Is this an integer?

What this sequence teaches

A number is an integer if and only if it is a whole number — no fractional part, no decimal point — regardless of sign. Negatives count, provided they are themselves whole. Zero is an integer (covered in the testing card, not here).

Two critical features carry the work:

• Whole-numberness — intuitive; students recognise non-whole forms easily.
• Sign tolerance — non-intuitive; students often assume integers must be positive.

The teaching sequence

Example 1 — Is 0.2 an integer? (No.) Opens with the most common non-integer surface form: a decimal less than 1. Frames the criterion: this isn’t an integer because it isn’t whole.

Example 1 → Example 2. The decimal part is removed via cross-fade, leaving 2 behind. Minimal change isolating the whole-numberness criterion.

Example 2 — Is 2 an integer? (Yes.) First positive: minimal change locks in the “whole, no fractional part” feature.

Example 2 → Example 3. Varied positive. A clean fade marks the new example.

Example 3 — Is 53076 an integer? (Yes.) Pushes magnitude hard. Integers don’t have an upper bound; a five-digit number is no less an integer than a single-digit one.

Example 3 → Example 4. Varied positive introducing the non-intuitive critical feature: a negative sign appears for the first time.

Example 4 — Is −3 an integer? (Yes.) The non-intuitive feature. The minus sign is something integers permit, not something that disqualifies a number.

Example 4 → Example 5. Cross-fade with the minus sign held visually constant. The “3” body morphs into the fraction stack “1/3”. Single-feature change: whole becomes fractional.

Example 5 — Is −1/3 an integer? (No.) Attacks the “all negatives are integers” over-generalisation. The requirement was negative and whole; this fails the whole part.

Example 5 → Example 6. Cross-fade with the minus sign still constant. The fraction body dissolves into a decimal body. Same rule, different surface form.

Example 6 — Is −2.5 an integer? (No.) Generalises across surface forms. The pattern in Examples 4→5→6 is “negative + whole, negative + fraction, negative + decimal” with the minus invariant — the operand carries the verdict.

What this sequence doesn’t address

Zero is deliberately deferred to testing. Most students accept it as an integer with a moment’s thought; it’s a boundary case rather than a non-intuitive critical feature.

Fractions and decimals that equal integers (such as 6/2 or 5.0) are excluded throughout. Their verdict depends on whether the question is about surface form or underlying value — a fraction-simplification skill outside what’s being taught.

Positive non-integers other than 0.2 are deferred to testing, where students probe whether the rule transfers across surface forms and sign.

Running the sequence

Pause on each example before revealing. The pause is the prediction moment — students commit to an answer mentally before the marker appears.

Example 4 is where the harder thinking happens. Expect a slower reveal; this is the non-intuitive critical feature. Some students will say “no” because the number is negative — surface their reasoning before showing the verdict.

The Replay button restarts the animation into the current frame, useful for Examples 2, 5, and 6 where a single-feature minimal change is the lesson. Replay doesn’t appear on Examples 3 and 4 (varied positives) — there’s no continuous transformation to replay; the change is a new example.

If a class struggles with Example 5 → 6, ask “what’s stayed the same?” and direct attention to the minus sign. The invariance is the lesson.

About the testing sequence

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

What each item is diagnosing

7 — baseline. A positive whole number; confirms core understanding before probing edge cases.

0. Boundary case: zero is an integer. Some students treat it as “not really a number” or as belonging to neither positives nor negatives.

−89. Negative integer under realistic conditions, away from the supported atmosphere of the teaching sequence.

89412 and −5043. Large magnitudes. There is no largest or most-negative integer; students who associate “integer” with small familiar numbers are surfaced here.

4.6. Decimal greater than 1. The teaching sequence only used 0.2 (less than 1) and −2.5 — this catches students who think decimals less than 1 are the only “non-integer decimals.”

3/4. Positive proper fraction. The teaching sequence used a stacked fraction only with a negative sign attached; this confirms the rule transfers to positive fractions.

7/2. Improper fraction. The numerator larger than the denominator may tempt students to read this as “a big whole-looking number” — but 7/2 is 3.5.

−7.5. Negative decimal of different magnitude than the teaching sequence’s −2.5. Confirms the rule isn’t size-specific.

−5/8. Negative fraction at a different denominator than the teaching sequence’s −1/3. Confirms the rule isn’t denominator-specific.

Common confusions to watch for

“Integers must be positive” — probed by −89 and −5043. Students who say ✗ on either are still treating sign as disqualifying.

“Zero isn’t really a number” — probed by 0. A surprisingly common slip.

“Decimals less than 1 fail; decimals greater than 1 are integers” — probed by 4.6 and −7.5. Catches over-fitting to the teaching sequence’s 0.2.

“Improper fractions are integers” — probed by 7/2. The visual layout is identical to a proper fraction, but the larger numerator may suggest “whole.”

“Fractions are only non-integers when negative” — probed by 3/4 (reinforced by −5/8). The teaching sequence used a negative fraction; this catches asymmetric generalisation.

Discussion prompts

1. “In your own words: what is an integer?” — tests whether students can articulate the criterion without leaning on examples.
2. “Why is zero an integer but ¾ isn’t?” — surfaces the whole-number criterion explicitly.
3. “Is there a smallest integer? A largest integer? Why or why not?” — surfaces that integers extend without bound in both directions.

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 number actually is an integer.

If a student misses specific items consistently, the pattern points to the gap: missed negative items (−89, −5043) mean sign tolerance hasn’t been internalised; missed zero means the boundary case needs revisiting; missed 4.6 or −7.5 means “decimal” hasn’t generalised; missed 7/2 means the improper-fraction surface form is causing trouble.