The unit starts with a fraction whose top and bottom contain letters. Factorizing both parts shows which factors can be cancelled and which values of x the original fraction can never take. Multiplying and dividing then follow the rules for numbers, and adding and subtracting need a common denominator built from the factorized bottoms.
The graph of y = k/x comes next: two branches, the axes as asymptotes, and a domain and range that each leave out one value. Translating that graph moves the asymptotes with it, so a single fraction such as (ax + b)/(cx + d) is read as a shifted reciprocal graph.
The last two lessons use all of this. Equations with fractions turn into linear or quadratic equations, and any value that makes a denominator zero is rejected. Average cost and mean speed are then modelled as rational functions, where the horizontal asymptote becomes a floor that the model approaches but never reaches.
The lessons, in order
- Lesson 1 · Simplifying algebraic fractions Cancelling a common factor, factorizing a quadratic or a difference of two squares first, stating the excluded values, and finding an unknown from a cancelled form. Slide x along the number line and compare a fraction with its cancelled form, including the value that breaks one of them. No calculator
- Lesson 2 · Multiplying and dividing algebraic fractions Multiplying and dividing algebraic fractions after factorizing, turning a division into a multiplication by the reciprocal, and finding every excluded value including the divisor's zeros. Multiply or divide two fractions at one x at a time and compare the result with the cancelled form. No calculator
- Lesson 3 · Adding and subtracting algebraic fractions A common denominator from factorized bottoms, adding and subtracting with brackets, simplifying the numerator, and using the sum in a rate context. Set two numerators over x + 1 and x + 2 and watch the combined numerator, checked at x = 2. No calculator
- Lesson 4 · The reciprocal function The shape of y = k/x for positive and negative k, the two asymptotes, domain and range, finding k from a point, and an inverse-proportion rate context. Choose k and its sign, then move x towards 0 and far away to see the two asymptotes. No calculator
- Lesson 5 · Shifting the reciprocal graph Reading asymptotes from y = k/(x − p) + q and from (ax + b)/(cx + d), writing one form as the other, domain and range, and intercepts. Slide the reciprocal graph across and up and watch the dashed asymptotes and the single-fraction form follow. GDC optional
- Lesson 6 · Equations with fractions Clearing fractions by the common denominator, solving the linear or quadratic equation that results, rejecting excluded values, and a rate context. Move a horizontal line against a graph with a gap, and see which answers the algebra gives and which the graph allows. No calculator
- Lesson 7 · Rate and average-cost models An average cost as F/n + v, its floor, solving for n from a target average, a mean speed over two equal distances, and finding unknown costs from two readings. Set a fixed cost and a cost per item, then raise the number of items and watch the average fall towards a floor. GDC optional
Assessing this unit
Your teacher chooses which of these tasks to set, and when: nobody does all of them. Each sheet says where it fits in the unit, how long it takes and how your work is judged. To print a sheet, open it first and then print the page.
| Task | Criteria | When | Time |
|---|---|---|---|
| A1: Quick check: simplify, combine, find the missing value | A | After lesson 3 | 30 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 7 | 50 minutes |
| B1: Adding a list of fractions | B | Mid-unit, after lesson 4 | 50 minutes |
| C1: Put Tomas's working right and explain it | C | After lesson 6 | 40 minutes |
| D1: Printing cloth bags for a market stall | D and C | End of unit, after lesson 7 | 50 minutes |
| D2: A courier's mean speed | D and C | End of unit, after lesson 7 | 50 minutes |
D1 and D2 are alternatives. Both assess criteria D and C, in two different real-life contexts, one built on average cost and one on mean speed: set one, not both.
Task A1: Quick check: simplify, combine, find the missing value (criterion A)
When: After lesson 3. Time: 30 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
Answer every question you can and show your working: a correct method earns credit even if the final answer slips. Questions 7 and 8 are meant to make you think, so try something rather than leaving them blank.
Part 1. Simplifying
- Simplify (x² − 9)/(x² + 5x + 6). (3 marks)
- For which values of x is the original fraction in question 1 not defined? (2 marks)
Part 2. Products and quotients
- Simplify (2x + 6)/(x² − 4) × (x − 2)/(x + 3). (3 marks)
- Simplify (x² − x − 12)/(x + 1) ÷ (x − 4)/(3x + 3). (3 marks)
Part 3. Sums and differences
- Write 2/(x − 1) + 3/(x + 2) as a single fraction. (3 marks)
- Write 5/(x² − 1) − 2/(x + 1) as a single fraction. (3 marks)
Part 4. Think it through
- Dalia writes: (x + 4)/(x + 6) = 4/6 = 2/3, “because the x cancels”. Choose a value of x that shows she is wrong, and say what she should have noticed about the top and the bottom. (2 marks)
- The fraction (x² + kx + 6)/(x² − 4) can be simplified for two different whole-number values of k. Find both values and simplify the fraction each time. (4 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You answer some of questions 1 to 6 correctly, for example one factorization or one common denominator. |
| 3–4 | You answer most of questions 1 to 6 correctly, with the factorizing shown, and name an excluded value in question 2. |
| 5–6 | You answer questions 1 to 6 correctly with clear working, give every excluded value, and make a good start on question 7 or 8. |
| 7–8 | You answer questions 1 to 6 correctly and solve questions 7 and 8, with reasoning that shows why each answer works. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
Three sections, from familiar questions to ones you have not met in this form. Show your working everywhere: it earns credit even when an answer slips. Check every answer to an equation in the original equation.
Section 1. Familiar questions
- Simplify (3x² + 6x)/(x² − 4). (2 marks)
- Write down the vertical and horizontal asymptotes of y = 4/(x + 1) − 5. (2 marks)
- Write 2/(x + 3) + 1/(x − 1) as a single fraction. (3 marks)
- y is inversely proportional to x, and y = 8 when x = 3. Find y when x = 12. (2 marks)
Section 2. More complex questions
- Solve 3/(x − 1) = 2/(x + 2). (3 marks)
- Solve 2/x + 3/(x + 1) = 2. (4 marks)
- Write y = (2x + 1)/(x − 3) in the form k/(x − p) + q, and state its two asymptotes. (3 marks)
- A community hall costs 300 currency units to hire, plus 12 currency units for each guest. (a) Write the average cost per guest for n guests as a rule in n. (b) What is the least whole number of guests for which the average cost per guest is no more than 20 currency units? (3 marks)
Section 3. Unfamiliar questions
- One pipe fills a tank in x hours working alone. A second pipe fills it in 3 hours longer. Together they fill it in 2 hours. Find how long each pipe takes alone, and explain why one solution of your equation is not used. (5 marks)
- Gunther claims: “Plot y = (x² − 1)/(x − 1) and you will see the straight line y = x + 1, nothing else.” Is he right? Explain. (3 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You answer some Section 1 questions correctly, for example an asymptote or a simplification. |
| 3–4 | You answer most of Section 1 correctly and make progress in Section 2, solving at least one equation with fractions. |
| 5–6 | You answer Section 1 and most of Section 2 correctly with each step shown, checking answers in the original equation, and make a start in Section 3. |
| 7–8 | You answer Sections 1 and 2 correctly and solve most of Section 3, rejecting values that make a bottom zero and explaining why your answers make sense. |
Task B1: Adding a list of fractions (criterion B)
When: Mid-unit, after lesson 4. Time: 50 minutes. Calculator: no calculator. Criterion: B, investigating patterns.
Here is a list of fractions, each with 1 on the top: 1/(1 × 2), 1/(2 × 3), 1/(3 × 4), 1/(4 × 5), and so on, where the bottom of each is two whole numbers in a row multiplied together. Call the sum of the first fraction S1, the sum of the first two S2, the sum of the first three S3, and so on. This task is about finding what SN is, without adding every fraction.
Part 1. Collect results
- Work out S1 to S5, each as a single fraction in its simplest form. Record each sum in a table you design yourself.
Part 2. Describe what you find
- Look at your table. Describe what you notice, in words, and write a rule that gives SN for any N.
- Use your rule to predict S6. Then work it out by adding the next fraction to S5, to check your prediction.
Part 3. Use and test it
- Without adding, predict S10 and S99. Show how your rule gives each answer.
- Now change the list to 1/(1 × 3), 1/(2 × 4), 1/(3 × 5), 1/(4 × 6), and so on, and call the sums T1, T2, …. Collect T1 to T4. Say what is the same and what is different, and what you think happens to the sum if the list goes on for ever.
Part 4. Explain
- Explain, using algebra on the fractions and not more examples, why your rule for SN works for every value of N.
- Do you need to know all of S8 to find S9? Explain your answer.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you add some of the fractions, write the sums as fractions and say something about them. |
| 3–4 | With help, you find S1 to S5 correctly as simplest fractions and suggest a rule in words for the sums. |
| 5–6 | You organize your sums, describe a rule that fits them, check it on S6, and use it for S10 and S99. You collect T1 to T4 and say what is the same and what differs. |
| 7–8 | You organize your sums, describe a rule that fits all your findings, check and use it, investigate the second list, and justify your rule for every N with algebra rather than examples. |
Task C1: Put Tomas's working right and explain it (criterion C)
When: After lesson 6. Time: 40 minutes. Calculator: no calculator. Criterion: C, communicating.
Lin was away for lessons 5 and 6. She copied this working from a classmate, Tomás, and it has mistakes in it.
Tomás’s working
(a) Solve 1/(x − 2) + 3 = x/(x − 2).
Multiply by (x − 2): 1 + 3 = x, so x = 4.
(b) The graph of y = 2/(x − 1) + 1.
The vertical asymptote is x = −1 and the horizontal asymptote is y = 1. The curve meets y = 1 when x = 3.
Write a one-page explanation for Lin. She should be able to follow it on her own, without asking anyone.
Part 1. Put it right
- Find every mistake in Tomás’s working. For each one, say what he did, why it is wrong, and write the correct working.
Part 2. Teach the method
- Write a worked solution to (a) that Lin could follow, with each step on its own line and the answer checked in the original equation.
- Draw a sketch of y = 2/(x − 1) + 1 with both asymptotes dashed and labelled and both axis intercepts marked. Under it, write two sentences saying how you found the asymptotes from the rule.
Part 3. Check your explanation
- Make up one equation with a fraction in it that has a value which must be rejected. Solve it and explain the rejection to Lin in two sentences.
- Read your page as if you were Lin. Check that your equals signs join equal things, your brackets are in place, and you used the words asymptote, denominator, check and reject.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | Some of your steps can be followed, but the order is hard to see and the sketch or the notation is missing or unclear. |
| 3–4 | Your working for the equation is in a sensible order and your sketch shows both asymptotes; some labels or notation are wrong or missing, so Lin has to guess in places. |
| 5–6 | Each line follows from the one before and the check is written out. Your sketch labels the asymptotes and intercepts and matches your words, and your notation is mostly correct. |
| 7–8 | Your page is complete and concise: every step is justified in words or symbols, the sketch, working and sentences agree, notation is correct throughout, and your rejected-value example is explained so that Lin could use it alone. |
Task D1: Printing cloth bags for a market stall (criteria D and C)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: no calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A group prints cloth bags by hand to sell at a weekend market. Setting up the screens and ink costs 240 currency units, however many bags are printed. Each bag then costs 8 currency units in cloth and ink. They will sell every bag at 14 currency units.
Part 1. Average cost
- Write a rule for the total cost of printing n bags. Then write the average cost per bag, C, as a rule in n.
- Find the average cost per bag when 10, 30, 60 and 120 bags are printed. Describe how quickly the average falls as more bags are printed.
Part 2. Reading the rule
- Describe the graph of C against n, including its asymptotes, and say what each asymptote means for the group.
- Can the average cost per bag ever be 8? Can it be 7? Explain using your rule.
Part 3. Decisions
- How many bags must the group print so that the average cost per bag is no more than the selling price of 14?
- A second printer quotes C = 150/n + 9. For which numbers of bags is the second printer cheaper per bag? Write an equation and solve it.
Part 4. Does it make sense?
- The group expects to sell about 55 bags, but it could be 100. Which printer should they use? Work out the total cost at both numbers and say how sure you are.
- Name one real thing the rule C = 240/n + 8 leaves out. Then write a short note to the group with your recommendation, so that someone who has not seen your working can understand it.
How your work is judged
| Level | Criterion D in task D1: what it looks like |
|---|---|
| 1–2 | You find some average costs for given numbers of bags, but you do not link them to a decision. |
| 3–4 | You write the average cost rule, find correct values and the number of bags for 14 per bag, and say whether your answers are sensible. |
| 5–6 | You use the rule correctly for every part, solve for the 14 per bag and the second quote, explain the lowest the average can reach, and say how far the model can be trusted. |
| 7–8 | You choose and use suitable methods throughout, justify each answer, show why the average has a floor, compare the printers across the whole range, and explain what the model leaves out and how that changes your advice. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how you reached the average costs or the number of bags. |
| 3–4 | Your working for the averages and the comparison can be followed, with some correct notation and units. |
| 5–6 | Your working is clear and complete, usually with correct units (currency units, bags) and notation, and your note to the group is mostly well organized. |
| 7–8 | Your working is clear, complete and well ordered, units and notation are always correct, and your note to the group makes sense on its own. |
Task D2: A courier's mean speed (criteria D and C)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: no calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A bicycle courier rides 15 km to a customer at a steady 30 km/h, then rides straight back along the same road at a steady speed of v km/h. The courier wants to know the mean speed for the whole round trip.
Part 1. Set it up
- Write an expression for the total time of the round trip, in hours, in terms of v.
- Show that the mean speed for the round trip is M = 60v/(v + 30) km/h.
- Find M when v = 15, v = 30 and v = 60.
Part 2. Reading the rule
- Write M in the form k/(v + p) + q, and give its asymptotes. Say which of them tells the courier something real.
- Can the courier’s mean speed for the round trip ever reach 60 km/h? Explain using the rule.
- The courier says: “My mean speed is the average of my two speeds.” Test this claim when v = 60.
Part 3. Decisions
- The courier wants a mean speed of 40 km/h. Find the return speed v that is needed.
- On this road a cyclist cannot hold more than about 45 km/h. What is the highest mean speed the courier can reach?
Part 4. Does it make sense?
- The 15 km is only known to the nearest kilometre. Explain whether this changes your mean speeds.
- Write a short reply to the courier giving your answers, how reliable they are, and one real thing the model leaves out.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You find some of the right numbers, for example the time for the outward ride, but do not reach a mean speed. |
| 3–4 | You find the mean speed for each return speed, find the return speed for a mean of 40, and say whether your answers are sensible. |
| 5–6 | You use suitable methods for every part, read the asymptotes from the rule, explain why the mean cannot reach 60, test the courier’s claim, and say how reliable your numbers are. |
| 7–8 | You choose and use suitable methods throughout, justify each answer from the rule, explain the ceiling and why the claim fails, show why the distance does not matter, and explain what the model leaves out and how it changes your advice. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how you found the total time or the mean speeds. |
| 3–4 | Your working for the times and speeds can be followed, with some correct notation and units (h, km/h). |
| 5–6 | Your working is clear and complete, usually with correct units and notation, and your reply to the courier is mostly well organized. |
| 7–8 | Your working is clear, complete and well ordered, units and notation are always correct, and your reply to the courier makes sense on its own. |
Teachers: the assessment teacher page has the markschemes, exemplar answers, the MYP descriptors for this year and support for each task.
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.