One dot for each reading
Every day at noon, a class read the temperature outside and marked it on a graph. The days go along the bottom, and the temperature in degrees goes up the side, one degree for each line. Each reading is one dot, above its day and level with its temperature.
The dots are joined in order, Monday to Tuesday to Wednesday to Thursday. A graph drawn this way is a line graph.
Monday's dot is level with 3, Tuesday's with 5, Wednesday's with 4 and Thursday's with 7.
Reading a dot
To read Tuesday, go up from Tuesday to its dot, then straight across to the scale. The dot is level with 5, so the temperature on Tuesday was 5 degrees.
Go up from the day you are asked about. Wednesday's dot, at 4, is right next to Tuesday's, and reading the dot of the next day is the usual slip. So is landing one line too high or too low: follow the line of the scale all the way across.
The line shows the change
The line between two dots shows how the temperature changed from one reading to the next. From Tuesday to Wednesday the line goes down, from 5 to 4, so it got colder, by 5 − 4 = 1 degree. From Wednesday to Thursday the line goes up, from 4 to 7, so it got warmer, by 7 − 4 = 3 degrees.
A change is the difference between two readings, so subtract the smaller from the bigger. Adding them, 4 + 7 = 11, does not give a change, and neither reading on its own is the change.
Steep and flat
The steeper the line, the bigger the change from one reading to the next. In another week, Monday and Tuesday were both 4 degrees, so the line between them is flat: a flat piece means no change.
That week, from Tuesday to Wednesday the line climbs 3 degrees, from 4 to 7, and from Wednesday to Thursday it drops 5 degrees, from 7 to 2. The drop is the steeper piece, so it is the bigger change.
The flat piece from Monday to Tuesday is no change. The steepest piece, from Wednesday to Thursday, is a fall of 5 degrees.
Between the dots
Only the dots are readings. The line joins them because the temperature was there all the time in between, rising or falling. A point on the line between two dots is an estimate of the temperature at that time, not a reading, because nobody measured it.
Worked example: Temperature Graph: The Greatest Rise
Question Ravi reads the temperature every two hours and draws a line graph. The readings are 24 degrees at 6 am, 25 at 8 am, 28 at 10 am, 32 at 12 noon, 33 at 2 pm, 30 at 4 pm and 27 at 6 pm. (a) Between which two readings next to each other does the temperature rise the most, and by how many degrees? (b) What is the difference between the highest and the lowest temperature?
1.The line goes up from 6 am to 2 pm, so work out the rise for each part of the line there.
The line goes up from 6 am to 2 pm. Work out the rise for each part. 2.From 6 am to 8 am: 25 − 24 = 1. From 8 am to 10 am: 28 − 25 = 3. From 10 am to 12 noon: 32 − 28 = 4. From 12 noon to 2 pm: 33 − 32 = 1.
The rises are 1, 3, 4 and 1 degrees. 3.(a) The greatest rise is 4 degrees, between 10 am and 12 noon. This is the steepest part of the line going up.
(a) The greatest rise is 4 degrees, from 10 am to 12 noon, the steepest part going up. 4.The highest point is 33 degrees at 2 pm. The lowest point is 24 degrees at 6 am.
The highest point is 33 degrees at 2 pm. The lowest point is 24 degrees at 6 am. 5.(b) 33 − 24 = 9, so the difference is 9 degrees.
(b) 33 − 24 = 9 degrees.
Answer: (a) Between 10 am and 12 noon, by 4 degrees; (b) 9 degrees
Common mistakes
- Choosing 12 noon to 2 pm because the line is highest there. The highest point is not the greatest rise. The temperature rises only 1 degree in that part.
- Taking the first reading from the last reading for part (b), 27 − 24 = 3. The highest temperature is 33 degrees at 2 pm, not the last reading of the day.
Worked example: Plant Height Graph: A Value Between Two Points
Question Siti measures a bean plant every two weeks and draws a line graph. The plant is 2 cm tall at the start, 6 cm in week 2, 12 cm in week 4, 16 cm in week 6 and 18 cm in week 8. (a) Use the graph to estimate the height of the plant in week 3. (b) Why is this only an estimate, and between which two heights must the real height in week 3 be?
1.Siti did not measure in week 3. Week 3 is halfway between week 2 and week 4, so look at the line between those two dots.
Week 3 has no dot. It is halfway between the dots for week 2 and week 4. 2.The height is 6 cm in week 2 and 12 cm in week 4. The plant grew 12 − 6 = 6 cm in those two weeks.
From week 2 to week 4 the plant grew 12 − 6 = 6 cm. 3.Halfway along the line is half of that growth: 6 ÷ 2 = 3 cm, and 6 + 3 = 9 cm.
Halfway along the line is 6 + 3 = 9 cm. 4.(a) The height in week 3 is about 9 cm.
(a) The height in week 3 is about 9 cm. 5.(b) Only the dots are measurements. The straight line joins two of them, and the plant may have grown more in one week than in the other. A plant does not get shorter, so the real height is between 6 cm and 12 cm.
(b) Week 3 was not measured. The real height is between 6 cm and 12 cm.
Answer: (a) About 9 cm; (b) it was not measured in week 3, and the real height is between 6 cm and 12 cm
Common mistakes
- Reading the nearest dot and answering 6 cm or 12 cm. Those are the heights in week 2 and week 4. For week 3, go up from week 3 to the line and then across to the scale.
- Giving 9 cm as a measured height. Siti measured only in weeks 0, 2, 4, 6 and 8. A height read from the line between two dots is an estimate.