Cheryl RoddickAssociate Professor, San Jose State University

roddick@math.sjsu.edu

Christina Silvas-Centeno Mathematics Curriculum Specialist, SJUSD

Christina_centeno2000@yahoo.com

Developing Fraction Sense with

Pattern Blocks

Developing Fraction Sense with

Pattern Blocks

Agenda:

1) Introducing Fractions: “The Doorbell Rang” A) Skills Trace across NCTM and California Content Standards for Fractional understanding.

2) Pattern Blocks, Fractional Parts, and Fair Trades3) Arithmetic of Fractions4) Providing Meaning to the Algorithms

In this workshop, pattern blocks are used as the foundation for understanding all aspects of fraction concepts as well as computations. Participants will learn how to use the blocks to: 1) develop the idea of fractional parts, 2) make fair trades with the pattern blocks to create equivalent fractions, 3) solve real world problems involving addition, subtraction, multiplication, and division of fractions, and 4) provide meaning for the algorithms for arithmetic of fractions.

Models of Division

• Sharing Model (“How many in a group?”)

There are 9 cookies and 6 children. How many cookies does each child get?

• Measurement Model (Repeated Subtraction or “How many groups?”)

There are 12 cookies and you eat ½ cookies each day. How many days can you eat cookies?

Skills Trace through Content Standards for the conceptual development of fractions:

Equivalency, Equal Exchanges, Solving Problems

Skills Trace through Content Standards for the conceptual development of fractions:

Equivalency, Equal Exchanges, Solving Problems

Kindergarten: NS 1.1

1st Grade: NS 1.3

2nd Grade Content Standards2nd Grade Content StandardsNS

2nd Grade Content Standards2nd Grade Content StandardsNS

NS

NS

NS

NS

3rd Grade - Content Standards3rd Grade - Content StandardsNS

3rd Grade - Content StandardsNS

NS

NS

NS

3rd Grade - Content Standards3rd Grade - Content Standards

NS

4th Grade Content Standards4th Grade Content Standards

NS

NS

NS

4th Grade Content Standards4th Grade Content Standards

NS

MG

SDAP

5th Grade Content Standards5th Grade Content Standards

NS

NS

5th Grade Content Standards5th Grade Content Standards

NS

NS

NS

NS

NS

SDAP

6th Grade Content Standards6th Grade Content Standards

NS

NS

NS

NS

NS

6th Grade Content Standards6th Grade Content Standards

NS

NS

NS

NS

6th Grade Content Standards6th Grade Content Standards

NS

AF

AF

6th Grade Content Standards6th Grade Content Standards

SDAP

SDAP

SDAP

SDAP

The Basics of Fractions

• Let two yellow hexagons = 1.

• 1. What fraction is represented by each of the following pattern blocks?

a. b.

c. d.

Possible Extensions

• How do you represent 1 1/2? Draw three different ways.

• Create your own pattern block picture to represent 1, and make 1/2 of it blue. Draw both your original picture and what ½ looks like.

Finding the Fractional Part

Let the flower = 1.

1. Draw 1/2 of this shape.

2. Draw 1/4 of this shape.

3. Draw 2/4 of this shape.

4. Draw 1/8 of this shape.

Please do the following activity.

Let the king’s crown = 1.

• Make 1/3 of this shape with pattern blocks. ____________________

• Make 1/2 of this shape with pattern blocks. ____________________

• Make 2/3 of this shape with pattern blocks. ____________________

• Can you make 1/5 of the crown? Why? ___________________________

G2 NS 4.1, 4.2, 4.3 G3 NS 3.1

€

1

€

2

€

3

€

4

Please do the following activity.

Let the fish = 1.

• Make 1/5 of this shape with pattern blocks. ____________________

• Make 1/2 of this shape with pattern blocks. ____________________

• Make 3/10 of this shape with pattern blocks. ____________________

• Can you make draw 1/3 of the fish? Why? ___________________________

G2 NS 4.1, 4.2, 4.3 G3 NS 3.1

Multiplication of Fractions

1. Amber’s friend Jessica pulls out 12 candies from the bag, and divides them into four equal groups.

a. How many candies are in each group?

b. What fraction is one of the groups?

Multiplication

• Amber has 12 starburst candies and she eats 1/4 of them. How many candies has she eaten?

• Amber has 12 starburst candies and she eats 3/4 of them. How many candies has she eaten?

Addition and Subtraction of Fractions

(Estimation)

• José and Minh went to the store together and bought some candy. José bought 2/3 lb. of gummy worms and he gave 1/2 lb. of the candy to Minh. How much candy (in lbs.) does José have left?

The Doorbell Rang Revisited

Recall the story The Doorbell Rang, where cookies were being shared among friends. If you have more friends than cookies, then each friend can only have a fraction of a cookie. (Let’s say these are big cookies!) Using pattern blocks, solve the different scenarios below to determine how many friends you can give cookies to.

Number of cookies

Fraction of whole cookie to give to each friend

Number of friends you can give a portion to

3

3

3

6

5

3

1

2

1

€

1

6

€

2

3

€

5

6

Why Invert and Multiply?

• Use patterns blocks to model the solution to

5 ÷ 5/6.• You need to divide each hexagon into 6 equal

pieces this is 5 x 6 = 30.• Then take the pieces 5 at a time. How many

groups do you have? 30 ÷ 5 = 6. Thus the answer to 5 ÷ 5/6 is 5 x (6/5) = 30/5 = 6.

Where do we go from here?

• After developing a foundation for fractional understanding, students should be allowed to use these methods to solve fraction word problems.

In the 4th grade, of the students are boys. If there are 36 girls in 4th grade, how many students are there altogether?

€

3

7

G4 NS 1.7, SDAP 2.1 G5 NS 2.1, NS 2.3, NS 2.4, NS 2.5, SDAP 1.3

In the 4th grade, of the students are boys. If there are 36 girls in 4th grade, how many students are there altogether?

€

3

7

Girls: 36 =

Boys: X =

The group, set, or whole aredivided into 7 equal parts.

Do not forget that the parts are equal pieces!

36 = 4 units36 4 = 9 students

Therefore:1 unit = 9 students

€

3

7= 3 9 = 27

27 + 36 = 63 students

There are 63 students in the 4th grade.

Kiley has 1,200 markers. of them were red, of them were green, and of them remaining amount was blue. The rest of the remaining markers were purple. How many purple markers were there?

€

2

6

€

3

6

€

1

5

G4 NS 1.7, SDAP 2.1 G5 NS 2.1, NS 2.3, NS 2.4, NS 2.5, SDAP 1.3 G6 NS 1.2, NS 1.3, NS 2.1, AF 2.1, AF 2.2

Kiley has 1,200 markers. of them were red, of them were green, and of them remaining amount was blue. The rest of the remaining markers were purple. How many purple markers were there?

€

2

6

€

3

6

€

1

5

€

2

6Are red Are green

Equal Parts

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.€

3

6

€

1

6 = 200 markers€

6

6= 1200 markers

Kiley has 1,200 markers. of them were red, of them were green, and of them remaining amount was blue. The rest of the remaining markers were purple. How many purple markers were there?

€

2

6

€

3

6

€

1

5

€

2

6Are red Are green

€

1

5

Equal Parts

6 units = 1200 markers1 unit = 200 markers

5 units = 200 markers1 unit = 40 marks4 units = 160 markers

160 markers are purple

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

€

3

6

= 200

200 = 5 equal parts