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Change charts

Change and motion · level 2, lesson 3 of 6

A question to start

On a spring day, Ella checked the thermometer outside her window every two hours, from 6 in the morning to 8 at night. She wants to know two things:

  1. When was it warmest?
  2. When did it warm up fastest?

Most people think these have the same answer: surely it warmed up fastest just before it was warmest? But on Ella's day, the answers are hours apart. In this lesson you will learn to read change from a table and a chart, and see why "warmest" and "warming fastest" are different questions.

The idea

A table of temperatures

Here is what Ella wrote down.

Time 6 am 8 am 10 am 12 pm 2 pm 4 pm 6 pm 8 pm
Temperature (°C) 4 6 11 15 17 16 12 8

A table puts the numbers in order, so you can read them one by one. The top row is when, the bottom row is how warm. (12 pm is noon, the middle of the day.)

From the table you can answer the first question at once: the biggest temperature is 17 °C, at 2 pm. That was the warmest time.

Adding a row of changes

For the second question we need the change from each reading to the next, as in lesson 1. Add a third row:

Time 6 am 8 am 10 am 12 pm 2 pm 4 pm 6 pm 8 pm
Temperature (°C) 4 6 11 15 17 16 12 8
Change since 2 hours before up 2 up 5 up 4 up 2 down 1 down 4 down 4

Now look along the new row. The biggest rise is up 5, from 8 am to 10 am. So it warmed up fastest between 8 and 10 in the morning, when the sun was climbing and starting to heat the ground. That was four hours before the warmest time!

Check the change row the way we learned in lesson 1: the small changes should add up to the big change. Up 2, 5, 4 and 2 makes up 13, and down 1, 4 and 4 makes down 9. Up 13 and down 9 is up 4. And from 6 am (4 °C) to 8 pm (8 °C) really is up 4. ✓

Why the warmest time is not the fastest-warming time

Think of a child running up a hill. The top of the hill is the highest point. But the child is not climbing fastest at the top. At the top, the child stops climbing! The fastest climbing happens on the steepest part of the slope, on the way up.

Temperature is the same. Near 2 pm, the temperature is high, but it is only creeping up (up 2), and soon it stops and starts to fall. The fastest warming was earlier, on the "steep part" of the day.

A bar chart

A bar chart draws each temperature as a bar. The taller the bar, the warmer it was.

0510 1520 461115 1716128 6am8am10am12pm 2pm4pm6pm8pm time of day °C

The tallest bar (orange) is 2 pm: the warmest time. To see the fastest warming in a bar chart, look at how much taller each bar is than the one before it. The jump from the 8 am bar to the 10 am bar is the biggest step up.

The numbers on the left side are the scale. Each line across is 5 degrees more. Always read the scale before you read the bars.

A line chart

A line chart puts a dot for each temperature, at the same height as the top of the bar, and joins the dots with straight lines.

0510 1520 6am8am10am12pm 2pm4pm6pm8pm steepest warmest °C

A line chart is especially good for showing change, because you can see the change as steepness:

  • where the line goes up, it was getting warmer;
  • where the line goes down, it was getting cooler;
  • where the line is steep, the change was fast;
  • where the line is nearly flat, the change was slow;
  • where the line is flat, nothing changed.

The steepest piece going up (orange) is from 8 am to 10 am. That is the fastest warming, the same answer as the table. This is the same idea as the steep and gentle ramps of level 1, but now the "ramp" is made of numbers.

The two steepest pieces going down are from 4 pm to 6 pm and from 6 pm to 8 pm: both fell 4 degrees. They are equally steep. The day cooled fastest in the evening, after the sun went low.

Why only line charts get joined up

Ella only checked the thermometer every two hours. What happened in between, at 9 am, say? We did not measure it. Joining the dots with straight lines is a good guess: temperature changes smoothly, without jumping, so at 9 am it was probably somewhere between 6 and 11 degrees. That is why line charts are used for things that change smoothly over time, like temperature, height or distance.

For things that do not have an "in between", like the number of children in each class, we use a bar chart and do not join anything up.

Worked examples

Example 1. Reading the chart. Using the chart above, what was the temperature at 12 pm? How much did it rise from 6 am to 12 pm?

  1. Find 12 pm at the bottom and go up to the dot: it sits exactly on the 15 line, so it was 15 °C.
  2. At 6 am it was 4 °C.
  3. From 4 to 15: count on 1 to 5, then 10 more to 15. 1 + 10 = 11 degrees.

Example 2. A cooling cup of cocoa. Tom measures his cocoa every 5 minutes.

Minutes 0 5 10 15 20
Temperature (°C) 80 64 53 46 41

When did it cool fastest?

  1. Changes: 80 to 64 is down 16; 64 to 53 is down 11; 53 to 46 is down 7; 46 to 41 is down 5.
  2. The biggest fall is down 16, in the first 5 minutes.
  3. The cocoa cools fast at first and then slower and slower. (A hot drink loses heat fastest when it is much hotter than the room.)

Example 3. Matching words to line pieces. A line chart has four pieces: steep up, gentle up, flat, steep down. Match: "the sun came out strongly", "a cloud stayed over the sun", "the sun went behind the hill at evening", "a thin haze".

  1. Steep up: the sun came out strongly (fast warming).
  2. Gentle up: a thin haze (slow warming).
  3. Flat: a cloud stayed over the sun (no change).
  4. Steep down: the sun went behind the hill (fast cooling).

Try it

Use this table for questions 1 to 5. It shows the temperature on a cold winter day.

Time 6 am 9 am 12 pm 3 pm 6 pm 9 pm
Temperature (°C) −6 −4 1 3 −1 −5
  1. What was the warmest temperature, and when?
  2. What was the coldest temperature, and when?
  3. Write the change for each 3 hours (up or down, and how much).
  4. When did it warm up fastest? When did it cool fastest?
  5. Was it warmer or colder at 9 pm than at 6 am, and by how much?
  6. On a line chart, what does a flat piece of line mean?
  7. On a line chart, two pieces both go up. One is steep and one is gentle. Which one shows faster warming?
  8. Ella says: "The warmest time must always be the time when it warmed fastest." Explain why she is wrong.
Answers
  1. 3 °C at 3 pm.
  2. −6 °C at 6 am.
  3. 6 am to 9 am: −6 to −4, up 2. 9 am to 12 pm: −4 to 1, that is 4 up to zero and 1 more, up 5. 12 pm to 3 pm: 1 to 3, up 2. 3 pm to 6 pm: 3 to −1, that is 3 down to zero and 1 more, down 4. 6 pm to 9 pm: −1 to −5, down 4.
  4. Warmed fastest: 9 am to 12 pm (up 5). Cooled fastest: 3 pm to 6 pm and 6 pm to 9 pm (down 4 each, a tie).
  5. From −6 to −5 is up 1: it was 1 degree warmer at 9 pm. (Check: up 2, 5, 2 is up 9; down 4 and 4 is down 8; up 9 and down 8 is up 1. ✓)
  6. No change: the temperature stayed the same during that time.
  7. The steep piece.
  8. The warmest time is the top of the "hill" on the chart. At the top, the temperature has stopped going up. The fastest warming happens earlier, on the steepest part of the way up.

Common mistake

Mixing up "highest" and "rising fastest". The highest point of a chart answers "when was it warmest?". The steepest piece going up answers "when did it warm up fastest?". These are different questions with different answers.

Not reading the scale. If each line across is 5 degrees, a bar that reaches 2 lines is 10 degrees, not 2. Always look at the numbers on the side first.

Where it leads