At a glance. One lesson of about 60 minutes. Students expand (x + a)(x + b) with negatives from lesson 2. This lesson treats two special cases: a bracket squared, and a bracket times its partner with the opposite sign.
Calculator: no calculator. Nothing here needs one. Questions 5 and 8 are mental arithmetic on purpose: the identities make them quick.
What students will learn
Know
- (x + a)² = x² + 2ax + a² and (x − a)² = x² − 2ax + a².
- x² − a² = (x + a)(x − a).
- x² + a² does not split into two brackets of this kind.
Understand
- The 2ax is two equal strips of the square, which (x + a)² = x² + a² leaves out.
- The slip x² + a² is out by more as x grows.
- An L shape can be rebuilt as a rectangle with the same area, which proves the difference of two squares.
Do
- Expand (x + a)², (x − a)² and (px − a)².
- Expand (x + a)(x − a) in one step.
- Use the difference of two squares for mental products and differences of squares.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the two identities and their use (questions 1 to 9).
- Criterion B. Investigating patterns: how far out is x² + a² as x changes? On the first figure, record the gap for a = 2 and x = 1 to 5 and find the rule.
- Criterion C. Communicating: describe the cut-and-move picture in words, with the sides of the rectangle labelled in terms of x and a.
- Criterion D. Applying mathematics in real-life contexts: the tile in the hook and the photo frame in question 6.
The one thing to do with the figure
First figure: start from Yusuf. Set a = 4 and ask students to predict the gap at x = 1 and x = 10 before sliding. The “out by” line grows, which rules out “it is only a little bit wrong”.
Second figure: before pressing “Strip moved”, ask what shape the pieces could make. Then ask for its sides in terms of x and a.
A possible order
| 5 min | Warm-up from the student page, then answers. |
|---|---|
| 10 min | Hook: test Yusuf at x = 1 and x = 10, then the first figure. |
| 10 min | Key idea, then the difference of two squares with the second figure. |
| 10 min | Worked examples, including 52 × 48; the faded example in pairs. |
| 5 min | Question 1, the check. If many choose A, go back to the first figure's strips. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 49 and 49; x² − 5x − 24; 144 cm². |
|---|---|
| Worked examples | x² − 10x + 25; x² − 64; 2496. |
| Faded example | x² − 14x + 49. |
| 1. Squaring a bracket | C, x² + 10x + 25. A leaves out the strips; B counts one strip; D adds 5 and 5 for the corner. |
| 2. The x term of a square | 14. x² + 14x + 49. |
| 3. The difference of two squares | −81. x² − 81. |
| 4. Squaring x minus a number | −6. x² − 6x + 9. |
| 5. Mental arithmetic with two squares | 899. 30² − 1². |
| 6. Same perimeter, different area | 9. x² − (x² − 9). |
| 7. Subtracting two squares | 16. 8x − (−8x). |
| 8. Two squares in reverse | 3400. 100 × 34. |
| 9. Squaring 2x minus a number | D, 4x² − 12x + 9. A squares only the x; B counts one strip; C gets the corner's sign wrong. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 0 in question 2 | Squared each term: the strips are missing. | The first figure's “out by” line. |
| B in question 1; 7 in question 2; −3 in question 4 | Counted one strip. | Write the square as two brackets and draw the full grid. |
| D in question 1 | Added for the corner. | The corner is a square: a × a. |
| 81, 0 or −18 in question 3 | Lost the sign, gave the x term, or added. | The corner of (x + 9)(x − 9) is 9 × (−9). |
| 6 or 9 in question 4 | Lost the strips' sign, or gave q. | Label each cell with its sign. |
| 900 or 901 in question 5 | Forgot or added the 1². | Write (30 + 1)(30 − 1) and use the pattern. |
| 0 or 6 in question 6 | Assumed the changes cancel, or added 3 + 3. | The second figure: the missing piece is a 3 by 3 square. |
| 0 or 8 in question 7 | Left out the strips, or kept one square's strips. | Bracket the second expansion before subtracting. |
| 1156 or 10000 in question 8 | Squared the difference or the sum. | Write x² − a² = (x + a)(x − a) with x = 67 and a = 33. |
| A, B or C in question 9 | Squared only the x, counted one strip, or made the corner negative. | Write (2x − 3)(2x − 3) and draw the grid. |
Questions worth asking
- Is Yusuf ever right? Only at x = 0, where the strips have no area.
- Why is the corner of (x − a)² positive? It is (−a) × (−a).
- What is the shape after the strip moves, and how do you know it is a rectangle? Both pieces are x − a wide, so they stack edge to edge.
- Can x² + 9 be written as two brackets with whole numbers? Not as (x + a)(x + b): ab = 9 and a + b = 0 cannot both hold.
Differentiation
A way in for anyone stuck
- Keep the full grid: write (x + 5)² as (x + 5)(x + 5) every time.
- Cut the L shape from squared paper and let students move the strip by hand.
- For question 5, start with 21 × 19 = 20² − 1² = 399.
Stretch
- The what-if: 99² and 101² without a calculator.
- Show that the difference between the squares of two consecutive whole numbers is always odd.
Students learning in English as an additional language
“Squared” means multiplied by itself. “Difference” means the result of a subtraction.
| Word | Plain meaning | Example |
|---|---|---|
| squared | multiplied by itself | (x + 1)² means (x + 1)(x + 1) |
| perfect square | an expression that is something squared | x² + 2x + 1 is (x + 1)² |
| difference of two squares | one square take away another | x² − 25 = (x + 5)(x − 5) |
| identity | a statement that is true for every value of x | (x + 1)² = x² + 2x + 1 |
| strip | a long thin rectangle in the grid | the two strips of (x + 1)² are x and x |
Sentence starters for explaining aloud or in writing:
- (x + a)² is not x² + a², because …
- The two strips are … and …
- This is a difference of two squares, so it equals …
Two things not to say
“Square the first, square the last.” It is the slip in a sentence.
“The minus disappears.” Say why the corner is positive: a negative times a negative.
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.
- SquareExpand (x + 6)².
x² + 12x + 36. - From lesson 2Expand (x − 2)(x + 7).
x² + 5x − 14.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Two squaresExpand (x − 11)(x + 11).
x² − 121. - MentalWork out 43 × 37.
1591: 40² − 3².
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that (x + 5)² − (x + 3)² is always a multiple of 4.
It is 4x + 16 = 4(x + 4). - Squares in reverseWork out 85² − 15².
7000: 100 × 70.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard: the buttons take Enter or Space, the sliders take the arrow keys. Nothing moves on its own; the second figure changes only when a button is pressed, and “Show me the steps” gives every state in words.
No device? Cut the L shape from squared paper; the strip really does fit along the top.