At a glance. One lesson of about 60 minutes. Students can expand a single bracket from Year 8; this lesson multiplies two brackets with positive numbers, using a grid, and checks each expansion with a value of x.
Calculator: no calculator. Nothing here needs one. The numbers are small so that the attention goes on the four products and on collecting.
What students will learn
Know
- (x + a)(x + b) has four products before collecting: x², ax, bx and ab.
- The x term is (a + b)x and the number term is ab.
- x × x is x², not 2x.
Understand
- Multiplying two brackets is the area of a rectangle cut into four parts.
- An expansion and the original are equal for every value of x, so a number is a check.
- x² + ab leaves out two strips of the rectangle, which is why it is too small.
Do
- Expand (x + a)(x + b) and (px + a)(x + b) with a grid and collect like terms.
- Find a missing number in a bracket from the expanded form.
- Check an expansion by substituting 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 double brackets with a grid (questions 1 to 8).
- Criterion B. Investigating patterns: on the figure, keep a + b fixed and change a and b. Which part of the expansion changes? Ask for a rule.
- Criterion C. Communicating: a labelled grid, then the collected line, then a check with a number written beside it.
- Criterion D. Applying mathematics in real-life contexts: the courtyard in the hook and the enlarged photo in question 7.
The one thing to do with the figure
Take the hook's two answers to the figure. Ask everyone to commit to Amira or Jonas, then set a = 3 and b = 2 and read the four cells. Ask which cells Amira counted, then compare her total with the rectangle at x = 4.
Then change only x and ask why the “short by” number changes but the expansion does not.
A possible order
| 5 min | Warm-up from the student page, then answers. Linger on warm-up 3 (x × x against x + x). |
|---|---|
| 10 min | Hook with a vote, then the figure at a = 3, b = 2. Students say which cells each person counted. |
| 10 min | Key idea and “why four parts”: the grid as two single brackets. |
| 10 min | Worked examples, including 3x + 2; the faded example in pairs. |
| 5 min | Question 1, the check. If many choose A, go back to the figure and read the “short by” line. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 8. |
The answers
| Warm-up | 4x + 28; 7x + 11; 36 and 12. |
|---|---|
| Worked examples | x² + 10x + 21; 3x² + 17x + 10. |
| Faded example | x² + 11x + 24. |
| 1. Expanding with a grid | B, x² + 8x + 12. A leaves out the strips; C adds the brackets; D adds 6 and 2 for the corner. |
| 2. The x term of a double bracket | 10. x² + 4x + 6x + 24 = x² + 10x + 24. |
| 3. An area at a value of x | 70. 7 × 10; also 4 + 26 + 40. |
| 4. Find a missing number in a bracket | 7. 2 + k = 9 and 2k = 14 agree. |
| 5. A bracket with 2x | B. A adds 3 and 4 for the x term; C leaves out the strips; D adds 3 and 4 for the corner. |
| 6. Work back from the expansion | 42. a = 7, c = 7 × 6. |
| 7. How much an area grows | 95. 8x + 15 at x = 10; or 195 − 100. |
| 8. Spot the slip in a grid | C. A accepts 3 + 7 for the corner; B drops the x from the strips; D leaves out the strips. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 44 in question 3; 15 in question 7 | Firsts times firsts and lasts times lasts: the strips are missing. | Point at the strips on the figure and ask what area they are. |
| C in question 1 | Added the brackets instead of multiplying them. | Ask what (x + 6)(x + 2) means when x = 1: 7 × 3, not 7 + 3. |
| D in question 1 or 5; A in question 8 (accepting 5 + 9 = 14) | Added the numbers for the corner. | The corner is a rectangle a by b: area multiplies. |
| 24 or 6 in question 2 | Took the corner as the x term, or counted one strip. | Label each cell of the grid with its letter part before collecting. |
| 34 or 17 in question 3 | Perimeter or a sum of the sides instead of the area. | Ask for the area in words first: length times width. |
| 11, 12 or 14 in question 4 | Undid the strips or the corner with the wrong operation. | Write both facts: 2 + k = 9 and 2 × k = 14. |
| 7, 78 or 13 in question 6 | Stopped at a, multiplied the x term, or gave the x term. | Ask what c stands for in the grid before working it out. |
| 80 or 195 in question 7 | Left out the corner, or gave the whole new area. | Shade the old photo in the grid and count only the rest. |
Questions worth asking
- Why are there four parts and not two? Each of two terms meets each of two terms: 2 × 2 products.
- Amira's answer works for one value of x. Which one? x = 0, where the strips have no area. Any other x shows the gap.
- Does the order of the brackets matter? No: (x + 2)(x + 3) and (x + 3)(x + 2) are the same rectangle turned round.
- How could you check an expansion without a teacher? Put the same x into the brackets and into the expansion.
Differentiation
A way in for anyone stuck
- Give a printed blank 2 by 2 grid with the headings to fill in.
- Use algebra tiles or squared paper: an x by x square, x by 1 strips and 1 by 1 squares.
- Start with a = 1 on the figure so the strips are easy to see.
Stretch
- Question 6, then: find every pair of whole numbers a and b with (x + a)(x + b) = x² + 13x + c, and list the values of c.
- Expand (x + 2)(x + 3)(x + 1) by expanding two brackets first. How many cells would a three-bracket “grid” have?
Students learning in English as an additional language
“Expand” means “multiply out the brackets”. “Cell” is one box of the grid; “strip” is a long thin rectangle.
| Word | Plain meaning | Example |
|---|---|---|
| expand | multiply out the brackets | (x + 1)(x + 5) becomes x² + 6x + 5 |
| term | one part of an expression, joined to the rest by + or − | x², 6x and 5 are the terms of x² + 6x + 5 |
| like terms | terms with the same letter part, which can be collected | 4x and 5x make 9x |
| grid | a table with one row and one column for each term | a 2 by 2 grid for two brackets of two terms |
| double brackets | two brackets multiplied together | (x + 1)(x + 5) |
Sentence starters for explaining aloud or in writing:
- The four cells of the grid are …
- I collect the x terms: … + … = …
- I check at x = …: the brackets give … and my expansion gives …
Two things not to say
“First, outer, inner, last.” A list of letters hides why there are four products; the grid shows it.
“The x squared is just there.” Say where it comes from: x times x, the big square.
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.
- GridExpand (x + 9)(x + 1).
x² + 10x + 9. - Earlier workExpand 5(x + 9).
5x + 45.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- 2xExpand (2x + 1)(x + 6).
2x² + 13x + 6. - ContextA rug is (x + 4) m by (x + 9) m. Write its area without brackets.
x² + 13x + 36 square metres.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoning(x + a)(x + b) = x² + 11x + 30 with whole numbers a and b. Find a and b.
5 and 6 (in either order): 5 + 6 = 11 and 5 × 6 = 30. - Two 2x bracketsExpand (2x + 1)(2x + 3).
4x² + 8x + 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The three sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” under the figure gives the same states in words, with no movement.
No device? Draw the grid on squared paper with an x by x square of any size and strips of 1, 2, 3 squares.