At a glance. One lesson of about 60 minutes. It assumes expanding brackets and collecting like terms. Students write even, odd and consecutive numbers with letters, simplify, show a factor and finish with a sentence, and decide which arguments are proofs.
Calculator: no calculator. The lesson is algebra on the page, so no calculator is needed.
What students will learn
Know
- Even numbers are 2n, odd numbers are 2n + 1 and consecutive numbers are n, n + 1, n + 2 for a whole number n.
- A proof for all whole numbers uses a letter and ends with a factor.
- Different numbers need different letters.
Understand
- Examples give evidence; a letter covers every case at once.
- The factor that shows in the final line is the reason the statement is true.
- A proof is a short argument written for a reader, with a sentence at the end.
Do
- Write consecutive, even and odd numbers with a letter.
- Expand, collect and factorize to show a multiple.
- Judge whether an argument is a proof or only examples.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: forming and simplifying expressions (questions 1 to 7).
- Criterion B. Investigating patterns: the figure. Students record totals for several n and d, describe the pattern in words and then explain why it holds with algebra.
- Criterion C. Communicating: a proof as a short chain of lines with a closing sentence (worked examples, faded example, exit ticket).
- Criterion D. Applying mathematics in real-life contexts: using a general rule to answer a specific question quickly (hook and extra questions).
The one thing to do with the figure
Ask for the prediction before moving anything. Then slide n with d = 1 and ask what is the same each time. Ask students to find the number that the total is a multiple of, then set d to something else and ask whether the pattern survives. It does, and the middle number is always the multiplier.
Ask: “Why must the total be three times the middle?” The outer two are the same distance either side of it, so they share out evenly. That is the proof in words; the algebra says the same thing.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: pick three consecutive numbers and add them; collect predictions. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, return to the key card and the gaps of 2. |
| 20 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 | 3n + 12; 2n + 2; odd. |
|---|---|
| Hook | Every total is a multiple of 3; it is three times the middle number. |
| Worked examples | 3(n + 1); 2(a + b + 1); 2n + 1, which is odd. |
| Faded example | 2(ab), which is even. |
| 1. Writing consecutive even numbers | A. B uses consecutive whole numbers; C keeps one 2n; D multiplies the constants. |
| 2. Adding consecutive numbers | 6. 0 + 1 + 2 + 3. |
| 3. Consecutive odd numbers | 3. −1 + 1 + 3. |
| 4. Using the middle number | 5. The plain numbers cancel. |
| 5. Showing an odd product | 1. 2(2ab + a + b) + 1 is odd. |
| 6. Counting cases from a proof | 20. All of them. |
| 7. A difference of squares as a proof | 12. 6n + 6n. |
| 8. Which argument is a proof | B. A tests examples; C adds the same odd number twice; D proves only that the sum is even. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Used consecutive whole numbers, kept one 2n, or multiplied the constants. | Write the three numbers in a column first: 2n, 2n + 2, 2n + 4. Then add down. |
| 10 or 3 in question 2 | Started the constants at 1, or used the last constant only. | Ask: what is the first number? Write n + 0 so that the 0 is visible. |
| 4 or 5 in question 3 | Dropped the −1, or changed its sign. | Put the first term in brackets: (2n − 1). Write the constants in a row. |
| 10 in question 4 | Used the first number n as the starting point. | Write the numbers either side of m: m − 2, m − 1, m, m + 1, m + 2. |
| 0 or 2 in question 5 | Lost the +1 or confused the factor with the remainder. | Rewrite 4ab + 2a + 2b + 1 as (4ab + 2a + 2b) + 1 and factorize the bracket. |
| 10 in question 6 | Counted only even n. | Test n = 3 and n = 5: 12 and 30. |
| 0 or 6 in question 7 | Changed only the first sign in the second bracket, or kept one 6n. | Write the second expansion in a bracket with the minus sign outside, then change every sign. |
| A, C or D in question 8 | Offered examples, a wrong pair, or only parity. | Ask: does this argument cover every pair? Where is the factor 4? |
Questions worth asking
- Why do we use a different letter for each odd number? They may be different; using n twice would force them to be the same number.
- How do you know a proof is finished? Listen for: the last line shows a factor and says what it means.
- Does 2n + 1 give every odd number? Yes, for n = 0, 1, 2, and so on. Ask for the odd numbers it gives.
- Why is a counterexample enough to disprove but not enough to prove? One failing case ends a claim; one working case says nothing about the others.
Differentiation
A way in for anyone stuck
- Give the structure as sentence frames: “Let … be …. Then … = … = …. So …”
- Start with the dot pictures: an even number is pairs, an odd number is pairs with one extra.
- Write each number on its own line in a column, and add down.
Stretch
- Prove that the product of two consecutive even numbers is a multiple of 8.
- Prove that the sum of the squares of two consecutive whole numbers is always odd.
Students learning in English as an additional language
“Consecutive” means one after another with no gaps. “Multiple of 3” means 3 times a whole number. “Prove” means to show with a reason that a statement is always true. Say “n plus one” and “two n plus one” aloud to separate them.
| Word | Plain meaning | Example |
|---|---|---|
| consecutive | following one after another with no gaps | 8, 9, 10 |
| even | a multiple of 2, written 2n | 14 |
| odd | one more than a multiple of 2, written 2n + 1 | 27 |
| multiple | a number you get by multiplying by a whole number | 5 × 7 |
| proof | an argument that a statement is true for every case | 7(2n + 1) is a multiple of 7 |
| expression | numbers and letters with operations, but no equals sign | 2n + 5 |
Sentence starters for explaining aloud or in writing:
- Let the numbers be … Then the total is …
- This is a multiple of … because …
- The examples suggest … but a proof needs …
Two things not to say
“Plug in numbers until you believe it.” Say instead: use a letter so that it covers all of them.
“Let n be any number” without saying whole number; the statement is about whole numbers.
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.
- ExpressWrite the next three consecutive whole numbers after n.
n + 1, n + 2, n + 3. - ExpandExpand and simplify n + (n + 2) + (n + 4).
3n + 6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA fruit seller in Kumasi stacks oranges in three rows, the rows holding n, n + 1 and n + 2 oranges. Show that the total is always a multiple of 3.
n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). - Earlier work (Year 8)Expand and simplify 2(n + 3) + 3(n − 1).
5n + 3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the sum of any three consecutive even numbers is a multiple of 6.
2n + (2n + 2) + (2n + 4) = 6n + 6 = 6(n + 1). - Working backwardsThe sum of four consecutive odd numbers is 80. Find the numbers.
2n + 1 … add to 8n + 16 = 80, so n = 8: 17, 19, 21, 23.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works 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? Number cards on a table work: three equally spaced cards and a total.