A short bridging unit. Year 8 multiplied one bracket by a number; here two brackets are multiplied together, and the answer is drawn as a rectangle cut into four parts. That picture explains the quick pattern for (x + a)(x + b), why (x + a)² is not x² + a², and why a square with a corner cut away can be rebuilt as a rectangle.
Every idea is something you can move: resize the cells of a grid, flip the signs, test a simplification at nine values at once, widen a path round a pool, and slide a rod along the side of a right-angled triangle. The last lesson brings back index laws, Pythagoras' theorem and linear equations, the Year 9 ideas that every Year 10 unit builds on, and joins them in one problem.
The lessons, in order
- Lesson 1 · Area grids for double brackets Expanding (x + a)(x + b) and (px + a)(x + b) with an area grid, collecting like terms, and checking with a number. Resize the four cells of an area grid and compare the whole rectangle with the slip x² + ab. No calculator
- Lesson 2 · Signs in double brackets Expanding (x + a)(x + b) with negative numbers, the pattern x² + (a + b)x + ab and why it works, and checking with a value. Change the signs in a multiplication grid and watch the x terms combine and the corner change sign. No calculator
- Lesson 3 · Perfect squares and the difference of two squares Expanding (x + a)² and (x − a)², the difference of two squares x² − a² = (x + a)(x − a), why each is true, and using them for mental arithmetic. See the two strips in a squared bracket, then cut a corner from a square and move a strip to make a rectangle. No calculator
- Lesson 4 · Expanding, simplifying and checking Expanding and simplifying expressions with two products and a subtraction, and checking a simplification by substituting values. Test two candidate simplifications against the original at nine values of x and see which ones only agree by luck. No calculator
- Lesson 5 · Area and perimeter with algebra Areas and perimeters of rectangles, squares, paths and borders written with brackets, expanded, simplified and used to solve problems. Resize a pool and the path round it, and watch the path's area stay a simple expression in x. No calculator
- Lesson 6 · Year 9 ideas that Year 10 builds on Retrieval of the index laws, Pythagoras' theorem and multi-step linear equations, joined by expanding a squared bracket. Slide x until a rod as long as x + m fits the long side of a right-angled triangle exactly. No calculator
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: double brackets and special products | A | Mid-unit, after lesson 3 | 25 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 6 | 50 minutes |
| B1: Investigation: four numbers in a row | B | After lesson 4 | 45 minutes |
| C1: Explain it to someone who missed it | C | After lesson 4 | 35 minutes |
| D1: Tiling a courtyard round a fountain | D and C | After lesson 5 | 50 minutes |
| D2: Picture frames for a market stall | D and C | After lesson 5 | 45 minutes |
D1 and D2 are alternatives. Both assess criteria D and C, in two different real-life contexts: one is set, not both. D1 leans on a cost and a budget; D2 on comparing two areas and on whether an exact answer exists.
Task A1: Quick check: double brackets and special products (criterion A)
When: Mid-unit, after lesson 3. Time: 25 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
Answer every question you can and show your working: a grid, a pattern or a check with a number all earn credit. Questions 8 and 9 are meant to make you think, so try something rather than leaving them blank.
Part 1. Expand
- Expand (x + 7)(x + 5). (2 marks)
- Expand (x − 9)(x + 2). (2 marks)
- Expand (2x + 5)(x − 3). (3 marks)
- Expand (x − 10)². (2 marks)
- Expand (3x + 1)(3x − 1). (2 marks)
Part 2. Use and explain
- Work out 103 × 97 without a calculator. Show your method. (2 marks)
- Saba says (x + 6)² = x² + 36. Show that she is wrong by choosing a value of x, and use a diagram to explain what she has left out. (3 marks)
Part 3. Think it through
- (x + a)(x + b) = x² + x − 30, where a and b are whole numbers and a is bigger than b. Find a and b. (2 marks)
- Show that (n + 1)² − (n − 1)² is a multiple of 4 for every whole number n. (2 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You expand one or two of the brackets in Part 1, for example with a grid. |
| 3–4 | You expand most of Part 1 correctly, including a bracket with a minus sign, and make a start on Part 2. |
| 5–6 | You answer Parts 1 and 2 correctly, with a grid, a pattern or a check, and make a good start on question 8 or 9. |
| 7–8 | You answer Parts 1 and 2 correctly and solve question 8 or 9 with reasoning that works for every case, not only the one you tried. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 6. 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.
Section 1. Familiar questions
- Expand and simplify (x + 8)(x − 5). (2 marks)
- Expand (x + 9)². (2 marks)
- Expand and simplify (x + 4)(x + 5) − (x + 2)(x + 3). (3 marks)
- Solve (x + 2)(x + 6) = (x + 1)(x + 3) + 25, and check your answer. (3 marks)
Section 2. Using the ideas
- A rug is (2x + 1) m by (x + 3) m. (a) Write its area without brackets. (b) Write its perimeter in terms of x. (c) Its perimeter is 26 m. Find x and the area of the rug. (5 marks)
- Work out 61² − 39² without a calculator. Show your method. (2 marks)
Section 3. Unfamiliar questions
- A right-angled triangle has shorter sides x cm and 16 cm, and its longest side is (x + 8) cm. Find the lengths of all three sides. (4 marks)
- Show that when two whole numbers differ by 2, their product is 1 less than the square of the number halfway between them. (3 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You expand one of the brackets in Section 1, or find the rug's area for one value of x. |
| 3–4 | You answer most of Section 1 and parts of Section 2, choosing a grid, a pattern or a substitution. |
| 5–6 | You answer Sections 1 and 2 correctly with working that shows your method, and make a good start on a Section 3 question. |
| 7–8 | You answer Sections 1 and 2 correctly and, in Section 3, set up and solve or prove something you have not seen in this form. |
Task B1: Investigation: four numbers in a row (criterion B)
When: After lesson 4. Time: 45 minutes. Calculator: no calculator. Criterion: B, investigating patterns.
Choose four whole numbers in a row, such as 3, 4, 5 and 6. Multiply the middle two numbers. Multiply the outer two numbers. Then take the second result from the first. Your job is to find out what happens, and why.
Part 1. Collect results
- Do this for at least six sets of four numbers in a row. Include one set that starts with a number bigger than 50, and one set that includes negative numbers.
- Record your results in a table you design yourself.
Part 2. Describe what you find
- Look at your table. Describe what you notice, as a rule another student could use.
- Use your rule to predict the result for 97, 98, 99 and 100, then check by working it out.
Part 3. Test and extend your rule
- Now use four numbers going up in twos, such as 3, 5, 7 and 9, and then four numbers going up in threes. Does your rule still work? If not, change it so that it does.
- Predict the result for four numbers going up in fives, then check one example.
Part 4. Explain
- Use letters for the first number and the size of the step, and use algebra to show that your rule works for every set of this kind, not only the ones you tried.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you work out some of the results and say something you notice about them. |
| 3–4 | You work out the results, record them in a table and suggest a rule that fits most of your numbers in a row. |
| 5–6 | You organize your results clearly, describe a rule that fits Parts 1 and 2, use it to predict, and test whether it still works for steps of 2 and 3. |
| 7–8 | You organize your results, describe a rule that fits every step size you tried, make correct predictions, change the rule for new steps and prove it with algebra. |
Task C1: Explain it to someone who missed it (criterion C)
When: After lesson 4. Time: 35 minutes. Calculator: no calculator. Criterion: C, communicating.
Tomi was away for lessons 1 to 4. He copied this working from a friend, Nia, and it has mistakes in it.
Nia’s working
(a) (x + 5)² = x² + 25
(b) (x − 4)(x + 7) = x² + 3x + 28
(c) (x + 3)(x + 2) − x(x + 4) = x² + 5x + 6 − x² + 4x = 9x + 6
Write a one-page explanation for Tomi. He should be able to follow it on his own, without asking anyone.
Part 1. Put it right
- Find the mistake in each line of Nia’s working. For each one, say what she did, why it is wrong, and write the correct working through to the answer.
Part 2. Teach the idea
- Draw a labelled area diagram for (x + 4)(x + 3) and use it to explain to Tomi why multiplying two brackets gives four products.
- Show Tomi how to check an answer by putting a number in place of x. Use your answer to part (c) and choose a value that really tests it.
Part 3. Check your explanation
- Read your explanation as if you were Tomi. Underline one word or symbol you used and explain it in a short sentence.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to tell which of Nia’s lines you are correcting or what the mistake was. |
| 3–4 | Your corrections can be followed, with some correct notation, and your area diagram has its parts labelled. |
| 5–6 | Your explanation is clear and mostly well ordered, brackets and signs are written correctly, and your diagram and your check both match the algebra. |
| 7–8 | Your explanation is clear and well ordered, the notation is always correct, the diagram explains the four products, and each step says why it is allowed. |
Task D1: Tiling a courtyard round a fountain (criteria D and C)
When: After lesson 5. Time: 50 minutes. Calculator: no calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A community centre in Nairobi is planning a courtyard. The courtyard is a rectangle (2x + 3) m long and (x + 4) m wide. A square fountain, x m by x m, sits inside it, and the rest of the courtyard is tiled. Tiles cost 50 shillings for each square metre, and the tiles budget is 5000 shillings.
Part 1. Models
- Sketch the courtyard and the fountain, and label every length in terms of x.
- Write the tiled area in terms of x, without brackets. Show your method.
- Write the length of the courtyard's outside edge (its perimeter) in terms of x.
Part 2. A decision
- Find the tiled area and the cost of the tiles when the fountain is 4 m wide.
- The planning group wants the biggest fountain, a whole number of metres wide, that keeps the tiles within budget. Find it, and the cost of the tiles.
Part 3. Does it make sense?
- Tilers usually buy 10% more tiles than the area, because some tiles are cut or broken. Does your answer to Part 2 still fit the budget? If not, what should the group do?
- Write a short note to the planning 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 of the right numbers, for example the courtyard's area for one fountain, but you do not reach a decision. |
| 3–4 | You write the tiled area in x, find the cost for a 4 m fountain and choose a fountain size, and say whether your answer makes sense. |
| 5–6 | You choose suitable methods for every part, reach correct answers, find the biggest fountain within budget, and discuss how the extra 10% changes the decision. |
| 7–8 | You choose and use suitable methods throughout, reach correct answers, justify why your fountain is the biggest that works, and explain whether your recommendation makes sense for real tiling. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see which area or which cost each number belongs to. |
| 3–4 | Your working for the area and the costs can be followed, with some correct notation and units. |
| 5–6 | Your working is clear and complete, usually with correct units (metres, square metres, shillings), a labelled sketch, and a note to the group that is mostly well organized. |
| 7–8 | Your working is clear, complete and well ordered, units and notation are always correct, the sketch is fully labelled, and your note makes sense on its own. |
Task D2: Picture frames for a market stall (criteria D and C)
When: After lesson 5. Time: 45 minutes. Calculator: no calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A frame maker at a market in Oaxaca makes square frames. The picture is x cm by x cm and the wooden border round it is 3 cm wide all the way round. The wood costs 1 peso for each square centimetre of border, and the glass over the picture costs 1 peso for every 4 square centimetres.
Part 1. Models
- Write the area of the wooden border in terms of x, without brackets. Show your method.
- Write the cost of the wood and the cost of the glass for one frame, in terms of x.
Part 2. Prices
- Find the cost of the wood and the glass for a picture 20 cm wide.
- The maker says: “For a bigger picture, the border costs more than the glass.” Test her claim for pictures 10, 30 and 50 cm wide.
Part 3. Does it make sense?
- Is there a picture size where the border's area is exactly the same as the picture's area? Find it as closely as you can, and say whether a frame maker could make it exactly.
- Write a short note to the frame maker about her claim, so that someone who has not seen your working can follow it.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You find some numbers, for example the border area for one picture, but do not test the claim. |
| 3–4 | You write the border area in x, find the costs for a 20 cm picture, test the claim at the three sizes and say whether your answers make sense. |
| 5–6 | You choose suitable methods for every part, reach correct answers, find where the two areas are equal to a sensible accuracy, and discuss what that means for the maker. |
| 7–8 | You choose and use suitable methods throughout, reach correct answers, explain with the algebra why the claim holds for small pictures and fails for big ones, and explain whether your conclusion makes sense for real frames. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see which area or cost each number is. |
| 3–4 | Your working for the border and the costs can be followed, with some correct notation and units. |
| 5–6 | Your working is clear and complete, usually with correct units (centimetres, square centimetres, pesos) and notation, and your note to the maker 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 maker 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.