For a table whose input values increase by equal amounts, constant successive differences in the output values indicate a linear relationship; the common output difference may be positive, negative, or zero. The learner distinguishes this change from the input increment when interpreting rate of change and connects the table pattern to a linear equation or graph, without extending to higher-order differences or generalizations involving unequally spaced inputs.
To recognize a constant first difference, compare each pair of neighboring input values and each pair of neighboring output values.
Example
| Input | Output |
|---|---|
Step 1: Check how the inputs change.
The inputs increase by equal amounts:
So, the input increases by each time.
Step 2: Find the successive output differences.
The outputs increase by each time. Since the output difference is constant, the table represents a linear relationship.
Step 3: Distinguish the output change from the input change.
The output changes by while the input changes by . The rate of change is
This means the output increases by for every increase of in the input.
Step 4: Connect the pattern to an equation.
The rule is
For example, when ,
and when ,
The points from the table would lie on a straight line when graphed. Therefore, equal input increases and constant output differences show a linear relationship.
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