Write the calculation for the missing angle

What this is

Students already know which angles are on a straight line, and that angles on a straight line add up to 180°. This step turns that into one line of working. They write the calculation and do not work it out: the response to 47°, x and 56° is x = 180 − 47 − 56, not 77.

The rule

Exactly as Show rule displays it: For the following examples: write 180. Take away every angle you know.

The teaching sequence

Example 1 — 47°, x and 56° on a sloping line. The rule in full: 180 first, then each known angle taken away.

Example 2 — one line added. A new line splits x, making a new angle of 26°. The old answer stays on screen and one new “take away” arrives. This is the example that teaches every angle you know.

Both examples are on a sloping line, on purpose. Students who only see a flat line at the bottom tie the rule to that picture. Level and upright lines, x at the end, and an angle bigger than 90° are all left to the testing sequence: the rule covers them, so students can work them out.

Running the sequence

• Keep it fast and uninterrupted, well under a minute. The testing sequence is where students take over.
• Reveal follows the rule in order. First the straight line lights up and “x = 180” appears. After a pause, each known angle lights in turn and its number moves from the diagram into the calculation.
• Moving to Example 2 is deliberately slow. The answer dims and is labelled “Previous answer”, the screen holds still, and only then does the new line grow and the 26° appear. Ask: “How little can you change your answer?”
• Replay plays the change again. Before Reveal it plays the change to the question; after Reveal, the change and then the answer.

Why nothing is worked out

A whiteboard showing 77 could be wrong for many reasons. A whiteboard showing 180 − 47 − 56 shows exactly what the student did. It also means a subtraction slip can’t hide a student who understood. Working it out is the next step in the routine.

Where it sits

Third of four steps in finding a missing angle on a straight line: decide which angles are on the line, recall the fact, write the calculation, then work it out.

Two check questions before moving on:

• x and 112° on a straight line. x = 180 − 112. (Catches 112 − 180 and 180 + 112.)
• 38°, 51°, x and 29° on a straight line. x = 180 − 38 − 51 − 29. (Catches stopping at two.)

About the testing sequence

Six items in a designed order. Students answer on mini-whiteboards; show the item, wait, then reveal. Any order of the known angles is fine, and so is 180 − (64 + 83). There is no scoring: the whiteboards are the data.

If a student writes the final number, ask for the calculation instead. It is not wrong, but it is not this step.

What a wrong whiteboard usually means

1. x, 64° and 83° on a level line. 64 + 83 means no 180. 360 − … means the wrong fact. A plus sign means the rule was lost after the first step.
2. The same picture, with x at the other end. A board that still has 83 in it means the student copied the last answer without looking at which angles are known.
3. One line removed: 33° and x. 33 − 180, or a blank, means the student thinks two numbers are needed.
4. x, 29°, 68° and 51° on a sloping line. Only two angles taken away means the student stopped early. The 51° is the angle furthest from x.
5. The same angles on the other side of the line. Any change from the last answer means the student thinks the side of the line matters.
6. 102°, x and 49° on an upright line. A board with 29, 68 or 51 on it means the student edited the last answer instead of starting from 180.

Discussion prompts

• “Say the rule, then use it on this one.”
• “Which angles do you know? Have you taken away all of them?”
• “The picture changed. Did your answer need to?”

About the expansion sequence

Seven items with no new teaching. Each adds one thing, and the rule still works. Don’t skip items or run them out of order.

Before you start, check students know that a square corner means a right angle, 90°. Item 5 relies on it.

What each item adds

1. A decimal. 102.5 is copied exactly as it is written.
2. The missing angle is called y. Only the letter changes.
3. A known angle is a letter, a. Classes most often stall here: “you can’t take away a”. You can write it.
4. Four known angles.
5. A right angle shown only by its square corner. The 90 is not written on the diagram, but it is still taken away.
6. No numbers at all. The response is the rule written in symbols: x = 180 − p − q.
7. Two angles are both x. This is a stretch item, and the only one the rule alone doesn’t reach. The rule gives 180 − b, which is what the two x angles make together. Reveal shows that line first, 2x = 180 − b, holds it, and then shows x on its own. A student who wrote 2x = 180 − b was one step away.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “Where is the 90 on the diagram?”
• “What does 180 − b tell you? One x, or two?”