At a glance. One lesson of about 60 minutes. It builds on factorizing from Year 9. Students factorize top and bottom, cancel common factors, state excluded values from the original denominator, and use a cancelled form to find an unknown.
Calculator: no calculator. No calculator: the factorizing is the skill, and every number is small.
What students will learn
Know
- A common factor can be cancelled; an added or subtracted term cannot.
- Excluded values come from the original denominator.
- A difference of two squares and a perfect square are the common special cases.
Understand
- Cancelling gives a fraction equal to the original wherever the original has a value.
- The cancelled form can have a value where the original does not, so the excluded value must be stated.
- Taking out a number first can reveal a difference of two squares.
Do
- Factorize and cancel to simplify an algebraic fraction.
- State the values of x the fraction cannot take.
- Use a cancelled form to find a missing coefficient or a dimension.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: factorizing and cancelling (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe where the two bars differ and why.
- Criterion C. Communicating: stating the excluded values with the simplified answer, in correct notation.
- Criterion D. Applying mathematics in real-life contexts: the plot with a factorized area (question 7).
The one thing to do with the figure
Ask for the prediction first. Set x = 3 and ask students to write both values before reading them. Then move to x = 4 and ask what has happened to the top bar.
Ask: “Which of the two fractions would you trust at x = 4, and why?” The original has no value there and the cancelled form does, so the excluded value has to be stated with the simplified fraction.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: Esi's cancelling; collect predictions about x = 4. |
| 12 min | Figure, key idea and warning card; test (x + 5) divided by 5 at a value. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, rewrite both parts as products and underline the common 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 + 3)(x + 4); (x − 7)(x + 7); 3/5. |
|---|---|
| Hook | No: they differ at x = 4, where the original has no value. |
| Worked examples | 2(x + 3) ÷ (3x); (x + 2) ÷ (x − 3); 2(x + 2) ÷ (x − 2). |
| Faded example | (x − 5) ÷ (x − 3), x ≠ −5 and x ≠ 3. |
| 1. Cancelling a factor | D. A cancels x squared; B cancels one x from each; C loses the x on the top. |
| 2. Reading a cancelled form | 9. 4/(x − 5). |
| 3. Simplify, then evaluate | 1.3. 13 ÷ 10. |
| 4. Excluded values | 2. 5 and −3. |
| 5. Factor out a number first | 8. 2(x − 3) ÷ (x + 3). |
| 6. Finding a missing coefficient | 1. (x + 4)(x − 3). |
| 7. A plot with a factorized area | 66. x = 19, length 22. |
| 8. Excluded values | C. A uses the zero of the top; B misses x = 0; D adds the zero of the top. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Cancelled x squared, cancelled one x from each, or lost the x on the top. | Write both parts as products and underline the bracket that appears twice. |
| −1 or 29 in question 2 | Read x + 5 as x − 5, or took q from 25. | Factorize the bottom and say which bracket was cancelled. |
| 0.3 or −3.5 in question 3 | Wrong sign in a factor, or cancelled x squared. | Check the factors by expanding them. |
| −2 or 3 or 8 in question 4 | Used the wrong signs, the zero of the top, or the difference. | Ask: which part decides what x cannot be? |
| 2 or 7 in question 5 | Sign slip on p, or left out the common factor 2. | Take out the number before looking for the difference of squares. |
| 7 or −1 in question 6 | Multiplied the cancelled form by the wrong second factor. | Ask: which factor was cancelled, and what is the top times that factor? |
| 33 or 34 in question 7 | Half the perimeter, or put the width in for x. | Sketch the plot and label both sides in terms of x. |
| A, B or D in question 8 | Used the zero of the top, or missed x = 0. | Factorize x squared minus 5x first. |
Questions worth asking
- Why can the 3 be cancelled in (3x + 6) ÷ 3 but not in (x + 3) ÷ 3? Listen for: 3 is a factor of 3x + 6, but in x + 3 it is added.
- What does the cancelled fraction not tell you? Listen for: the values that were excluded before cancelling.
- How can you check a simplification quickly? Listen for: put in a value of x that is not excluded and compare.
- Where would the excluded value show on a graph? Listen for: a gap in the graph.
Differentiation
A way in for anyone stuck
- Give the factorizing as a framed grid: product, sum, pair of numbers, brackets.
- Provide partly factorized fractions and ask students to complete and cancel.
- Give the frame: “The top is … and the bottom is …. The common factor is …. The fraction cannot take x = ….”
Stretch
- Find all the values of a for which (x2 + ax + 12) ÷ (x2 − 9) can be cancelled.
- Show that (x2 − 1) ÷ (x − 1) and x + 1 are equal for every x except one, and sketch both graphs.
Students learning in English as an additional language
“Cancel” means divide the top and the bottom by the same factor. “Factor” is a part of a product. “Excluded” means not allowed. “Denominator” is the bottom of a fraction.
| Word | Plain meaning | Example |
|---|---|---|
| algebraic fraction | a fraction with letters in the top, the bottom or both | (x + 2) / (x + 4) |
| factor | a part of a product, such as a bracket | x − 4 in (x − 4)(x + 2) |
| cancel | divide top and bottom by the same factor | 6x / 9 becomes 2x / 3 |
| excluded value | a value of x that makes the denominator zero | x = 2 for 1 / (x − 2) |
| denominator | the bottom of a fraction | x2 − 16 |
Sentence starters for explaining aloud or in writing:
- The top factorizes as … and the bottom as ….
- The common factor is ….
- The fraction cannot take x = … because ….
Two things not to say
“Cross out the x squared.” Say instead: cancel a factor that is in every term of both parts.
“Once cancelled, the excluded value disappears.” Say instead: it stays with the original fraction.
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.
- CancelSimplify (5x + 10) ÷ (x + 2).
5(x + 2) ÷ (x + 2) = 5. - FactorizeSimplify (x2 − 9) ÷ (x + 3).
x − 3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ExcludedState the values x cannot take in 4 ÷ (x2 − 7x + 10).
x ≠ 2 and x ≠ 5. - Earlier work (Year 9)Factorize 3x2 + 12x.
3x(x + 4).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs (x2 + 4) ÷ (x + 2) equal to x + 2? Test x = 1.
No: 5/3 and 3 are different. x squared plus 4 does not factorize. - Working backwards(x2 + ax − 10) ÷ (x2 − 4) simplifies to (x + 5) ÷ (x + 2). Find a.
The top is (x + 5)(x − 2) = x2 + 3x − 10, so a = 3.
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 the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.