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Decimals and Percentages

Marie Hirst, Numeracy Facilitator,

m.hirst_at_auckland.ac.nz Mathematics Lead

Teacher Symposium Waipuna Conference

Centre September 2011

To be a proportional thinker you need to be able

to think multiplicatively

- How do you describe the change from 2 to 10?

Additive Thinking Views the change as an

addition of 8 Multiplicative Thinking Views the

change as multiplying by 5

Proportional Thinking

A sample of numerical reasoning test questions as

used for the NZ Police recruitment

- ½ is to 0.5 as 1/5 is to
- a. 0.15
- b. 0.1
- c. 0.2
- d. 0.5

- 1.24 is to 0.62 as 0.54 is to
- a. 1.08
- b. 1.8
- c. 0.27
- d. 0.48

- If a man weighing 80kg increased his weight by

20, what would his weight be now? - a. 96kg
- b. 89kg
- c. 88kg
- d. 100kg

Developing Proportional thinking Fewer than half

the adult population can be viewed as

proportional thinkers And unfortunately. We do

not acquire the habits and skills of proportional

reasoning simply by getting older.

Objectives

- Understand common decimal place value

misconceptions and how to address these. - Develop content knowledge of how to add, subtract

and multiply decimals. - Develop content knowledge of calculating

percentages - Become familiar with useful resources

- At what stage of the Number Framework are

decimals first introduced to students?

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Decimals

Decimals are special cases of equivalent

fractions where the denominator is always a power

of ten.

Misconceptions with Decimal Place Value How do

these children view decimals?

- Bernie says that 0.657 is bigger than 0.7
- (decimals are 2 separate whole number systems

separated by a decimal point, 657 is bigger than

7, so 0.675 is bigger than 0.7) - 2. Sam thinks that 0.27 is bigger than 0.395
- (the more decimal places, the tinier the

number becomes, because thousandths are really

small) - 3. James thinks that 0 is bigger than 0.5
- (decimals are negative numbers)
- Adey thinks that 0.2 is bigger than 0.4
- (direct link to fractional numbers , i.e. ½

0.2, ¼ 0.4) - 5. Claire thinks that 10 x 4.5 is 4.50
- (when you multiply by 10, just add a zero)

Addressing Misconceptions

Use materials to develop an understanding of

decimal tenths and hundredths place value

- Use decipipes, candy bars, or decimats to

understand how tenths and hundredths arise and

what decimal numbers look like

3 5

3 chocolate bars shared between 5 children.

30 tenths 5 0 wholes 6 tenths each 0.6

0

6

Now try this 5 4

Connecting the Place Value

5 4 1 whole 2 tenths 5 hundredths

1

2

5

- Understand how tenths and hundredths arise
- express remainders as decimals

- BIG IDEA
- The CANON law in our place value system is that

ONE unit must be split into TEN of the next

smallest unit AND NO OTHER!

Read, Say, Make

Using Decipipes Book 7 p.38-41 (Understanding

how tenths and hundredths arise)

- What is 1 quarter as a decimal?

View childrens response to this task (30.40

33.30 0r 34.40)

- Make and compare decimals

- Which is bigger 0.6 or 0.43?
- How much bigger is it?

- Add and subtract decimals

- Rank these questions in order of difficulty.
- 0.8 0.3,
- 0.6 0.23
- 0.06 0.23,

Exchanging ten for 1 Mixed decimal place

values Same decimal place values

Add and Subtract decimals (Stage 7)

Place Value

Tidy Numbers

1.4 - 0.9

Reversibility

Equal Additions

Standard written form (algorithm)

Add and Subtract decimals (Stage 7)

Place Value

Tidy Numbers

1.6 - 0.98

Reversibility

Equal Additions

Standard written form (algorithm)

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Decimal Keyboard

- When you multiply the answer always gets

bigger. - True False

0.4 x 0.3 Which is the correct answer? 0.12

1.2 0.012

Multiplying Decimals by a whole number(Stage 7)

Tidy Numbers

Place Value

5 x 0.8

Proportional Adjustment

Convert to a fraction, e.g. x 0.25 ¼ of

Standard written form (algorithm)

Multiplying a decimal by a decimal (Stage 8)

using Arrays 0.4 x 0.3

0.3

0

1

0.4

Ww w

1

Using Arrays 0.4 x 0.3 0.12

0.3

0

1

0.12

0.4

Ww w

1

1.3 x 1.4

1

0.4

1

0.3

1.3 x 1.4

1

0.4

1.82

1

0.4

1

0.3

0.12

0.3

1.3 x 1.4

0.4

1

1

0.4

1

0.12

0.3

0.3

0.7 x 1.6

1

0.6

1.12

0.0

0

0

0.42

0.7

0.7

Why calculate percentages?

- It is a method of comparing fractions by giving

both fractions a common denominator i.e.

hundredths. - So it is useful to view percentages as

hundredths.

Applying Percentages

- Types of Percentage Calculations at Level 4

(stage 7)

- Estimate and find percentages of amounts,
- e.g. 25 of 80
- Expressing quantities as a percentage
- (Using equivalence)
- e.g. What percent is 18 out of 24?

- Estimate and find percentages of whole number

amounts. - 25 of 80

Using common conversions halves, thirds,

quarters, fifths, tenths

35 of 80

Using benchmarks like 10, and ratio tables FIO

Pondering Percentages NSAT 3-4.1(p12-13)

Find __________ (using benchmarks and ratio

tables)

100

Find 35 of 80

100

80

80

Find 35 of 80

100

80

80

Find 35 of 80

100

80

Find 35 of 80

35 28

100

80

- Now try this
- 46 of 90

100

90

46 of 90

46 of 90

100 10 40 5 1 6 46

90 9 36 4.50 0.90 5.40 41.40

Is there an easier way to find 46?

Estimating Percentages

16 of 3961 TVs are found to be faulty at the

factory and need repairs before they are sent for

sale. About how many sets is that? (Book 8 p.26

- Number Sense)

About 600

Decimal Games and Activities

- First to the Draw
- Four in a Row Decimals
- Beat the Basics
- Decimal Keyboard Games
- Target (Figure It Out)
- Decimal Jigsaw
- Percents
- Decimal Sort

What is this game aimed at? How could you adapt

it to make it easier / harder?

http//mathsleadteachers.wikispaces.com/

http//teamsolutions.wikispaces.com/

- Objectives
- Understand common decimal place value

misconceptions and how to address these. - Develop content knowledge of how to add, subtract

and multiply decimals. - Develop content knowledge of calculating

percentages - Become familiar with useful resources.

What do you know now that you didnt know

before? What parts of this workshop could you

share back with your staff?

Thought for the day

- A DECIMAL POINT
- When you rearrange the letters becomes
- I'M A DOT IN PLACE

Problem Solving from nzmaths