Are these angles on a straight line?
A categorical atom from mrbartonmaths.com
What this sequence teaches
A set of highlighted angles is on a straight line when two things are true together: the angles meet at one point, and together they fill one side of a straight line, with no gap and nothing left over. The two non-examples each break one half.
There are no letters, numbers or measurements anywhere. The decision is made from position alone. “They add up to 180°” is the fact that follows this decision, and it is deliberately not mentioned here.
The teaching sequence
Example 1 — two angles at two different points (not on a straight line). Each angle touches the same sloping line, but they are apart.
Example 1 → Example 2. The verdict clears, the stage holds still for a second, then the purple angle and its line slide along the straight line until the two points are one. Neither angle changes size or turns.
Example 2 — the same two angles at one point. The first yes is on a sloping line on purpose, so students do not fix on a level one.
Example 3 — an upright line, three angles, lines that cross through the point. A new example. One angle is very thin. The diagram is busy, as exam diagrams often are.
Example 4 — a level line, three angles underneath, lines that stop at the point. A new example. The point is off-centre.
Example 4 → Example 5. The verdict clears, the stage holds, then the highlight on one angle slowly fades. No line moves.
Example 5 — a gap (not on a straight line). The two highlighted angles still meet at one point on a straight line, and it is still a no. This is the contrast to leave them with.
What this sequence doesn’t address
An extra angle on the other side of the line, an opposite pair where two lines cross, angles in a corner with no straight line, and a gap in the middle are in the testing sequence. The rule decides all of them.
Four and five angles, angles all the way round a point, and a triangle with a side carried on 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 angles meet at one point and together fill one side of a straight line, they are angles on a straight line.
- Example 1 — “These angles are not on a straight line. How do I know? Because each angle touches the straight line — but they don’t meet at one point.”
- Examples 2, 3 and 4 — “These angles are on a straight line. How do I know? Because they meet at one point, and together they fill one side of a straight line.”
- Example 5 — “These angles are not on a straight line. How do I know? Because they meet at one point — but together they don’t fill one side of the line. There’s a gap.”
Whether students see the words is your call. Nothing appears unless you click. Show rule puts the rule above the stage and it stays until you turn it off. Why? appears after each reveal: it shows the justification and plays the check again.
What Reveal shows. Before the tick or cross appears, the check plays in the order the rule says it. The straight line lights up. The meeting point pulses. Then an orange line runs round each highlighted angle in turn. On a yes the orange pieces join into one half-turn that starts and ends on the straight line. On Example 1 two points pulse, and the orange pieces are in two places. On Example 5 the orange stops short of the line and the gap is left dotted.
The colours only tell the angles apart, so you can say “the purple one”. They carry no meaning and never show the answer.
Running the sequence
Pause before each reveal. The pause is the prediction moment.
Replay appears on Examples 2 and 5 and replays the change from the example before, with the same pauses. After Reveal it plays the check again as well.
About the testing sequence
Ten items in a new random order each time: four are on a straight line, six are not. The rule decides every one. There is no Show rule button here. After each answer the same check plays as in the teaching sequence: the straight line, the point, then each angle in turn. Why? shows the justification and plays it again.
What each item is diagnosing
The thin angle beside a very wide one, the two square corners on an upright line, and the three angles on a steep line. Whether students accept angles on a straight line that look different from each other.
The two crossing lines with two neighbouring angles highlighted. The angles fill one side of the long sloping line, not the short level one. Students who say no have picked the wrong line.
The two angles at different points on an upright line, and the three angles where one is at a different point. Meeting at one point. In the second, two of the angles are correct together, so the whole set has to be checked.
The item with an unhighlighted angle in the middle. Filling the side. In the teaching sequence the gap was next to the line; here it is between the angles.
The item with a third angle underneath the line. Too much: the angles go past the line.
The opposite pair where two lines cross. These are vertically opposite angles, which students often mix up with angles on a straight line.
The two angles in a corner. They are side by side at one point, but there is no straight line.
Discussion prompts
- “Say the rule. Now use it on the item with the angle underneath the line.”
- “On the two crossing lines: which line do these angles fill one side of?”
- “On the item with three angles at two points: which part of the rule does it break?”
Reading the summary
Each cell has two channels: a green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether the angles are on a straight line.
Saying yes to the items with angles at two points suggests students are checking only that the angles touch the line. Saying yes to the gap or the extra angle suggests they are not checking that one side is filled exactly. Saying yes to the opposite pair or the corner suggests they are answering from “angles at a point”.
About the expansion sequence
Six items a little beyond what the teaching and testing sequences showed, to see how far students’ understanding transfers. It works exactly like the testing sequence. Teaching and testing used two or three angles; here there are four and five.
The two kinds of item
The rule still works, on something unfamiliar. Four angles on a sloping line, four angles on an upright line with lines crossing through, and five angles on a steep line are all on a straight line. So are the inside and outside angles of a triangle with one side carried on: the straight line is the side of a shape.
Four angles is not enough on its own. In one item five angles fill the side and only four are highlighted, so there is a gap. In another, two lines cross and all four angles are highlighted, so they go all the way round the point.
Discussion prompts
- “Does it matter how many angles there are? Say the rule and check.”
- “On the triangle: where is the straight line?”
Reading the summary
The same two channels as the testing sequence. Saying no to the four- and five-angle items suggests the number of angles unsettled the rule. Saying yes to the gap or to the angles all the way round suggests students are judging by “lots of angles at a point on a line”.