Something to Think About

What are good activities for helping students to learn their basic facts?

Secondary Numeracy Project

Fraction, Decimal, Percents

Fraction Constructs

Fractions are needed when whole units are inadequate to get a job done.

Part-Whole: continuous & discrete

Continuous fraction model Fractions are met as cutting up an

entire object– e.g. one quarter of an apple

This single object soon extends to multiple objects to be shared– e.g. sharing two apples between 5

people Leads to the concept that

fractions are divisions– e.g.

5

252

Discrete fraction model Fractions are met through sharing a

collection of objects – e.g. 4 kids share 12 cookies

Problems initially lead to discrete, whole number answers

Later problems merge with the continuous as “remainder” objects are cut– e.g. 4 kids share 10 cookies

Handout: A Teaching Progression for Fractions

Stages – Continous & Discrete

Any surprises?

Look at Knowledge & Strategy Tests

Knowledge Test: Fractions Stage 4 (3 sec)

9: Write one half as a fraction. 10: Write one sixth as a fraction 11: Write one third as a fraction12: Write one quarter as a fraction

Stage 5 (5 sec)13: Which of these fractions is the smallest?14: Which of these fractions is the largest?

6

1

2

1

4

1

5

1

3

1

Stage 6 (5 sec)15: Which of these numbers is the

same as eight-sixths?

Stage 7 (5 sec)16: Which of these fractions is the

same as two-thirds?17: Which of these fractions is the

same as three-quarters?

14

21

6

21

8

6

9

3

12

9

12

8

8

6

6

2

Stage 818: Which of these fractions is the

smallest? (10 sec)

19: Which of these fractions is the largest? (10 sec)

16

5

10

3

5

2

10

7

9

5

3

2

Strategy Test: Fractions Stage 2-4 (materials)/Stage 5

(facts): This cake has been cut into thirds.

Here are twelve jellybeans to spread out evenly on top of the cake. You eat one third of the cake. How many jellybeans do you eat?

Stage 6:– What is of 28?

Stage 7:– 12 is of a number. What is the

number?

Stage 8: – There are 21 boys and 14 girls in Ana’s

class. What percentage of Ana’s class are boys?

– It takes 10 balls of wool to make 15 beanies. How many balls of wool does it take to make 6 beanies?

4

3

3

2

SNP Teaching Model

Use a concrete representation (Materials/Diagrams)

Encourage imaging and visualisation (Imaging)

Push to the inherent property and generalisation (Abstraction)

How can we use this with fractions?

Fraction Strips Work in groups of 8 To start each person has a different

colour paper Fold paper in eighths – make sure

everyone folds the same direction so that all eighths are the same size

Cut 1/8 strips (along fold lines) Share out your strips so that everyone

ends up with 8 strips of eight different colours

Cut strips as directed to match magnets

Uses of fraction strips

Order unit fractions and fractions with the same denominator and explain why they are larger or smaller – Which is bigger? Why?

Order fractions visually using materials, including improper fractions and explain what the numerator and denominator mean. – Make pairs of fractions. Which is bigger?

Which is the bigger fraction?

Many students miss the comparative nature of fractions – the relationship between the numerator and denominator. (Some research suggests that the failure to understand this is a reason why students have difficulty with fractions)

Complexifying the constructs What fraction is shaded pink?

3

4 48

64

48 = 3 x 16

64 = 4 x 16

What fraction is shaded pink?

Operational Construct ¼ of 36

– halve and halve again– relationship between times/division

and fractional strategies– ¾ of 36

0.25 of 36

25% of 36

Book 7

Changing up the way the question is asked.

A Teaching Progression for Percentages

Requires high levels of thinking

Decimals (decimal fractions)

Other handouts

Remember the long term goal – improving their algebraic thinking:

One thing you can do to help this every day is to ask questions in class:

Maths is......explaining...

What’s the answer?

How did you get the answer?

does it work?alwaysWhy

Fraction Teaching Resources Numeracy Folder – Number – Frac-Ratio

– Key Progressions for Fractions, Decimals, Percentages, Rates and Proportions

– A teaching progression for fractions– Intro to Fractions Worksheets– HIBS material– New Numeracy Books– No Nonsense Number Fractions, stages 5-6– NZMaths (DecimalFractionsOfASet,

AddingFractions) Rime 10 Ticks (Pack 4 Level 4 and Pack 5

Level 2)

Activity: In groups of 2 or 3

Create a brief outline for a lesson that gets students to learn without being taught an algorithm

How to convert to (and vice versa)

How to find 4 fractions equivalent to

How to simplify (a) (b)

3

11

3

4

7

2

72

3635

28

Activity 21) Pick one stage of the progression2) Think of as many relevant fraction

based activities as possible for that stage

Consult the Getting Started book Consult the activity books Consult Figure It Out Think of good activities that were used

in the past – and still fit Think them up