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TUNING UP FRACTIONS

LINDA WEST

LWEST@SINGAPOREMATHTRAINING.COM

SMARTTRAINING, LLC

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BIG IDEAS IN MATHEMATICS - COMPETENCIES

– Visualization– Making Connections (looking for patterns

in order to generalize)– Communication

Problem Solving Number Sense

• Based on the Theories of:– Jerome Bruner– Zoltan Dienes– Richard Skemp

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NUMERATOR

•An adjective•The counting number

•Tell how many•The “numberator”

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DENOMINATOR•A noun•A label•Tells what kind of unit

•The “deNAMEnator”•An Ordinal number

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What makes these types of numbers so difficult for

students?

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• Notation confusing• “bigger” numbers indicate “smaller amount”• Same number indicates different amount• Multiplication sometimes yields smaller

result while division sometimes yields a larger one

• The “whole” must always be held in the mind • Fractions, decimals and percents

traditionally taught as disparate topics

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FRACTIONS

FractionsQuantityProportion

Percentage is exclusively used for ProportionDecimal is exclusively used for Quantity

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Whole vs. Fractional Number Sense

Whole Numbers Fractional Numbers

Counting units Relationships to

other numbers• More/Less• 5 & 10 benchmarks• Part/Whole relationships

Counting units Relationships to

other numbers• More /Less• 0, ½, 1 Benchmarks• Part /Whole Relationships• Equivalency

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Whole Numbers Fractional Numbers• Equal units based on

10’s with relationships to each other

• Connect constructs with abstract notation and operations

• Whole universe of “units” that are less than 1, but are related to each other!

• Connect constructs with abstract notation and operations

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Keys to Understanding

• Concepts and Problem Solving before Rules and Drill

• Connections before Calculations

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The Dangerous Rush to Rules• None of the rules help students think about

the meaning• Rules give students no means of assessing

whether an answer is reasonable• Surface mastery of rules is quickly lost• Algorithm rules do not immediately apply to

every situation• Incredibly defeating for students

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2 ÷ = ?Please:

1. Solve.

2. Draw a picture.

3. Write a word problem.

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12 ÷ 4 = ____

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Our real question is:

How many halves are contained in 2 ¾?

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EXAMPLE:A recipe calls for 2cups of flour.

The only size measuring cup available is a cup measure. How many measuring cups are required to complete the recipe?

WOULD ANYONE LIKE TO SHARE THEIR WORD PROBLEM?

Jacqui ran 2 ¾ miles. This was ½ the distance that she runs each day. What is the total distance that Jacqui runs each day?

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NOW TRY THIS ONE2 ½ ÷ ⅓

• Use some of the pattern blocks on your table to solve this problem.• Verify with algorithm.• Write a word problem that can be solved using this equation.

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2 ½ ÷ 1/3

2 ½ ÷ 2/3

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FUN WITH FRACTIONS

 

+ =

Using digit tiles 1 – 9, once only in each equation, how many equations can you create?

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HOW DO WE DEVELOP VISUALIZATION SKILLS WITH FRACTIONS?

• Take out the yellow hexagon. Cover it with as many different pattern blocks as you can until the entire hexagon is filled.

• How can we express what we have done concretely in a number sentence?

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HOW DO WE USE

LogicTo explain

The fraction division algorithm?

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6 ÷ = ____WHAT DOES THAT MEAN?

Let’s go to an easier problem:6 ÷ 3 = ____ What does that mean?

• How can I read that differently?

• How many threes are in six?

• 6 is 3 of what number?

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6 ÷ = ____WHAT DOES THAT MEAN?

• Who can read the equation differently?

• How many ½’s are in 6?

• 6 is ½ of what number?

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DEVELOPING THE CONCEPT

• 6 ÷ 6 = ____

• 6 ÷ 3 = ____

• 6 ÷ 2 = ____ Does anyone see a pattern?

• 6 ÷ 1 = ____

• 6 ÷ = ____

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DO YOU SEE A PATTERN?

• How many halves are in 1? (2)• How many halves are in 2? (4)• How many halves are in 3? (6)• Let’s focus on thirds:

– How many thirds are in 1? (3)– How many thirds are in 2? (6)– How many thirds are in 3? (9)– How many thirds are in 4? (12)

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DO YOU SEE A PATTERN?

• LET’S FOCUS ON FOURTHS:– How many fourths are in 1? (4)– How many fourths are in 2? (8)– How many fourths are in 3? (12)– How many fourths are in 4? (16)

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DO YOU SEE A PATTERN?

• LET’S SWITCH IT UP A LITTLE BIT:– How many halves are in 1?– How many thirds are in 1?– How many fourths are in 1?– How many fifths are in 1?– How many halves are in 2?– How many thirds are in 2? etc., etc.

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AREA MODEL TO REAL LIFE

We need multiple embodiments. (Zhang 2012)

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DIVISION BY A FRACTION

• 1 =

0 1

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USING BAR MODELS WITH FRACTIONSDivide 3 by 2/3

2/3

There are four 2/3’s. and another half of a 2/3 in 3. So there are four-and-a-half 2/3’s in 3.

3 ÷ 2/3 = 3 x 3/2 = ?

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3 DIVIDED BY

• What is your word problem?

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A man travelled from Town X to Town Y in three days. He covered of the journey of the first day and of the remainder of the journey on the second day. What fraction of the total journey did he cover on the last day?

4

He travelled of the total journey on the last day.

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You painted of a room. of the area painted was green. What fraction of the room was green?

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A PICTURE IS WORTH …

• Find ¼ of 48.

• Find ¾ of 48.

• 1/3 of a number

is 48. Find the number.

• 4/5 of a number

is 48. Find the number.

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