At a glance. One lesson of about 60 minutes. It builds on lesson 1. Students multiply and divide algebraic fractions after factorizing, cancel across a product, and list the excluded values including the divisor's zeros.
Calculator: no calculator. No calculator: the structure of the factorizing and cancelling is the point.
What students will learn
Know
- To divide, multiply by the reciprocal of the second fraction.
- Factorize every part before cancelling across.
- Excluded values come from every denominator and from the zeros of the divisor.
Understand
- A division is a multiplication by the reciprocal, so the divisor's top becomes a bottom.
- The final fraction hides values that the original ruled out.
- Cancelling across a product is the same as cancelling within one fraction.
Do
- Multiply and divide algebraic fractions in simplest form.
- List all values of x that cannot be used.
- Find an unknown from a simplified result.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: factorizing, flipping and cancelling (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe where the result and the cancelled form differ.
- Criterion C. Communicating: writing the result with its excluded values.
- Criterion D. Applying mathematics in real-life contexts: the rice sacks (question 7).
The one thing to do with the figure
Ask for the prediction first. Set x = 3 and multiply, then ask students to calculate the left and right fractions on paper. Press divide and ask which fraction was flipped.
Ask: “At x = −2 the right fraction alone is 0. Give two reasons why x = −2 is ruled out of the division.” The left bottom is zero, and the divisor would be zero.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: Tomás's product; collect which values are ruled out. |
| 12 min | Figure, key idea and warning card; practise flipping with numeric fractions. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask which fraction the sign ÷ is applied to. |
| 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 | 3/2; 3 ÷ (x − 3); (x − 2)(x − 4). |
|---|---|
| Hook | x = −2, x = 4 and x = −4 are ruled out by the original; the cancelled form shows only x = −4. |
| Worked examples | 2(x − 3); x(x − 1); 1. |
| Faded example | 3(x − 4), x ≠ −2 and x ≠ −4. |
| 1. Dividing algebraic fractions | B. A flips the first fraction; C keeps an x that cancels; D does not flip. |
| 2. Multiplying and cancelling | 6. 2(x − 4) ÷ x. |
| 3. Dividing then evaluating | 15. x(x + 2). |
| 4. Solving after simplifying | 10. x/(x − 5) = 2. |
| 5. Counting excluded values | 3. 2, −3 and 4. |
| 6. Finding a missing coefficient | 2. (x − 3)(x + 5). |
| 7. A quotient in context | 5. x + 1. |
| 8. A quotient of equal tops | D. A cancels everything; B inverts the result; C multiplies without flipping. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Flipped the first fraction, cancelled only the numbers, or did not flip. | Rewrite as a multiplication first, then factorize. |
| −2 or 22 in question 2 | Took b with the wrong sign, or multiplied by the 3. | Underline which factors are on the top and which are on the bottom. |
| 3 or 5 in question 3 | Kept x − 2 instead of x + 2, or lost x. | Write the flipped divisor in full before cancelling. |
| −10 or 7 in question 4 | Kept the wrong bottom, or lost the x on top. | Write the simplified fraction and check it at x = 10. |
| 2 or 4 in question 5 | Missed the divisor's zero, or counted the zero of a top. | Flip the divisor, then list every bottom. |
| −2 in question 6 | Factorized with the signs swapped. | Check the product of the two factors by expanding. |
| 3 or 35 or 0.2 in question 7 | Kept x − 1, left x + 3 uncancelled, or divided the wrong way. | Ask: how many 2-tonne sacks are in 10 tonnes? Then write the same division. |
| A, B or C in question 8 | Cancelled everything, inverted the result, or did not flip. | Rewrite the division as a multiplication. |
Questions worth asking
- Which fraction is flipped in a ÷ b? Listen for: b, the second one.
- Why does the divisor's top give an excluded value? Listen for: it becomes a denominator after flipping.
- Why factorize before cancelling across? Listen for: only factors can be cancelled.
- How can you check the final answer? Listen for: substitute an allowed value into the original and the result.
Differentiation
A way in for anyone stuck
- Give a three-column table: original, flipped, factorized, with the first row done.
- Provide fractions already factorized and ask students to cancel and multiply.
- Give the frame: “I flipped …. The factors that cancel are …. The excluded values are ….”
Stretch
- Find a pair of algebraic fractions whose product is 1 for every allowed x, and describe the relationship.
- Show that dividing a fraction by itself gives 1 only where the fraction is defined and not zero.
Students learning in English as an additional language
“Reciprocal” means turned upside down. “Divisor” is the number or fraction you divide by. “Quotient” is the result of a division. “Flip” means turn upside down.
| Word | Plain meaning | Example |
|---|---|---|
| reciprocal | the fraction turned upside down | the reciprocal of 3/7 is 7/3 |
| divisor | the fraction you divide by | the second fraction in a ÷ b |
| quotient | the result of a division | 12 ÷ 4 |
| cancel across | remove a factor from one top and another bottom | the 5 in 5/9 × 4/5 |
| excluded value | a value of x that has to be left out | a zero of a bottom |
Sentence starters for explaining aloud or in writing:
- I flipped … and factorized ….
- The factors that cancel are ….
- The values x cannot take are … because ….
Two things not to say
“Turn the whole thing upside down.” Say instead: turn the second fraction upside down.
“The answer shows what x cannot be.” Say instead: read the excluded values from the original question.
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.
- MultiplySimplify (x + 1) ÷ 3 × 6 ÷ (x + 1).
2. - DivideSimplify 4x ÷ 5 ÷ (2x ÷ 15).
6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ExcludedState the values x cannot take in ((x + 2) ÷ (x − 5)) ÷ ((x − 1) ÷ x).
x = 5 and x = 0 from the denominators; x = 1 from the divisor's top, so x ≠ 0, 1, 5. - Earlier work (lesson 1)Simplify (x2 − 36) ÷ (x − 6).
x + 6.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs a ÷ b ÷ c the same as a ÷ (b ÷ c)? Test with 8, 4, 2.
No: (8 ÷ 4) ÷ 2 = 1 and 8 ÷ (4 ÷ 2) = 4. - Working backwards(x2 − 1) ÷ (x + a) × (x + 3) ÷ (x − 1) equals x + 1 for every allowed x. Find a.
a = 3, so that x + 3 cancels.
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 and the buttons, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.