L2 · Equivalent linear relations

Learning goal

“I can tell whether two linear relations are equivalent - they give the same value for every input - and I can prove it with values, not by how the relations look.”

Equivalent relations — two relations that produce the same value for every input (e.g. 2n + 3 and 3 + 2n).

Expression — a mathematical phrase built from numbers, variables and operations, with no equals sign (e.g. 5n + 2).

Substitute — replace the variable in an expression with a number to find its value at that input.

Constant — the part of a relation that does not change as the input changes (the start value, e.g. the +10 in 5w + 10).

Worked example

A counter pattern: Figure 1 has 5 counters, Figure 2 has 7, Figure 3 has 9. Writer 1 uses L1's method - growth 2, test the constant: c = 2n + 3. Writer 2 describes the build: "3 counters sit in the middle, and each figure adds a pair of 2": c = 3 + 2n. Step 1: They LOOK different. Looks prove nothing - substitute. Step 2: Substitute n = 1, 2, 3 into both: 2n + 3 gives 5, 7, 9. 3 + 2n gives 5, 7, 9. Step 3: Same values at every tested input, and the reason is real: addition can be done in either order, so 2n + 3 and 3 + 2n are the same expression written two ways. Say it plainly: equivalent relations are two descriptions of ONE pattern. To prove equivalence, substitute the same inputs into both and compare - at least two inputs, because one matching value can be luck.

Check yourself — explain it

Why does matching at ONE input not prove two relations are equivalent?

Paired problem — example still visible

Is 4n + 1 equivalent to 1 + 4n? Substitute n = 1 and n = 2 into both. -> Yes: both give 5 at n = 1 and 9 at n = 2.

Faded completion — last step is yours

"6n + 2 is equivalent to 2 + ___" -> 6n

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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