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```					Decimals and Percentages

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

Waipuna Conference Centre
September 2011
How do you describe the
change from 2 to 10?
Views the change as an addition of 8

Multiplicative Thinking:
Views the change as multiplying by 5

To be a proportional thinker you need
to be able to think multiplicatively
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?
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)
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!

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)

What is 1 quarter as a decimal?
Make and compare decimals

• Which is bigger: 0.6 or 0.43?

• How much bigger is it?

Rank these questions in order of difficulty.

a)0.8 + 0.3,       Exchanging ten for 1
b)0.6 + 0.23       Mixed decimal place values
c)0.06 + 0.23,     Same decimal place values
Add and Subtract decimals (Stage 7)

Tidy Numbers                   Place Value

1.4 - 0.9

Standard written
form (algorithm)
Add and Subtract decimals (Stage 7)

Tidy Numbers                   Place Value

1.6 - 0.98

Standard written
form (algorithm)
Decimal Keyboard
When you multiply the

True        False

0.4 x 0.3
0.12     1.2    0.012
Multiplying Decimals by a whole
number(Stage 7)

Tidy Numbers                      Place Value

5 x 0.8
Proportional              Convert to a fraction,
Adjustment                        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   0.3          1

0.4

Ww
1
w
Using Arrays
0.4 x 0.3 = 0.12

0      0.3             1

0.12
0.4

Ww
1
w
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
0.3       0.12
1.3 x 1.4    1    0.4
1     0.4
1

0.3    0.3   0.12
0.7 x 1.6       1    0.6
= 1.12
0    0     0.0

0.7   0.7   0.42
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 at
Types of Percentage Calculations
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.

Using common conversions
25% of \$80            halves, thirds, quarters, fifths, tenths

35% of \$80

Using benchmarks like 10%, and ratio tables
FIO: Pondering Percentages NS&AT 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
100%   10%       30%     5%   35%
\$80    \$8        \$24     \$4   \$28

\$4

\$8 \$8 \$8
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)

Decimal Games and Activities
1.   First to the Draw        What is this game
2.   Four in a Row Decimals   aimed at?
3.   Beat the Basics
How could you
4.   Decimal Keyboard Games
5.   Target (Figure It Out)   easier / harder?
6.   Decimal Jigsaw
7.   Percents
8.   Decimal Sort

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 didn’t know before?

What parts of this workshop could you share back
Thought for the day

A DECIMAL POINT

When you rearrange the letters becomes

I'M A DOT IN PLACE
Problem Solving from nzmaths

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