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  Know common factors & multiples  Recall of multiplication and division facts

Fabulous Folding The first step might be to….. Using a piece of paper, fold it and mark the fold lines. 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

Then progress to double number lines

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? So 8 students travel by bus 24

These lines can be simplified to What stage do students need to be to do this? Multiplicative? 0 1

Fraction Problems Use a number line to solve/explain: of 9 of = 18

Hot Shots Book 7 P Extending Hot Shots P

% Problems 20% of 150 is 20% of is 30 % of 150 is 30

0% 20% 100%  20% of 150 is  Question (in context) The local dairy farmer is selling 20% of his herd of 150 cows. How many is he selling? Rewrite in maths language 150

How do we use the lines to get the answer? 0% 20% 100%  x 5 = divided by 5 = 30 0% 20% 100%  150 Find 10% : 150 divided by 10 So 10% = 15 So 20% =30

0% 20% 100%  x x x 2 15 x 2

There are 30 students in Room % are girls. How many girls are there in the class? What is the maths? (Mathematize it)

0% 40% 100%  40% of 30 is  30 __________________________________________

How do we use the lines to get the answer? 0% 20% 40% 100%  x 5 = % = 6 So 40% = divided by 5 = 6 0% 20% 40% 100%  30 Find 20% : 30 divided by 5 20% = 6 so 40% =12

3 x x 1010 x 4 3 x % 100% 0  30

30% of the swimming team are girls. If there are 18 girls. How many are in the team altogether? 18 is 30% of 

0 30% 100% 18 3 x x 10 3 x 6 6 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 0% 80%100% $350

× 3 or add to 80 = 120 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?

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  Or divide 400 by 10 (to get 10%) and multiply by 12. X 4 0%120%100% $400 

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%120% $540 

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 =  $ % of $1 200 = 

0 100% 112.5% 8 x 12.5% 9 x12.5% 1200  8 x x 150 Will you pay more or less? 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 Materials Images Knowledge 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

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? × 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?