Concept 5

Recognize patterns, relationships, and invariants across representations.

Core Idea

Different representations can hide the same relationship. Look for what stays fixed, such as a constant rate of change, equal differences, or an unchanged geometric property.

Understanding

Look for what stays fixed. A table, graph, diagram, and equation can all show the same relationship.

For linear patterns, the constant feature is often the rate of change. For similar figures, it may be a ratio. For transformed graphs, one value may move while another stays the same.

Find the quantity that does not change.
Translate that invariant into the form the question uses.
Check one known value to confirm the pattern.

Step by Step

Look for a quantity that stays constant, such as equal differences or equal ratios.
Decide what that constant means in the current representation.
Translate the invariant into the form the question asks for.
Check one known value to make sure the translated rule still matches.

Misconceptions

Treating every row of a table as unrelated instead of looking for a shared pattern.
Confusing a constant difference with a constant ratio.
Writing an equation that matches one point but not the overall pattern.

Question

Worked Example

The table shows values of $𝑥$ and $𝑦$ .

When $𝑥 = 1, 3, 5,$ and $7$ , the corresponding values of $𝑦$ are $14, 8, 2,$ and $- 4$ , respectively.

Which equation represents the relationship between $𝑥$ and $𝑦$ ?

Select an answer to see the explanation