Skip to content

Steady and not steady

Change and motion · level 2, lesson 4 of 6

A question to start

Take a sheet of paper and fold it in half. Now it is 2 layers thick. Fold it in half again: 4 layers. Again: 8 layers.

Each fold is just one more fold, the same little action every time. So does the paper get thicker by the same amount with each fold?

Try it and count: 1, 2, 4, 8, 16, 32... After only 7 folds the paper is 128 layers thick, and it is so stiff that you can hardly fold it again. Something is going on here that is very different from adding the same amount each time. In this lesson you will learn to tell the two kinds of change apart: steady change and change that is not steady.

The idea

Steady change: the same amount each time

A bathtub has its plug in, and the tap is on. Every minute, the tap adds 2 litres of water. (A litre is the amount in a big milk carton.)

Minutes 0 1 2 3 4 5
Water in the tub (litres) 0 2 4 6 8 10
Change each minute up 2 up 2 up 2 up 2 up 2

The change row says the same thing every time: up 2. That is what steady means: the same change in every equal step of time. Another word for it is constant (it does not vary).

You can predict a steady change very easily. After 6 minutes: 12 litres. After 10 minutes: count by twos ten times, 20 litres. You just keep adding the same amount.

Not steady: the snowball

Now think about rolling a snowball across a snowy garden to make a snowman. Each time you roll it all the way across the garden, it picks up more snow. Here is a pretend snowball's weight after each trip across:

Trips across 0 1 2 3 4 5
Weight (kg) 1 2 4 7 11 16
Change each trip up 1 up 2 up 3 up 4 up 5

The change row is not the same each time. The snowball gains more on every trip: 1, then 2, then 3, then 4, then 5. This change is not steady. It is speeding up, growing faster and faster.

Why does the snowball grow faster and faster? A snowball picks up snow with its outside, the part that touches the ground. A small snowball has a small outside, so it picks up only a little. A big snowball has a big outside, so it picks up a lot. The bigger it gets, the faster it grows. That is the secret of every change that speeds up like this: the more there is, the more gets added.

Seeing the difference with jumps

On a number line, steady change makes equal jumps. Growing-faster change makes bigger and bigger jumps.

0246 810 1247 1116 tub: equal jumps snowball: bigger and bigger jumps

Seeing the difference on a chart

Put both on a dot chart: minutes (or trips) along the bottom, amount up the side.

051015 0123 45 tub snowball minutes or trips
  • The tub's dots (blue) lie on a straight line. A straight line has the same steepness everywhere, because it goes up the same amount at every step.
  • The snowball's dots (orange) make a curve that bends upwards. It gets steeper and steeper, because it goes up more at every step.

So here is a rule you can use for life: steady change makes a straight line; change that speeds up makes a curve that bends up.

Look at something surprising in the chart. At the start, the snowball is ahead (1 kg against 0 litres). At steps 1 and 2 they are level. After that, the snowball races away. Something that grows faster and faster will always, in the end, overtake something that grows steadily, however far behind it starts.

Back to the paper folds

Each fold doubles the layers: 1, 2, 4, 8, 16, 32, 64, 128.

Folds 0 1 2 3 4 5 6 7
Layers 1 2 4 8 16 32 64 128
Change each fold up 1 up 2 up 4 up 8 up 16 up 32 up 64

The change row grows even faster than the snowball's: it doubles too. Why? A fold puts a copy of all the layers on top of themselves. So the number of new layers added is always the same as the number already there. The more layers there are, the more get added. Just like the snowball, but even stronger.

Not steady the other way: slowing down

Change can also be not steady because it slows down. A young tree grows a lot each year when it is small, and less each year when it is old. A cup of hot cocoa cools fast at first and then more slowly (you saw this in lesson 3). The change row gets smaller each time, and the dots make a curve that bends over and flattens out.

So there are three kinds to look for:

Change row Name Picture
The same each time Steady Straight line
Bigger each time Speeding up (growing faster) Curve bending up
Smaller each time Slowing down Curve bending over, flattening out

Worked examples

Example 1. Pocket money. Zara gets 3 dollars every week and saves all of it. She starts with nothing. Is this steady? How much after 6 weeks?

  1. Each week adds 3 dollars: 3, 6, 9, 12, 15, 18.
  2. The change is up 3 every week: steady.
  3. After 6 weeks: 18 dollars.

Example 2. A growing stack. A pattern of squares has 1 square, then 3, then 6, then 10, then 15. Is it steady?

  1. Changes: 1 to 3 is up 2; 3 to 6 is up 3; 6 to 10 is up 4; 10 to 15 is up 5.
  2. The changes get bigger: not steady, speeding up.
  3. Next: up 6, so 15 + 6 = 21.

Example 3. A candle. A candle is 20 cm long. Every hour it burns down 2 cm. Steady or not?

  1. 20, 18, 16, 14, ... The change is down 2 every hour.
  2. That is steady (steady going down). On a chart it makes a straight line going down.
  3. It will be gone after 10 hours, because 10 lots of 2 cm is 20 cm.

Example 4. Telling them apart from a table. Which of these are steady?

  • A: 5, 10, 15, 20, 25. Changes: 5, 5, 5, 5. Steady.
  • B: 2, 4, 8, 16, 32. Changes: 2, 4, 8, 16. Speeding up (doubling).
  • C: 50, 40, 32, 26, 22. Changes: down 10, down 8, down 6, down 4. Not steady, slowing down.
  • D: 30, 27, 24, 21, 18. Changes: down 3 each time. Steady (going down).

Try it

  1. A tap fills a bucket with 3 litres every minute. Write how much water there is after 1, 2, 3, 4 and 5 minutes. Is it steady?
  2. Find the change row for 4, 7, 10, 13, 16. Is it steady?
  3. Find the change row for 1, 3, 7, 13, 21. Is it steady? What comes next?
  4. A plant is 10 cm, then 15 cm, then 18 cm, then 20 cm, then 21 cm. Is its growth speeding up or slowing down?
  5. A rumour: on day 1 one child knows a secret. Every day, every child who knows tells one new child. How many know on day 2, 3, 4, 5? Is this steady?
  6. Which makes a straight line on a dot chart: 2, 5, 8, 11 or 2, 4, 8, 16?
  7. Sam saves 10 dollars a week. Mia starts with 1 dollar and doubles her money every week. Who has more after 3 weeks? After 6 weeks?
  8. Why does a snowball grow faster as it gets bigger?
Answers
  1. 3, 6, 9, 12, 15 litres. The change is up 3 every minute: steady.
  2. Changes: 3, 3, 3, 3. Steady.
  3. Changes: 2, 4, 6, 8. Not steady (speeding up). The next change is 10, so 31 comes next.
  4. Changes: up 5, up 3, up 2, up 1. They get smaller: slowing down.
  5. Each child who knows tells one more, so the number doubles: day 1: 1, day 2: 2, day 3: 4, day 4: 8, day 5: 16. Not steady: it speeds up, like the paper folds.
  6. 2, 5, 8, 11: the change is up 3 every time, so it is steady and makes a straight line.
  7. Sam: 10, 20, 30, 40, 50, 60 dollars after weeks 1 to 6. Mia: 2, 4, 8, 16, 32, 64 dollars after weeks 1 to 6. After 3 weeks Sam has more (30 against 8). After 6 weeks Mia has more (64 against 60). Doubling wins in the end.
  8. A snowball picks up snow with its outside. A bigger snowball has a bigger outside touching the snow, so it picks up more on each roll.

Common mistake

Looking at the numbers instead of the changes. The numbers 1, 2, 4, 7, 11 all go up, and so do 2, 4, 6, 8, 10. To tell steady from not steady, you must look at the change row, not the numbers themselves.

Thinking that "going down" means "not steady". A candle burning down 2 cm every hour is steady. Steady means the change is the same each time, whether it is up or down.

Where it leads