At a glance. One lesson of about 60 minutes. It assumes expanding double brackets (Year 9) and common factors (Year 8). Students factorize x² + bx + c, a difference of two squares, and ax² + bx + c by splitting the middle term.
Calculator: no calculator. No calculator: the work is exact algebra, and every answer is checked by expanding the brackets.
What students will learn
Know
- x² + bx + c = (x + m)(x + n) when mn = c and m + n = b.
- a² − k² = (a + k)(a − k).
- ax² + bx + c can be factorized by splitting bx into two terms whose coefficients have product ac.
Understand
- Factorizing reverses expanding, and the answer is checked by expanding.
- The signs of the two numbers are decided by the signs of c and b.
- A sum of two squares does not factorize with real numbers.
Do
- Factorize quadratics with a leading coefficient of 1.
- Recognize and factorize a difference of two squares, taking out a common factor first.
- Factorize ax² + bx + c where a is not 1.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: factorizing three kinds of quadratic (questions 1 to 8).
- Criterion B. Investigating patterns: the what-if. Students list product pairs for x² + 4 and generalize why a sum of squares does not factorize.
- Criterion C. Communicating: writing factorized forms clearly, with signs, and checking by expanding (the worked examples).
- Criterion D. Applying mathematics in real-life contexts: the courtyard, where the area expression is factorized to find a side (question 5).
The one thing to do with the figure
Ask for a prediction before every slide. With c = −12 on the screen, ask: “What must be true about the two numbers if b is positive?” before anyone moves b.
Then set b = 3 and c = 4 and ask students to find a pair. When none exists, ask what the figure is saying about x² + 3x + 4.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tile panel. Collect predictions for the two numbers. |
| 12 min | The first figure with predictions; the key idea; settle the hook. |
| 10 min | The second figure, the warning card, and the worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the signs in the figure. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | x² + 8x + 15; 3(2x + 5); −28 and 3. |
|---|---|
| Hook | 3 and 7, so the sides are (x + 3) and (x + 7): 3 × 7 = 21 and 3 + 7 = 10. |
| Worked examples | (x + 4)(x − 6); (5x + 7)(5x − 7); (2x + 1)(x + 3). |
| Faded example | (x + 7)(x − 4). |
| 1. Factorizing x² + bx + c | B. A has the signs swapped; C has the wrong sum; D has the wrong sign of the product. |
| 2. Finding the larger number of a pair | 9. (x + 2)(x + 9). |
| 3. Difference of two squares | 8. (5x − 8)(5x + 8). |
| 4. Factorizing with a leading coefficient | 5. (2x + 1)(x + 5). |
| 5. Factorizing in a context | 3. (x + 9)(x − 3). |
| 6. Splitting the middle term | 4. (3x + 4)(2x − 1). |
| 7. Factorizing completely | 6. 3(x − 6)(x + 6). |
| 8. Factorizing with a leading coefficient | D. A, B and C each give the right first and last terms but the wrong middle term. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Swapped the signs, used the wrong sum, or ignored the sign of the product. | Always expand the answer and compare the middle term. |
| 2 or 18 in question 2 | Gave the smaller number, or a pair with the right product only. | Ask for the sum of 1 and 18, then of 2 and 9. |
| 64 in question 3 | Left the constant unrooted. | Write 25x² as (5x)² and 64 as 8² before using the pattern. |
| 1 or 11 in question 4 | Copied a number from the quadratic into the bracket. | Expand (2x + 1)(x + 1) and compare the middle term with 11x. |
| 9 or 27 in question 5 | Used the number already in the bracket, or the constant. | Ask what number times 9 gives −27. |
| 1 or −4 in question 6 | Matched the first and last terms but not the middle term. | Expand each candidate and read the middle term. |
| 36 or 108 in question 7 | Stopped before the square root, or did not take out the 3. | Divide every term by 3 first. |
| A, B or C in question 8 | Right first and last terms, wrong middle term. | Expand each option and read the middle term; the signs decide it. |
Questions worth asking
- Why must a negative c give a pair with opposite signs? Listen for “a positive times a negative is negative”.
- How can you tell before searching whether x² + bx + c might factorize? Look for a pair; the figure shows none exists for x² + 3x + 4. (Lesson 6 gives a test with the discriminant.)
- How is a² − k² = (a + k)(a − k) like a rectangle? The corner strip is moved to the side: a + k wide and a − k tall.
- Why check by expanding? It is quick and exact, and catches a sign slip.
Differentiation
A way in for anyone stuck
- Give a product and sum grid: a table of pairs with product c and a column for the sum.
- Use algebra tiles to build each x² + bx + c as a rectangle, and read the sides.
- Insist on expanding every answer and comparing the middle term.
Stretch
- Factorize x⁴ − 81 completely.
- Show that x² + bx + 36 factorizes with whole numbers for exactly five positive values of b, and list them.
Students learning in English as an additional language
“Factorize” means to write as a product of factors; the noun is “factor”. “Expand” is the opposite. “Difference” here means the result of subtracting. Say “x squared” for x².
| Word | Plain meaning | Example |
|---|---|---|
| factorize | write an expression as a product of brackets | x² + 9x + 20 as (x + 4)(x + 5) |
| expand | multiply out brackets | (x + 1)(x + 2) gives x² + 3x + 2 |
| coefficient | the number multiplying a letter | the 7 in 7x |
| constant term | the term with no letter | the 12 in x² + 7x + 12 |
| difference of two squares | one square subtracted from another | x² − 100 |
| common factor | a number or letter that divides every term | 7 in 14x² + 21 |
Sentence starters for explaining aloud or in writing:
- The product is … and the sum is …, so the numbers are …
- I checked by expanding: …
- This is a difference of two squares because …
Two things not to say
“Just guess and check.” Use the product and the sum to make the search finite, and check by expanding.
“x² + 9 factorizes as (x + 3)².” (x + 3)² has a middle term 6x; a sum of squares does not factorize.
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.
- FactorizingFactorize x² + 7x + 10.
(x + 2)(x + 5). - SquaresFactorize x² − 49.
(x − 7)(x + 7).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA rectangular rice terrace in Yuanyang has area x² − 5x − 24 square metres and a length (x − 8) m. Find the width.
(x + 3) m, because (x − 8)(x + 3) = x² − 5x − 24. - Earlier work (Year 9)Expand (x − 6)(x + 2).
x² − 4x − 12.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFactorize 4x² − 20x + 25 and say what is special about it.
(2x − 5)²: a perfect square, so both brackets are the same. - Leading coefficientFactorize 3x² + 10x + 8.
(3x + 4)(x + 2).
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. “Show me the steps” gives the same states in words.
No calculator is needed: every answer is exact and is checked by expanding.