use smaller number lines.
Extension: Ask the students to use the number line II to find the exact amount of
money that Tasha will earn is she babysits 6 ½ hours.
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Part 2:
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Part 3:
Multiplying Fractions on Number Lines
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The following additional word problems are provided as needed.
4.NF.4c Solve word problems involving multiplication of a fraction by a whole
number, e.g., by using visual fraction models and equations to represent the
problem.
SMP 5 Use appropriate tools strategically.
Teacher may use this next handout from the website
http://www.k-5mathteachingresources.com/ for extra practice. This website has
great resources.
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15 friends want to order pizza for dinner.
They predict that each person will eat 1/3 of a
pizza. How many pizzas should they order?
Tom picked 10 plums from a tree in the garden
and ate 3/5 of them before lunch. How many
plums did he eat before lunch?
Mike rode his bicycle 6 kilometers to school.
He stopped at his friend’s house after 2/3 of
the total journey. After how many kilometers
did Mike stop?
On Monday Maria spent 3 hours reading. Of
the time she spent reading, ½ was spent reading
magazines. For how many hours did Maria read
magazines?
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Fractions
Common Core State Standards
A restaurant uses 1/3 cup of mayonnaise in
each batch of salad dressing. How many cups
of mayonnaise will be used in 7 batches?
Peter went to the store and bought 8 pounds
of apples. If ¼ of the apples were cooking
apples, how many pounds of cooking apples did
Peter buy?
A farmer owns 4 acres of farmland. He grows
potatoes on 3/8 of the land. On how many
acres of land does the farmer grow potatoes?
A cookie factory puts 3/6 of a barrel of flour
into each batch of cookies. How much flour will
the factory use in 7 batches?
Developed by SC Leaders of Mathematics Education 2011-2012
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Fractions
Common Core State Standards
Sue is baking cherry pies for a family dinner.
She expects that each of the 15 guests will
eat 1/5 of a cherry pie. How many cherry pies
should she bake?
Chris had 25 marbles in his collection. He gave
2/5 of his marbles away to his friends. How
many marbles did Chris give away?
Two runners ran for 9 kilometers. They stopped
for water after 2/3 of the run. After how many
kilometers did the runners stop for water?
Mrs. Smith spent 8 hours in the kitchen. Of
the time she spent in the kitchen, ¾ was spent
making bread. For how many hours did Mrs.
Smith make bread?
Developed by SC Leaders of Mathematics Education 2011-2012
104
Fractions
Common Core State Standards
Appendix A
Sample Anchor Chart
One-half
1
2
4
 8 
Developed by SC Leaders of Mathematics Education 2011-2012
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