At a glance. One lesson of about 60 minutes. Taking out the highest common number factor from a linear expression, finishing the bracket, and checking by expanding.
Calculator: no calculator. The numbers are small so that finding the highest common factor is a thought, not a chore.
What students will learn
Know
- Factorizing writes a sum as a product.
- The highest common factor is the largest number dividing every term.
- Expanding the answer gives back the original.
Understand
- Factorizing and expanding are inverse processes.
- Any common factor is allowed, but only the highest one leaves a bracket with no common factor.
Do
- Find the highest common factor of two numbers.
- Factorize a linear expression fully, including a 1 left in the bracket.
- Check a factorization by expanding.
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 (questions 2 to 8).
- Criterion B. Investigating patterns: with the first figure, ask students to predict which pairs have a highest common factor of 1, and to explain what that means.
- Criterion C. Communicating: the written check by expanding, and the exit ticket's reason for rejecting 7(4x − 6).
- Criterion D. Applying mathematics in real-life contexts: a floor with a known area and length (question 7).
The one thing to do with the figure
On the second figure, ask for f = 3 first and let students say whether it is finished. Most say yes: it is correct. Move to f = 6 and ask what is different about the bracket.
Then ask: “Which values of f give a bracket at all?” The ones that divide both 12 and 18, which are 1, 2, 3 and 6.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 1, which sets up the check. |
|---|---|
| 10 min | Hook, prediction of the highest common factor, then the first figure. |
| 10 min | Key idea; the second figure and the slider. |
| 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 | 10x − 15; 12; 4. |
|---|---|
| Worked examples | 5(3x + 5); 7(2x − 3); 6(x + 1). |
| Faded example | 6(3a + 4). |
| 1. Fully factorized | A, 3(3x + 4). B divides the first term only; C takes out the larger number; D takes out an x that is not common. |
| 2. Fill the bracket | 5. 20 ÷ 4. |
| 3. Highest common factor | 9. |
| 4. Negative term | 3. 24 ÷ 8. |
| 5. Not finished | B. A is the full answer; C and D do not expand back to 18x + 30. |
| 6. A 1 in the bracket | 1. 7 ÷ 7. |
| 7. Area | 7. 35 ÷ 5. |
| 8. Negative factor | 3. (−6) ÷ (−2). |
| 9. Factorize fully | A. B stops early; C divides only the first term; D loses the minus sign. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Divided one term only, took out the larger number, or took out an x that is not common. | Ask them to expand their answer and compare. |
| 20, 16 or 4 in question 2 | Left the term, subtracted, or gave the factor. | Ask what the bracket holds: what is left after taking out the 4. |
| 3 or 90 in question 3 | Gave a common factor but not the highest, or the lowest common multiple. | List the factors of both numbers and circle the shared ones. |
| 16 or 192 in question 4 | Subtracted or multiplied. | Run it forward: 8 × 3 = 24. |
| A, C or D in question 5 | Did not check the answer, or picked the finished form. | Expand each option and compare with 18x + 30. |
| 0 or 7 in question 6 | Lost the 1, or copied the term. | Expand 7(x) and see what is missing. |
| 30 or 175 in question 7 | Subtracted or multiplied the area by the length. | Draw the rectangle and ask what operation turns an area into a width. |
| −3 or 12 in question 8 | Sign slip or multiplication. | Run it forward: (−2) × 3 = −6. |
Questions worth asking
- How do you know you have taken out the highest common factor? The numbers left in the bracket share only 1.
- Why does 6x + 6 become 6(x + 1) and not 6(x)? Listen for “expand it back and the second 6 is missing”.
- Does every expression factorize? Not with a number factor: 7x + 9 shares only 1. The figure shows this.
Differentiation
A way in for anyone stuck
- List the factors of both numbers in two columns and circle the common ones.
- Use the groups picture: 12 counters and 18 counters shared into equal groups.
- Give the common factor and ask students to fill the bracket, then to find the factor themselves.
Stretch
- Factorize 12x + 30 − 6 fully. (12x + 24 = 12(x + 2).)
- Factorize 8xy + 12x fully. (4x(2y + 3): the letter x is a common factor as well as the number 4.)
Students learning in English as an additional language
Factor, common, highest and bracket are the key words. Pair each with a gesture: hands together for common, a raised hand for highest.
| Word | Plain meaning | Example |
|---|---|---|
| factor | a number that divides another exactly | 6 is a factor of 18 |
| common factor | a factor of two or more numbers | 3 is a common factor of 12 and 18 |
| highest common factor | the largest of the common factors | 6 for 12 and 18 |
| factorize | write a sum as a product with a bracket | 12x + 18 = 6(2x + 3) |
| expand | multiply out a bracket | 6(2x + 3) = 12x + 18 |
Sentence starters for explaining aloud or in writing:
- The highest common factor of … and … is …
- I divided each term by … to fill the bracket.
- It is / is not finished because the numbers in the bracket share …
Two things not to say
“Just divide by the biggest number you can.” It must divide every term, or the bracket will hold a fraction.
“The answer has a bracket, so it is done.” 2(9x + 15) has a bracket and is not finished.
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 12x + 20.
4(3x + 5). - FactorizeFactorize 9y − 15.
3(3y − 5).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tailor in Chennai has a ribbon (24x + 36) cm long. Write its length as a whole number times a bracket, using the highest common factor.
12(2x + 3). - Earlier work, expandingExpand 7(2a − 3).
14a − 21.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs 20x + 8 fully factorized as 2(10x + 4)? Explain.
No. 10 and 4 share 2, so it is 4(5x + 2). - Earlier work, integersWork out (−3)(−4) + (−2)(5).
2: 12 − 10.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard and by touch. Under each, “Show me the steps” gives the same states in words.
No device? Use counters or squared paper: 12 and 18 shared into equal rows.