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All courses Math I · F-LE.1.a 28 of 59
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.

Use first differences to identify a linear pattern

Problem
Use the displayed table. Compute first differences and classify the pattern using the requested evidence.
Raw input/output table without computed differences, ratios, or classification cues. Open full size
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Hint

Subtract consecutive outputs to find the first differences: \(5-2\), \(8-5\), and \(11-8\).

A pattern is linear when the first differences are constant over equal input intervals.

Solution walkthrough

01

Find the first difference between the first two outputs

\[5-2~=~3\]

The first difference measures how much the output changes from one row to the next.

02

Find the remaining first differences

\[\begin{aligned} 8-5~=~3 \\ 11-8~=~3 \end{aligned}\]

Keep subtracting consecutive outputs. Each change is 3.

03

List the first differences

\[\text{differences}:~3,~3,~3\]

The first differences are all the same, so the pattern changes by a constant amount each step.

04

Classify the pattern

\[\text{classification}:~\text{linear};~\text{differences}:~3,~3,~3\]

Because the first differences are constant over equal input intervals, the table represents a linear pattern.

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Another way

  1. You can notice that the outputs go up by 3 each time: \(2 -> 5 -> 8 -> 11\). A constant increase means the pattern is linear.

!

Common mistake

A common mistake is to subtract in the wrong order and write \(-3, -3, -3\). Since the outputs are increasing, each next value minus the previous value is \(+3\).

First-differences pattern card showing outputs 2, 5, 8, 11 with cues to subtract consecutive outputs and check whether the differences match.
classification: linear; differences: 3, 3, 3