So a forgotten row costs twenty seconds, not a mark.
Build to row 5 and ask where 10 came from.
The orange lines show it: 4 + 6, the two entries above. A student who knows that can rebuild any row from scratch and never needs the triangle printed on anything.
The symmetry is visible rather than taught. nCr = nC(n-r) is just the observation that the row reads the same backwards, which is a much better reason to believe it than a factorial manipulation.
Substituting x = 1 into (x + 2)⁵ must give 3⁵ = 243, so the six coefficients must total 243. It checks every coefficient at once and costs five seconds.
Make it automatic. It is the only self-check available on an expansion, and students who have it catch their own errors instead of handing them in.
| 1. ⁵C₂ | 10. |
| 2. Coefficient of x³ in (x + 2)⁵ | ⁵C₂ × 2² = 40. |
| 3. Row 5 total | 32, which is 2⁵. |
1 markThe general term with its nCr written before substituting.
1 markBoth powers present and adding to n.
1 markThe whole second term raised, so (2x)³ is 8x³.
| They give | What it means |
|---|---|
| 5 (Q1) | Counted from r = 1. The row starts at r = 0. |
| 20 (Q1) | Gave ⁵P₂, where order matters. |
| 10 (Q2) | Gave the nCr alone and forgot the power of 2. |
| 80 (Q2) | Took the wrong r. The powers must add to 5, so x³ pairs with 2². |
| 243 (Q3) | Confused the row total with the x = 1 check on the expansion. |
"Do I need to memorise the triangle?" No, and you should not. Rebuild it: it takes twenty seconds and cannot be misremembered.
"Why is r not the power I want?" Because r counts the other bracket. Writing "powers add to n" first, every time, settles it.
"What about (2x + 3)?" Same theorem, and the trap is forgetting to raise the 2 as well as the x. Do one of these early.
| What is happening | |
|---|---|
| 1 | Build the triangle. Ask where each entry came from. |
| 2 | The theorem, with powers adding to n emphasised. |
| 3 | A full expansion, checked at x = 1. |
| 4 | Single-term questions, which is how it is actually examined. |
| 5 | A bracket like (2x + 3) where both parts carry a coefficient. |
Do not hand out a printed triangle. It removes the only thing that makes the row memorable.
Do not expand everything when a question asks for one term. It wastes exam time and invites arithmetic slips.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.