Strand 3: Number

• Half a cup of tea• Quarter of an hour• Three quarters full

Fractions are Easy!

00000 =2 4 of ?3 5

Point to Ponder!

Chief Examiners’ Reports

LC HL - 2000

"Candidates answering showed weaknesses in the following specific areas: handling fractions"

Chief Examiners’ Reports

LC HL - 2000

"Candidates answering showed weaknesses in the following specific areas: handling fractions"

Chief Examiners’ Reports

LC HL - 2005

“Incorrect cancelling in algebraic fractions.”

Chief Examiners’ Reports

LC HL - 2005

“Incorrect cancelling in algebraic fractions.”

Chief Examiners’ Reports

LC HL - 2005

“Incorrect cancelling in algebraic fractions.”LC FL - 2005

“Fractions again caused problems for many.” “..it would appear that fractions are not well understood.”

Chief Examiners’ Reports

LC FL - 2005

“Fractions again caused problems for many.” “..it would appear that fractions are not well understood.”

Chief Examiners’ Reports

LC FL - 2005

“Fractions again caused problems for many.” “..it would appear that fractions are not well understood.”

JC OL - 2006

“Common errors were: mishandling either fraction or square root; adding before multiplying.”

Chief Examiners’ Reports

JC OL - 2006

“Common errors were: mishandling either fraction or square root; adding before multiplying.”

Chief Examiners’ Reports

JC OL - 2006

“Common errors were: mishandling either fraction or square root; adding before multiplying.”

JC HL - 2006

“Areas of weakness in candidate performance:simplifying algebraic fractions.”

Chief Examiners’ Reports

JC HL - 2006

“Areas of weakness in candidate performance:simplifying algebraic fractions.”

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Cara has 4 pizzas for her party.3She decides that a serving will be of a pizza.5

2 2Will she get 6 or 6 servings from the 4 pizzas?3 5

Unifix Cubes

Cara has 4 pizzas for her party.3She decides that a serving will be of a pizza.5

2 2Will she get 6 or 6 servings from the 4 pizzas?3 5

Unifix Cubes

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Fraction Wall

Fraction Strips & Fraction Circles

Spoken Symbols

ManipulativeModels

Real Life Situations

Written Symbols

Pictures

Lesh’s Translation Model

Prior knowledge

Overview of Fractions

Diagnostic Test

Prior knowledge

Overview of Fractions

00000

3Which of these rectangles has shaded in? 4

Is it more than one rectangle?

A. B. C. D.

Answer:

Diagnostic Test

00000

What fraction of the fraction circle is marked ‘f’?

a

bc

d

e

f

Diagnostic Test

Answer:

00000

(a) (b) (c) (d)

(a) (b) (c) (d)

The shaded part of this diagram could represent the numbers:1 5 15  2     12 8 4

Identify the unit in each case by drawing:     

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Equivalence

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Add Subtract

Equivalence

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Multiply & Divide

Add Subtract

Equivalence

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Decimals

Multiply & Divide

Add Subtract

Equivalence

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Percentages

Decimals

Multiply & Divide

Add Subtract

Equivalence

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Ratio

Percentages

Decimals

Multiply & Divide

Add Subtract

Equivalence

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

Ratio

Percentages

Decimals

Multiply & Divide

Add Subtract

Equivalence

Ordering

Partitioning

Diagnostic Test

Concepts

Operations

Connections

Prior knowledge

Overview of Fractions

AlgebraStrand 4

• Partitioning

• Ordering

• Equivalence

Reasons for rules ~Checking tools

Concepts

00000

3Is always the same?4

Concepts

00000

3Is always the same?4

Only if the unit is the same.

Concepts

00000

3Is always the same?4

Only if the unit is the same.

Concepts

00000

3Is always the same?4

Only if the unit is the same.

Concepts

3 5Which is bigger: or ?7 7

2 2Which is bigger: or ?3 5

5 3Which is bi

5 1 becau

gger:

se there are

or ?9 7

more 7 7

1 1 2 23 5 3 5

5 1 3 1 5 3 and 9 2 7 2 9 7

> ⇒ >

> < ⇒ >

Concepts

3 5Which is bigger: or ?7 7

2 2Which is bigger: or ?3 5

5 3Which is bi

5 1 becau

gger:

se there are

or ?9 7

more 7 7

1 1 2 23 5 3 5

5 1 3 1 5 3 and 9 2 7 2 9 7

> ⇒ >

> < ⇒ >

Concepts

3 5Which is bigger: or ?7 7

2 2Which is bigger: or ?3 5

5 3Which is bi

5 1 becau

gger:

se there are

or ?9 7

more 7 7

1 1 2 23 5 3 5

5 1 3 1 5 3 and 9 2 7 2 9 7

> ⇒ >

> < ⇒ >

Concepts

3 5Which is bigger: or ?7 7

2 2Which is bigger: or ?3 5

5 3Which is bi

5 1 becau

gger:

se there are

or ?9 7

more 7 7

1 1 2 23 5 3 5

5 1 3 1 5 3 and 9 2 7 2 9 7

> ⇒ >

> < ⇒ >

Concepts

3 5Which is bigger: or ?7 7

2 2Which is bigger: or ?3 5

5 3Which is bi

5 1 becau

gger:

se there are

or ?9 7

more 7 7

1 1 2 23 5 3 5

5 1 3 1 5 3 and 9 2 7 2 9 7

> ⇒ >

> < ⇒ >

Concepts

3 5Which is bigger: or ?7 7

2 2Which is bigger: or ?3 5

5 3Which is bi

5 1 becau

gger:

se there are

or ?9 7

more 7 7

1 1 2 23 5 3 5

5 1 3 1 5 3 and 9 2 7 2 9 7

> ⇒ >

> < ⇒ >

Concepts

Concepts

3 2Which is bigger: or ?5 3

=

=

> ⇒ >

3 6 1If we double , we get 15 5 52 4 1If we double , we get 13 3 3

1 1 2 3 .3 5 3 5

Whole Number Multiplication

Whole Number Multiplication

Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?

Whole Number Multiplication

Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?

Whole Number Multiplication

Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?

Whole Number Multiplication

Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?

Whole Number Multiplication

Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?

• 3 multiplied by 4 or 3 times 4

Whole Number Multiplication

Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?

• 3 multiplied by 4 or 3 times 4

• 4 + 4 + 4 (Repeated Addition)

Whole Number Multiplication

Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?

• 3 multiplied by 4 or 3 times 4

• 4 + 4 + 4 (Repeated Addition)

• 3 groups of 4

Whole Number Multiplication

Multiplying a Whole Number by a Fraction

Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3

How is this like the last problem? Draw a picture to model this situation.

(a)(b)

(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3

Multiplying a Whole Number by a Fraction

Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3

How is this like the last problem? Draw a picture to model this situation.

(a)(b)

(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3

Multiplying a Whole Number by a Fraction

Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3

How is this like the last problem? Draw a picture to model this situation.

(a)(b)

(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3

Multiplying a Whole Number by a Fraction

Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3

How is this like the last problem? Draw a picture to model this situation.

(a)(b)

(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3

Multiplying a Whole Number by a Fraction

Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3

How is this like the last problem? Draw a picture to model this situation.

(a)(b)

(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3

Picture/Words

Multiplication sentence

...groups of...

Repeated addition

Poster/ White Boards

00000If I multiply what incorrect answer do you think I might get?

Pair Work

×243

00000

Pair Work

4 2 8 1 3 3How come 4 2 24 ?1 3 3

× = + =

and

00000

Fraction x Fraction

2Cara has of her rectangular birthday cake left 5

over from her party. 3She ate of the leftover cake.4

How much of the full cake was this?