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How much did it change?

Change and motion · level 2, lesson 1 of 6

A question to start

Maya planted a bean. On Monday the little plant was 4 cm tall. Every day that week it grew, and on Friday it was 9 cm tall.

Her brother Theo planted a bean too. His was 10 cm tall on Monday and 14 cm tall on Friday.

Theo says: "Mine is taller, so mine grew more this week." Is he right?

Take a moment to think. Theo's plant is taller. But "taller" and "grew more" are two different questions. In this lesson you will learn how to find out exactly how much something changed, and why that is a different number from how big it is.

The idea

The change is the difference

When something changes, we can measure it before and after. The change is how much it went up or down between those two times. It is the difference between the two numbers.

For Maya's plant: it was 4 cm on Monday and 9 cm on Friday. To find how much it grew, count on from 4 to 9.

0123 4567 8910 Monday Friday 5 jumps

From 4 to 9 is 5 jumps of 1 cm. So Maya's plant grew 5 cm. You can also take away: 9 − 4 = 5. Counting on and taking away give the same answer, because they are two ways of finding the gap between two numbers.

For Theo's plant: from 10 to 14 is 4 jumps, so it grew 4 cm. (14 − 10 = 4.)

So Theo is wrong. His plant is taller, but Maya's plant grew more this week: 5 cm against 4 cm.

How big and how much it changed are different questions

It is easy to mix these up, so here they are side by side.

Question Maya Theo
How tall on Friday? 9 cm 14 cm (taller)
How much did it grow this week? 5 cm (more) 4 cm

Theo's plant was taller already on Monday. It had a head start, like a runner who starts ahead in a race (level 1, lesson 2). Being ahead at the end does not mean you ran further. To know who grew more, look at the change, not at the size.

Going up and going down

A change can go up or down.

  • If the "after" number is bigger, the thing went up (it grew, it got warmer, it got more).
  • If the "after" number is smaller, the thing went down (it shrank, it cooled, it got less).
  • If the two numbers are the same, there was no change.

Always say which way. "The temperature changed by 6 degrees" is only half the story. "It rose by 6 degrees" or "it fell by 6 degrees" tells the whole story.

Temperatures below zero

Temperature is measured in degrees. In Canada we use degrees Celsius, written °C. Water freezes at 0 °C. On a very cold day the temperature can go below zero: −3 °C (say "minus three degrees") means 3 degrees colder than freezing.

A thermometer is like a number line standing up. Numbers below zero are under the 0 mark.

1050−5 3 up to zero 5 more morning: −3 °C afternoon: 5 °C

One winter morning it is −3 °C. In the afternoon it is 5 °C. How much warmer did it get?

Count up in two parts, using zero as a stepping stone:

  1. From −3 up to 0 is 3 degrees.
  2. From 0 up to 5 is 5 more degrees.
  3. Altogether: 3 + 5 = 8 degrees. The temperature rose by 8 °C.

Be careful here: the answer is not 5 − 3 = 2. The numbers are on opposite sides of zero, so the gap is the two parts added together. The number line (or thermometer) keeps you honest: just count the steps.

Changes add up

Here is Maya's plant every day of the week:

Day Mon Tue Wed Thu Fri
Height (cm) 4 5 6 8 9
Growth since the day before 1 1 2 1

Add up all the daily changes: 1 + 1 + 2 + 1 = 5 cm. That is exactly the change from Monday to Friday!

Why must it always work? Think of walking along the number line from 4 to 9. You can walk in one big jump, or in little jumps with stops at 5, 6 and 8. Either way you end up at the same place, so the little jumps must add up to the big jump. Small changes add up to the big change.

And if a change goes up and then down, the up and the down can cancel. If the temperature rises 4 degrees in the morning and falls 4 degrees in the evening, the change over the whole day is zero: it ends where it started.

Worked examples

Example 1. A puppy's weight. A puppy weighed 3 kg in March and 7 kg in June. How much weight did it gain?

  1. Count on from 3 to 7: 4, 5, 6, 7. That is 4 steps.
  2. Or take away: 7 − 3 = 4.
  3. The puppy gained 4 kg.

Example 2. A melting snowman. A snowman was 120 cm tall in the morning and 85 cm tall in the evening. How much did it change?

  1. The evening number is smaller, so it went down.
  2. Count on from 85 to 120: 85 to 90 is 5, 90 to 100 is 10, 100 to 120 is 20. Total 5 + 10 + 20 = 35.
  3. Or take away: 120 − 85 = 35.
  4. The snowman shrank by 35 cm.

Example 3. A cold night. At 6 pm it was 2 °C. By midnight it was −4 °C. How much did it change?

  1. It went from above zero to below zero: it got colder.
  2. From 2 down to 0 is 2 degrees.
  3. From 0 down to −4 is 4 more degrees.
  4. Total: 2 + 4 = 6. The temperature fell by 6 °C.

Example 4. Working backwards. A sunflower grew 15 cm in one week. At the end of the week it was 62 cm tall. How tall was it at the start?

  1. The start is 15 cm less than the end, because it grew to get to 62.
  2. 62 − 15 = 47.
  3. It was 47 cm at the start. Check: 47 + 15 = 62. ✓

Example 5. Up and down. A lift (elevator) starts on floor 3. It goes up 5 floors, then down 6 floors. Where is it now, and what was the change from the start?

  1. Up 5 from floor 3: floor 8.
  2. Down 6 from floor 8: floor 2.
  3. Change from the start: from 3 to 2, down 1 floor. (Up 5 and down 6 add up to down 1.)

Try it

  1. A candle was 20 cm long. After burning, it is 13 cm long. How much did it change, and which way?
  2. A child was 118 cm tall last year and 125 cm tall this year. How much did she grow?
  3. Ben's tower was 6 blocks tall, then 11 blocks, then 9 blocks. What was the change each time? What was the change from the start to the end?
  4. In the morning it is −2 °C. At lunch it is 6 °C. How much warmer is it at lunch?
  5. At 3 pm it is 4 °C. At 9 pm it is −5 °C. How much colder is it at 9 pm?
  6. Plant A went from 12 cm to 20 cm. Plant B went from 25 cm to 31 cm. Which plant is taller now? Which grew more?
  7. A balloon seller had 40 balloons. She sold 25 and then blew up 10 more. How many does she have now? What was the change from 40?
  8. The temperature went up 3 degrees, then up 2 degrees, then down 7 degrees. Did it end up warmer or colder than at the start, and by how much?
Answers
  1. 20 − 13 = 7. It got shorter by 7 cm.
  2. Count on from 118 to 125: 118 to 120 is 2, 120 to 125 is 5. 2 + 5 = 7 cm.
  3. 6 to 11: up 5. 11 to 9: down 2. Start to end, 6 to 9: up 3 (and up 5 with down 2 is up 3 ✓).
  4. −2 up to 0 is 2; 0 up to 6 is 6. 8 degrees warmer.
  5. 4 down to 0 is 4; 0 down to −5 is 5. 9 degrees colder.
  6. Plant B is taller (31 cm against 20 cm). But Plant A grew more: 20 − 12 = 8 cm, while Plant B grew 31 − 25 = 6 cm.
  7. 40 − 25 = 15, then 15 + 10 = 25 balloons. The change from 40 to 25 is down 15 (down 25 and up 10).
  8. Up 3 and up 2 is up 5. Then down 7: up 5 and down 7 is down 2. It ended 2 degrees colder.

Common mistake

Using the size instead of the change. "Mine is taller, so it grew more" is the mistake from the start of the lesson. To find how much something grew, always use the before and the after, and find the gap between them.

Taking away across zero. From −3 °C to 5 °C is 8 degrees, not 2. When the two temperatures are on opposite sides of zero, count to zero first and then count on.

Where it leads