Up and Off

For the ratios to have meaning, it is important that the students see the relationship between the speeds of the different aircraft. For example, if the ATR travels 10 kilometres in a given time, then the Saab will fly 8.8 kilometres in the same time. Expressed as a ratio, Saab : ATR is 8.8 : 10. The students will need to use this ratio to create an equation that calculates the Saab’s distance in relation to the ATR’s 200 kilometres. Ask the following questions to help the students to develop the numerical intuition they need to see that Saabs fly slower than ATRs:
“If an ATR flies 200 kilometres in a given time, will a Saab fly a longer or shorter distance in the same time?”
“Will it take a Saab more or less time than an ATR to fly 200 kilometres?”

Number lines are a useful graphical tool for helping students to visualise problems of this sort. The diagram below shows the graphical comparison of an ATR and Saab where the ATR has flown 10 kilometres.

The students can then be asked to mark the position of the Saab when the ATR has flown 20 kilometres. The ATR has flown twice as far, so the Saab will also have flown twice as far, 17.6 kilometres. The students can then extend their number line so that they can plot the ATR at 200 kilometres. Ask “Where will the Saab be then?” The ATR has flown 10 times as far as its last position (20 kilometres), so the Saab will have flown 10 times its previous distance, which is 176 kilometres.

By graphing the comparative paths of the two aircraft, the students will see how the ratio linking them is maintained. They could check this link by comparing the ratios at the 10, 20, and 200 kilometre marks. A table may help:

By converting this ratio into a fraction or a decimal, we can easily compare distances travelled by the ATR and the Saab in the same time period. For example, if the ATR has flown 350 kilometres, then the Saab has flown 8.8/10 of that distance or 350 x 0.88 = 308 kilometres. Number lines can be drawn to show the relationship between the speeds of the ATR and the other aircraft, or better still, the relative positions of all the planes can be plotted on a single horizontal bar chart (a kind of multiple number line).
A possible bar chart is shown here:

Question 2 raises the inverse relationship between speed and time. This can be a stumbling block for many students. You may need to help them gain the intuition to determine whether a quantity is being increased or decreased by a given ratio. They also need to know that multiplying by a number less than 1 decreases the original number and multiplying by a number more than 1 increases the original number.
The following questions may be helpful:

• Does a 737 fly faster than an ATR?
  (Yes, in the ratio 15.3 : 10.)
• In that case, will the ATR take more or less time to fly 640 kilometres than a 737? (More)
• Is the ratio 15.3 : 10 used to increase or decrease 50 minutes? (Increase)

So the time needed for an ATR to fly 640 kilometres is 50 x (15.3/10) minutes. Asking similar questions will show that a 767 will fly the distance in less time than the 737. So the 50 minutes is decreased by the ratio 15.3 : 16.4, or the 767 time = 50 x (15.3/16.4) minutes.

It may be helpful to illustrate this graphically by having the students plot a straight line graph comparing the flight times of two aircraft. This is shown below.

The line representing the 737 is drawn by linking the origin to the point (50,640). When the 737 is at 153 kilometres, the ATR will be at 100 kilometres. When the 737 is at 306 kilometres, the ATR will be at 200 kilometres. The line for the ATR can now be drawn by linking these two points with the origin and projecting the line past 640 kilometres. The time for this flight can be read off the horizontal axis. It should be approximately 77 = 50 x (15.3 ÷ 10) minutes.

The concept of increasing or decreasing by a given ratio is explored further in question 3. The ratio of the speed of the Beechcraft to the speed of the ATR is 9.8 : 10. This comes from our original information. An ATR flies faster than a Beechcraft, so the speed of 490 km/h must be increased by the given ratio (multiplied by a factor of more than 1). The speed of the ATR is 490 x (10/9.8) = 500 km/h. Similarly, a Saab flies slower than a Beechcraft, so 490 kilometres will be decreased by the ratio 8.8 : 9.8. That is, the speed of the Saab is 490 x (8.8/9.8) = 440 km/h.

Another approach to this type of problem is as follows:
For every 9.8 kilometres the Beechcraft flies, the Saab will fly 8.8 kilometres. If the Beechcraft flies 490 kilometres in 1 hour, how many units of 9.8 kilometres has it flown? (490 ÷ 9.8 = 50) So, in 1 hour, the Saab will fly 50 units of 8.8 km, which is 50 x 8.8 = 440 kilometres.

Answers to Activity


(Times have been rounded to the nearest min, distances to the nearest km, and speeds to the nearest km/h. Given information is shaded.)