Decimals and Percentages

Marie Hirst, Numeracy Facilitator, m.hirst@auckland.ac.nz

Mathematics Lead Teacher SymposiumWaipuna Conference CentreSeptember 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. 1.08b. 1.8c. 0.27d. 0.48

If a man weighing 80kg increased his weight by 20%, what would his weight be now?

a. 96kgb. 89kgc. 88kgd. 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?

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?

1. 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)

4. 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 decipipes, candy bars, or decimats to

understand how tenths and hundredths

arise and what decimal numbers ‘look like’

Use materials to develop an understanding of decimal tenths and hundredths place value

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

1 2

5 ÷ 4 = 1 whole + 2 tenths + 5 hundredths

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

What is 1 quarter as a decimal?

Using Decipipes: Book 7 p.38-41

(Understanding how tenths and hundredths arise)

View children’s 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.

a)0.8 + 0.3,

b)0.6 + 0.23

c)0.06 + 0.23,

Exchanging ten for 1

Mixed decimal place values

Same decimal place values

Add and Subtract decimals (Stage 7)

1.4 - 0.9

Tidy Numbers Place Value

Equal Additions Reversibility

Standard written form (algorithm)

Add and Subtract decimals (Stage 7)

1.6 - 0.98

Tidy Numbers Place Value

Equal Additions Reversibility

Standard written form (algorithm)

Decimal Keyboard

When you multiply the answer always gets bigger.

True False

0.4 x 0.3Which is the correct answer?

0.12 1.2 0.012

Multiplying Decimals by a whole number(Stage 7)

5 x 0.8

Tidy Numbers Place Value

Proportional Adjustment

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

Standard written form (algorithm)

Ww

w

0 1

1

0.3

0.4

Multiplying a decimal by a decimal (Stage 8) using Arrays

0.4 x 0.3

Ww

w

0 1

1

0.3

0.4

Using Arrays0.4 x 0.3 = 0.12

0.12

1.3 x 1.4 1

1

0.3

0.4

1.3 x 1.4 1

1

0.3

1

0.3

0.4

0.12

= 1.82

0.4

1.3 x 1.4 1

0.3

1

0.3

0.4

0.12

0.4

1

0.7 x 1.6

0.7 0.7

0.0

0.42

0.6

0

1

0= 1.12

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.

Why calculate percentages?

=

Applying PercentagesTypes 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

35% of $80

Using benchmarks like 10%, and ratio tablesFIO: Pondering Percentages NS&AT 3-4.1(p12-13)

Using common conversions halves, thirds, quarters, fifths, tenths

100%

Find __________ (using benchmarks and ratio tables)

100%

$80

Find 35% of $80

$80

100%

$80

Find 35% of $80

$80

100%

$80

Find 35% of $80

100%

$80

Find 35% of $80

$8

10%

$8

35%

$28

$4

5%

$4

$8$8

30%

$24

Now try this…46% of $90

100%

$90

46% of $90100% 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%?

46% of 90

Estimating Percentages16% 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

1. First to the Draw2. Four in a Row Decimals3. Beat the Basics4. Decimal Keyboard Games5. Target (Figure It Out)6. Decimal Jigsaw7. Percents8. 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/

What do you know now that you didn’t know before?

What parts of this workshop could you share back with your staff?

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.

Thought for the day

A DECIMAL POINT  

When you rearrange the letters becomes

I'M A DOT IN PLACE

Problem Solving from nzmaths