At a glance. One lesson of about 60 minutes. Expanding a single bracket with a positive or negative multiplier, simplifying after expanding, and testing equivalence by substitution.
Calculator: no calculator. The numbers are small on purpose: the work is in seeing what the number outside the bracket reaches.
What students will learn
Know
- a(n + b) = an + ab, and a(n − b) = an − ab.
- Expressions with the same value for every x are equivalent.
- A negative multiplier changes the sign of every term it reaches.
Understand
- Expanding does not change the value: it re-describes the same area.
- A single substitution can show two expressions differ, but not that they always agree.
Do
- Expand a single bracket with a positive or negative multiplier.
- Expand and collect like terms.
- Test a suspected equivalence with a value of x.
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 2 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to find how the shortfall depends on a and c, and to state a rule.
- Criterion C. Communicating: writing each step, and justifying a choice in the exit ticket by substituting a number.
- Criterion D. Applying mathematics in real-life contexts: areas of beds and floors written as products with a bracket.
The one thing to do with the figure
Before moving a slider, ask for the area. Set a = 3, n = 4, b = 5 and ask them to say both areas from the picture before the lines of working appear.
On the second figure, ask: “If I change x, does the gap change?” Many expect it to grow. It does not: the slip leaves out the same c × (a − 1) every time.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 3 (a negative times a negative). |
|---|---|
| 10 min | Hook and the rectangle: both areas by hand for n = 2, then the first figure. |
| 10 min | Key idea; the slip and the second figure. |
| 10 min | Worked examples; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 6a + 2; 17; 21. |
|---|---|
| Worked examples | 12y + 30; 15x − 20; −2x + 12; 5x + 10. |
| Faded example | 7w + 8. |
| 1. Expanding | B, 4x + 28. A multiplies only the first term; C only the second; D adds the 4 and the 7. |
| 2. Expand, positive | 24. 8 × 3. |
| 3. Expand, subtraction | 15. 3 × 5. |
| 4. Context, area | 12. 4 × 3. |
| 5. Negative multiplier | 12. (−4) × (−3). |
| 6. Expand and collect | A. B multiplies only the first term; C adds the a; D loses the minus sign. |
| 7. Two brackets | 10. 12 − 2. |
| 8. Subtracting a bracket | 18. 10 + 8. |
| 9. Equivalent pair | B. A and C multiply only one term; D leaves the sign of the 3 alone. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Multiplied one term only, or added the numbers. | Draw the rectangle and ask what the 4 reaches. |
| 3 or 11 in question 2 | Left the second term alone, or added. | Substitute x = 1 into both sides: 8 × 4 = 32 against 8 + 3 = 11. |
| 5 or 8 in question 3 | Left the −5 alone, or added the 3 and 5. | Ask what 3 groups of (2y − 5) are. |
| 3 or 7 in question 4 | Left the 3 alone, or added 4 and 3. | Draw the bed as two rectangles and find the area of the maize strip. |
| −12 or 7 in question 5 | Negative times negative taken as negative. | Return to Unit 3, lesson 4: same signs give a positive. |
| B, C or D in question 6 | Multiplied one term, added the a, or lost the minus. | Expand 5(2a − 3) on its own first. |
| 14 or 11 in question 7 | Mishandled 2 × (−1). | Write 2 × (−1) = −2 beside the line. |
| 2 or 8 in question 8 | Mishandled −4 × (−2). | Say it aloud: the bracket is subtracted, and a minus of a minus adds. |
Questions worth asking
- Why is 3(x + 4) not 3x + 4? Listen for “the 3 also multiplies the 4” or a substitution that shows they differ.
- If two expressions agree at x = 2, are they equivalent? No: one value is not enough. Ask for a pair that agrees at x = 2 but not at x = 3.
- What does the rectangle tell you that the rule does not? Why: cutting does not change the area.
Differentiation
A way in for anyone stuck
- Draw an area model with a grid: every cell is one unit square, so 3 × 5 is 15 cells.
- Write the multiplier in the margin with an arrow to each term before computing.
- Start with positive numbers only, and bring in negative multipliers once those are secure.
Stretch
- Show that 2(x + 3) + 4(x − 1) and 6x + 2 are equivalent. (Both give 6x + 2.)
- Find a and c so that a(x + c) = 4x + 20. (a = 4, c = 5. Is there another? No: the x term forces a = 4. 2(2x + 10) also equals 4x + 20, but it is a different shape.)
Students learning in English as an additional language
Expand, bracket, term and equivalent are new in this sense. Pair each with a picture: the rectangle for expand, the brackets drawn round a group for bracket.
| Word | Plain meaning | Example |
|---|---|---|
| expand | multiply out a bracket so it is gone | 3(x + 2) = 3x + 6 |
| bracket | the ( ) that group terms together | 3(x + 2) |
| term | one part of an expression, with its sign | in 3x − 5, the terms are 3x and −5 |
| equivalent | has the same value for every x | 2(x + 1) and 2x + 2 |
| substitute | put a number in place of a letter | x = 4 in 3x + 1 gives 13 |
Sentence starters for explaining aloud or in writing:
- The number outside the bracket multiplies …
- I tested with x = …, and the two expressions gave … and …
- These are / are not equivalent because …
Two things not to say
“Multiply the bracket by the number.” It sounds like one multiplication. Say “multiply each term”.
“It works for x = 2, so it is right.” One value cannot prove equivalence; two different expressions can agree at one value.
“x is the beans.” x is the width of the bean strip in metres, a number.
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.
- ExpandExpand 9(n + 8).
9n + 72. - ExpandExpand 4(5k − 2).
20k − 8.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA hall has 12 rows. Each row has x seats on the left and 6 on the right. Write the total seats as an expression with a bracket and expand it.
12(x + 6) = 12x + 72. - Earlier work, collecting like termsSimplify 6p + 3 − 2p + 5.
4p + 8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExpand and simplify 5(2x − 1) − 3(x + 4).
7x − 17: 10x − 5 − 3x − 12. - Earlier work, integersWork out (−6) × (−5) − (−4).
34: 30 + 4.
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. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper: draw a rectangle 3 tall, cut it at 4 and 5 across, and count squares.