Is this a trapezium?

What this sequence teaches

A trapezium is decided by one thing: a pair of its sides is parallel. Orientation, symmetry, which side is longer and how the slanting sides lean don’t matter.

House position. This page uses the inclusive definition: a trapezium has at least one pair of parallel sides, so parallelograms, rectangles and squares are trapeziums too. Sources differ, and some textbooks say “exactly one pair”. The teaching and testing cards only use shapes with one pair or none, so every verdict there is the same under either definition. The expansion card is where the difference shows.

Notation. Arrows mark a parallel pair. Every diagram is accurate, and any pair of sides without arrows is clearly not parallel. Ticks and right-angle marks appear too, but they never decide the verdict.

The teaching sequence

Example 1 (not a trapezium). A sloping four-sided shape with no parallel sides.

Example 1 → Example 2. One corner turns until the top side is parallel to the bottom, and the arrows appear as it lands. The shape is still sloping, so a student who said “no, it’s tilted” to Example 1 watches the tilt survive into a yes. The parallel sides were the only change.

Example 3 (trapezium). The parallel pair stands upright, one end overhangs, and the parallel sides are very different lengths — as far from the usual picture as a trapezium gets.

Example 4 (trapezium). The usual picture, with the two equal slanting sides ticked.

Example 4 → Example 5. One top corner turns about the corner beneath it. The equal sides stay equal and the ticks stay; only the top tilts, and the arrows fade.

Example 5 (not a trapezium). It looks like a trapezium and has the “trapezium” ticks, but it isn’t one. It’s last because it is the one students are most likely to get wrong.

What this sequence doesn’t address

Right-angled trapeziums, kites, and marks that don’t decide the verdict are in the testing card. Shapes with two parallel pairs, and shapes where parallel has to be worked out rather than read from arrows, are on the expansion card.

The rule and the script

The rule, exactly as the Show rule button displays it: For the following examples: if a pair of sides is parallel, the shape is a trapezium. It says “a pair”, not “one pair”. That silence matters: it keeps the rule true for the parallelograms on the expansion card.

• Examples 2, 3 and 4 — “This is a trapezium. How do I know? Because a pair of its sides is parallel.”
• Example 1 — “This is not a trapezium. How do I know? Because no pair of its sides is parallel.”
• Example 5 — “This is not a trapezium. How do I know? Because it has two equal sides — but no pair of its sides is parallel.”

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. On a trapezium, the parallel pair and its arrows turn orange and pulse. On a non-example, each pair of opposite sides pulses in turn.

A word to watch. Many books call the parallel sides the “bases”. The word suggests “the bottom” — exactly the misconception this page works against — so it doesn’t appear here. If students meet it later, tell them the bases can point any way.

Where this atom sits

Before it: knowing what parallel means and reading arrow marks. After it: the area atoms — which lengths tell the area of a trapezium. The inclusive definition pays off there: the trapezium area formula works for parallelograms too.

Running the sequence

Pause on every example before revealing, and ask for a prediction. After Example 5, “What changed?” gets the whole lesson in a sentence. Replay appears on Examples 2 and 5 and repeats the turn.

About the testing sequence

Ten items in a new random order each time: five trapeziums and five not. There is no rule on screen. Every item can be decided by the rule, and none has two parallel pairs. The items are static: the animations in the teaching sequence mark the change between two examples, and a still shape is a fair test of whether a pair of sides is parallel.

What the items diagnose

“Only the usual picture counts.” The trapeziums include an upright pair, a pair on a diagonal, one that is nearly a triangle, and the usual shape upside down.

“The parallel sides are the top and bottom.” One trapezium has a flat bottom and a tilted top that are not parallel; its parallel pair is the two slanting sides. It’s the most revealing item on the card.

“It looks like one, so it is one.” The usual shape with its top tilted, and a shape whose two sides lean the same way without being parallel.

“Equal slanting sides make a trapezium.” A kite, and a turned shape with two equal ticked sides and no parallel pair. On the positive side, one trapezium also has ticks.

“Marks mean yes.” A shape with one right angle marked and no parallel sides.

Common confusions to watch for

The kite is the shape most often called a trapezium. A student who says no to the diagonal or upright trapeziums is still holding the usual picture.

Discussion prompts

• “Say the rule, then use it on this one.”
• “Which sides would you check first? Is there another pair?”
• “This one has equal sides marked. Does that help decide?”

Reading the summary

The tick or cross in the corner shows what each shape is. The green or red background shows whether the answer was right.

About the expansion sequence

Six items just outside the rule’s scope (“For the following examples”). Use it to see how far understanding stretches; it’s best after the testing card goes well.

Two kinds of item

The rule still works on an unfamiliar surface. One shape has two pairs of parallel sides, both arrowed. The rule says yes. Students who answer no have probably picked up “exactly one pair” from the earlier cards — that’s the conversation to have, and it’s why the rule says “a pair”.

The rule needs something extra. On each of these, applying the rule blindly gives the wrong answer: two right angles on the same side and no arrows (yes — both sides meet the same side at right angles, so they’re parallel); a square with no arrows (yes — four right angles make both pairs parallel); a five-sided shape with an arrowed pair (no — a trapezium has four sides); and two shapes on a grid, where parallel is read by counting squares (3 across and 1 up, against 3 across and 2 up).

Who it’s for

The grid items assume students can count squares along a line. The square will start an argument. That’s the point: under this page’s definition, a square is a trapezium.

The justifications

• Two pairs — “…Because a pair of its sides is parallel — two pairs, in fact. Having more than one pair doesn’t stop it being a trapezium.”
• Right angles, no arrows — “…There are no arrows, but both of those sides meet the same side at a right angle, so they are parallel.”
• Square — “…There are no arrows, but with four right angles, each pair of opposite sides is parallel.”
• Five sides — “This is not a trapezium. How do I know? A pair of its sides is parallel — but it has five sides, and a trapezium has four.”
• Grid, yes — “…On the grid the bottom side goes 3 across and 1 up, twice, and the top side goes 3 across and 1 up, once.”
• Grid, no — “…On the grid, the bottom side goes 3 across and 1 up, twice, but the top side goes 3 across and 2 up.”

Discussion prompts

• “The rule only talked about arrows. How could you tell these sides are parallel without them?”
• “Is a square a trapezium? What does the rule say?”

Reading the summary

As on the testing card: the corner mark shows what each shape is; the background shows whether the answer was right.