Is this an integer?
A categorical atom from mrbartonmaths.com
What this sequence teaches
An integer is a number with no decimal part and no fraction part. That is the only thing that decides it. How big the number is doesn’t matter, and neither does whether it is positive or negative.
Students rarely struggle to see that 4.7 isn’t an integer. What they get wrong is everything around it: they think integers must be positive, or that “integer” means “negative number”, or that a number with a whole part (7.3, 5¼) is close enough. The sequence is built around those three errors.
The teaching sequence
Example 1 — 4.7 (not an integer). A decimal with a whole part. A student who thinks “it has a 4, so it’s an integer” says yes.
Example 1 → Example 2. The decimal part fades away. The 4 does not move.
Example 2 — 4 (an integer). The only thing that changed was the decimal part leaving.
Example 2 → Example 3. A new number, shown with a clean fade.
Example 3 — 53,086 (an integer). Size is irrelevant. The number has a zero inside it and is far bigger than anything students would call a “whole number” by habit.
Example 3 → Example 4. A minus sign fades in. Nothing else changes.
Example 4 — −53,086 (an integer). The sign made no difference. This is the example most students would get wrong without being shown.
Example 4 → Example 5. A half fades in. Nothing else changes.
Example 5 — −53,086½ (not an integer). A fraction part, however small next to the rest of the number, is enough. Being negative doesn’t make a number an integer, and having a whole part doesn’t either.
Why the sequence is shaped this way
- One feature, two ways to fail it. A decimal part opens the sequence and a fraction part closes it. Both appear in the same place, to the right of the ones digit, which sits in the same column in every example.
- Both non-examples keep a whole part. 4.7 and −53,086½ are more tempting than 0.2 or ¾, which are in the testing sequence.
- The minus sign arrives on its own, on a number students have just agreed is an integer, so they can see it changed nothing. It is then still there on the final non-example, so the sign can’t be the reason that one fails.
- The last example is deliberately strange. Students can’t recognise −53,086½ from memory. They have to use the rule.
What this sequence doesn’t address
Zero is in the testing sequence. The rule decides it.
Plain fractions and decimals (¾, 0.5) and top-heavy fractions (7/2) are in the testing sequence.
Numbers written in a way that hides their value (5.0, 6/2, 3², π) are outside this sequence’s scope. They are in the expansion sequence.
The rule and the script
The rule, exactly as the Show rule button displays it: For the following examples: if the number has no decimal part and no fraction part, it is an integer.
The spoken justification for each example:
- 4.7 — “This is not an integer. How do I know? Because it has a decimal part.”
- 4, 53,086 and −53,086 — “This is an integer. How do I know? Because it has no decimal part and no fraction part.”
- −53,086½ — “This is not an integer. How do I know? Because it has a fraction part.”
Whether students see the words is your call. Nothing appears unless you click. Show rule puts the rule above the stage, and you can turn it on before Example 1. Why? appears after each reveal. It shows the justification and highlights what decided it: the decimal part or the fraction part on a non-example, and the whole number on an integer, where there is nothing extra to point at.
Vocabulary. The page avoids “whole number” in the rule, because many students hear that as 0, 1, 2, 3 and so on, which would rule out −3. Once the sequence is secure you might say: “The integers are the whole numbers, their negatives, and zero.”
Running the sequence
Pause on each example before revealing, so students commit to an answer.
When you click Next on Examples 1, 3 and 4, the tick or cross clears, the stage holds still for a second, and then one thing changes slowly. Tell students to watch for what changes. Replay shows the change again.
If a class is unsure about Example 5, ask: “What stayed the same? What arrived?”
About the testing sequence
Ten items in a new random order each time: four integers and six that aren’t. The rule decides every one. There is no Show rule button here, because students apply the rule without it on screen. Why? works as in the teaching sequence, after each answer.
What each item is diagnosing
0. Some students think zero is “nothing” and so can’t be an integer. It has no decimal part and no fraction part, so it is one.
−1. Integers must be positive.
−7,300. Large and negative together.
2,000,000. Integers are small, familiar numbers.
0.5. The plain decimal, with no whole part to tempt anyone.
8.01. “It’s nearly 8.” The zero straight after the point makes the decimal part easy to dismiss.
−9.6. “Integer” means “negative number”.
3/4. A fraction looks as if it is made of two integers, so some students call it one.
7/2. A top-heavy fraction. Being bigger than 1 doesn’t remove the fraction part.
5¼. A whole part is enough. The teaching sequence showed this with a negative number; here it is positive.
Common confusions to watch for
“Integers are positive” shows up on −1 and −7,300.
“Integer means negative” shows up on −9.6, and sometimes on 2,000,000.
“It has a whole part” shows up on 8.01 and 5¼.
“It’s made of integers” shows up on 3/4 and 7/2.
Discussion prompts
- “Say the rule. Now use it on 0.”
- “Which part of 5¼ stops it being an integer?”
- “−1 is an integer and −9.6 isn’t. What is the difference between them?”
- “Change 8.01 as little as you can to make it an integer.”
Reading the summary
At the end, all ten items are shown together. The green or red tint shows whether the student answered correctly. The ✓ or ✗ in the corner shows whether the number is an integer.
About the expansion sequence
Six items that sit just outside what the teaching and testing sequences covered, to see how far students’ understanding stretches. It works exactly like the testing sequence.
The rule begins “For the following examples”. In those examples, how a number was written matched what it was worth. Here it doesn’t always. The point of this card is that being an integer is about the number’s value, not the way it is written.
Two pairs and two new surfaces
5.0 and 2.50. Both end in a zero. 5.0 is just 5, so it is an integer: the decimal part is zero. 2.50 is 2.5, so it isn’t. A student using the rule on sight says no to both.
6/2 and 10/4. Both are written as fractions. 6 ÷ 2 is exactly 3, so 6/2 is an integer. 10 ÷ 4 is 2.5, so 10/4 isn’t. Students have to work out the value before they can decide.
3². A power. Its value is 9, which is an integer.
π. A symbol with no digits showing. Its value is 3.14159…, which has a decimal part.
This page takes the position that 5.0 and 6/2 are integers. If your scheme of work treats them differently, skip this card or use it to open that discussion.
Who it’s for
Students who are secure on the testing sequence and who can simplify 6/2 and know roughly what π is.
The justifications
The Why? reasons go beyond the rule and name what is new. For example: 6/2 — “This is an integer. How do I know? Because 6 ÷ 2 is exactly 3. It is written as a fraction, but its value has no fraction part.”
Why? still highlights the decimal part or the fraction, as in the other cards. On 5.0 and 6/2 that is deliberate: it points at what the rule looks at, while the words say why that doesn’t settle it.
Discussion prompts
- “5.0 is an integer and 2.50 isn’t. What is the difference?”
- “Give me another fraction that is an integer. Give me one that isn’t.”
- “For which of these items did you have to work something out before you could decide?”
Reading the summary
The summary uses the same two channels as the testing card. A student who says “not an integer” to 5.0 and 6/2 is using the rule on how the number is written. That is the moment to go back to “For the following examples”.