Is this a factor?
A categorical atom from mrbartonmaths.com
What this sequence teaches
When one number is a factor of another. There are two conditions: the first number is a whole number, and you can multiply it by a whole number to make the second number.
The sequence also takes on three habits students bring with them: mixing up which way round the question works (factor and multiple); deciding by looking at the last digits; and counting a decimal as a factor because it “goes in exactly”.
The teaching sequence
Example 1 — Is 30 a factor of 6? (No.) It opens with a no. Many students say yes, because 30 and 6 are “connected” through the 6 times table.
Example 1 → Example 2. The two numbers swap places. Nothing else changes.
Example 2 — Is 6 a factor of 30? (Yes.) Same numbers, opposite answer: the order of the question matters.
Example 2 → Example 3. The 6 stays put; the second number becomes 6,000.
Example 3 — Is 6 a factor of 6,000? (Yes.) Factors aren’t limited to times tables up to 12, and the same number can be a factor of many different numbers.
Example 3 → Example 4. “,000” fades away and the 6 slides across.
Example 4 — Is 6 a factor of 6? (Yes.) A number is a factor of itself. Many students believe a factor has to be smaller.
Example 4 → Example 5. “0.” appears in front of the first 6.
Example 5 — Is 0.6 a factor of 6? (No.) It still multiplies to make 6 (0.6 × 10), but it isn’t a whole number, so it isn’t a factor.
Example 5 → Example 6. The “0.” goes, the page pauses on “Is 6 a factor of 6?”, and then a 1 appears in front of the second 6.
Example 6 — Is 6 a factor of 16? (No.) The last digits still match, just as they did in Example 4 — and this time the answer is no. Matching last digits can’t be what decides it. This is the contrast students should leave with.
Why the sequence is shaped this way
- The first number stays at 6 from Example 2 to Example 6. Every change happens in one place, so every step can animate, and students see one number being a factor of 30, of 6,000 and of 6. The testing card then varies the first number widely.
- The pause before Example 6 is deliberate. Going straight from 0.6 to 16 would change both conditions at once. Returning to “6 of 6”, a yes the class has just seen, means only one thing changes at a time.
- Both numbers line up on the right, so in Examples 4 to 6 the first 6 sits directly above the second number’s 6. The last-digit habit is in plain view exactly where Example 6 knocks it down.
What this sequence doesn’t address
These are left to the testing card: 1 as a factor; a first number other than 6; a first number more than half the second; a decimal bigger than 1; “even goes into even”; and a second number that contains the first.
Negative numbers and 0 aren’t covered anywhere, because the conventions for them vary.
The rule and the script
The rule, exactly as the Show rule button displays it: For the following examples: if the first number is a whole number, and you can multiply it by a whole number to make the second number, then the first number is a factor of the second number.
The spoken justification for each example:
- 30 and 6 — “30 is not a factor of 6. How do I know? Because 30 is a whole number — but you can’t multiply it by a whole number to make 6. 30 × 1 is already 30.”
- 6 and 30 — “6 is a factor of 30. How do I know? Because 6 is a whole number, and you can multiply it by a whole number to make 30 — 6 × 5 = 30.”
- 6 and 6,000 — “…to make 6,000 — 6 × 1,000 = 6,000.”
- 6 and 6 — “…to make 6 — 6 × 1 = 6.”
- 0.6 and 6 — “0.6 is not a factor of 6. How do I know? Because you can multiply 0.6 by a whole number to make 6 — 0.6 × 10 = 6 — but 0.6 isn’t a whole number.”
- 6 and 16 — “6 is not a factor of 16. How do I know? Because 6 is a whole number — but you can’t multiply it by a whole number to make 16. 6 × 2 = 12 and 6 × 3 = 18.”
Whether students see the words is your call. For some classes the rule and the justifications help; for others the examples should do the talking. 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: it shows the justification, and at the same moment a working line such as “6 × 5 = 30” pulses in under the question. The working line puts on screen the number the rule depends on but the question doesn’t show: the whole number you multiply by. It is worth asking “What would you multiply by?” before you click.
Vocabulary. The page says “factor” and never “multiple”. Once the idea is secure, you might add: “You know how 6 × 5 makes 30? We say 30 is a multiple of 6, and 6 is a factor of 30. Same fact, two words, opposite directions.” Saved until then, the second word names a distinction students already hold, rather than adding to the confusion.
Where this atom sits
Before it: multiplication facts, and the difference between whole numbers and decimals.
After it: the expansion card below; then listing all the factors of a number, multiples, primes and highest common factors.
Running the sequence
Pause on each example before revealing. The pause is the prediction moment — students commit to an answer before the marker appears.
Examples 1 and 6 are where you’ll hear objections. Let them surface.
The Replay button appears on every example after the first, because every step is a single change. It is most worth using on Example 2 (the swap) and Example 6 (the return to 6 and 6, then the new digit).
About the testing sequence
Ten items in a new random order each time: five factors, five not. 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, including the working line.
The items are static. The animations in the teaching sequence mark the change between two examples; they don’t carry the meaning of factor, so a still question is a fair test.
What each item is diagnosing
24 and 8. Factor or multiple: which way round. It mirrors 8 and 24, so a student who says yes to both is reading the question as “are these numbers connected?”
4 and 14. Matching last digits, in its plainest form.
8 and 24. A factor whose last digit doesn’t match.
30 and 130. “The second number contains the first” — the harder form of the last-digit habit. 30 × 4 = 120 and 30 × 5 = 150.
40 and 4,000. It also contains its first number, and this time it is a factor. It also tests whether large numbers put students off.
9 and 18, and 17 and 17. “A factor must be much smaller”: 9 is half of 18, and 17 is 17 itself.
1 and 47. Whether students count 1 as a factor.
2.5 and 5. A decimal that multiplies to make the second number. It separates “not a whole number” from “0.6 was just too small”.
4 and 10. “Even goes into even.”
Common confusions to watch for
Right on Example 5 of the teaching sequence, wrong on 2.5 and 5. The student probably decided 0.6 was too small, not that it isn’t a whole number.
Yes to both 8 and 24 and 24 and 8. The student is checking that the numbers are related, not which way round.
Wrong on 30 and 130, right on 40 and 4,000. The student may be deciding by what they see written, not by multiplying.
Discussion prompts
- “Say the rule, then use it on 30 and 130.”
- “Which condition does 2.5 and 5 fail? Which one does it pass?”
- “What whole number would you multiply 9 by to make 18?”
- “8 and 24 is yes; 24 and 8 is no. What’s the only difference?”
Reading the summary
At the end, all ten items are shown together. Each cell has two channels of information: a green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether the first number really is a factor. A red cell with a ✓ means “this was a factor, and it was missed”.
About the expansion sequence
Six items that step just beyond the teaching and testing sequences, to see how far students’ understanding transfers. It works exactly like the testing sequence: random order, Factor or Not a factor, feedback, Why?, and a summary.
The teaching sequence’s rule begins “For the following examples”. This card is where students meet what lies just beyond those examples.
The items
On every item the rule still works, but on a surface students haven’t seen:
- Times tables students don’t know: 17 and 51 (a factor), 17 and 71 (not).
- Very large numbers: 13 and 1,300,000 (a factor), 3 and 1,000,000 (not — big and round doesn’t make it a factor).
- A second number written as a calculation: 7 and 7 × 99 (a factor), 7 and 7 × 99 + 1 (not). Both can be answered without working anything out. A student who reaches for division is showing that the rule hasn’t yet become a way of seeing the relationship.
There are no items where the rule stops working. The whole-number condition means it gives the right answer for every positive number; the only cases where it wouldn’t are 0 and negative numbers, whose conventions vary.
Who it’s for
Students who are secure on the testing card. The large numbers and unfamiliar times tables need confident multiplication; the two “7 × 99” items need none.
The justifications
They follow the same template and name what’s new:
- 17 and 51 — “…17 × 3 = 51. The rule works for any times table, not just the ones you know.”
- 7 and 7 × 99 — “…it’s already written that way, so there’s nothing to work out.”
- 7 and 7 × 99 + 1 — “…7 × 99 is just below it and 7 × 100 is just above it.”
Discussion prompts
- “Could you answer 7 and 7 × 99 + 1 without working anything out? How?”
- “17 and 51 is yes; 17 and 71 is no. Say the rule for each.”
Reading the summary
The same two channels as the testing card: green or red tint for whether the student was right, ✓ or ✗ for whether the first number is a factor. Missing the “7 × 99” pair suggests students are calculating rather than seeing; missing the large numbers suggests size is unsettling a rule they otherwise own.