Is this a factor?

What this sequence teaches

Students learn to decide whether a number is a factor of another by checking three critical features:

1. The candidate must divide exactly into the target (no remainder).
2. The candidate must be the right way round — a factor of n is something n can be divided by, not something n divides into.
3. The candidate must be a whole number (an integer).

The question form — “Is X a factor of Y?” — carries the notation. There is no implicit notation switch in this resource (in contrast to “Is this a square?”, where some examples use labelled dimensions and others use tick marks). All teaching and testing items use the same form.

The teaching sequence

Example 1 — Is 3 a factor of 13? (No.) Opens with a divides-with-remainder case, setting up the “divides exactly” feature by failing it.

Example 1 → Example 2. Cross-fade isolating the target: 13 becomes 12. Now 3 divides exactly. The candidate is unchanged.

Example 2 — Is 3 a factor of 12? (Yes.) The first positive: minimal change locks in the “divides exactly” feature.

Example 2 → Example 3 and Example 3 → Example 4. Maximally-varied positives. Both numbers change at once (3 & 12 → 2 & 10 → 5 & 20). These transitions are clean fades, not animations — they’re marking a change of example, not transforming anything.

Example 4 → Example 5. Swap: the candidate (5) and the target (20) exchange positions. The numbers themselves don’t change — only their roles. This isolates the “right way round” feature.

Example 5 — Is 20 a factor of 5? (No.) 20 is a multiple of 5, not a factor.

Example 5 → Example 6. Cross-fade isolating the candidate: 20 becomes 2.5. The target stays at 5.

Example 6 — Is 2.5 a factor of 5? (No.) Although 5 ÷ 2.5 = 2 exactly, 2.5 is not a whole number. This isolates the integer requirement.

What the teaching sequence does and doesn’t address

All three critical features have minimal-change transitions in the sequence: divides exactly (1→2), direction (4→5), and integer requirement (5→6). Unlike the square sequence, no critical feature is held back for testing.

The testing sequence’s job is to probe specific misconceptions within these three features — particularly the “factor must be much smaller” intuition and the boundary cases (1 as a factor, a number as a factor of itself).

Running the sequence

Example 4 → Example 5 (the swap from “Is 5 a factor of 20?” to “Is 20 a factor of 5?”) is the most important transition — the direction reversal is the lesson. Replay it if students miss the moment, and consider asking “is this the same question as the previous one?” before revealing. Example 6 (Is 2.5 a factor of 5?) often catches students by surprise; pause longer before revealing, and surface the Why? panel afterwards to make the integer requirement explicit. If students are confidently calling every example correctly without slowing down, push back — the goal is reasoning, not pattern-matching. The testing card’s teacher notes list the specific misconceptions to watch for during the testing sequence.

About the testing sequence

Ten items, randomised on each load. Five are factors and five are not, each chosen to probe a specific misconception or boundary case rather than be obvious filler.

What each item is diagnosing

4 is a factor of 12 — baseline. A standard divides-exactly case to confirm core understanding.

9 is a factor of 18. Probes the “factor must be much smaller” misconception. 9 is half of 18 — students who think factors must be small may resist this.

1 is a factor of 9. Boundary case — 1 is a factor of every whole number. Some students forget or never internalise this.

7 is a factor of 7. Boundary case — every whole number is a factor of itself. Students who treat “factor” as strictly smaller will miss this.

20 is a factor of 100. A factor outside the times tables that students typically rehearse. 100 ÷ 20 = 5 is straightforward arithmetic but unfamiliar packaging.

10 is a factor of 5 and 12 is a factor of 4. Direction errors. Both items should feel obviously wrong once students apply the “divides into” test — you can’t divide 5 by 10 and get a whole number.

4 is a factor of 14. Last-digit trap. 14 ends in 4, but 14 ÷ 4 = 3 remainder 2. Catches students who reason from surface features rather than dividing.

5 is a factor of 12. Divides-exactly fails without any surface trap. Pure check that the student is actually doing the division.

3.5 is a factor of 7. Integer requirement. 7 ÷ 3.5 = 2 cleanly, so the divides-exactly test passes — but 3.5 isn’t a whole number. Catches students who’ve internalised the divides-exactly rule but missed the integer constraint.

Common confusions to watch for

Direction confusion (treating “factor” as if it could mean “multiple”) — probed by 10 of 5 and 12 of 4. Students who say ✓ on either haven’t internalised the divides-into direction.

“Factor must be much smaller” — probed by 9 of 18, 1 of 9, and 7 of 7. Three angles on the same misconception: 9 doesn’t feel small enough; 1 is often forgotten; 7 of 7 is a boundary case students resist instinctively.

Last-digit trap — probed by 4 of 14. Catches students reasoning from surface features (14 ends in 4) rather than dividing.

Integer requirement missed — probed by 3.5 of 7. Catches students who have internalised “divides exactly” but skipped the whole-number constraint.

“Factors only come from times tables” — probed by 20 of 100. Probes students who recognise 5 × 20 = 100 but don’t readily flip it to say 20 is a factor of 100.

Discussion prompts

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

1. “In your own words: what does it mean for one number to be a factor of another?” — tests whether students can articulate the rule without leaning on examples.
2. “Why can’t 2.5 be a factor of 10, even though 10 ÷ 2.5 = 4 exactly?” — surfaces the integer requirement explicitly.
3. “We say ‘10 is a multiple of 5’ but ‘5 is a factor of 10’. Why do these point in opposite directions?” — surfaces the direction asymmetry students need to internalise.

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 relationship actually is a factor relationship.

If a student misses specific items consistently, the pattern points to the gap: missed direction items (10 of 5, 12 of 4) mean the “which way round” check isn’t happening; missed 3.5 of 7 means the integer constraint hasn’t been picked up; missed boundary cases (1 of 9, 7 of 7) mean the student is using an over-restrictive definition of “factor.”