Proportional Thinking
Using Double Number Lines
Jill Smythe
With thanks to Phil Doyle
The Power of Number Lines
 Fraction
Problems
 Algebra Problems
 Ratio Problems
 Percentage Problems
What do students need to be
able to do before we solve
percentage problems?
Have a knowledge of percentages
Recall of multiplication and division facts
Know common factors & multiples
Fabulous Folding
The first step might be to…..
Using a piece of paper, fold it and mark the fold lines.
0
1
2
What is the meaning of the denominator and numerator?
FIO Number Level 2-3 Page 18. (Teacher guide - notes)
Purpose - build up 2 double no lines
1
Then progress to double number
lines
0
1
4
1
2
3
4
1
To find the fraction of a quantity..
eg. one quarter of the class of 32 students travel to
school by bus,
how many of the class travel by bus?
0
1
4
8
1
2
3
4
1
16
24
32
So 8 students travel by bus
These lines can be simplified to
0
0
1
4
1
8
32
What stage do students need to be to
do this?
Multiplicative?
Fraction Problems
Use a number line to solve/explain:
2
•
of 9
3
2
• 3 of
= 18
Hot Shots
Book 7 P 47 - 49
Extending Hot Shots P 56 - 60
% Problems
 20%
of 150 is
 20%
of

% of 150 is 30
is 30
Question (in context)
The local dairy farmer is selling
20% of his herd of 150 cows.
How many is he selling?
20% of 150 is 

0%
20%
Rewrite in maths
language
150
100%
How do we use the lines to get the answer?
150 divided by 5 = 30

0%
20%
20 x 5 = 100
Find 10% :
20%
100%
150 divided by 10

0%
150
150
So 10% = 15
So 20% =30
100%
15 x 2

0%
15 x 10
150
20%
10 x 2
100%
10 x 10
There are 30 students in Room 16.
40% are girls.
How many girls are there in the
class?
What is the maths?
(Mathematize it)
40% of 30 is 

30
__________________________________________
0%
40%
100%
How do we use the lines to get the answer?
30 divided by 5 = 6

0%
20%
30
40%
100%
20 x 5 = 100
20% = 6
So 40% = 12
Find 20% :
30 divided by 5

0%
20%
30
40%
20% = 6
100%
so 40% =12
3x4
0

0
40%
10 x 4
3 x 10
30
100%
10 x 10
30% of the swimming team are
girls. If there are 18 girls . How
many are in the team altogether?
18 is 30% of 
0
3x6
18
6 x 10
30%
3 x 10
100%
10 x 10
Sarah went shopping for a new bike which cost
$350. When she got to town there was a sale and
she got 20% off the price. What did she pay?
Did she pay more or less?
How much less?
So instead of paying 100% she only paid?
Show all this on the number lines
$350
0%
80%
100%
Don’t forget to use “reverse”
problems.
Jim watched 2 thirds of a DVD. If he watched for 80
minutes, how long was the DVD?
1
3
2
3
40
80
× 3 or add to 80 = 120
120
The value of a $400 antique vase has been
increased by 20%. What is its value now?
What questions do we ask?
120% of 400 is 
0%
$400

X 4 100%
X 4120%
Or divide 400 by 10 (to get 10%) and multiply by 12.
After an increase in his weekly wage
of 20% Joe has $540.What was his
wage before the increase?
$540 is 120% of 

0%
100%
$540
120%
How about looking at GST?
Problem
 A plasma TV costs $1 200 before GST.
How much GST will have to be paid on this?

What is the maths?

112.5% of $1 200 = 
$1 200 + 12.5% of $1 200 = 

Will you pay more or less?
0
8 x 150
1200
9 x 150

100%
8 x 12.5%
112.5%
9 x12.5%
When GST is raised to 15%, how much more will you pay?
Moving to number properties
20% of 150 is ?
Now is time to link what they know about % with
decimal fractions.
How else can we write this?
What does 20% actually mean?
How could we do this without the number line?
For some students this stage will be a long time
coming! For others they will tell you.
Now might be the time to bring in a calculator
and some more “awkward” q’s
Teaching
progression
Start by:
Using materials, diagrams to
illustrate and solve the
problem
Progress to:
Developing mental images to
help solve the problem
Extend to:
Working abstractly with the
number property
Materials
Images
Knowledge

Sian has 2 packs of sweets, each
with the same number of sweets.
She eats 6 sweets and has 14 left.
How many sweets are in a pack?
A possible way…..

As double number line
14
6
Don’t forget to always use
“reverse” problems
Jim watched 2/3 of a DVD. If he watched for 80
minutes how long was the DVD?
1
3
2
3
8
0
120
40
× 3 or add to 80 = 120

Ameeta has 3 packs of biscuits, and 4
extra loose biscuits. Sam has one pack
of biscuits and 16 loose biscuits. If they
both have the same number of biscuits,
how many biscuits are in a pack?
Can you draw a picture to show the
problem?
4
16
28 is  % of 50
30% of the swimming team are
girls. If there are 18 girls . How
many are in the team altogether?