Are these like terms?

What this sequence teaches

Two terms are like terms when they are exactly the same apart from the number at the front. The number at the front never matters: not its size, not its sign, not whether it is a whole number, and not whether it is there at all. Everything else has to match — the same letters, each with the same power.

Two things about this aren’t intuitive. The first is that the number really doesn’t matter: students hesitate over 8x and −3514x because the numbers look so different, or over ay² because it has no number at all. The second is that powers count: y and y² “both have y”, so many students call them like terms. The sequence is built to break “sharing a letter is enough”.

The teaching sequence

Example 1 — 8x and −3514y (not like terms). Almost everything is different: the numbers, the signs and the letters. A student who thinks the numbers have to match says no, for the wrong reason.

Example 1 → Example 2. Only the y changes, becoming x. The numbers and signs stay exactly where they are.

Example 2 — 8x and −3514x (like terms). The numbers are wildly different and one is negative, and yet these are like terms. So the numbers can’t have been what decided Example 1.

Example 2 → Example 3. A clean fade: a new example, not a transformation.

Example 3 — ⅗m⁷ and 26m⁷ (like terms). A fraction against a whole number, both positive, and a power that matches. Powers are fine when they are the same.

Example 3 → Example 4. A clean fade.

Example 4 — ay² and −40ay² (like terms). Two letters, a squared letter, and no number at the front of ay². That’s fine: the rule sets the number aside, and what’s left is ay² in both. This example sets up the last one.

Example 4 → Example 5. Only the power on y changes: it goes. The a stays in both terms.

Example 5 — ay and −40ay² (not like terms). Both terms have an a and both have a y, and they are still not like terms, because ay and ay² are different. This is the contrast students should leave with.

Why the sequence is shaped this way

• The two animated pairs are the lesson. In each, one thing changes and the verdict flips. The three like pairs in the middle are as different from each other as possible, to show the range of what like terms can look like: the numbers at the front run from an invisible 1 to 3514, positive and negative, whole and fractional.
• The layout does some of the looking for students. The two terms are stacked, with the numbers at the front ending in the same place and what’s left starting in the same place. What’s left in one term sits directly above what’s left in the other, ready to compare.
• Letters are always written in alphabetical order. That is how terms are usually written, and the rule relies on it. Letters in a different order are on the expansion card.

What this sequence doesn’t address

Left to the testing card, where the rule decides every item: the same number at the front but not like terms (7x and 7y); decimals at the front; and a power on a different letter (x²y and xy²).

Left to the expansion card: numbers on their own (7 and −2); letters written in a different order (4ab and 2ba); and a number written somewhere other than the front (x/2).

The rule and the script

The rule, exactly as the Show rule button displays it: For the following examples: if the two terms are exactly the same apart from the number at the front, then they are like terms. “For the following examples” matters: every example here has its letters in alphabetical order, and the expansion card is where that stops being true.

The spoken justification for each example:

• 8x and −3514y — “These are not like terms. How do I know? Because the two terms aren’t exactly the same apart from the number at the front — what’s left is x in one and y in the other.”
• 8x and −3514x — “These are like terms. How do I know? Because the two terms are exactly the same apart from the number at the front — what’s left is x in both.”
• ⅗m⁷ and 26m⁷ — “… what’s left is m⁷ in both.”
• ay² and −40ay² — “… what’s left is ay² in both.”
• ay and −40ay² — “These are not like terms. How do I know? Because the two terms aren’t exactly the same apart from the number at the front — what’s left is ay in one and ay² in the other.”

Whether students see the words is your call; nothing appears unless you click. Show rule puts the rule above the stage — you can turn it on before Example 1 — and it stays until you turn it off. Why? appears after each reveal: at the same moment the numbers at the front fade to grey, and what’s left in each term turns orange, pulses and is underlined, lined up one above the other. On the two fades (Examples 3 and 4), where no animation has drawn the eye, it is worth asking “What decided it?” before you click.

Vocabulary. The page says “the number at the front” throughout. Once students are fluent, you might add: “You know how I’ve been saying ‘the number at the front’? Its proper name is the coefficient.” It is worth pointing out then that ay² has a coefficient of 1, even though no 1 is written.

Where this atom sits

Before it: knowing what a term is, and reading 3x as 3 × x and y² as y × y.

This atom: that like terms are the same apart from the number at the front.

After it: the expansion card below. Beyond that comes collecting like terms, where the reason it works is the point: 8x and −3x are 8 lots of x and 3 fewer lots of x. A possible later atom is “Is this expression fully simplified?”

Running the sequence

• Pause on each example before revealing. The pause is the prediction moment — students commit to an answer before the marker appears.
• The Replay button appears on Examples 2 and 5, where a single change is the lesson.
• If a class struggles with Example 5, go back to Example 4 and replay the change: “What’s stayed the same? What changed?”

About the testing sequence

Ten items in a new random order each time: four pairs of like terms and six that aren’t. In every item each term has its letters in alphabetical order, so the rule decides every one. There is no Show rule button here — students apply the rule without it on screen. Why? works as in the teaching sequence, after each answer.

The items are static. The animations in the teaching sequence mark the change between two examples; they don’t carry the meaning of like terms, so a still pair is a fair test.

What each item is diagnosing

7x and 7y; −3k² and −3k. The same number at the front, and not like terms. The first is the plain form. The second is harder, because everything matches, minus sign included, except one power.

−6pq and −6pq. Identical terms, which are like terms. It is here so that “same number” can’t decide the verdict in either direction.

w and 0.6w. No number at the front, against a decimal.

−⅜h⁹ and 12h⁹. A negative fraction and a large power.

x²y and 130x²y, with x²y and 4xy². A deliberate pair. Both use x and y, but in the second the squared letter has moved. Students who check which letters are present, but not which letter carries the power, say yes to both.

x² and 2x. The “x² is 2x” confusion. When Why? opens, the 2 at the front of 2x greys out while the 2 in x² stays orange, so the two 2s are visibly doing different jobs.

2.5pq and −3p. “They both have p.”

a² and 250b². “They’re both squared.”

Common confusions to watch for

• “The numbers have to match” — shows up on w and 0.6w, −⅜h⁹ and 12h⁹, and x²y and 130x²y.
• “The same number means like terms” — shows up on 7x and 7y, and −3k² and −3k.
• “Sharing a letter is enough” — shows up on 2.5pq and −3p, −3k² and −3k, x² and 2x, and x²y and 4xy².
• “The same power is enough” — shows up on a² and 250b².
• “No number means it can’t match” — shows up on w and 0.6w, and x²y and 130x²y.

Discussion prompts

• “Say the rule. Now use it on x² and 2x.”
• “Cover the numbers at the front of −3k² and −3k. What’s left?”
• “Change one thing in 7x and 7y so they are like terms. What could you change?” — either letter, and the numbers don’t need to change at all.
• “Read x²y and 4xy² letter by letter. Where’s the difference?”

Reading the summary

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

A pattern of misses points to the gap. Missing 7x and 7y, or −3k² and −3k, suggests the number at the front is still being used to decide. Missing x² and 2x, 2.5pq and −3p, or x²y and 4xy² suggests “sharing a letter is enough”. Missing w and 0.6w, or −⅜h⁹ and 12h⁹, suggests unusual numbers are unsettling the rule.

About the expansion sequence

Six items that sit just outside what the teaching and testing sequences covered, to see how far students’ understanding transfers. It works exactly like the testing sequence: random order, Like terms or Not like terms, feedback, Why?, and a summary.

The rule begins “For the following examples”. Every one of those examples had letters, written in alphabetical order, with the number at the front. This card breaks each of those in turn.

Two kinds of item

The rule still works.

• 7 and −2 (like terms). There’s nothing left in either term. Numbers on their own are always like terms, which is why 7 + (−2) simplifies to 5.
• 7 and 7x (not like terms). Nothing is left in one term and x is left in the other. The matching 7s make it tempting.
• a³bk⁵w and −62a³b²k⁵w (not like terms). A long letter part with one power hidden in the middle. When Why? opens and the two parts line up, the b² visibly pushes everything after it out of line.

The rule stops working.

• 4ab and 2ba (like terms). The terms aren’t written exactly the same, so the rule on its own says no. But ab means a × b, and b × a is the same thing.
• x/2 and 3x (like terms). There’s no number at the front of x/2, so the rule can’t be applied as written. But x/2 is half of x, the same as ½x.
• ab and 6ba² (not like terms). The order doesn’t matter, but one term has a² and the other only a. It is here so that “written in a different order means like terms anyway” can’t score perfectly.

The first two rule-breaking items can only be answered from what the letters mean, not from the rule. They are the most revealing items on the page.

Who it’s for

The first two items suit anyone who has done the testing card. The rest need students who read ab as a × b and x/2 as half of x. With classes new to algebra, use this card selectively.

The justifications

Here the Why? reasons go beyond the rule and name what’s new, because the rule alone can’t decide the last three:

• 7 and −2 — “These are like terms. How do I know? Because the two terms are exactly the same apart from the number at the front — there’s nothing left in either. Two numbers on their own are always like terms.”
• 4ab and 2ba — “These are like terms. How do I know? Because what’s left is ab in one and ba in the other — a different order, but b × a is the same as a × b.”
• x/2 and 3x — “These are like terms. How do I know? Because x/2 has no number at the front — but x/2 means half of x, which is ½x. So what’s left is x in both.”

As on the other cards, Why? points at what the rule looks at. On 7 and −2 an empty dashed slot appears in each row, showing that nothing is left in either.

Discussion prompts

• “The rule says 4ab and 2ba aren’t like terms. Why is the rule wrong here?”
• “Is 7 like 7x? Is 7 like −2? What’s the difference?”
• “Rewrite x/2 so the number is at the front.”

Reading the summary

The summary uses the same two channels as the testing card: a green or red tint for whether the student was right, and ✓ or ✗ for whether the pair are like terms.

Missing 7 and −2 suggests a belief that terms need letters. Answering “not like terms” to 4ab and 2ba, or to x/2 and 3x, means the rule is being applied outside the examples it was built for — a good moment to return to “For the following examples”, and to what ab and x/2 mean. Answering “like terms” to ab and 6ba² means “a different order is fine” has been over-learned.