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Rate of change from a table worksheets

How much does y move for every ONE x? These free printable worksheets read the rate of change and the starting value straight off a table.

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Ready to printRate of change from a table
Rate of change from a table worksheet — Tables, rate and start, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Rate of change is a division, and the whole difficulty is the denominator. On a table whose x column steps by one, the rate IS the difference in y — and a student who has only ever seen those tables reads 12 off a column that steps by three, because 12 is the number the subtraction gave them. It is the right subtraction and the wrong answer.

Why this sheet works

So both kinds are on every page, drawn separately rather than left to the seed. A sheet of only unit steps teaches subtraction and calls it rate of change; a sheet of only larger steps never shows what the easy case is a special case of. Seeing them side by side is what makes the division visible, and the walks write it out even when the divisor is 1 — that is the one place the habit can be formed, because dividing by one leaves no trace in the answer.

How this one works

One question from this sheet, worked through a step at a time.

x: 4, 5, 6, 7 y: 20, 16, 12, 8

x4567y2016128
x steps111y steps−4−4−4
−4÷1=−4
x = 4→x = 0
4 each step× 4
at x = 036
  1. Rate of change means how much y moves for every ONE x. So it is a division, and the two things being divided are both differences: how far y went, over how far x went.
  2. Take both rows of steps. x moves 1, 1, 1 — the same each time — and y moves −4, −4, −4. They are steady, so this table really does have one rate rather than a different one in every gap.
  3. Now divide: −4 ÷ 1 = −4. Dividing by one changes nothing, which is exactly why it is worth writing down — on a table whose x steps by more, it will not.
  4. The table starts at x = 4, not at 0, so the value there is not written down — step back to it. Going one x backwards adds 4 to y each time, and there are 4 to go.
  5. 20 + 16 = 36. So at x = 0 the value would be 36 — at x = 4 the value is 20, and stepping back 4 of them at −4 each gives 36.

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