Line graphs: Year 7 collecting and presenting data

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

(Unit 3, lesson 3) The position rule of a sequence is 3n + 1. What is the 6th term?
Answer
19. 3 × 6 = 18, then 18 + 1 = 19.
(Unit 2, lesson 6) A price goes from 40 to 50. What is the percentage increase?
Answer
25%. The change is 10, and 10 ÷ 40 = 0.25.
(Lesson 1 of this unit) Is temperature discrete or continuous?
Answer
Continuous. Temperature is measured and can take any value in a range.

Words used in this lesson

line graph: a graph that joins points to show change over time. For example: the height of a plant by week.
axis: a line with a scale along one side of a graph. For example: the horizontal axis shows weeks.
steep: rising or falling quickly. For example: the line is steepest from week 2 to week 4.
rate of change: how much a quantity changes in one unit of time. For example: 4 cm per week.
predict: say what will happen beyond the data. For example: 52 cm at week 9.
trend: the general direction the data is going. For example: rising, then levelling off.

Change, one measurement at a time

The garden club measures a bamboo shoot at the end of each week. For weeks 0 to 5 the readings are 0, 4, 10, 18, 26 and 33 cm. A table of numbers hides how the growth feels. Predict first: after week 5, will the line keep getting steeper, flatten off but still rise, or turn downwards? Then plot it week by week and see weeks 6 to 8.

Weeks plotted 4

26height at the last week (cm)

8rise in the last week (cm)

At week 4 the shoot is 26 cm tall, up 8 cm from week 3.

The figure needs JavaScript. In words: a bamboo shoot is 0 cm at week 0 and then 4, 10, 18, 26, 33, 39, 44 and 48 cm at the end of weeks 1 to 8. The weekly rises are 4, 6, 8, 8, 7, 6, 5 and 4 cm. The line climbs most steeply between weeks 2 and 4 and flattens a little after.

Show me the steps

Week 2: 10 cm, up 6 from week 1. The line is rising and getting steeper.
Week 4: 26 cm, up 8 from week 3. The rise has been 8 cm for two weeks running, the steepest part.
Week 6: 39 cm, up 6 from week 5. The line is still rising but less steeply.
Week 8: 48 cm, up 4 from week 7. The weekly rise is back to what it was in week 1.

A line graph shows how something changes over time. Time goes on the horizontal axis, the measured quantity goes on the vertical axis, and the points are joined in order. For a quantity such as height or temperature the value exists at every moment between the readings. For totals or averages per month, such as rainfall or sales, the line shows the trend, and a reading between two months means nothing.

Steeper means faster change. A segment that rises 8 cm in a week is steeper than one that rises 4 cm. A flat segment means no change, and a falling one means a decrease.

Back to the bamboo: the climb is steepest between weeks 2 and 4, at 8 cm a week, and eases to 4 cm a week by week 8.

Between the points, and past the end

A line graph lets you read a value at a time nobody measured. That works between plotted points, and it stops being reliable past the last one. Move the marker, then extend the line.

Read at week 2.5

Extend the line no prediction

2.5week

14height read from the line (cm)

noneprediction

Read at week 2.5: halfway between 10 cm and 18 cm, which is 14 cm.

The figure needs JavaScript. In words: the line passes through 10 cm at week 2 and 18 cm at week 3, so at week 2.5 it reads 14 cm, halfway between them. Reading between points assumes the shoot grows steadily between measurements. Past week 8 the last rise of 4 cm a week would give 52 cm at week 9 and 56 cm at week 10, but that is only a prediction.

Show me the steps

Week 0.5: halfway between 0 cm and 4 cm, so 2 cm.
Week 2.5: halfway between 10 cm and 18 cm, so 14 cm.
Week 7.5: halfway between 44 cm and 48 cm, so 46 cm.
Extend to week 9: 48 + 4 = 52 cm, the last weekly rise repeated. It is a prediction, not a measurement.

Between points you can read a value by going halfway: halfway between 10 and 18 is 14. This assumes the quantity changes steadily between measurements. Past the last point you can only predict, and the further past the data, the less sure the prediction.

A common mistake: “Join the dots on any chart.” A line between “Football” and “Hockey” says there is something between them, and there is not. Only join points when the horizontal axis is time (or another measurement) and the points follow in order.

Worked example: reading and predicting

Water in a tank, in litres, at minutes 0, 2, 4 and 6: 10, 18, 28, 36. How much did it rise from minute 2 to minute 4? What is the reading at minute 3? If the last rise repeats, predict minute 8.

Rise from minute 2 to 4: 28 − 18 = 10 litres.
Minute 3 is halfway between 18 and 28: (18 + 28) ÷ 2 = 23 litres.
The last rise, from minute 4 to 6, is 36 − 28 = 8 litres per two minutes. Minute 8: 36 + 8 = 44 litres. 10 litres; 23 litres; about 44 litres.

Your go. The first step is given. A snail's distance from its start, in centimetres, at minutes 0, 5, 10 and 15: 0, 12, 30, 42. In which five-minute period did it move fastest?

From minute 0 to 5 it moved 12 − 0 = 12 cm.
Now find the other two periods yourself and compare, then check.

Check the answer

Minutes 5 to 10: 30 − 12 = 18 cm. Minutes 10 to 15: 42 − 30 = 12 cm. The fastest was minutes 5 to 10, the steepest part of the line.

Check before you go on

1. Which data is best shown on a line graph?

The favourite fruit of 30 students The temperature of a room at every hour of a day The number of students in each of four clubs The colours of 20 cars

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A town's mean temperature in months 1 to 5 is 6, 9, 14, 19 and 21 degrees Celsius. By how many degrees did it rise from month 2 to month 5?

3. A candle is 18 cm tall after 2 hours and 10 cm tall after 4 hours, burning steadily. How tall is it after 3 hours?

Level 2 · Building

4. A line graph shows a river's depth: 2 m on day 1, 6 m on day 3, 6.4 m on day 7 and 3 m on day 9. Which statement matches the graph?

The river rose quickly at first, stayed almost level, then fell. The river rose slowly at first, then quickly from day 3 to day 7. The river was deepest on day 1 and became shallower after that. The river fell steadily from day 1 to day 9, getting shallower each day.

5. A pot plant is 12 cm tall in week 1 and 28 cm tall in week 5, growing steadily. How many centimetres does it grow each week?

6. A pool fills steadily: 30 litres at minute 2 and 45 litres at minute 5. How many litres will it hold at minute 9, if it keeps filling at the same rate?

Level 3 · Stretch

7. Plant A is 5, 9, 13 and 17 cm tall at weeks 1 to 4. Plant B is 3, 8, 13 and 18 cm. In which week are the plants the same height?

8. A line graph covers weeks 1 to 8. A student extends it to week 40 to predict a height. Why is this prediction risky?

Because line graphs cannot be extended past the last point at all Because the pattern may not continue that far Because 40 is not a multiple of 8, the number of weeks drawn Because predictions are always wrong, however close to the data

Something to think about

A gardener extends the bamboo line to week 20 and predicts 96 cm. Another says bamboo shoots stop growing at some height. Who is more likely to be right, and what does that tell you about extending a line graph?

One way to think about it

The second gardener, because real growth slows and stops, while a straight extension keeps going for ever. A prediction a little past the data (week 9) is reasonable. A prediction far past the data (week 20) rests on the pattern not changing, which nothing in the data can promise.

Exit ticket

9. A line graph shows a pot plant's height at the end of each week. Which statement is right?

Time goes on the vertical axis, because it is what changes. Time goes on the horizontal axis, and joining the points suggests heights between the weeks. The points should not be joined, because heights are separate. It should be drawn as a pie chart.