Is this an odd number?
A categorical atom from mrbartonmaths.com
What this sequence teaches
To decide whether a whole number is odd, you only need its last digit. If the last digit is 1, 3, 5, 7 or 9, the number is odd; if it isn’t, the number isn’t odd. Every other digit is irrelevant — however many digits there are, and however odd or even they look.
The deciding feature is intuitive for students who already know which numbers up to 10 are odd. What isn’t intuitive is that nothing else matters. Students often judge by the first digit (“56 starts with a 5, so it’s odd”) or by what most of the digits are (“7,530 is mostly odd digits”). The sequence is built to break both habits.
The teaching sequence
Example 1 — 56 (not odd). The first digit is odd, the number isn’t. A student who decides by the first digit says yes.
Example 1 → Example 2. Only the last digit changes: 6 becomes 7, and the 5 stays exactly where it is. 57 is one more than 56 — 56 splits into pairs exactly, and one more leaves one over — which is worth saying if you want to tie the rule back to what odd means.
Example 2 — 57 (odd). The first odd number. The change that made it odd was the last digit, and only the last digit.
Example 2 → Example 3. A clean fade: a new example, not a transformation.
Example 3 — 684,201 (odd). Six digits, an even first digit, a zero inside, and every even digit (0, 2, 4, 6 and 8) somewhere in the number — including the 6 and the 0 that end the two non-examples. So no even digit can make a number “not odd” just by being in it. Students can decide this one without being able to read it aloud.
Example 3 → Example 4. A clean fade.
Example 4 — 7,539 (odd). Every digit is odd. This sets up the final example.
Example 4 → Example 5. Only the last digit changes: 9 becomes 0.
Example 5 — 7,530 (not odd). Three odd digits, and still not odd: most of the digits being odd doesn’t decide it. The final 0 does the deciding, which is the harder half of this example: students often treat a 0 on the end as nothing rather than as a digit, and 0 isn’t 1, 3, 5, 7 or 9. This is the contrast students should leave with.
Why the sequence is shaped this way
- It opens with a non-example, even though the page’s main audience is young. Every example has a prediction pause before the reveal, so the first question gets full attention. Opening with 57 instead would force the next odd number to be another number in the fifties, wasting an example.
- The two animated pairs are the lesson. In 56 → 57 and 7,539 → 7,530 the only thing that changes is the last digit, and the verdict flips. The three odd numbers in the middle are as different from each other as possible, to show the range of what an odd number can look like.
- Every number is right-aligned, so the last digit sits in the same place in every example and students’ eyes learn where to look. Nothing marks the digit unless you ask for it (see below).
What this sequence doesn’t address
A last digit of 0, and the number 0 itself, are in the testing card. The rule decides both.
Numbers whose other digits mix odd and even are also left to the testing card.
Negative numbers and decimals are outside this sequence’s scope, which is whole numbers of zero or more. They sit in the expansion card, where students test how far their understanding stretches.
The rule and the script
The rule, exactly as the Show rule button displays it: For the following examples: if the last digit is 1, 3, 5, 7 or 9, then the number is odd. “For the following examples” matters: the rule works for whole numbers, and the expansion card shows where it stops working.
The spoken justification for each example:
- 56 — “This is not an odd number. How do I know? Because the last digit isn’t 1, 3, 5, 7 or 9 — it’s 6.”
- 57 — “This is an odd number. How do I know? Because the last digit is 1, 3, 5, 7 or 9 — it’s 7.”
- 684,201 — “This is an odd number. How do I know? Because the last digit is 1, 3, 5, 7 or 9 — it’s 1.”
- 7,539 — “This is an odd number. How do I know? Because the last digit is 1, 3, 5, 7 or 9 — it’s 9.”
- 7,530 — “This is not an odd number. How do I know? Because the last digit isn’t 1, 3, 5, 7 or 9 — it’s 0.”
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 the last digit pulses and is underlined, so the picture and the words arrive together. On the two fades (Examples 3 and 4), where no animation has pointed at the last digit, it is worth asking “What decided it?” before you click.
Vocabulary. The page says “last digit” throughout. Once students are fluent, you might add: “You know how I’ve been saying ‘the last digit’? It has a proper name — the ones digit, because it tells you how many ones there are.” That also opens the door to why the rule works.
Where this atom sits
Before it: knowing which small numbers are odd through pairing — a number is odd if, when you put it into pairs, there is one left over. This atom assumes that. The meaning of odd lives there, not here.
This atom: that only the last digit decides. The sequence shows that it’s true. Why it’s true is for you to share when students are ready: every ten, every hundred, every thousand splits exactly into pairs, so the only part of a number that can leave one over is its ones.
After it: the expansion card below (negatives and decimals). A possible later atom for secondary is “Is this expression always odd?” (2n + 1, 2n + 4, n + 1, …).
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. It doesn’t appear on Examples 3 and 4: those are new examples, not transformations.
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: five odd, five not. Every item is a whole number of zero or more, 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 odd, so a still number is a fair test.
What each item is diagnosing
21 and 72. The first-digit habit in its plainest form: 21 starts even and is odd; 72 starts odd and isn’t.
5,000,000. The same habit, harder to resist: a 5 at the front and nothing but zeros after. Many students — and adults — hesitate over five million.
99,994. Deciding by what most digits are: four 9s and still not odd.
1,350. A last digit of 0 after three odd digits, at a different size from the taught example: does the teaching transfer?
0. The number zero, where the whole number is the digit students doubt. Its last digit is 0, which isn’t 1, 3, 5, 7 or 9, so it isn’t odd. Some students think zero is “nothing”, so neither odd nor even; the rule settles it.
40,003. Zeros inside the number, and an even first digit. Neither matters.
55,525. One even digit hidden among odd ones. Students who think any even digit makes a number even say no.
8,888,889. Every digit but the last is 8: the same belief in its plainest form, at seven digits.
6,208,647. Seven digits, odd and even mixed, with a zero inside. Big numbers need no working out — only the last digit.
Common confusions to watch for
“Look at the first digit” — shows up on 21, 72 and 5,000,000.
“Mostly odd digits means odd” — shows up on 99,994 and 1,350.
“An even digit anywhere makes it even” — shows up on 55,525, 8,888,889 and 6,208,647.
“Zero doesn’t count” — shows up on 0, 1,350, 5,000,000 and 40,003.
Discussion prompts
- “Say the rule. Now use it on 5,000,000.”
- “Which digit decided 99,994? Which digits didn’t matter?”
- “Change one digit of 55,525 so it isn’t odd. Which digits could you change?” — only the last one works, and it must become 0, 2, 4, 6 or 8.
- “Is 0 odd? Use the rule.”
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 number actually is odd.
A pattern of misses points to the gap: missing 21, 72 or 5,000,000 suggests the first-digit habit; missing 99,994 or 1,350 suggests the majority habit; missing 55,525, 8,888,889 or 6,208,647 suggests “any even digit spoils it”; missing 0 or 40,003 suggests uncertainty about zero.
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, Odd or Not odd, feedback, Why?, and a summary.
The teaching sequence’s rule begins “For the following examples”. This card is where students meet the edge of those examples.
Two kinds of item
Negative numbers — the rule still works. −47 and −1,009 are odd; −56 isn’t. The last digit decides, exactly as before. These test whether an unfamiliar minus sign unsettles a rule students already own. (Odd numbers carry on below zero: …, −5, −3, −1, 1, 3, 5, …)
Decimals — the rule stops working. 3.5, 0.7 and −2.5 all have a last digit of 5 or 7, so a student applying the rule without thinking says “odd”. None of them is odd. Only whole numbers can be odd or even: you can’t put three and a half into pairs with exactly one left over. These can only be answered from what odd means, not from the rule. They are the most revealing items on the page.
In this card, “whole number” means a number with no decimal or fraction part, so −47 counts as a whole number.
Who it’s for
The negatives suit students who have met numbers below zero. The decimals need students who read decimals confidently — older primary, or secondary classes revisiting odd and even. With younger classes, use this card selectively or not at all.
The justifications
Here the Why? reasons go beyond the rule and name what’s new, because the rule alone can’t decide a decimal:
- −47 — “This is an odd number. How do I know? Because the last digit is 1, 3, 5, 7 or 9 — it’s 7. The minus sign doesn’t change that.”
- 3.5 — “This is not an odd number. How do I know? Because the last digit is 5 — but 3.5 isn’t a whole number, and only whole numbers can be odd.”
As in the other cards, the last digit pulses when Why? opens. On the decimals that is deliberate: it points at the digit the rule looks at, while the words say why it doesn’t settle the question.
Discussion prompts
- “The last digit of 3.5 is 5. Why isn’t it odd?”
- “Is −1 odd? What about −2? How do you know?”
- “For which of these items did the rule give the right answer? For which did it give the wrong one?”
Reading the summary
The summary uses the same two channels as the testing card: green or red tint for whether the student was right, ✓ or ✗ for whether the number is odd.
Missing the negatives suggests the minus sign is unsettling a rule the student otherwise owns. Answering “odd” to the decimals 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 odd means.