This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.
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use multiplication to solve simple rate problems
Number Framework Links
Use this activity to help students consolidate and apply their knowledge of ratios and proportion (stages 6 and 7).
A calculator (optional)
FIO, Level 3+, Proportional Reasoning, Speed Read, page 10
In this activity, students need to calculate how long it will take Jack and each of his friends to read 630 pages, given information about their reading speeds. To help them, all speeds except Jack’s are visualised as double number lines.
The book suggests that a calculator could be used for this activity, but there is nothing in it that students couldn’t reasonably be expected to manage using familiar number strategies. If you decide that calculators are not to be used, you will need to make this clear from the beginning.
Each part of question 1 can be solved by working out the number of pages the person reads in 1 hour and dividing 630 by this number. But the parts can all be solved in a variety of ways, and your students are more likely to develop their proportional reasoning skills if you don’t give them a formula to follow. Here are examples of possible strategies:
Jack: 30 x 20 = 600 and 30 x 1 =30. Time taken = 21 hours.
Atama: 20 x 30 = 600 and 20 x 1 = 20, leaving 10, which is half of 20. Time taken = 31.5 hours.
Mikey: 45 x 10 = 450 and 630 – 450 = 180, 45 x 2 = 90, so 45 x 4 =180. Time taken = 14 hours.
Charlotte: 630 ÷ 9 = 70, so the time taken would be half of this, which is 35 hours.
Miranda: 630 ÷ 90 = 7, so she must take 7 x 4 = 28 hours.
Students with more advanced proportional reasoning skills may realise that it is possible to use the reading time for one person to find the reading time for another. For example, Mikey reads 90 pages in 2 hours, which is twice Miranda’s speed, so Miranda must take twice as long as Mikey: 2 x 14 = 28 hours. Jack reads 1 times as fast as Atama, so Atama must take 1 1/2 times as long as Jack: 1 1/2 x 21 = 31 hours.
The Answers demonstrate one way of solving question 2. Another is to enter the known information in a table and fill in the empty cells. The result is a triple number line, slightly disguised. A strip diagram would be equally suitable.
| 1 hour | 2 hours | 3 hours | 4 hours | |
| Jack | 30 | 60 | 90 | 120 |
| Atama | 20 | 40 | 60 | 80 |
| Mikey | 45 | 90 | 135 | 180 |
From the table, we can see that after 3 hours, Jack has read 90 pages, Atama has read 60, and Mikey has read 135. When Mikey is on page 180, Jack is on page 120, and Atama is on page 80.
Answers to Activity
1. Jack will take about 21 hours. (630 ÷ 30 = 21)
Atama will take about 31.5 hours. (630 ÷ 20 = 31.5)
Mikey will take about 14 hours. (630 ÷ 45 = 14)
Charlotte will take about 35 hours. (54 ÷ 3 = 18, 630 ÷ 18 = 35)
Miranda will take about 28 hours. (630 ÷ 90 = 7, 7 x 4 = 28)
2. a. Atama will be on page 60, and Mikey will be on page 135. (It would take 3 hours for Jack to reach page
90. In 3 hours, Atama would get to page 60 [3 x 20] and Mikey would get to page 135 [3 x 45].)
b. Mikey will take 4 hours to reach page 180. After 4 hours:
Jack will be on page 120 (4 x 30)
Atama will be on page 80 (4 x 20)