At a glance. One lesson of about 60 minutes. It assumes expanding double brackets from Year 9 and the idea of a counterexample from lesson 1. Students write a pattern for the grid box and the difference of squares in general, and finish with a proof as a chain that ends in a sentence.
Calculator: no calculator. The lesson is by hand. Proof depends on the algebra, and a calculator can only test cases.
What students will learn
Know
- n for any whole number, 2n even, 2n + 1 odd, n, n + 1, n + 2 consecutive.
- Two different numbers need two letters.
- (n + d)² − (n − d)² = 4nd and an s by s box on a grid with rows of w has difference (s − 1)² w.
Understand
- A proof ends in the line where the free choices have cancelled or a factor shows.
- A general result can be used to find numbers, not only to check them.
- A condition such as even or consecutive is built into the choice of letters.
Do
- Write a statement about numbers as an expression with letters.
- Expand, collect and factorize to show a property for every n.
- Use a proved identity to solve for unknown numbers.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: expanding double brackets and collecting terms (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure and the box questions. Students find the difference for several boxes, state a rule in the box size and the row length, and then prove it.
- Criterion C. Communicating: a chain of steps that ends in a sentence about every n (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the classroom grid in the hook, and using a proved identity to recover numbers (question 7).
The one thing to do with the figure
Make the pattern feel certain before it is proved. Ask students to move the 3 by 3 box to as many places as they can and to keep their eyes on the difference. Then ask what would have to be true of the four corners for the difference not to move, and write the corners with a letter on the board.
In the second figure, ask for a pair (n, d) that gives a ring of 200, then ask students to explain why they can find it without a calculator: 4nd = 200.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: will boxes in different places give different answers? Collect guesses. |
| 12 min | First figure; key idea; settle the hook with letters; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C or D, return to the key card. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 2n; n² − 6n + 9; a case that meets the conditions and breaks the conclusion. |
|---|---|
| Hook | 10 every time: (a + 1)(a + 10) − a(a + 11) = 10. |
| Worked examples | 8k; 4w; 3(n² + 2n + 1) + 2. |
| Faded example | 6(n + 1). |
| 1. Writing numbers with letters | B. A includes an even number; C lists multiples; D repeats one number. |
| 2. Difference of squares with a letter | 4. 4n. |
| 3. A box on a wider grid | 12. The row length. |
| 4. Odd numbers, difference of squares | 8. 8n. |
| 5. A larger box on another grid | 63. 3 × 3 × 7. |
| 6. Working back to the row length | 6. 96 ÷ 16. |
| 7. Using an identity to find numbers | 21. 21 × 23 = 483. |
| 8. Recognizing a proof | C. A is cases; B reuses one number; D proves a weaker result. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Wrote consecutive odd numbers wrongly. | Ask: how far apart are consecutive odd numbers? Is 2n + 2 odd? |
| 2 or 0 in question 2 | Dropped middle terms or expanded (n − 1)² wrongly. | Expand (n − 1)² on its own line, with the −2n. |
| 13 or 144 in question 3 | Gave a single step or the square of the row length. | Expand fully; the a² and a terms cancel. |
| 0 or 4 in question 4 | Dropped middle terms or lost the minus. | Write each expansion on its own line before subtracting. |
| 21 or 112 in question 5 | Used s − 1 once, or used s. | Use letters for the four corners and expand. |
| 24 or 3.84 in question 6 | Divided by s − 1 or by s squared. | Write 96 = (s − 1)² w and put s = 5 in. |
| 22, 11 or 23 in question 7 | Gave 2n, n, or the larger number. | Write the two numbers as 2n − 1 and 2n + 1 and read the smaller. |
| A, B or D in question 8 | Accepted cases, reused one number, or proved a weaker result. | Ask: does this cover every pair? Is it about multiples of 4? |
Questions worth asking
- Why does a disappear in the box proof? Listen for: the a² and the a terms in both products are the same, so subtracting cancels them.
- Why two letters for two odd numbers? Listen for: they may differ, and one letter would make them equal.
- Does a proof for 2 by 2 boxes prove it for 3 by 3? Listen for: no, a new box needs its own letters, though the method is the same.
- Is a proof a better test than 100 examples? Listen for: yes, it covers every case and shows why.
Differentiation
A way in for anyone stuck
- Give a card with the letter forms: 2n even, 2n + 1 odd, n and n + 1 consecutive, with one worked expansion on it.
- Provide a proof frame: “Let …”, “Then …”, “So …, which is … for every n.”
- Start with a real grid: students mark boxes in colour and write the four products in a table before using letters.
Stretch
- Prove that the difference for an s by s box on a grid with rows of w is (s − 1)² w.
- Show that the product of any four consecutive whole numbers, plus 1, is a perfect square.
Students learning in English as an additional language
“Consecutive” means one after another with nothing missing. “Multiple of 8” means 8 times a whole number. “Simplify” means collect like terms. “Prove” means show it is true for every case, which is more than showing it for some.
| Word | Plain meaning | Example |
|---|---|---|
| consecutive | following one after another with no gap | n, n + 1 and n + 2 |
| expression | numbers, letters and operations written together | 4n(n + 1) |
| identity | a statement that is true for every value of its letters | (n + 3)² − n² = 6n + 9 |
| general case | a case with letters, standing for every case at once | the box with top-left corner a |
| multiple | a number you get by multiplying by a whole number | the multiples of 9 are 9, 18, 27, and on |
| factor | a number that divides exactly into another | 5 is a factor of 40 |
Sentence starters for explaining aloud or in writing:
- Let the numbers be … so …
- This is a multiple of … because it is … times a whole number.
- This is not a proof because it only shows …
Two things not to say
“Just try a few numbers.” Trying numbers is how a conjecture is found; it is not how it is proved.
“Use n for both.” Two numbers that may differ need two letters.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- WriteWrite the sum of two consecutive odd numbers with a letter and simplify.
(2n + 1) + (2n + 3) = 4n + 4. - ExpandExpand (n + 2)² − (n − 2)².
8n.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- GridA 3 by 3 box is drawn on a grid with rows of 9. What is the difference of the diagonal products?
4 × 9 = 36. - Earlier work (lesson 1)Give a counterexample to “n² + 3 is prime for all n”.
n = 1 gives 4; n = 3 gives 12.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningProve that the product of any three consecutive whole numbers is a multiple of 6.
One of three is a multiple of 3 and at least one is even, so 6 divides the product. - Working backwardsTwo consecutive even numbers have the product 224. Find them.
(2n)(2n + 2) = 224 gives 4n² + 4n = 224, n = 7: 14 and 16.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? A printed 100-grid and a coloured pencil are enough for the box investigation. No calculator is needed.