Fractions Louise Addison. Fraction starter One player is dots the other is crosses Number line from...

Fractions

Louise Addison

Fraction starter

• One player is dots the other is crosses• Number line from 0 to 6• Roll 2 dice and form a fraction, place this on

number line (use materials if necessary)• Aim is to get 3 marks uninterrupted by your

opponent’s marks on the number line.• If a player chooses a fraction that is equivalent

to a mark that is already there they lose a turn.

Fractions in the new curriculum Level 4

• Number strategies and knowledge• NA4-2 Understand addition and subtraction of fractions, decimals, and

integers.• NA4-3 Find fractions, decimals, and percentages of amounts expressed

as whole numbers, simple fractions, and decimals.• NA4-4 Apply simple linear proportions, including ordering fractions.• NA4-5 Know the equivalent decimal and percentage forms for everyday

fractions.• NA4-5 Know the relative size and place value structure of positive and

negative integers and decimals to three places.

• Probability• S4-4 Use simple fractions and percentages to describe probabilities.

Level 5

• Number strategies and knowledge• NA5-3 Understand operations on fractions, decimals, percentages, and

integers.• NA5-4 Use rates and ratios.• NA5-5 Know commonly used fraction, decimal, and percentage

conversions.

• Patterns and relationships• NA5-8 Generalise the properties of operations with fractional numbers

and integers.

• Probability• S5-4 Calculate probabilities, using fractions, percentages, and ratios.

Level 6

• Number strategies and knowledge• NA6-2 Extend powers to include integers and

fractions.

• Patterns and relationships• NA6-6 Generalise the properties of operations with

rational numbers, including the properties of exponents.

A closer look…

• NA4-2 Understand addition and subtraction of fractions, decimals, and integers.

• NA4-3 Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.

• NA5-3 Understand operations on fractions, decimals, percentages, and integers.

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QuickTime™ and a decompressor

are needed to see this picture.

Provide sufficient

opportunities to learnQuickTime™ and a decompressor

Facilitate shared

learning

Inquire into the teaching-

learning relationship

Create a supportive

learning environment

E-Learning and

Pedagogy

are needed to see this picture. Encourage reflective

thought and action

are needed to see this picture. Make connections to prior

learning and experience

Effective PedagogyTeacher actions promoting

student learning

Enhance the relevance

of new learning

5 views of fractions

73 over 73 : 7

3 out of 7 3 ÷ 7

3 sevenths

Implications for each…

• 3 pizzas are divided amongst 4 people. How much of a pizza does each person get?

3 out of 7

How would each of the views solve this problem?

3 over 7

3 out of 7

3 ÷ 7

3 sevenths

3 out of 7 of 42 - ok if factor of 7

0.42857143 of 42

3 over 7 of 42?

1 seventh is 6 so 3 sevenths are 18

3:7 split 12.6 : 29.4

3 over 7

3 out of 7

3 ÷ 7

3 sevenths

0.42857143 + 0.21428571

6 out of 21

3:7 + 3: 14 = 6:21

3 over 7 + 3 over 14

3 over 7

3 out of 7

3 ÷ 7

3 sevenths

1 - 0.42857143

1-3 out of 7 = -2/7 (or 2/7)

1 - 3:7?

1 - 3 over 7 = -2/7 (or 2/7)

1 - 3 sevenths = 4 sevenths

3 over 7

3 out of 7

3 ÷ 7

3 sevenths

0.42857143 of 0.333333333

3 out of 7 of 1 out of 3

3:7 of 1:3

3 over 7 of 1 over 3

3 sevenths of 1 third1 third, split into 7 pieces gives ‘21ths’

So is three 21ths (3/21)

Key ideas of fractions

Fractional vocabulary

One half

One third

One quarter

Don’t know

Implications for teaching

• Use words (pattern and meaning needs to be taught)

• Always refer to the ‘whole’

• Modelling with covered unifix cubes

• This needs to be understood before decimal fractions can be taught

Provide sufficient

Facilitate shared

learning

Create a supportive

E-Learning and

Pedagogy

thought and action

student learning

of new learning

• NA5-5 Know commonly used fraction, decimal, and percentage conversions.

• 3 4

• 3 8

Provide sufficient

Facilitate shared

learning

Create a supportive

E-Learning and

Pedagogy

thought and action

student learning

of new learning

Use fraction strips to work out:

QUARTER

Two more questions…

1 THIRD

Also…

3÷ 4 =

QUARTER

HUNDREDTHS

Algebraic generalisation

• NA4-8 Generalise properties of multiplication and division with whole numbers.

• NA5-8 Generalise the properties of operations with fractional numbers and integers.

• NA6-6 Generalise the properties of operations with rational numbers, including the properties of exponents.

The EGG technique

Explain the strategy

Give other examples

Generalise using Algebra

Example:

Task 1

9 + 9 + 9 + 5 + 5 + 5 = 3 × 9 + 3 × 5

6 + 4 + 6 + 4 + 6 + 4 = 3 × (6 + 4) 9 + 9 + 9 – 5 – 5 – 5 = 3 × 9 – 3 × 5 6 – 4 + 6 – 4 + 6 – 4 = 3 × (6 – 4)

Lightbulb Moments

• How useful is the algebra this generates?• How could you use this in your classroom?• Who could you use this with?• What connections between ideas can you

make?• What thinking is involved?• What issues could arise...

Task 2

15 + 16 = 15 + 15 +1 = 2 15 + 1

19 + 20 = 20 + 20 – 1 = 2 20 – 1

9 + 10 + 11 = 9 + (9+1) + (9+2) = 3 9 + 3

9 + 10 + 11 = (10–1) + 10 + (10+1) = 3 10

9 + 10 + 11 = (11–2) + (11–1) + 11 = 3 11 – 3

Task 3

12 × 13 = 12 × 12 + 12 × 1 =12 2 + 12

13 × 12 = 13 × 13 – 13 × 1 =13 2 – 13

Task 4

7 × 32 = 7 × 30 + 7 × 2 7 × 39 = 7 × 40 – 7 × 1

Task 5

32 42 = 30 40 + 2 40 + 30 2 + 2 2

32 48 = 30 50 + 2 50 + 30 -2 + 2 -2

39 42 = 40 40 + -1 40 + 40 2 + -1 2

39 49 = 40 50 + -1 50 + 40 -1 + -1 -1

Task 6

9 9 9 9 9 9 9 = 97

92 95 = (9 9) (9 9 9 9 9) = 97

97 = 9 9 9 9 9 9 995 9 9 9 9 9 = 92

(94)3 = 94 94 94

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Fractions Louise Addison. Fraction starter One player is dots the other is crosses Number line from...

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