Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Year 8, Unit 1, lesson 6) A proportional graph passes through (3, 18). What is k in y = kx?
Answer
6. k = y ÷ x = 18 ÷ 3. - (Lesson 2) A car travels 150 km in 2.5 hours. What is its speed?
Answer
60 km per hour. 150 ÷ 2.5 = 60. - (Year 7, Unit 6, lesson 6) On a line graph, a point is 4 across and 20 up. Write it as a coordinate pair.
Answer
(4, 20). The number across comes first.
Words used in this lesson
- rate of change
- how much one quantity changes for one step of another. For example: 20 litres per minute.
- steep
- rising quickly for each step across. For example: the steeper line is the faster.
- rise
- how far a line goes up between two points. For example: 15 litres.
- run
- how far a line goes across between two points. For example: 3 minutes.
- constant
- staying the same. For example: a constant speed.
- origin
- the point (0, 0). For example: where the axes meet.
What steeper means
Two friends set off at the same time and travel steadily. After 3 hours Kofi is 36 km from the start and Ana is 27 km from the start. Predict first: draw each journey as a straight line from the origin, with hours across and kilometres up. Which line is steeper? How many kilometres further does that friend go in each hour?
The figure draws two lines through the origin. Slide the second point along line A and watch the rise and the run.
Line A rises 30 km over a run of 2 hours: 30 ÷ 2 = 15 km per hour. Line B is 10 km per hour, so line A is steeper.
Show me the steps
- Line A at 15 km per hour, run 2 hours: rise 15 × 2 = 30 km, and 30 ÷ 2 = 15. Line B at 10 km per hour is less steep.
- Same line, run 4 hours: rise 15 × 4 = 60 km, and 60 ÷ 4 = 15. A longer run, a longer rise, the same rate.
- Line A at 5 km per hour, run 3 hours: rise 5 × 3 = 15 km, and 15 ÷ 3 = 5. With line B at 10 km per hour, line B is the steeper one.
- Both lines at 10 km per hour: they lie on top of each other, and the rate is the same.
On a graph of a steady rate, the rate is the steepness: the rise divided by the run between any two points on the line. Its unit is the unit up the side per the unit across: kilometres per hour.
On the same scales, the steeper line has the greater rate. Move the second point and the rise changes, but rise ÷ run does not, because a steady rate is the same at every point on the line.
Back to the friends: Kofi's line is 36 ÷ 3 = 12 km per hour and Ana's is 27 ÷ 3 = 9 km per hour. Kofi's is steeper, and he goes 3 km further in each hour.
A common mistake: “The line ends at 90, so the rate is 90.” The height of the end point is the total after all the time, not the amount for one hour. A line from (0, 0) to (6, 90) has a rate of 90 ÷ 6 = 15 km per hour.
Worked example: two points
A straight line through the origin passes through (4, 18) and (10, 45), with minutes across and litres up. Find the rate.
- Rise: 45 − 18 = 27 litres. Run: 10 − 4 = 6 minutes.
- Rate = rise ÷ run = 27 ÷ 6 = 4.5.
- 4.5 litres per minute. Check from the origin: 4.5 × 4 = 18, which matches the first point.
Using the rate. The same line: how many litres after 14 minutes, and after how many minutes are there 36 litres?
- After 14 minutes: 4.5 × 14 = 63 litres.
- 36 litres: 36 ÷ 4.5 = 8 minutes.
- 63 litres, and 8 minutes.
Your go. The first step is given. A straight line through the origin passes through (5, 35) and (9, 63), with minutes across and litres up. Find the rate in litres per minute.
- Rise: 63 − 35 = 28 litres.
- Now find the run, divide, and check against the origin.
Check the answer
Run: 9 − 5 = 4 minutes, and 28 ÷ 4 = 7, so 7 litres per minute. Check: 7 × 5 = 35, which matches the point (5, 35).
Check before you go on
1. A straight line on a graph of distance (km) against time (hours) passes through (0, 0) and (3, 42). What is the speed?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A straight line through the origin passes through (6, 78) on a graph of litres against minutes. What is the rate in litres per minute?
3. A straight line on a graph of distance in metres against time in seconds rises 18 m over a run of 6 s. What is the rate?
Level 2 · Building
4. Line P and line Q both go through the origin on the same graph. P passes through (2, 18) and Q passes through (3, 18). Which is the steeper line?
5. A straight line through the origin passes through (2, 34) and (6, 102). What is its rate?
6. A tap fills a tank at a steady rate. On a graph of litres against minutes the line goes through the origin and (3, 24). How many minutes does it take to reach 72 litres?
Level 3 · Stretch
7. On one graph, line A goes through the origin and (5, 40). Line B also goes through the origin and rises 3 less than line A for each step across. What is the height of line B at x = 5?
8. A graph of distance in metres against time in minutes is a straight line through the origin and (5, 1500). What is the speed in kilometres per hour?
Something to think about
Line A has a rate r. Line B has a rate 2r. At what x does line B reach the height that line A reaches at x = 6? Decide, then say what that shows about steepness.
One way to think about it
Line A reaches 6r at x = 6. Line B reaches 2r × x, which is 6r when x = 3. So line B gets there in half the time, which is what twice as steep means: for every step across it rises twice as far.
Exit ticket
9. Two straight lines through the origin show distance in km against time in hours. Line A ends at (8, 64) and line B ends at (4, 48). Gita says line A has the greater rate because it ends higher. Which reply is right?