At a glance. One lesson of about 60 minutes. Students can expand (x + a)(x + b) with positive numbers from lesson 1 and multiply integers from Year 8. This lesson puts negative numbers in the brackets and names the quick pattern.
Calculator: no calculator. Nothing here needs one. The work is the signs, and a calculator would hide them.
What students will learn
Know
- (x + a)(x + b) = x² + (a + b)x + ab, where a and b keep their signs.
- x − 4 is x + (−4).
- A negative times a negative is positive; a negative times a positive is negative.
Understand
- The pattern is the grid written short: the x cells collect and the corner multiplies.
- When a + b = 0 the x terms cancel.
- A value of x checks an expansion, sign and all.
Do
- Expand double brackets with one or two negative numbers.
- Find a missing number in a bracket from the expanded form.
- Choose a value to check an expansion, including a negative value.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: expanding with negatives (questions 1 to 8).
- Criterion B. Investigating patterns: on the figure, keep a × b the same and change a and b. What happens to the x term? The what-if extends this.
- Criterion C. Communicating: write each sign in its cell, then the collected line, then a check.
- Criterion D. Applying mathematics in real-life contexts: ask for a situation that (x − 2)(x + 5) could describe, such as a square plot made 2 m shorter one way and 5 m longer the other.
The one thing to do with the figure
Use the hook's x = 5 test, then the grid. Set a = −4 and b = 6 and ask which cells are red and why the corner is red but the x term ends up positive.
Then ask students to find a setting where the x term is 0 before anyone touches the slider.
A possible order
| 5 min | Warm-up from the student page, then answers. |
|---|---|
| 10 min | Hook: everyone tests x = 5 and votes. Then the figure at a = −4, b = 6. |
| 10 min | Key idea: the pattern and why. Students say the x term and number term for three brackets you write. |
| 10 min | Worked examples, then the faded example in pairs. |
| 5 min | Question 1, the check. If many choose D, return to warm-up 1. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 8. |
The answers
| Warm-up | 21; 5x; x² + 11x + 18. |
|---|---|
| Worked examples | x² − 4x − 21; x² − 8x + 12. |
| Faded example | x² + 5x − 24. |
| 1. Two negative numbers | A, x² − 11x + 24. B subtracts the numbers; C loses both signs; D gets the corner's sign wrong. |
| 2. The number term | −14. 7 × (−2). |
| 3. The x term with two negatives | −9. x² − 9x + 20. |
| 4. Check with a negative value | −42. (−6) × 7; also 4 − 10 − 36. |
| 5. Find a missing number | 7. k − 3 = 4; 7 × (−3) = −21. |
| 6. Make the x term vanish | 1.5. 4b − 6 = 0. |
| 7. Two facts, one bracket | −40. a = 5. |
| 8. x term with 2x | B, x. A loses the sign of −5; C ignores the 2; D gives the number term. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1; −1 in question 3 | Subtracted the numbers for the x term. | Ask for the two x cells and add them. |
| C in question 1; 9 in question 3 | Dropped the minus signs. | Rewrite x − 5 as x + (−5) before starting. |
| D in question 1; 14 in question 2; 42 in question 4 | Sign of a product. | Warm-up 1 again: (−3) × (−7) and 3 × (−7). |
| 5 in question 2; 20 in question 3 | Gave the other term. | Label the answer: x term or number term? |
| −32 or −50 in question 4 | Left out the x terms, or squared −2 as −4. | Work out each bracket first; (−2)² = (−2) × (−2). |
| 1 or −7 in question 5 | Undid k − 3 = 4 the wrong way, or dropped the sign of −3. | Write both facts and check k in both. |
| 6, −1.5 or 24 in question 6 | Ignored the 4, a sign, or multiplied. | Write the two x cells with b in them and set their sum to 0. |
| 40, 5 or 88 in question 7 | Sign of the corner, stopped at a, or solved a − 8 = −3 wrongly. | Check a in the x term before using it. |
| A, C or D in question 8 | Lost the sign of −5, ignored the 2, or gave the number term. | Fill in a full grid for (2x − 5)(x + 3). |
Questions worth asking
- Why do we add for the x term but multiply for the number? The x term collects two cells; the number term is one cell, a product.
- When does the x term vanish? When a + b = 0, so the numbers are opposites.
- Lena's answer is wrong. Is there any x where it gives the right value? Where the two lines agree: 2x − 24 = −2x + 24, so x = 12.
- Which value of x is the best check, and why? Not 0 (it only tests the number term) and not 1 (x and x² look alike there); a value like 2 or −3 is better.
Differentiation
A way in for anyone stuck
- Keep the grid for every question; the pattern can wait.
- Give a sign card: + × + = +, + × − = −, − × − = +.
- Use the figure to check each answer before writing it.
Stretch
- The what-if: every pair with product −12 and the x term it makes.
- Expand (x − 3)(x + 3)(x + 1). Which two brackets is it easiest to expand first, and why?
Students learning in English as an additional language
Say “minus four” for −4 and “take away four” only for a subtraction. “Sign” means plus or minus.
| Word | Plain meaning | Example |
|---|---|---|
| sign | whether a number is positive or negative | −6 has a negative sign |
| product | the result of multiplying | the product of −2 and 5 is −10 |
| x term | the part of an expansion with x but not x² | in x² − 7x + 10 the x term is −7x |
| number term | the part with no letter | in x² − 7x + 10 it is 10 |
| cancel | add to zero, so that nothing is left | 6x − 6x = 0 |
Sentence starters for explaining aloud or in writing:
- The two numbers are … and …
- They add to …, so the x term is …
- They multiply to …, so the number term is …
Two things not to say
“Two minuses make a plus.” Only when multiplying: −3 + (−8) is −11.
“The sign goes with the x.” The sign belongs to the number in front of it.
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.
- PatternExpand (x + 6)(x − 1).
x² + 5x − 6. - Earlier workWork out (−6) × 4.
−24.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Two negativesExpand (x − 9)(x − 2).
x² − 11x + 18. - From lesson 1Expand (2x + 5)(x + 1).
2x² + 7x + 5.
3Challenge
Reasoning, generalizing and working backwards.
- Backwards(x + a)(x + b) = x² − x − 20. Find a and b.
4 and −5 (in either order). - CancelFind a value of b that makes (3x + 6)(x + b) have no x term.
b = −2: 3b + 6 = 0.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The sliders work with the keyboard and by touch, and the colours are backed by the signs written in each cell, so colour is never the only clue. “Show me the steps” gives the same states in words.
No device? A 2 by 2 grid on paper with a red pen for negative cells does the same job.