At a glance. One lesson of about 60 minutes. It assumes lesson 6 and factorizing a common factor. Students collect terms, factorize, divide by a bracket, clear fractions and substitute into the result.
Calculator: no calculator. All the substitutions work with small whole numbers or simple fractions, so no calculator is needed.
What students will learn
Know
- Terms in the subject go on one side and the rest on the other.
- A common factor of the subject is taken out and then divided away.
- A fraction is cleared by multiplying both sides by its denominator.
Understand
- Factorizing turns two appearances of the subject into one.
- The bracket divides as a whole.
- A quadratic in the subject cannot be solved by this method.
Do
- Collect, factorize and divide.
- Clear fractions, expand, then collect.
- Substitute into the rearranged formula and check.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: collecting, factorizing and substituting (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record a, b, x and the total for several settings and state the factoring rule.
- Criterion C. Communicating: each step labelled (collect, factorize, divide).
- Criterion D. Applying mathematics in real-life contexts: comparing plans and lens formulas (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then keep x fixed and change a and b, asking what happens to the total. Ask what number the total is divided by to get back to x.
Ask: “Why can we join the two rectangles?” Because they have the same height x, which is exactly what a common factor means.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the two gym plans; write the equation. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C, ask what divides by a bracket. |
| 20 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 | x = (y − 9) ÷ 4; 3(2x + 5); x = 5. |
|---|---|
| Hook | 40 + 5n = 10 + 8n gives 30 = 3n, so n = 10: the plans cost the same after 10 classes. Plan B has the lower monthly fee, so it is cheaper for fewer than 10 classes, and plan A is cheaper for more. |
| Worked examples | x = (d − b) ÷ (a − c); p = T ÷ (q + 1); x = (2y + 1) ÷ (y − 3). |
| Faded example | a = c ÷ (b + d). |
| 1. Collecting and factorizing the subject | A. B leaves x on the right; C divides one term; D keeps the sign of x when it moves. |
| 2. Using a factorized formula | 6. 72 ÷ 12. |
| 3. Subject on both sides | 7. 4x = 28. |
| 4. Subject in a fraction | 3. 2x = 6. |
| 5. Factorizing to find P | 1300. 1560 ÷ 1.2. |
| 6. Subject in a denominator | 3. 1/v = 1/3. |
| 7. A square in the formula | 2. a = 2(s − ut) ÷ t². |
| 8. Collect, factorize, divide | C. A subtracts c; B adds d; D divides by b only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Left x on both sides, divided one term, or kept the sign of x when moving it. | Test k = 3 (so x = 3) in each option. |
| 9 or 2.25 in question 2 | Divided by a only, or by ab. | Write 8x + 4x = 12x first. |
| 5 or 2.8 in question 3 | Subtracted q instead of adding, or added the x coefficients. | Draw the balance: x terms left, numbers right, signs flip. |
| 6 or 0 in question 4 | Stopped before dividing, or moved a term without changing sign. | Write each line and say the operation aloud. |
| 7800 or 1559.8 in question 5 | Divided by rt only, or subtracted rt. | Ask: is P multiplied by 1, by rt, or by their sum? |
| −4 or 1.5 in question 6 | Treated reciprocals as numbers, or added. | Use fractions with common denominators. |
| 1 or 8 in question 7 | Forgot to double, or divided by t. | Ask: what is t squared when t = 4? |
| A, B or D in question 8 | Subtracted c, added d, or divided by b only. | Substitute small numbers into both sides of each option. |
Questions worth asking
- Why must you collect before factorizing? Listen for: the letter has to be in a single bracket.
- What if the subject cancels? Then the formula may not define it; a rare case.
- Why does a fraction need clearing first? Listen for: the subject is trapped in the denominator.
- How can you check a result with numbers? Listen for: substitute values that satisfy the original.
Differentiation
A way in for anyone stuck
- Provide the three headings Collect, Factorize, Divide as a three-column table.
- Use coloured pens for the subject's terms.
- Give the frame: “Terms with … on the left: …. Factorize: …. Divide: …”
Stretch
- Make t the subject of s = t(u + v) − 4t.
- Explain why you cannot make x the subject of y = x² + x using collect, factorize and divide.
Students learning in English as an additional language
“Collect” means bring together. “Factorize” means write as a product, here taking the common letter out of a bracket. “Denominator” is the bottom of a fraction. “Common factor” is a number or letter that divides every term. “Reciprocal” means one divided by the number.
| Word | Plain meaning | Example |
|---|---|---|
| collect | bring terms together on one side of a formula | put the t terms on the left |
| factorize | write as a product with a common factor taken out | 10k + 15 = 5(2k + 3) |
| common factor | a letter or number that divides every term | m in km + 3m |
| denominator | the bottom number or expression of a fraction | the w − 1 in 4 ÷ (w − 1) |
| reciprocal | 1 divided by a number | 1⁄8 is the reciprocal of 8 |
Sentence starters for explaining aloud or in writing:
- The subject is in … places, so I will …
- Factorizing gives …, then I divide by …
- I cleared the fraction by multiplying by …
Two things not to say
“Swap sides and flip.” Say instead: do the same operation to both sides.
“Cancel the letter.” Say instead: factorize the letter out of the bracket, then divide.
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.
- RearrangeMake x the subject of 7x + 2x = 63.
x = 63 ÷ 9 = 7. - RearrangeSolve 3x + 1 = x + 11.
x = 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo cinema plans in Lima cost 6 + 2n and 2 + 3n for n films. For which n are they equal?
6 + 2n = 2 + 3n gives n = 4. - Earlier work (lesson 6)Make y the subject of x = 3y − 7.
y = (x + 7) ÷ 3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningMake x the subject of y = (2x + 3) ÷ (x − 1).
yx − y = 2x + 3, so x(y − 2) = y + 3 and x = (y + 3) ÷ (y − 2). - Working backwardsa = (c + e) ÷ (b − d). Find e when a = 4, b = 6, d = 2 and c = 5.
4 × 4 = 16 = 5 + e, so e = 11.
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 device? Cut two paper rectangles of equal height and join them.