At a glance. One lesson of about 60 minutes. Students can expand double brackets, squares and the difference of two squares from lessons 1 to 3. This lesson puts two products in one expression, with a subtraction, and builds the checking habit.
Calculator: no calculator. Nothing here needs one. The substitution checks use small whole numbers on purpose.
What students will learn
Know
- A minus sign before a bracket changes the sign of every term inside.
- Only like terms collect: x² terms, x terms and numbers separately.
- A value where two expressions disagree proves they are different.
Understand
- Bracketing each expansion before subtracting prevents most sign slips.
- One agreeing value does not prove two expressions equal; a slip can agree by luck.
- Tests at 0 or 1 are weak: 0 hides the x terms and 1 cannot tell x² from x.
Do
- Expand and simplify expressions with two products and a subtraction.
- Check a simplification by substituting a well-chosen value.
- Use a test to find and name a mistake in someone's working.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: expanding and simplifying (questions 1 to 8).
- Criterion B. Investigating patterns: on the figure, which slips agree at exactly one value, which at two, which at none? The what-if asks for a general rule.
- Criterion C. Communicating: each expansion in a bracket, then the collected line, then a check line written as a sentence.
- Criterion D. Applying mathematics in real-life contexts: lesson 5 applies this routine to areas and borders.
The one thing to do with the figure
Use the figure to break the “one test is enough” belief. Open on the opening state (a slip that disagrees), then switch to candidate A. Then choose the third expression, candidate A, and set x = 3: it agrees. Ask whether it is right, then look at the row of dots.
Students should then explain, in a sentence, what one agreeing value does and does not show.
A possible order
| 5 min | Warm-up from the student page, then answers. |
|---|---|
| 10 min | Hook: everyone tests x = 0, notices all three agree, then picks a better value and votes. |
| 10 min | Key idea and the figure: the third expression's slip that agrees twice. |
| 10 min | Worked examples, then the faded example in pairs. |
| 5 min | Question 1, the check. If many choose B, ask what x² − x² is. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 8. |
The answers
| Warm-up | x² − 12x + 36; −3x + 12; 2x² + 3x. |
|---|---|
| Worked examples | x² + 3x + 5; −6x + 13. |
| Faded example | 4x + 5. |
| 1. Expand, then take away | D, 3x − 4. A ignores the sign of −1; B adds the x²; C loses the sign of the number term. |
| 2. Expand and take away x² | 8. 8x + 15. |
| 3. Check a value both ways | −20. 5 × (−4); also 9 − 15 − 14. |
| 4. Simplifies to a number | 16. 36 − 20. |
| 5. A number outside two brackets | −50. 2(x² − 25) − 2x². |
| 6. When do two products agree? | −11. 2x − 15 = 3x − 4. |
| 7. Two products, one subtraction | 5. x² + 5x + 2. |
| 8. What a test can prove | B. A trusts one test at 0; C expects an x² term; D leaves the + 12 out of Lin's value (it is 38 at x = 2). |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Sign of −1 in the x term or the number term. | Write the numbers 4 and −1 before using the pattern. |
| B in question 1 | Added x² instead of taking it away. | Ask what x² − x² is. |
| 15 or 2 in question 2 | Gave the number term, or subtracted. | Label p and q in the answer. |
| 20, −5 or 10 in question 3 | Sign, missing x terms, or a sign slip in the x term. | Evaluate the brackets first, then the expansion, and compare. |
| 56 or −16 in question 4 | Did not bracket the second expansion, or subtracted the wrong way. | Write the brackets in before removing them. |
| −25 or 50 in question 5 | Multiplied only one term by 2, or lost the sign. | Expand the brackets first, then the 2. |
| 11 or 19 in question 6 | Sign slips while solving. | Write the operation done to both sides beside each line. |
| 9 or 7 in question 7 | Took away only the x², or forgot the second product. | Bracket the second expansion. |
| A, C or D in question 8 | Trusts one test, expects an x², or substituted carelessly (13 × 2 without the + 12). | The figure's third expression; then substitute into every term. |
Questions worth asking
- Why bracket an expansion before taking it away? So that the minus reaches every term, not only the first.
- If a check agrees, are you done? Not with one value; a second value, not 0 or 1, makes a slip much less likely to hide.
- Why does x = 0 hide mistakes in the x terms? Every x term is 0 there, whatever its coefficient.
- Can two different expressions with x² agree at three values? No: see the what-if. That is why three good tests settle it.
Differentiation
A way in for anyone stuck
- A routine card: expand, bracket, remove the bracket, collect, check.
- Colour the terms: x² terms, x terms and numbers in three colours before collecting.
- Use the figure to test each answer before writing it.
Stretch
- The what-if: why three agreeing values settle it when x² is the highest power.
- Find two different products of brackets whose difference is always 10.
Students learning in English as an additional language
“Simplify” means “write it with fewer terms”. “Substitute” means “put a number in place of the letter”.
| Word | Plain meaning | Example |
|---|---|---|
| simplify | write with as few terms as possible | 4x + 3 + x becomes 5x + 3 |
| substitute | put a number in place of a letter | x = 2 in 3x + 1 gives 7 |
| equivalent | equal for every value of x | (x + 1)² and x² + 2x + 1 |
| counter-example | one case that shows a claim is false | x = 2 shows x² and 2x + 1 are different: 4 and 5 |
| expression | numbers and letters joined by operations, with no equals sign | 3x² − x + 7 |
Sentence starters for explaining aloud or in writing:
- I expand each product and keep it in a bracket: …
- The minus changes every sign in the bracket, so …
- At x = … the first line gives … and my answer gives …, so …
Two things not to say
“It works for x = 1, so it is right.” One test proves nothing; say what it does show.
“The minus only goes on the first one.” It goes on every term of the bracket.
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.
- SimplifySimplify (x + 1)(x + 7) − x².
8x + 7. - From lesson 3Expand (x + 10)(x − 10).
x² − 100.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Two productsSimplify (x + 4)² − (x + 1)(x + 7).
9. - Spot itKemal says (x − 2)² − x² = 4. Test x = 3 and correct him.
At x = 3: 1 − 9 = −8, not 4. The answer is −4x + 4.
3Challenge
Reasoning, generalizing and working backwards.
- Find the gapFind two products of brackets whose difference is always 10.
For example (x + 3)(x + 6) − (x + 1)(x + 8) = 18 − 8 = 10: same sum, products 10 apart. - SolveSolve (x + 2)(x + 9) = (x + 3)(x + 5).
x = −1: 11x + 18 = 8x + 15 gives 3x = −3. Check: 1 × 8 = 8 and 2 × 4 = 8.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The sliders take the arrow keys and the candidate buttons take Enter or Space. The row of dots is backed by the written count, “agree at n of 9”, so colour is never the only clue. “Show me the steps” gives the states in words.
No device? A table with columns x, the expression and the candidate, for x = −2 to 3, does the same job on paper.