Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Work out 7 − 12.
Answer
−5. - Which quadrant is the point (−3, 4) in?
Answer
Quadrant II: x is negative and y is positive. - A recipe uses 3 cups of rice for every 2 people. How many cups for 6 people?
Answer
9 cups: 6 people is three lots of 2 people, and 3 × 3 = 9.
Words used in this lesson
- relationship
- a rule that links one quantity to another. For example: the fare depends on the distance.
- linear
- having a constant change, so the graph is a straight line. For example: y = 2x + 1 is linear.
- rate of change
- how much y changes for each step of 1 in x. For example: 400 naira for each km.
- constant
- staying the same every time. For example: a constant change of 5.
- table of values
- pairs of x and y written in rows. For example: x = 0, 1, 2 and y = 3, 5, 7.
- change in y
- the difference between two neighbouring y values. For example: 12 − 7 = 5.
Two clean-up crews
After a storm, two crews clear plastic from a beach in Zanzibar. Each one records the total kilograms collected at the end of each hour.
Team A: 0, 40, 80, 120 kg after 0, 1, 2 and 3 hours.
Team B: 0, 70, 110, 130 kg after 0, 1, 2 and 3 hours.
Predict first: if you plotted each crew's totals against the hours, which would give points on a straight line? Can you tell from the numbers alone, without drawing anything? Then try six relationships below and watch what the “change” row does.
y = 2x + 1: y goes 1, 3, 5, 7, 9. The change is +2 every time, so the points lie on the dashed straight line.
Show me the steps
- y = 2x + 1: y is 1, 3, 5, 7, 9. The change is +2 every time, so the points lie on a straight line.
- y = x²: y is 0, 1, 4, 9, 16. The changes are +1, +3, +5, +7. They are not constant, so the points bend away from the dashed line.
- y = 10 − 3x: y is 10, 7, 4, 1, −2. The change is −3 every time: linear, and the line falls.
- y = 5: y is 5 every time. The change is 0 every time: linear, a flat line.
- y doubles each step: 1, 2, 4, 8, 16. The changes are +1, +2, +4, +8: not linear.
- y = 4x − 6: y is −6, −2, 2, 6, 10. The change is +4 every time: linear, and the line crosses y = 0 between x = 1 and x = 2.
A relationship is linear when equal steps in x always give the same change in y. Then the points lie on one straight line, and that constant change is the rate of change.
Team A's changes are 40, 40, 40: linear, at 40 kg per hour. Team B's are 70, 40, 20: the pace keeps dropping, so the points do not lie on a line.
The change can be negative or zero. A falling line has a negative change, and a flat line has a change of 0. Both are still linear.
A common mistake: “y keeps going up, so it is linear.” Look at y = x²: it goes up at every step, but by +1, then +3, then +5. A linear relationship needs the same change each time, not only the same direction.
Worked example: a taxi fare
A taxi in Lagos charges these fares: 0 km costs 500 naira, 1 km costs 900, 2 km costs 1300 and 3 km costs 1700. Is the fare linear in the distance?
- The distance goes up by 1 km each row, so the steps in x are equal.
- The changes in fare are 900 − 500 = 400, 1300 − 900 = 400 and 1700 − 1300 = 400.
- The change is always 400, so the relationship is linear. The fare rises by 400 naira for each km. The 500 is what you pay before you travel anywhere.
Your go. The first step is given. A rain barrel holds 90 litres at the start, then 75, 60 and 45 litres after 1, 2 and 3 hours. Is the amount linear in the time?
- The hours go up by 1 each row, so the steps in x are equal.
- Now work out the changes and decide, then check.
Check the answer
The changes are 75 − 90 = −15, 60 − 75 = −15 and 45 − 60 = −15. It is linear: the barrel loses 15 litres every hour, so it is empty after 90 ÷ 15 = 6 hours.
Check before you go on
1. A table has x = 0, 1, 2, 3 and y = 20, 17, 13, 8. Is the relationship linear?
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. x = 0, 1, 2, 3, 4 and y = 7, 12, 17, 22, 27. By how much does y change each time x goes up by 1?
3. Which table shows a linear relationship? In each, x = 1, 2, 3, 4.
Level 2 · Building
4. A pond's depth is 120 cm, then 105 cm, 90 cm and 75 cm at hourly readings. What is the change in depth each hour, in cm? Use a minus sign for a fall.
5. A table is linear. For x = 0, 1, 2, 3, 4, the y values are 4, 11, ?, 25, 32. Find the missing value.
6. A linear table starts with y = 3 when x = 0 and the change is +4 for each step of 1 in x. What is y when x = 5?
Level 3 · Stretch
7. x = 0, 1, 3, 6 and y = 4, 7, 13, 22. The relationship is linear. By how much does y change when x goes up by 1?
8. A linear relationship has y = 6 when x = 2 and y = 18 when x = 5. What is y when x = 0?
Something to think about
A table has two rows only: x = 1 gives y = 4 and x = 5 gives y = 16. If the relationship is linear, how many different straight lines could it be, and what is y when x = 3?
One way to think about it
Exactly one line passes through two points, so there is only one linear relationship. The change from x = 1 to x = 5 is 12 over 4 steps, which is 3 per step, so y at x = 3 is 4 + 2 × 3 = 10. Two points are enough because the change is constant: the steps in between must share it equally.
Exit ticket
9. Priya says the table x = 1, 2, 3, 4, 5 and y = 4, 7, 10, 13, 17 is linear “because the change is nearly always 3”. What is the right reply?