Slope is the step

The idea

For the line representing a linear pattern against its term number, the slope equals the constant difference between consecutive terms of the sequence.

The tile line

The tiles in the growing pattern make the sequence 1, 3, 5, 7, … Each term is 2 more than the one before. Plot the stage number along x and the number of tiles along y: the points lie on the line y = 2x − 1.

What do you think the slope of that line is?

• 1
• 2
• 7

Change the step

Change the difference d between terms. The pattern starts at 1. Watch the table and the slope of the line.

Slope = difference

For the line that shows a linear pattern against its term number, the slope equals the constant difference between consecutive terms.

The tiles 1, 3, 5, 7, … go up by 2 each time. The line y = 2x − 1 has slope 2.

Notes

In a linear pattern, the slope of its line equals the constant difference between consecutive terms.

Check yourself

A linear pattern goes 5, 8, 11, 14, … The line that shows it (against term number) has what slope?

Answer: 3

The terms go up by 3 each time, so the slope is 3.

The line y = 4x − 3 shows a pattern. What is the constant difference between consecutive terms?

Answer: 4

The slope is 4, so the terms go up by 4 each time (1, 5, 9, 13, …).

A pattern goes 20, 17, 14, 11, … What is the slope of its line?

The terms of a pattern differ by 6 each time. Which line could show it (term number x, term y)?

• −3 — correct. Yes. The terms go down by 3 each time, so the slope is −3.
• 3. The terms go down, so the slope is negative.
• 20. 20 is the first term, not the slope.
• y = 6x + 1 — correct. Yes. The slope is 6, the same as the difference.
• y = x + 6. Its slope is 1, so the terms go up by 1 each time.
• y = 6. This line is flat: the terms never change.