Developing Fraction Sense with Pattern Blocks Cheryl Roddick Associate Professor, San Jose State University roddick@math.sjsu.edu Christina Silvas-Centeno Mathematics Curriculum Specialist, SJUSD Christina_centeno2000@yahoo.com

Agenda: Introducing Fractions: “The Doorbell Rang” A) Skills Trace across NCTM and California Content Standards for Fractional understanding. Pattern Blocks, Fractional Parts, and Fair Trades Arithmetic of Fractions 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. Intro 5 minutes - Cheryl

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? Cheryl will model the equal shares and measurement (equal groups) models 5 minutes

Chris - quick synopsis of the trace through NCTM standards and Content standards - 5 minutes

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 Standards NS

2nd Grade Content Standards NS NS NS NS NS

3rd Grade - Content Standards NS

3rd Grade - Content Standards NS NS NS NS

3rd Grade - Content Standards NS

4th Grade Content Standards NS NS NS

4th Grade Content Standards NS MG SDAP

5th Grade Content Standards NS NS

5th Grade Content Standards NS NS NS NS NS SDAP

6th Grade Content Standards NS NS NS NS NS

6th Grade Content Standards NS NS NS NS

6th Grade Content Standards NS AF AF

6th 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. chris model 1 hex. Run through the fair exchanges for one hexagon. Group does 2 hex. 10 minutes

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. Chris - Don’t do, but say it is extension and why

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. Chris model: 5 mins

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? ___________________________ Chris will model flower and participants will actually work on king’s crown - 15 minutes Students had trouble determining whether 1/5 of the crown was possible. G2 NS 4.1, 4.2, 4.3 G3 NS 3.1

45 minute mark - 1/2 of presentation

Please do the following activity. G2 NS 4.1, 4.2, 4.3 G3 NS 3.1 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? ___________________________ Not doing - just an example for an extension

Multiplication of Fractions 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? Cheryl - Sharing problem and then connect to second slide for fractions 5 minutes

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? Cheryl will have participants do this problem - 5 minutes. Students do not make the connection that the equal sharing problem is the same as the fraction problem. Not understanding the relationship between denominator and numerator. Students could find 1/3 of three objects but had difficulty finding 1/3 of 12 objects (in the sunburst problem).

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? Cheryl - Participants try this problem. REMEMBER - to ask participants to estimate before solving - walk them through it. 10 minutes - we are at 65 minutes

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. Cheryl - talk set up the table 10 minutes

Fraction of whole cookie to give to each friend Number of cookies Fraction of whole cookie to give to each friend Number of friends you can give a portion to 3 6 5 75 minutes

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. 5 minutes

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 In the 4th grade, of the students are boys. If there are 36 girls in 4th grade, how many students are there altogether? 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 In the 4th grade, of the students are boys. If there are 36 girls in 4th grade, how many students are there altogether? The group, set, or whole are divided into 7 equal parts. Girls: 36 = Boys: X = Do not forget that the parts are equal pieces! 36 = 4 units 36  4 = 9 students Therefore: 1 unit = 9 students = 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? 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? Are red Are green Equal Parts = 1200 markers = 200 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? Are red Are green 6 units = 1200 markers 1 unit = 200 markers Equal Parts 5 units = 200 markers 1 unit = 40 marks 4 units = 160 markers = 200 200 = 5 equal parts 160 markers are purple