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Comparing and Ordering Fractions

Which is the larger number?

34

23

Which symbol >, < or = makes this true?

13

25

Place the following in order from smallest to largest.

35

810

34

712

12

Some Key Questions to Ask When Teaching Fractions

CMC South 2011: Barbara Schallau—Making Sense of Fractions and Operations with Fractions

Introduction of Fractions Bars ©Math Teachers Press, Inc.

Adding and Subtracting Fractions with Like Denominators

Problem Notes and drawing with fraction bars

26 + 3

6

410 + 3

10

410 - 3

10

1112 + 7

12

1112 - 7

12

Some More Key Questions to Ask When Teaching Fractions

Very important for students to create their own strips so they will be able to re-create it when needed.

Sample of completed multiples strips

Multiples Strips

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Multiples Strips

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

7 14 21 28 35 42 49 56 63 70 77 84 91 98 105

8 16 24 32 40 48 56 64 72 80 88 96 104 112 120

9 18 27 36 45 54 63 72 81 90 99 108 117 126 135

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

11 22 33 44 55 66 77 88 99 110 121 132 143 154 165

12 24 36 48 60 72 84 96 108 120 132 144 156 168 180

13 26 39 52 65 78 91 104 117 130 143 156 169 182 195

14 28 42 56 70 84 98 112 126 140 154 168 182 196 210

15 30 45 60 75 90 105 120 135 150 165 180 195 210 225

Barbara Schallau—Making Sense of Fractions and Operations with Fractions 5

Adding and Subtracting Fractions with Unlike Denominators

Problem Notes and drawing with fraction bars

13 +2

5

23 - 3

6

34 +2

3

34 −2

3


Using Multiples Strips for Equivalent Fractions © Math Teachers Press, Inc.

©M

ath Teachers Press, Inc.R

eproduction by any means strictly prohibited.

157NS 2.2

Adding and Subtracting Unlike Fractions

Lisa did !" of her weekly practice on

Monday. She did !# of her practice on

Tuesday. How much of her weekly

practice time has she completed?

Add.

Lisa completed $%& of her practice time.

13. Jane ate $%& of a candy bar and Ray

ate '( of the same candy bar.

Explain if this is possible.

____________________________

14. Jess ate !" of a pizza. Jack ate !) of

the same pizza. Dennis ate !* of the

same pizza. Explain if this is possible.

____________________________

1.

+

1__3

1__2

2.

+

1__5

1__2

3.

+

2__3 4.

+

3__10

2__5

5.

+

1__5

2__3

6.

+

1__5

3__4

7.

+

1__81__2

8.

+

1__9

2__3

9.

+

1__3

1__12

10.

+

3__10

4__10

11.

+

1__6

+, +- +&

3__4

12.

+

5__14

2__7

1__4

+ = ?1__4

1__6

1__4

1__6

3__12

2__12

5__12

The least common multiple for 4 and 6 is 12.

=

=+

$./

$%& $./ $0,

$./+$ +&

+$ 12 +& 3/

+$ +&

45

46 78

yes, because they each ate less than half no, because + + is more thanone-whole.

Objective: To add fractions with unlikedenominators.

Materials: Fraction Bars®, Inch Graph Paper(Master 3)

The following activities prepare students todiscover the patterns or rules for finding thelowest common denominator and changingfractions into equivalent fractions.Adding with Fraction Bars

Write on the board:You are making a pizza topping with 2⁄3 cup ofwhite cheese and 1⁄4 cup of yellow cheese. Howmuch cheese in all?Give each small group a set of fraction bars.

Allow time for students to discuss possible waysto solve the problem. Guide students to discoverthe Golden Rule of Fractions: You cannot add orsubtract fractions unless they are the same color(same denominator).

What common color can we change 2⁄3 and1⁄4 to? (orange) Find the equivalent fractions inorange. (2⁄3 = 8⁄12 and 1⁄4 = 3⁄12)

Show the addition on the board:

+ =

Addition with Multiple StripsDemonstrate the same problem with strips of

multiples. Have students prepare a multiplicationtable by taping together two copies of Inch GraphPaper (Master 3).

Cut the multiplication table horizontallyinto “multiple strips.” Use the strips to findthe lowest common denominator andequivalent fractions for each pair offractions. To add 2⁄3 + 1⁄4, place the 2 multiplestrip over the 3 multiple strip and the 1multiple strip over the 4 multiple strip.

11!12

3!12

8!12

What is the smallest common number inthe bottom rows of each fraction in the 3row and the 4 row? (12) What number isabove the 12 in the “2” row? (8) 8⁄12 is anothername for 2⁄3.

What number is above the 12 in the “1”row? (3) 3⁄12 is another name for 1⁄4.

Write on the board:

=

+ =

Have students use fraction bars for problems1–4 and multiple strips for problems 5–8.

Skill Builders Pages 89–93 (Pre 7NS 2.2-1 to 2.2-5)

11!12

3!12

1!4

8!12

2!3

15

1 1 2 3 4 5 6 7 8 9

2 2 4 6 8 10 12 14 16 18

3 3 6 9 12 15 18 21 24 27

4 4 8 12 16 20 24 28 32 36

5 5 10 15 20 25 30 35 40 45

6 6 12 18 24 30 36 42 48 54

7 7 14 21 28 35 42 49 56 63

8 8 16 24 32 40 48 56 64 72

9 9 18 27 36 45 54 63 72 81


Division

What does 12 ÷ 4 mean?

What does 23 ÷ 1

6 mean?

Problem In words and then drawing with fraction bars Solution

23 ÷ 1

12

510 ÷ 1

5

1÷ 1

2

1÷2

3


Problem In words and then drawing with fraction bars Solution

1÷ 3

4

810 ÷ 1

10

2÷ 1

6

14 ÷ 3

4


Mathematics | Grade 3 Number and Operations—Fractions5 3.NF Develop understanding of fractions as numbers. 1. Understand a fraction 1/b as the quantity formed by 1

part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram

by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on number line.

b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if

they are the same size, or the same point on a number line.

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Mathematics | Grade 4 Number and Operations—Fractions3 4.NF Extend understanding of fraction equivalence and ordering. 1. Explain why a fraction a/b is equivalent to a fraction

(n × a)/(n × b) 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.

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 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record results of comparisons with symbols >, =, or <, justify the conclusions, e.g., by using a visual fraction model.

Build fractions from unit fractions by applying & extending previous understandings of operations on whole numbers. 3. Understand a fraction a/b with a > 1 as a sum of fractions

1/b. a. Understand addition/subtraction of fractions as

joining/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.

d. Solve word problems involving addition &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.

3 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.


Mathematics | Grade 5 Number and Operations—Fractions 5.NF

Use equivalent fractions as a strategy to add and subtract fractions. 1. Add and subtract fractions with unlike denominators

(including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-‐lb sack of rice equally by weight, how many lbs of rice should each person get? Between what two whole numbers does your answer lie?

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a

partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.


5. Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one

factor on the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n b) to the effect of multiplying a/b by 1.

6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

a. Interpret division of a unit fraction by a non-‐zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 =1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit fractions by non-‐zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally?

1 Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

[email protected] Download handouts

for this presentation at...

http://my-ecoach.com/project.php?id=18773

Making Sense of Fractions and Operations with Fractions

Barbara Schallau [email protected]

East Side Union High School District in San José, CA

It is not important what the teacher covers, but what the student uncovers.

“Elementary school students’ knowledge of fractions and division predicts their algebraic knowledge in high school even after controlling

for family education and income, intellectual capacity, and knowledge

of whole number arithmetic.”

Siegler, R.S. et al (2012) Early predictors of high school mathematics achievement. Psychology Sci. 23 691-697

!

What is the correct order, from smallest to largest, of the three fractions below?

!!

!!!!!!!!Fifty percent of U.S. 8th graders could not correctly order these fractions on NAEP.

![Martin, W.G. et al., eds (2007) The Learning of Mathematics, 69th NCTM Yearbook]

!!!

27

, 1

12,

59 !

!

What is the approximate value, to the nearest whole number, of the sum !

!!!!!!!!!

A) 1 B) 2 C) 42 D) 45

1920 +

2325

On an international test, more than half of U.S. 8th graders chose 42 or 45.

[Steen, ASCD Educational Leadership, November, 2007]

What if students could make sense of the numbers or

visualize the fractions when asked the same question?

!

What is the approximate value, to the nearest whole number, of the sum !

!!!!!!!!!

A) 1 B) 2 C) 42 D) 45

1920 +

2325

8


Mathematics | Grade 3 Number and Operations—Fractions5 3.NF Develop'understanding'of'fractions'as'numbers.'1.## Understand#a#fraction#1/b"as#the#quantity#formed#by#1#

part#when#a#whole#is#partitioned#into#b"equal#parts;#understand#a#fraction#a/b"as#the#quantity#formed#by#a"parts#of#size#1/b.#

'''

2.## Understand#a#fraction#as#a#number#on#the#number#line;#represent#fractions#on#a#number#line#diagram.#a. Represent#a#fraction#1/b"on#a#number#line#diagram#

by#defining#the#interval#from#0#to#1#as#the#whole#and#partitioning#it#into#b"equal#parts.#Recognize#that#each#part#has#size#1/b"and#that#the#endpoint#of#the#part#based#at#0#locates#the#number#1/b"on#number#line.#

b. Represent#a#fraction#a/b"on#a#number#line#diagram#by#marking#off#a"lengths#1/b"from#0.#Recognize#that#the#resulting#interval#has#size#a/b"and#that#its#endpoint#locates#the#number#a/b"on#the#number#line.#

''''''''

3.## Explain#equivalence#of#fractions#in#special#cases,#and#compare#fractions#by#reasoning#about#their#size.#a.# Understand#two#fractions#as#equivalent#(equal)#if#

they#are#the#same#size,#or#the#same#point#on#a#number#line.#'

b.# Recognize#and#generate#simple#equivalent#fractions,#e.g.,#1/2#=#2/4,#4/6#=#2/3.#Explain#why#the#fractions#are#equivalent,#e.g.,#by#using#a#visual#fraction#model.#

c. Express#whole#numbers#as#fractions,#and#recognize#fractions#that#are#equivalent#to#whole#numbers.#Examples:#Express#3#in#the#form#3#=#3/1;#recognize#that#6/1#=#6;#locate#4/4#and#1#at#the#same#point#of#a#number#line#diagram.#

d. Compare#two#fractions#with#the#same#numerator#or#the#same#denominator#by#reasoning#about#their#size.#Recognize#that#comparisons#are#valid#only#when#the#two#fractions#refer#to#the#same#whole.#Record#the#results#of#comparisons#with#the#symbols#>,#=,#or#<,#and#justify#the#conclusions,#e.g.,#by#using#a#visual#fraction#model.#

'

5#Grade#3#expectations#in#this#domain#are#limited#to#fractions#with#denominators#2,#3,#4,#6,#and#8.

Grade 3 Number & Operations—FractionsStandard 1

Understand a (unit) fraction as the quantity formed by 1 part when one whole is partitioned into b EQUAL parts...

1b

Creating a Number Line from a Fraction Bar

0 1

One of the sections is one tenth since the whole was partitioned into 10 equal parts.

Grade 3 Number & Operations—Fractions

Standard 1

Understand a fraction as the quantity formed by 1 part when one whole is partitioned into 10 EQUAL parts...

110


Standard 1 (continued)

...understand a fraction as the quantity formed by m parts of size

1b

mb

Creating a Number Line from a Fraction Bar

0 1 110

210

310

410

510

610



...understand a fraction as the quantity formed by 6 parts of size

*Grade 3 expectations are limited to fractions with denominators 2, 3, 4, 6, and 8

610 1

10


Standard 3

...compare fractions by reasoning about their size.

Fraction Bars

!

Important Step with any Manipulative–Time to Investigate Attributes–

!

Fraction Bars...

What is the same and what is different about the sets of bars at your table?

Comparing FractionsWhich is the larger number? Why?

!

! 34

14


!

! 24

23

Grade 4 Number & Operations—FractionsStandard 2

Compare two fractions with different numerators and different denominators... by comparing to a benchmark fraction such as ½ …

*Grade 4 expectations are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Conversation begins around equivalent fractions

...Recognize that equivalences are only valid 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.


!

! 34

23

Comparing FractionsWhich symbol >, < or = makes this true?

!

13

25

Comparing Fractions!

Place the following in order from smallest to largest.

!

35

810

34

712

12

Key Questions to Ask When Teaching Fractions

✓ Is the fraction more than the number one or less than one?

✓ If it is less than one, is the fraction close to one or close to zero? Or is it close to one half?

Sample Question from Grade 3 Smarter Balanced Practice Test

Adding Fractions with Like Denominators

26 + 3

6 = Should the sum be a number more than one?

Less than one?

Adding Fractions with Like Denominators

26 + 3

6 =

26

36

56

410 + 3

10 =Should the sum be a

number more than one? Less than one?

410 + 3

10 =

Subtracting Fractions with Like Denominators

410 – 3

10 =

Adding & Subtracting Fractions Like Denominators

1112 + 7

12 = 1112 – 7

12 =

Adding Fractions Unlike Denominators

13 +2

5 =

Should the sum be a number more than one? Less than one?

THE Coolest Activity EVER!!

Very important for students to create their own strips.

They must be able to recreate them, if needed.

Multiples Strips!!

Students Create a Times Table!

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Students Create a Times Table!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 4 8 12 16 20

5

6

7

8

9

10

11

12

13

14

15

Students then Cut the Rows HORIZONTALLYMultiples Strips

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

7 14 21 28 35 42 49 56 63 70 77 84 91 98 105

8 16 24 32 40 48 56 64 72 80 88 96 104 112 120

9 18 27 36 45 54 63 72 81 90 99 108 117 126 135

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

11 22 33 44 55 66 77 88 99 110 121 132 143 154 165

12 24 36 48 60 72 84 96 108 120 132 144 156 168 180

13 26 39 52 65 78 91 104 117 130 143 156 169 182 195

14 28 42 56 70 84 98 112 126 140 154 168 182 196 210

15 30 45 60 75 90 105 120 135 150 165 180 195 210 225

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

7 14 21 28 35 42 49 56 63 70 77 84 91 98 105

8 16 24 32 40 48 56 64 72 80 88 96 104 112 120

13 Becomes...

remember this was 1 times 6

remember this was 3 times 6

25 Becomes...

So 13 +2

5 =

becomes...

13

+ 25

515

+ 615

1030

+ 1230

1545

+ 1845

13

+ 25

515

+ 615

1030

+ 1230

1545

+ 1845

1115

2230

3345

1115

2230

3345

How can we have three answers for the same problem?

Try one!

23–3

6 =

34 –

23 =

34 +

23 =

or

How do you think multiplication strips will work in your classroom?

Things to remember: !

Students need to create their own strips. !

Key questions to get sense of answer.

Division with Fractions

What in the heck does mean? 23÷16

How in the world do we get “2” with ? !

1÷12

Division of Fractions

What do we mean when we are dividing two numbers?

!

!

Start with the familiar...

“What does 12 ÷ 4 mean?”

“What does 12 ÷ 4 mean?” !

One way: Partitioning 12 items into 4 groups. !

!

!

!

!

!

My answer tells me how many are in each group.

Text


Another way: Partitioning 12 items into groups of “size 4.”

!

!

!

!

!

My answer tells me how many groups of 4.

Text


Another way: Repeated subtraction. !

My answer tells me how many times I can subtract 4 items from 12 items.

!

Also, the inverse of multiplication.

4 12 12÷4 12/4 12

4Many symbols used:

So....What does mean? 23÷16

23÷ 1

6 16

23How many groups of size are in ?

Try These!

23 ÷ 1

12 =

510 ÷ 1

5 =

Try These!

1÷ 12

1÷23

Try These!

1÷ 3

4 =

810 ÷ 1

10 =

Try These!

2÷ 1

6 =

14 ÷ 3

4 =

65


Mathematics | Grade 5 Number and Operations—Fractions 5.NF

Use$equivalent$fractions$as$a$strategy$to$add$and$subtract$fractions.$1.## Add#and#subtract#fractions#with#unlike#denominators#

(including#mixed#numbers)#by#replacing#given#fractions#with#equivalent#fractions#in#such#a#way#as#to#produce#an#equivalent#sum#or#difference#of#fractions#with#like#denominators.#For$example,$2/3$+$5/4$=$8/12$+$15/12$=$23/12.$(In#general,$a/b$+$c/d$=$(ad$+$bc)/bd.)#

#

2.## Solve#word#problems#involving#addition#and#subtraction#of#fractions#referring#to#the#same#whole,#including#cases#of#unlike#denominators,#e.g.,#by#using#visual#fraction#models#or#equations#to#represent#the#problem.#Use#benchmark#fractions#and#number#sense#of#fractions#to#estimate#mentally#and#assess#the#reasonableness#of#answers.#For#example,$recognize$an$incorrect$result$2/5$+$1/2$=$3/7,$by$observing$that#3/7$<$1/2.#

#

Apply$and$extend$previous$understandings$of$multiplication$and$division$to$multiply$and$divide$fractions.$

3.## Interpret#a#fraction#as#division#of#the#numerator#by#the#denominator#(a/b$=#a$÷#b).#Solve#word#problems#involving#division#of#whole#numbers#leading#to#answers#in#the#form#of#fractions#or#mixed#numbers,#e.g.,#by#using#visual#fraction#models#or#equations#to#represent#the#problem.#For$example,$interpret$3/4$as$the$result$of$dividing$3$by$4,$noting#that$3/4$multiplied$by$4$equals$3,$and$that$when$3$wholes$are$shared#equally$among$4$people$each$person$has$a$share$of$size$3/4.$If$9$people#want$to$share$a$50Mlb$sack$of$rice$equally$by$weight,$how$many#lbs$of$rice$should$each$person$get?$Between$what$two$whole$numbers#does$your$answer$lie?##

$

4.## Apply#and#extend#previous#understandings#of#multiplication#to#multiply#a#fraction#or#whole#number#by#a#fraction.#a.# Interpret#the#product#(a/b)#×#q$as#a$parts#of#a#

partition#of#q$into#b$equal#parts;#equivalently,#as#the#result#of#a#sequence#of$operations#a$×#q$÷#b.#For$example,$use$a$visual$fraction$model$to$show$(2/3)$×$4$=$8/3,$and$create$a$story$context$for$this$equation.$Do$the$same$with$(2/3)$×$(4/5)$=$8/15.$(In$general,$(a/b)$×$(c/d)$=$ac/bd.)$

b.# Find#the#area#of#a#rectangle#with#fractional#side#lengths#by#tiling#it#with#unit#squares#of#the#appropriate#unit#fraction#side#lengths,#and#show#that#the#area#is#the#same#as#would#be#found#by#multiplying#the#side#lengths.#Multiply#fractional#side#lengths#to#find#areas#of#rectangles,#and#represent#fraction#products#as#rectangular#areas.#

#

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