At a glance. One lesson of about 60 minutes. It assumes expanding two brackets. Students take out common factors, factorize with a not equal to 1 by splitting the middle term, use the difference of two squares and perfect squares, explain why the method works, and decide when no whole-number brackets exist.
Calculator: no calculator. Factorizing is exact work, so no calculator is used. A GDC would not help find a pair of whole numbers and would hide the structure.
What students will learn
Know
- The common factor comes out first.
- A pair with product ac and sum b splits the middle term.
- A difference of two squares factorizes; a sum of two squares does not.
Understand
- The pair has product ac because (ps)(qr) = (pr)(qs), the product of the outer and inner terms.
- No pair means no whole-number brackets, which the graph shows as zeros that are not rational or are missing.
- Special patterns are cases of the general method.
Do
- Factorize any quadratic that factorizes over the whole numbers, completely.
- Explain why the method works using the expansion of two brackets.
- Decide, with a reason, that a quadratic does not factorize.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: factorizing and checking by expansion (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if. Students use product and sum to explain why a sum of two squares has no whole-number brackets.
- Criterion C. Communicating: the proof of the pair rule in the key card, and clear grouped steps in the worked examples.
- Criterion D. Applying mathematics in real-life contexts: the tile and the strip of cloth, where a factor is a side length (question 4).
The one thing to do with the figure
Ask for the pair before showing it. Set a, b and c and say: “What must the pair multiply to, and what must it add to?” Only then read the pair off the screen.
Then find a quadratic that does not factorize and ask students to explain, from ac and b, why no pair exists. Link it to the graph: there are no whole-number or fractional zeros.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tile. Collect predictions for what the sides multiply to. |
| 12 min | The figure with predictions; the key idea and the proof; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or C, return to the common factor. |
| 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 − 5x − 36; 2x² + 9x − 18; −8. |
|---|---|
| Hook | Their product is 6x² + 7x + 2: ac = 12 and the pair is 3 and 4, giving (2x + 1)(3x + 2). |
| Worked examples | (2x − 5)(3x + 4); 2(3x + 5)(3x − 5); no whole-number brackets. |
| Faded example | (2x + 3)². |
| 1. Complete factorization | A. B and C are correct but incomplete; D squares the wrong number. |
| 2. Finding a bracket constant | 14. 1 × p = 14. |
| 3. Difference of two squares | 7. k² = 49. |
| 4. Brackets when a is not 1 | 13. (−1) × m = −13. |
| 5. Factorizing an area in context | 5. (3x − 1) at x = 2. |
| 6. Perfect square condition | 12. 2 × 6. |
| 7. Which numbers factorize | 2. k = 9 and k = 11. |
| 8. Spotting no whole-number brackets | C. ac = 28 has no pair with sum 9; A, B and D factorize. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Left a common factor, left a difference of two squares unfinished, or squared the wrong number. | Ask: can anything else come out of each bracket? Expand option D to see 7x² − 112. |
| 28 or 29 in question 2 | Gave ac or b instead of the bracket constant. | Match only the constants: 1 × p = 14. |
| 49 or 24.5 in question 3 | Did not square-root, or halved. | Write k × k = 49. |
| −13 or 26 in question 4 | Copied the constant, or gave the number from the split. | Expand the brackets and compare the constants. |
| 7, 13 or 65 in question 5 | Wrong sign in the bracket, gave the length, or gave the area. | Ask what each quantity is. |
| 6 or 72 in question 6 | Gave the square root of 36, or doubled 36. | Write (x + t)² and expand. |
| 3 or 6 in question 7 | Counted a k outside the range, or counted factors. | List each factor pair and its sum. |
| A, B or D in question 8 | Believed a quadratic with a not equal to 1 cannot factorize. | Find a pair for each. |
Questions worth asking
- Why do the pair multiply to ac and not to c? The outer term ps and the inner term qr multiply to pr × qs, which is a × c.
- Why does a sum of two squares have no whole-number brackets? It would need a pair with sum 0 and a positive product, and opposites multiply to a negative.
- What does the graph show when a quadratic has no pair? Its zeros are not rational, or it has no zeros at all.
- Does it matter which pair order you split the middle term in? No: both groupings give the same brackets, in a different order.
Differentiation
A way in for anyone stuck
- Give an area-model grid (algebra tiles) for 2x² + 7x + 3 and let students arrange the tiles into a rectangle.
- Provide a flow chart: common factor, then difference of two squares, then pair with product ac and sum b.
- Write the factor pairs of ac in a column with their sums beside them.
Stretch
- Show that if ax² + bx + c has a pair of whole numbers with product ac and sum b, the discriminant b² − 4ac is a perfect square.
- Find all the values of c, between −20 and 20, for which x² + 4x + c factorizes into brackets with whole numbers.
Students learning in English as an additional language
“Factorize” means write as a product of brackets. A “coefficient” is the number multiplying a letter. “Split the middle term” means rewrite bx as two terms. Use “factorized” and say each form aloud with its purpose.
| Word | Plain meaning | Example |
|---|---|---|
| common factor | a number or letter that divides every term | 6 in 6x² + 18x |
| difference of two squares | one square minus another, which factorizes | x² − 100 = (x + 10)(x − 10) |
| perfect square | a quadratic that is a bracket squared | x² + 18x + 81 = (x + 9)² |
| coefficient | the number multiplying a letter | the 8 in 8x |
| split the middle term | write bx as two terms with product ac and sum b | 9x as 6x + 3x |
| factorize completely | keep going until no bracket has a common factor | 2x(x + 3), not x(2x + 6) |
Sentence starters for explaining aloud or in writing:
- The pair must multiply to … and add to …
- I took out the common factor … first because …
- It does not factorize because no pair of whole numbers …
Two things not to say
“Just guess brackets until it works.” Use the pair; a failed search is also an answer.
“x² + 81 is (x + 9)².” The middle term 18x would be there; a sum of two 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.
- FactorizeFactorize x² + 9x + 20.
(x + 4)(x + 5). - Common factorFactorize 5x² − 20.
5(x − 2)(x + 2).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA rectangular rice-paper sheet in Hanoi has area (6x² + 13x + 6) cm² and length (3x + 2) cm. Write the width as an expression.
(2x + 3) cm. - Earlier work (Year 9)Expand (3x − 5)².
9x² − 30x + 25.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFactorize 12x² − x − 6.
(3x + 2)(4x − 3). - ProofShow that x² + 6x + 10 has no whole-number brackets and that its graph never reaches the x-axis.
Pairs of 10 add to 11 or 7, never 6; and (x + 3)² + 1 is at least 1.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works 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. Factorizing is exact work and every answer is reached by hand.