Some problems associated with learning fractions Whole number confusion Language confusions Concrete models/abstract concepts Materials used for teaching.

Some problems associated with learning fractions

• Whole number confusion

• Language confusions

• Concrete models/abstract concepts

• Materials used for teaching

• Point of reference

• Restricted vision

• Restricted numbers

FRACTIONS – Whole number confusion

1. Transfer of whole number arithmetic processes to fractions

2+3 = 5

754332

75

43

32

FRACTIONS –Language confusions

e.g. the third element in a set versus a third of a sharing item

1st One third of the whole2nd 3rd

FRACTIONS – Development of the concept

Process Operation Concept

5

3

Process:- Fractions begins life in the process of sharing e.g.

FRACTIONS

5

3

•Begins as a process of counting or sharing

•Eventually welding together to a concept

•Adds, not replaces, a layer to the process of understanding.

FRACTIONS-Discrete vs continuous material

Circle two-sevenths of the faces

Mark off two -sevenths of the plank

FRACTIONS - Problems associated with materials used for teaching

Perceptual distractors

½ + ½ = 1

Why might this be a distractor?

Inconsistent cue (need to ignore all lines and reconstruct the diagram)

FRACTIONS – Perceptual distractors

Shade three quarters of this shape…

Complete cue

Incomplete cue (need to add lines)

Irrelevant cue (need to ignore some lines)

Err

or r

ate

inc r

eas e

s pr

o gre

ssiv

ely

Problems with materials used for teaching Structured vs Unstructured Materials

Many commonly used fraction ‘kits’

•are continuous quantity materials

•approximate the idea of equal units that students have so much trouble with

Context - what to focus on?

Draw lines to cut the cake into 21 equal pieces

Focus on the lines drawn to explore what misconceptions this student may have

Context - what is the big picture?

Colour 2/5 of these hearts

When interviewed, this student saw 5 hearts as the ‘whole unit’. She found 1/5 of each row and combined them to obtain 2/5

Fractions - open ended tasks:

Name a fraction•between ½ and 5/8•between ½ and ¾with a denominator of 12•between 0 and 1/3 with a numerator › 1

•The answer is 3½, what is the question?•The answer is 4x + 1 what is the

(x+1)(x-2) question?

FRACTIONS – what is it that makes each of these shaded areas worth one half of the

whole shape?

FRACTIONS – Point of reference

Approximation to zero, a half or one.

one thangreater bemust sum so

5

2.553

and 3

1.532

?53

32

Estimation(Note mix of decimal and fraction thinking)

FRACTIONS - Restricted vision

0 1 2

Typically work with fractions between 0 and 1

But what happens beyond 1?

FRACTIONS - Restricted numbers

Typically work with “round” fractions

43

,41

,32

,31

,21

May limit vision of fractions

Limited understanding of more “difficult looking” numbers such as 3/13 (found in probability of cards)

FRACTIONS –Multiplication

341

41

3

34

43

What do these mean visually?

The processes used for whole numbers apply equally to fractions

32

23

321

21

3

Two thirds

FRACTIONS –Multiplication

2/3

FRACTIONS –Multiplication

½

Two thirds x one half = two sixths

Represent these equations as diagrams

2½ ÷ ¼ =

3 ¼ ÷ ¾ =

FRACTIONS – Division “A NEW AND EXCITING WAY!!

105

106

21

53

105

106

56

Look at Using equivalent fractions find fractions with common denominator

Now introduce this new step!!

10 divided by 10 gives one …. so

511