Are these like terms?

What this card teaches

Two criteria for two algebraic terms to be “like”: (1) they have the same set of letters, and (2) the exponent on each letter matches. Coefficients (positive, negative, fractional, anything) are irrelevant. Two constants share an empty set of letters and are therefore like terms.

Example by example

Example 1 — 3x and 5y. Different letters. Opens with a negative case to set up the “same letters” criterion by failing it.

Example 1 → Example 2. Minimal change: y becomes x. The 5 stays put; the 3x on the left is unchanged. Students should notice that the only thing that changed is the letter.

Example 2 — 3x and 5x. Both x. The first positive: minimal change has locked in “same letters” while demonstrating that different coefficients are fine.

Example 2 → Example 3. Maximally-varied transition. Both terms become constants. This is a clean fade, not a continuous animation — it’s marking a change of example, not transforming anything.

Example 3 — 7 and 2. Two constants ARE like terms. Often surprising — students may resist this. Worth pausing on, and surfacing the Why? panel to anchor the “empty set of letters” framing.

Example 3 → Example 4. Maximally-varied transition into negative coefficients, a new letter, and exponents.

Example 4 — −5p² and 2p². Both p². Demonstrates that negative coefficients are fine, and matching exponents are part of the criterion.

Example 4 → Example 5. Minimal change: the ² disappears from term B. The 2 and p stay put.

Example 5 — −5p² and 2p. Same letter (p) but different exponents. Not like terms — exponents must match too.

Example 5 → Example 6. Minimal change attacking the “matching coefficients” misconception: term A’s −5 becomes 2. The p² stays put on the left; both terms now have coefficient 2.

Example 6 — 2p² and 2p. Matching coefficients but different exponents. Still not like terms. Directly attacks the misconception that matching coefficients is enough.

What’s not covered (saved for testing)

Fractional or non-integer coefficients; multi-variable terms (terms with more than one letter); letter order in multi-variable terms (commutativity). The testing card extends into all three.

Running the sequence

The three minimal-change transitions (1→2, 4→5, 5→6) are the foundational moments. Pause to discuss what changed and what stayed the same; replay if students miss the moment. Example 3 (constants ARE like terms) often surprises students — pause longer before revealing, and surface the Why? panel afterwards to make “empty set of letters” explicit. Example 6 is the most challenging — students who confidently called Example 5 negative may relapse on Example 6 because the matching coefficients pull them back toward saying ✓. The testing card’s teacher notes list the specific misconceptions to watch for in the testing sequence.

About this testing card

Ten items, shuffled fresh each session. Five are like terms; five are not. Every item probes a specific misconception or extends into territory the teaching card didn’t cover (fractional coefficients, multi-variable terms, letter order). Two items (the same-letter baseline and the constants pair) deliberately overlap with the teaching examples to confirm the core criteria transferred.

What each item is diagnosing

8x and 3x — baseline. Same letter, different positive integer coefficients. A confidence check that the core same-letters criterion transferred from teaching.

−¾y and 8y. Extends “coefficients are irrelevant” into fractional and negative coefficients. Catches students whose mental rule is “coefficients must be whole and positive.”

3xy and 7xy. First multi-variable item. Both terms have the same letter set (x and y, each to the implicit power of 1).

4ab and 2ba. Letter order is irrelevant. Multiplication is commutative, so ab and ba are the same product. Catches students who treat letter order as significant.

−6 and 11. Two constants of opposite sign. Reinforces the teaching that constants are like terms, with a wider variation than Example 3.

4x and 4y. Matching coefficients but different letters. The signature attack on the “matching coefficients means like terms” misconception.

2x² and 5x. Same letter (x) but different exponents. Each letter’s exponent must match too.

2xy and 3xz. Partial overlap — both have x, but the other letters differ (y vs z). Every letter must match, not just one of them.

5 and 3x. One is a constant, the other has a letter. Their sets of letters don’t match.

x²y and xy². Same letters but the powers are distributed differently. Each letter’s exponent must match in both terms.

Common confusions to watch for

“Matching coefficients means like terms” — probed by 4x and 4y. Teaching Example 6 addressed this directly; students who still answer ✓ here haven’t internalised that letters matter more than coefficients.

“Same letter is enough (exponents don’t matter)” — probed by 2x² and 5x and by x²y and xy². Two angles on the same misconception — the exponent on each letter must match.

“Any common letter is enough” — probed by 2xy and 3xz. Students who say ✓ here are reasoning “both have x” without checking that every letter appears in both.

“Coefficients must be whole and positive” — probed by −¾y and 8y, reinforced by −6 and 11. The teaching covered negative integers (Example 4); fractional coefficients are new ground.

“Letter order matters” — probed by 4ab and 2ba. Students who say ✗ here are treating the letters as a positional sequence rather than a multiplicative set.

Discussion prompts

Three short prompts to extend the work after the testing sequence:

1. “In your own words: what does it mean for two algebraic terms to be ‘like’?” — tests whether students can articulate the rule without leaning on specific examples.
2. “Can 5x³ and x³ be like terms? What about 5x³ and 5x²? What rule are you using?” — surfaces the dual criteria (same letters, matching exponents) explicitly.
3. “Why is 4ab the same as 2ba as far as being ‘like’ is concerned? What if it were 4a²b vs 2ba²?” — surfaces commutativity and how it interacts with exponents.

Reading the summary

At the end, all ten items are shown together. Each cell has two channels of information: a green or red tint indicates whether the student answered correctly; a ✓ or ✗ in the corner indicates whether the two terms actually are like terms. The two channels stay independent.

If a student misses specific items consistently, the pattern points to the gap: missed 4x and 4y means matching coefficients is still feeling decisive; missed 2x² and 5x or x²y and xy² means the exponent criterion hasn’t landed; missed 2xy and 3xz means partial overlap is reading as full overlap; missed 4ab and 2ba means letter order is feeling significant; missed −6 and 11 means constants haven’t been accepted as like terms despite teaching Example 3.