TEACHING DIVISION OF FRACTIONS

Teruni Lamberg, Ph.D.University of Nevada, Reno

1

Discussion

What are some challenges you have experienced teaching division of fractions and multiplication of fractions?

2

Common Core Division of Fractions

• CCSS.Math.Content.5.NF.B.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.

• CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

3

How divison of fractions is typically learned

The invert and multiply method (Sharp &Adams, 2002; Ball, 2007).

Even though this method is widely taught and used, individuals still struggle with understanding why this method works Ball (1990;1997).

“Invert and multiply” method is taught as rote procedures in school (Borko, Eisenhart, Brown, Underhill, Jones & Agard, 1992)

4

The challenges of teaching division of fractions for conceptual understanding

Teaching division of fractions is difficult because visualizing division of fractions and reconciling the algorithm is not easy (Perlwitze, 2005; Ball 1990).

5

Misconceptions of division of fractions“When you divide whole

numbers, the resulting answer gets smaller”

4÷2=2

“When you divide fractions the resulting answer gets bigger.”

1/2÷1/4 =2

6

Unitizing and its role in solving fractions

Unitizing involves identifying the referent unit and re-conceptualizing the unit during the process of problem solving (Behr, Khoury, Harel Post and Lesh, 1997, 1993; Lamon, 1999).

7

Whole Number DivisionWhat do you do when you divide

whole number? 9 ÷3=Partitive Division

Measurement Division

8

Dividing Fractions1÷ 3 = ?

Think of a context problem?

How would you solve it?

9

Try this! Think of a context and how would you solve it?

3÷2 =?

10

Levels of Analysis and Sense making

Phase 1: Making Thinking Explicit

(Explaining Reasoning)

Phase 2: Analyzing Each Other's Solution

(Analyzing Low Level to More Sophisticated Reasoning)

Phase 3: Developing New Mathematical Insights

(Abstract Mathematical Concepts)

11

Source: PDToolkit Lamberg

Continuum: Levels of Understanding and Student Strategies

Inefficient strategies

Efficient Strategies

11111 11111 11111 11111 5+5+5+5

5x4=20 20 ÷5=4

 

Simpler Representations (Concrete)

Abstract Representations

** + **

2 + 2

2 apples and 2 apples two groups of two apples

two plus two

12

.

Whole Number Divided by Fractions

13

• CCSS.Math.Content.5.NF.B.7b 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.

http://www.youtube.com/watch?v=wzbZeALhnrs

Number linesWhy invert and multiply?

14

Fraction Divided by a Whole NumberCCSS.Math.Content.5.NF.B.7a 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.

Try this! ¼ ÷ 4

http://www.youtube.com/watch?v=Xda7b7wq2-w 15

16

17

18

Try this! Create a model, connect model to algorithm

½ ÷ 2 =

19

Fraction bar and Numberline Toolhttp://

www.mathsisfun.com/numbers/fraction-number-line.html

http://www.mathplayground.com/ThinkingBlocks/thinking_blocks_modeling%20_tool.html

20

Try This!

21

2 ½ ÷ ¼ = ?

2 ½ ÷ ¼

22

2/3÷2/6=

23

Area Model

¾ ÷2/3=

24

¾ ÷ 2/3=?Identify a/b in relation to whole

unit“What does ¾ look like?”

3/4

Identify the “unit of measure” :What does 2/3 of the same unit used in step 1 look like? 2/3 25

¾ (a/b) become new referent unitTransition to multiplicative thinkingPartition (a/b) into c/d“How many 2/3 size unitsare there in ¾?”¾ x3/2How many 1/8 in 2/3?How many 1/8 in ¾?How many 2/3’rd are there in ¾Figuring out common denominator

26

Answer in relation to c/d unit as a reference unit 1/1/8

27

28

Try the following problems! Draw Area Model and connect to algorithm.Try also using a number line?

4/6÷1/3=

1 ½ ÷ 1/8=

29

AREA MODEL FOR

MUTIPLYING FRACTIONS

Multiplication of Fractionshttp://

nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=search.html

32

Thank youTeruni Lamberg, Ph.D University of Nevada, Reno terunil@unr.edu

Blog: http://www.mathdiscussions.wordpress.com

33