Is one quarter shaded?

What this sequence teaches

One quarter shaded means one of four equal parts is shaded. So before calling it a quarter, check that the four parts are all the same size. Four parts with one shaded isn’t enough.

The difficulty here is noticing, not judging. Every example makes it easy to see whether the parts are the same size; the question is whether students think to look. The habit the sequence is built to break is counting: “four parts, one shaded, so it’s a quarter.”

The teaching sequence

Example 1 — a rectangle cut into four strips, with the second strip shaded (not one quarter). Four parts, one shaded, so the counting habit says yes. But the shaded strip is only half as wide as it should be.

Example 1 → Example 2. Only one line moves: the one on the right of the shaded strip slides across. The shaded strip widens; the outline and the other lines stay exactly where they are.

Example 2 — four strips of the same size (one quarter). The change that made it a quarter was the size of the parts, and nothing else.

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

Example 3 — a tilted triangle cut at the middle of each side, with the centre triangle shaded (one quarter). The middle triangle is upside down, but it’s the same size as the other three. The shaded part touches no edge, and none of the lines crosses the middle of the shape.

Example 3 → Example 4. A clean fade.

Example 4 — a circle cut by two straight lines through its centre (one quarter). The familiar picture, tilted away from the usual “+”.

Example 4 → Example 5. Only one cut moves: it slides away from the centre, and the shading grows with it.

Example 5 — the same circle with one cut off-centre (not one quarter). The shaded part is now about two fifths of the circle. And the part below it has grown to exactly the same size. The shaded part matches its neighbour, but the parts aren’t all the same size. 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. The counting habit says yes to Example 1, so most students predict wrong, and the misconception is confronted at the very first reveal. The pause before each reveal makes every example a prediction, so the first one gets full attention.
• The two animated pairs are the lesson. In each, a single line moves and the answer flips. The three quarters in the middle are as different from each other as possible: a rectangle, a triangle and a circle; strips, triangles and slices; the shaded part in the middle, in the centre and at the edge.
• The two non-examples fail in opposite directions. In Example 1 the shaded part is too small; in Example 5 it’s too big. So “the shaded part mustn’t be small” isn’t something students can take away.
• No non-example is ever close. In every one, on every card, the shaded part is at most half a quarter or at least one and a half quarters. Nobody has to estimate.
• There’s no squared grid. A grid turns the question into counting squares and invites “6 twenty-fourths”, which is equivalence, and a different question from the one this page asks. Grids appear only on the expansion card.

What this sequence doesn’t address

• Parts that are the same size but different shapes. That’s about what equal means, and it belongs to the earlier atom “Are these parts equal?”. Here, equal parts are always the same shape.
• More than four parts, or more than one part shaded. Later atoms pick these up, and the expansion card previews them.
• Other familiar pictures (a square cut on its diagonals) and the harder non-examples (evenly spaced lines in a triangle, lines meeting off-centre) are on the testing card.

The rule and the script

The rule, exactly as the Show rule button displays it: For the following examples: if the shape is split into 4 parts that are all the same size, and 1 part is shaded, then the shaded part is one quarter of the shape.

“For the following examples” matters. The rule decides every example with four parts and one shaded, and the expansion card shows what lies beyond, including a case where the rule gets it wrong.

The spoken justifications:

• Examples 2, 3 and 4 — “The shaded part is one quarter of the shape. How do I know? Because the shape is split into 4 parts, 1 part is shaded, and the parts are all the same size.”
• Examples 1 and 5 — “The shaded part is not one quarter of the shape. How do I know? Because the shape is split into 4 parts and 1 part is shaded — but the parts aren’t all the same size.”

The page says “The shaded part is one quarter of the shape” rather than “One quarter is shaded”. The negative of that, “One quarter is not shaded”, is ambiguous: in every positive example, a quarter-sized part isn’t shaded.

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 copy of the shaded part moves onto each of the other parts in turn. Onto a neighbour, it folds over the line between them, like folding paper to check two halves match. Onto a part further away, it slides or turns. On a quarter, the copy fits every part exactly. On Example 1, it’s too small for every part. On Example 5, it fits the part below and overhangs the two thin ones.
• On the two fades (Examples 3 and 4), nothing has moved to draw the eye, so ask “How could we check the parts are the same size?” before you click Why?.

Vocabulary. The page says “the same size” throughout, because some students hear “equal” as “identical”. Once students are fluent, you might add: “You know how I’ve been saying ‘the same size’? In maths, parts that are the same size are called equal parts. A quarter is one of four equal parts.”

Where this atom sits

Before it: “Are these parts equal?” — what equal means, including that parts can be the same size without being the same shape. This atom doesn’t require it: every example here makes equality easy to see, so it stands on its own.

This atom: checking that the parts are equal before naming a quarter.

After it: “Are three quarters shaded?” (how many parts are shaded changes) and “Is one third shaded?” or “Is one fifth shaded?” (how many parts there are changes). Equivalent fractions first appear on the expansion card below.

Running the sequence

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

Replay appears on Examples 2 and 5, where a single change is the lesson. It doesn’t appear on Examples 3 and 4, which are new examples, not transformations.

If a class struggles with Example 5, go back to Example 4 and replay the change: “What moved? Which parts got bigger?” Then: “Is the shaded part the same size as the part next to it? Is it the same size as all of them?”

About the testing sequence

Ten items in a new random order each time: five show one quarter shaded, five don’t. Every item has four parts with one shaded, so the rule decides every one. There’s no Show rule button here: students apply the rule without it on screen. Why? works as in the teaching sequence, after each answer, with the same fold-and-slide copy.

The items are static. The animations in the teaching sequence mark the change between two examples; they don’t carry the meaning of a quarter, so a still picture is a fair test.

What each item is diagnosing

One quarter shaded:

• A tall rectangle cut across, with the third strip shaded. Familiar strips, turned the other way, with the shaded part not at an end.
• A square standing on its corner, cut along both diagonals. The shape is turned and the parts are triangles.
• A circle with the usual cross. The most familiar picture of all. It checks that students haven’t decided circles are always trick questions.
• A square cut by two tilted lines through its centre. The parts don’t look like any standard quarter and each is turned a different way, but they’re the same size.
• An L-shape cut into four small Ls. An unusual whole, with the parts turned different ways.

Not one quarter shaded:

• A rectangle with its left half shaded and the rest cut into three equal strips. The counting habit in its plainest form. Students who say “that’s a half” have the right reason for the right answer.
• A circle with three thin slices and one big part, with a thin slice shaded. The shaded slice matches two of the others. Only “all the same size” rules it out.
• A square cut by a cross away from the centre, with the small corner shaded. The exam classic.
• A triangle cut by evenly spaced lines, with the bottom part shaded. The lines are evenly spaced, but the parts aren’t the same size: from the top, they’re 1, 3, 5 and 7 sixteenths of the triangle. This tests whether students check the parts or just the lines.
• A square with lines from a point below the centre to each corner, with the top triangle shaded. It looks like the familiar diagonals picture. Two of the unshaded triangles are exactly a quarter each, but the shaded one isn’t.

Common confusions to watch for

• “Four parts and one shaded makes a quarter” — shows up on every non-example, most plainly on the half-shaded rectangle and the off-centre cross.
• “It’s the same size as the part next to it” — shows up on the circle with three thin slices.
• “Evenly spaced lines make equal parts” — shows up on the triangle.
• “It looks like the usual picture” — shows up on the square with lines to a point below the centre.
• “Quarters have to look like quarters” — shows up as a wrong no on the tilted square and the L-shape.

Discussion prompts

• “Say the rule. Now use it on the circle with three thin slices.”
• “The lines in the triangle are evenly spaced. Why aren’t the parts the same size?”
• “In the square with lines to a point, which parts are the same size? Is that enough?”
• “How could you show that the four parts of the L-shape are the same size?”

Reading the summary

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

A pattern of misses points to the gap: missing the half-shaded rectangle or the off-centre cross suggests the counting habit; missing the three-slices circle suggests checking only against a neighbour; missing the triangle suggests trusting the lines rather than the parts; missing the square with lines to a point suggests recognising pictures rather than checking sizes; missing the tilted square or the L-shape suggests a narrow idea of what a quarter can look like.

About the expansion sequence

Eight items 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, Yes or No, feedback, Why?, and a summary.

The teaching sequence’s rule begins “For the following examples”. The rule decides shapes with four parts and one shaded. This card goes beyond them.

Two kinds of item

The rule gets it wrong. Each of these has four parts with one shaded, and the parts aren’t the same size, so the rule says no. But the shaded part is exactly one quarter: Example 2’s rectangle with the same strip shaded in the same place and only the lines between the unshaded parts moved; and a circle with a right-angle slice shaded and the rest cut unevenly.

These can only be answered from what a quarter means: four of the shaded part would fill the whole shape. They show that “equal parts” is a way of checking for a quarter, while a quarter itself is a size. They’re the most revealing items on the page.

More than four parts. The parts are all the same size, but there are 8 or 12 of them. One quarter shaded: 2 of 8 strips side by side; 2 of 8 slices apart; 3 of 12 squares on a grid. Not one quarter shaded: 1 of 8; 3 of 8; 5 of 12.

The route is to group the equal parts into four equal groups. The rule can’t be applied here, and the counting habit (“that’s 2 out of 8, not 1 out of 4”) gives the wrong answer on every yes item. The no items make sure “more parts means yes” doesn’t become the pattern.

This is the page’s first contact with equivalent fractions, deliberately kept off the other two cards. It’s a natural starting point for your next lesson.

Who it’s for

The first kind suits any class that has worked through the teaching sequence, taken as discussion. The second kind needs students who can group equal parts; 2 of 8 slices apart is the hardest item on the page. 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:

• The rule gets it wrong — “The shaded part is one quarter of the shape. How do I know? Not from the rule — the parts aren’t all the same size. But the shaded part is exactly one quarter: 4 of it would fill the whole shape.”
• More parts, yes — “The shaded part is one quarter of the shape. How do I know? Because the shape is split into 8 parts that are all the same size, and 2 are shaded. 2 of the 8 parts make one quarter: 4 of the shaded part would fill the whole shape.” (For the slices apart, it adds: “They don’t touch, but together they still count.”)
• More parts, no — “The shaded part is not one quarter of the shape. How do I know? Because the shape is split into 8 parts that are all the same size, and 1 is shaded. One quarter would be 2 of the 8 parts.”

Why? behaves differently on this card. Four copies of the shaded region are laid down one after another, and they stay. On a yes they fill the shape exactly; on a no they leave gaps or overlap. On this card the drawn parts no longer decide the answer, so the pointer shows the meaning of a quarter itself: four of it make the whole.

Discussion prompts

• “The rule says no for this rectangle. Is the rule right? How do you know?”
• “How many copies of the shaded slice would fill the circle?”
• “Two slices of the circle are shaded, but they’re apart. Could you move one next to the other? What would you have?”
• “For which of these items did the rule give the right answer? For which did it give the wrong one? For which couldn’t it decide at all?”

Reading the summary

The summary uses the same two channels as the testing card. Answering no to the first kind means the rule is being applied beyond the examples it was built for — a good moment to return to “For the following examples”, and to what a quarter means. Answering no to the yes items with 8 or 12 parts suggests counting parts rather than grouping them. Answering yes to the no items suggests “more parts means yes”.