Is this a square?

What this sequence teaches

To decide whether a shape is a square, students check two things: all the sides are the same length, and all the angles are right angles. Both must be true.

Neither property is ever judged by eye. Every side carries a length label, and every corner carries a mark: a small box means a right angle; a curve means the angle is not a right angle. Each shape is drawn in proportion to its own labels. It’s worth telling students the box-and-curve convention before you start.

The teaching sequence

Example 1 — not a square. A shape standing on one corner, all sides 5 cm, with curves in the corners. The sides pass; the angles fail.

Example 1 → Example 2. The shape opens out while the side labels never change. Each curve becomes a box only as its angle arrives at a right angle. Only the angles change — this is the lesson of the first half.

Example 2 — a square, still standing on its corner. Many students call this “a diamond”. It keeps the look of Example 1, so the look can’t be what decided either verdict.

Example 2 → Example 3. A clean fade: a new example, not a transformation.

Example 3 — a square sitting flat, drawn small, labelled 300 m. A small drawing with a big number: drawn size, units and the value don’t matter.

Example 3 → Example 4. A clean fade.

Example 4 — a square at an awkward tilt, drawn large, labelled 20 with no units.

Example 4 → Example 5. The left side stays exactly where it is while one pair of sides grows from 20 to 21. The boxes stay throughout. Only one pair of lengths changes, by the smallest whole-number step.

Example 5 — not a square. The angles pass; the sides don’t all match. It still looks like a square, so the labels decide it, not the eye. This is the boundary students should walk out remembering.

Why the sequence is shaped this way

• One non-example for each condition. The first attacks the angles, the last attacks the sides. Between them, three squares that are as different from each other as possible: on a corner, flat and tilted; medium, small and large; cm, m and no units; 5, 300 and 20.
• The two animated pairs are the lesson. In each, one thing changes and the verdict flips.
• The non-examples are never named on screen. A square is a special rhombus and a special rectangle, so the page never says “this is a rectangle, not a square”.

What this sequence doesn’t address

Shapes that aren’t four-sided are in the testing card, as a regular pentagon. The rule deliberately doesn’t say “four”: a shape with straight sides whose angles are all right angles has to have four sides, so the rule already covers it.

Tick marks, parallel arrows, angles written as 90°, decimal lengths and mixed units are in the expansion card.

The rule and the script

The rule, exactly as the Show rule button displays it: For the following examples: if all the sides are the same length and all the angles are right angles, then the shape is a square.

The spoken justification for each example:

• Example 1 — “This is not a square. How do I know? Because all the sides are the same length — they’re all 5 cm — but not all the angles are right angles.”
• Examples 2, 3 and 4 — “This is a square. How do I know? Because all the sides are the same length — they’re all 5 cm / 300 m / 20 — and all the angles are right angles.”
• Example 5 — “This is not a square. How do I know? Because all the angles are right angles — but not all the sides are the same length. Two are 20 and two are 21.”

Whether students see the words is your call. 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 side labels pulse, then the corner marks, in the order the sentence names them. Every example gets the same treatment, whatever its verdict. On the two fades, where no animation has drawn the eye, it’s worth asking “What decided it?” before you click.

Vocabulary. Once students are secure, you might add: “The shape in Example 1 has a name — a rhombus — and Example 5 is a rectangle. A square is a special rhombus and a special rectangle, which is why we never say a square isn’t a rectangle.”

Where this atom sits

Before it: knowing what a right angle is. After it: the expansion card below, and possible later atoms “Is this a rectangle?” and “Is this a rhombus?”, where the square appears as a positive example.

Running the sequence

Pause on each example before revealing. The pause is the prediction moment.

The Replay button appears on Examples 2 and 5, where a single change is the lesson. Replay Example 1 → Example 2 if students miss the moment the curves become boxes.

On Example 5, if students say “square” before the reveal, don’t argue about how it looks — ask what the labels say. Push back on any verdict given by eye.

About the testing sequence

Ten items in a new random order each time: five squares and five non-squares. Every item uses the teaching conventions — every side labelled with a whole number, every corner marked with a box or a curve — 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 a square, so a still shape is a fair test.

What each item is diagnosing

The square standing on its corner (64 m). “A diamond isn’t a square.”

The large flat square labelled 2 cm, and the small tilted square labelled 150 cm. Judging by drawn size rather than by the labels.

The squares labelled 9 (no units) and 11 mm. Awkward tilts and different units shouldn’t unsettle the rule.

The flat, leaning shape with equal 8 cm sides and curves. The plainest form of “equal sides is enough”.

The shape standing on its corner, all sides 12, with curves. The same belief, harder to resist: it looks almost exactly like the corner-standing square, and only the curves say otherwise.

The regular pentagon with 6 cm sides. Equal sides on a shape that isn’t four-sided. Its curves mean the rule rejects it.

The long thin rectangle, 4 m by 15 m. The plainest form of “right angles is enough”.

The flat rectangle, 99 mm by 100 mm. The same belief, harder to resist. Students who judge by eye will call it a square.

Common confusions to watch for

Judging by eye — shows up on the 99 mm by 100 mm rectangle and on the corner-standing shape with curves.

Treating orientation as a feature — shows up on the corner-standing square.

“Equal sides is enough” — shows up on both equal-sided shapes with curves, and on the pentagon.

“Right angles is enough” — shows up on both rectangles.

Discussion prompts

1. “Say the rule. Now use it on the 99 mm by 100 mm rectangle.”
2. “Which part of the rule does the pentagon pass? Which part does it fail?”
3. “Put the corner-standing square next to the corner-standing shape with curves. What is the one difference that matters?”

Reading the summary

At the end, all ten items are shown together. Each cell has two channels: the ✓ or ✗ in the corner shows whether the shape is a square; the green or red tint shows whether the student answered correctly.

Missing either shape with curves or the pentagon suggests “equal sides is enough”; missing either rectangle suggests “right angles is enough” or judging by eye; missing the corner-standing square suggests orientation is getting in the way.

About the expansion sequence

Seven items that sit just outside what the teaching and testing sequences covered, to see how far students’ understanding stretches: three squares and four non-squares. It works exactly like the testing sequence: random order, Square or Not a square, feedback, Why?, and a summary.

The teaching sequence’s rule begins “For the following examples”. This card goes beyond those examples.

Who it’s for

Tick marks, parallel arrows and angles written as 90° assume students have met that notation, typically in Years 4–6. The mixed-unit items need students who know that 1 m = 100 cm and 1 cm = 10 mm. A Year 3 class may skip this card, or use it selectively.

Two kinds of item

The rule still works on an unfamiliar surface.

• A square with a single dash on every side and boxes. Matching dashes mean matching lengths.
• A leaning shape with a single dash on every side, parallel arrows, and curves. The arrows sound persuasive, but the angles still fail.
• A near-square with single dashes on one pair of sides, double dashes on the other, and boxes. It looks like a square; only the dashes say otherwise.
• A square with 90° written in each corner and 2.5 cm sides. A written right angle, and a decimal length.

Checking the numbers blindly gives the wrong answer.

• A square labelled 1 m, 100 cm, 1 m, 100 cm. Matching the numbers blindly says “not a square”.
• A long, thin shape labelled 10 cm, 10 mm, 10 cm, 10 mm. Matching the numbers blindly says “they’re all 10”.
• A shape with four rounded corners and four straight 5 cm sides. The sides condition passes, and there are no angles at all to check, so a student running the rule mechanically says “square”. The route in is what a square is: its sides are straight and they meet at corners. Note that no corner carries a box or a curve, because a rounded corner has no angle — that absence is the item. All four corners are rounded so that the four straight parts really are equal and the labels stay honest; with only two rounded, they wouldn’t be.

These two are the most revealing items on the card: students have to think about what the labels mean, not just whether the numbers match.

The justifications

The Why? reasons follow the usual template, then name what’s new:

• “…they all have one dash, and matching dashes mean matching lengths…”
• “…The arrows only show which sides are parallel.”
• “…Two sides have one dash and two have two dashes, so they’re different lengths.”
• “…Writing 90° is another way of showing a right angle.”
• “…1 m and 100 cm are the same length…”
• “…The numbers match, but 10 cm is ten times as long as 10 mm.”
• “…but the corners are curved, so it hasn’t got four right angles. A square’s sides are straight and meet at corners.” When Why? opens on this one, the curved corners light up in place of the angle marks.

Discussion prompts

1. “What do the dashes tell you that the numbers used to? Use the rule on the dashed near-square.”
2. “The shape labelled 10 cm and 10 mm: say the rule, then say why checking the numbers alone gets it wrong.”
3. “Which part of the rule do the arrows help with? Which part do they not help with at all?”
4. “The rounded shape: which part of the rule can you use on it, and which part can’t you use at all?” Rounded-corner “squares” are everywhere — app icons, tiles, photo frames — so it’s worth asking what we’d have to change to make it a square.

Reading the summary

The same two channels as the testing card. Missing the dashed items suggests the notation is new rather than the idea; missing the mixed-unit items suggests the student is matching numbers rather than lengths — a good moment to return to “For the following examples”.