How to use this menu
Six original tasks: two for criterion A (a short check and an end-of-unit test), one investigation for B, one communication task for C, and two real-life tasks for D that also assess C. Set at least one task for each criterion across this short unit; nobody needs all six. The timings spread the evidence across the unit.
Descriptors: Year 9 is assessed with the MYP Year 3 descriptors, paraphrased below each task. Check them against your school's current guide.
The menu at a glance
| Task | Criteria | When | Time | Format |
|---|---|---|---|---|
| A1: Quick check: double brackets and special products | A | Mid-unit, after lesson 3 | 25 minutes | Individual, written, in class. 20 marks. |
| A2: End-of-unit test | A | End of unit, after lesson 6 | 50 minutes | Individual, written, in class, one lesson. 24 marks in three sections. |
| B1: Investigation: four numbers in a row | B | After lesson 4 | 45 minutes | Individual, written, in class. Students may use any method, including a grid. |
| C1: Explain it to someone who missed it | C | After lesson 4 | 35 minutes | Individual, written, in class or as homework. One page of words and diagrams. |
| D1: Tiling a courtyard round a fountain | D and C | After lesson 5 | 50 minutes | Individual, in class. A written solution with a sketch and a short note to the planning group. |
| D2: Picture frames for a market stall | D and C | After lesson 5 | 45 minutes | Individual, in class. A written solution with explanations and a short note to the frame maker. |
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.
The unit's inquiry
- Key concept. Form: the same quantity written as a product and as a sum, and what each form shows.
- Related concepts. Equivalence, Representation, Generalization
- Global context. Globalization and sustainability: planning shared spaces such as gardens, courtyards and pools.
- Statement of inquiry. Representing a product of brackets as an area shows why two forms are equivalent and lets us generalize about how areas change.
MYP Year 3 descriptors, paraphrased
Each task's clarification, on its student sheet below, says what these levels look like in that task.
| Level | Criterion A: Knowing and understanding |
|---|---|
| 1–2 | Selects suitable mathematics for simple problems in familiar situations and applies it with some success. |
| 3–4 | Selects suitable mathematics for more complex problems in familiar situations and applies it successfully. |
| 5–6 | Selects suitable mathematics for challenging problems in familiar situations, applies it successfully and generally solves these problems correctly in a range of contexts. |
| 7–8 | Selects suitable mathematics for challenging problems in both familiar and unfamiliar situations, applies it successfully and generally solves these problems correctly in a range of contexts. |
| Level | Criterion B: Investigating patterns |
|---|---|
| 1–2 | With support, applies problem-solving techniques to find simple patterns and states predictions that fit them. |
| 3–4 | Applies problem-solving techniques to find simple patterns and suggests relationships or general rules that fit the findings. |
| 5–6 | Applies problem-solving techniques to find more complex patterns, describes them as relationships or general rules that fit the findings and verifies them. |
| 7–8 | Selects and applies problem-solving techniques to find complex patterns, describes them as relationships or general rules that fit correct findings, and verifies and justifies them. |
| Level | Criterion C: Communicating |
|---|---|
| 1–2 | Uses limited mathematical language and limited forms of representation; the reasoning is hard to follow. |
| 3–4 | Uses some appropriate mathematical language and forms of representation; the reasoning can be followed and the work is adequately organized. |
| 5–6 | Usually uses appropriate language, moves between forms of representation with some success, and the reasoning is complete and coherent in a mostly logical structure. |
| 7–8 | Consistently uses appropriate language, moves effectively between forms of representation, and the reasoning is complete, coherent and concise in a logical structure. |
| Level | Criterion D: Applying mathematics in real-life contexts |
|---|---|
| 1–2 | Identifies some relevant elements of the real-life situation and applies mathematical strategies to reach a solution with limited success. |
| 3–4 | Identifies the relevant elements, selects adequate strategies and applies them to reach a solution, and describes whether it makes sense in the context. |
| 5–6 | Identifies the relevant elements, selects adequate strategies and applies them to reach a valid solution, explains the degree of accuracy and explains whether the solution makes sense. |
| 7–8 | Identifies the relevant elements, selects appropriate strategies and applies them successfully to reach a correct solution, justifies the degree of accuracy and justifies whether the solution makes sense. |
Task A1: Quick check: double brackets and special products
Why set it. A short check of lessons 1 to 3 while there is time to reteach: grids, signs, squares and the difference of two squares. Questions 8 and 9 are unfamiliar, so the top band is open to everyone.
When: Mid-unit, after lesson 3. Time: 25 minutes. Format: Individual, written, in class. 20 marks.
The student sheet, with its task-specific clarification
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. |
Markscheme
| Task A1 question | Answer and marking |
|---|---|
| 1 | x² + 12x + 35. 1 mark for the four products, 1 for collecting. x² + 35 is the usual slip (no marks). |
| 2 | x² − 7x − 18. 1 mark for −7x, 1 for −18. |
| 3 | 2x² − x − 15. Cells 2x², −6x, 5x, −15. 1 mark for 2x², 1 for −6x and 5x, 1 for collecting to −x. |
| 4 | x² − 20x + 100. 1 mark for −20x, 1 for +100. |
| 5 | 9x² − 1. 1 mark for 9x², 1 for −1 with no x term. Credit 9x² − 3x + 3x − 1 collected. |
| 6 | 9991. 103 × 97 = (100 + 3)(100 − 3) = 10000 − 9. 1 mark for the method, 1 for the answer. |
| 7 | Any value other than 0, for example x = 1: (1 + 6)² = 49 but 1 + 36 = 37. 1 mark for the counter-example; 1 for a square of side x + 6 split into x², two strips 6x and a corner 36; 1 for saying Saba leaves out the two strips, 12x, so the answer is x² + 12x + 36. |
| 8 | a = 6, b = −5. a + b = 1 and ab = −30. 1 mark for both conditions, 1 for the pair. Credit a check: 6 + (−5) = 1 and 6 × (−5) = −30. |
| 9 | (n + 1)² − (n − 1)² = n² + 2n + 1 − (n² − 2n + 1) = 4n, which is 4 times a whole number. 1 mark for both expansions, 1 for 4n and the conclusion. Trying values alone earns 1 mark at most. |
| Marks to a level | Total 20. 1 to 5 marks: 1 to 2. 6 to 11: 3 to 4. 12 to 16: 5 to 6. 17 to 20, with question 8 or 9 fully correct: 7 to 8. Use the marks as a guide, then check the profile against the clarification. |
Exemplar answers
| Task A1 response | What it looks like |
|---|---|
| Level 3 to 4 profile | Questions 1 and 2 right (4 marks); question 3 gets 2x² − 15 only (1 mark); question 4 gives x² − 20x − 100 (1 mark); question 5 blank; question 6 right (2 marks); question 7 a value only (1 mark); nothing on 8 or 9. 4 + 1 + 1 + 2 + 1 = 9 marks, and the profile matches 3 to 4. |
| Level 7 to 8 answer to question 9 | “(n + 1)² = n² + 2n + 1 and (n − 1)² = n² − 2n + 1. Taking one from the other, the n² and the 1 cancel and 2n − (−2n) = 4n. 4n is 4 times a whole number, so it is a multiple of 4 for every n, not only the ones I tried (n = 3 gives 16 − 4 = 12).” |
Differentiation
A way in for anyone stuck
- Give printed 2 by 2 grids for Part 1.
- For question 8, suggest listing pairs of whole numbers that multiply to −30.
- Read questions 7 to 9 aloud on request. Reading is not what is being assessed.
Stretch
- Question 9 with (n + 3)² − (n − 3)². What is it a multiple of? (12, because it is 12n.)
- Find every pair of whole numbers a, b with (x + a)(x + b) = x² + kx − 30 for some k. Which values of k are possible? (±29, ±13, ±7, ±1.)
Students learning in English as an additional language
| Word in task A1 | Plain meaning |
|---|---|
| expand | multiply out the brackets |
| collect | add or subtract like terms |
| counter-example | one case that shows a claim is false |
| multiple of 4 | 4 times a whole number |
| whole number | a number with no fraction part, such as −4, 0 or 6 |
Sentence starters for task A1:
- The four products are …
- Saba is wrong because when x = …
- This is a multiple of 4 because …
Task A2: End-of-unit test
Why set it. The whole unit at three levels of demand. Section 3 is unfamiliar on purpose: it is where the evidence for 7 to 8 comes from.
When: End of unit, after lesson 6. Time: 50 minutes. Format: Individual, written, in class, one lesson. 24 marks in three sections.
The student sheet, with its task-specific clarification
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. |
Markscheme
| Task A2 question | Answer and marking |
|---|---|
| 1 | x² + 3x − 40. 1 mark for 3x, 1 for −40. |
| 2 | x² + 18x + 81. 1 mark for 18x (two strips), 1 for 81. |
| 3 | 4x + 14. (x² + 9x + 20) − (x² + 5x + 6). 1 mark for each expansion, 1 for 4x + 14. Check at x = 1: 30 − 12 = 18 and 4 + 14 = 18. |
| 4 | x = 4. x² + 8x + 12 = x² + 4x + 28, so 4x = 16. 1 mark for both expansions, 1 for 4, 1 for a check: 6 × 10 = 60 and 5 × 7 + 25 = 60. |
| 5 | (a) 2x² + 7x + 3. (b) 6x + 8. (c) x = 3, area 42 m². 6x + 8 = 26 gives x = 3; the rug is 7 m by 6 m. 1 mark for (a), 1 for (b), 1 for x = 3, 1 for 42, 1 for units (m and m²). |
| 6 | 2200. (61 + 39)(61 − 39) = 100 × 22. 1 mark for the method, 1 for 2200. |
| 7 | 12 cm, 16 cm and 20 cm. x² + 256 = x² + 16x + 64, so 16x = 192 and x = 12. Check: 144 + 256 = 400 = 20². 1 mark for the equation, 1 for the expansion, 1 for x = 12, 1 for all three sides. |
| 8 | Call the numbers n − 1 and n + 1; the number halfway is n. (n − 1)(n + 1) = n² − 1, which is 1 less than n². 1 mark for naming the numbers with one letter, 1 for the expansion, 1 for the conclusion. Examples alone (5 × 7 = 35 = 36 − 1) earn 1 mark. |
| Marks to a level | Total 24. 1 to 4 marks: 1 to 2. 5 to 10: 3 to 4. 11 to 18: 5 to 6. 19 to 24, with at least 5 of the 7 Section 3 marks: 7 to 8. |
Exemplar answers
| Task A2 response | What it looks like |
|---|---|
| Level 5 to 6 profile | Questions 1 to 3 right (7 marks); question 4 reaches x = 4 with no check (2 marks); question 5 (a) and (b) right and x = 3, but no area (3 marks); question 6 right (2 marks); question 7 sets up the equation and expands but stops at 16x = 192 (2 marks); question 8 not attempted. 7 + 2 + 3 + 2 + 2 = 16 marks, with only 2 from Section 3, so the level is 5 to 6. |
| Level 7 to 8 answer to question 8 | “Let the number halfway be n. The two numbers are n − 1 and n + 1, which differ by 2. (n − 1)(n + 1) is a difference of two squares, n² − 1. So the product is always 1 less than n², for any whole number n, not only for examples like 9 × 11 = 99 = 100 − 1.” |
Differentiation
A way in for anyone stuck
- Give a labelled sketch of the rug for question 5.
- For question 7, give the start: x² + 16² = (x + 8)².
- For question 8, suggest trying three pairs of numbers before using letters.
Stretch
- Question 8 for two numbers that differ by 6. (The product is 9 less than the square of the middle number.)
- In question 7, change 16 to k and 8 to m. Find x in terms of k and m. (x = (k² − m²) ÷ 2m.)
Students learning in English as an additional language
| Word in task A2 | Plain meaning |
|---|---|
| simplify | write with as few terms as possible |
| perimeter | the distance all the way round |
| halfway | exactly in the middle of two numbers |
| product | the result of multiplying |
| longest side | in a right-angled triangle, the side opposite the right angle |
Sentence starters for task A2:
- Expanding both sides gives …
- The x² terms cancel, so …
- Let the number halfway be …
Task B1: Investigation: four numbers in a row
Why set it. Criterion B on its own: collect results, find a relationship, test it in new cases and justify it with algebra, in a number pattern the lessons have not met.
When: After lesson 4. Time: 45 minutes. Format: Individual, written, in class. Students may use any method, including a grid.
The student sheet, with its task-specific clarification
In MYP Year 3, support you give (a prompt, a table outline) is allowed at 1 to 2: note it on the student's work.
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. |
Markscheme
| Task B1 question | Answer and marking |
|---|---|
| The pattern (do not share it before the task) | Middle product minus outer product is 2 for numbers in a row; for a step of d it is 2d² (8 for steps of 2, 18 for steps of 3, 50 for steps of 5). |
| 1 | Six correct results, each 2. For example 3, 4, 5, 6: 20 − 18 = 2; 51, 52, 53, 54: 2756 − 2754 = 2; −2, −1, 0, 1: 0 − (−2) = 2. |
| 2 | A clear table: the four numbers, both products and the result. Headings chosen by the student. |
| 3 | Any correct description, for example “the answer is 2 whatever four numbers in a row you start with”. A rule that only says “the answers are small” is evidence for 3 to 4. |
| 4 | 2. 98 × 99 = 9702 and 97 × 100 = 9700. |
| 5 | Steps of 2 give 8; steps of 3 give 18. For 3, 5, 7, 9: 35 − 27 = 8. For 4, 7, 10, 13: 70 − 52 = 18. The first rule fails; a changed rule links the result to the step: 2 × step × step. |
| 6 | 50. For example 1, 6, 11, 16: 66 − 16 = 50. |
| 7 | With first number n and step d: (n + d)(n + 2d) − n(n + 3d) = n² + 3dn + 2d² − n² − 3dn = 2d². With d = 1 this proves the Part 2 rule. Accept the d = 1 proof, n(n + 3) against (n + 1)(n + 2), at 5 to 6; the general step at 7 to 8. |
Exemplar answers
| Task B1 response | What it looks like |
|---|---|
| Level 5 to 6 | “3, 4, 5, 6 gives 20 − 18 = 2. 10, 11, 12, 13 gives 132 − 130 = 2. −2, −1, 0, 1 gives 0 − (−2) = 2. Every set gives 2, so 97 to 100 gives 2, and 9702 − 9700 = 2. With steps of 2 I got 8, and with steps of 3 I got 18, so the rule is different for steps but I could not say how. (n + 1)(n + 2) = n² + 3n + 2 and n(n + 3) = n² + 3n, so the answer is 2.” |
| Level 7 to 8 | “For numbers in a row the answer is 2. For steps of 2 it is 8 and for steps of 3 it is 18: 2, 8, 18 are 2 × 1, 2 × 4, 2 × 9, so it is 2 times the step squared, and steps of 5 give 50, which I checked with 1, 6, 11, 16. Proof: call the numbers n, n + d, n + 2d, n + 3d. The middle product is n² + 3dn + 2d² and the outer is n² + 3dn. The n² and 3dn cancel, leaving 2d² whatever n is.” |
Differentiation
A way in for anyone stuck
- Give a table outline with four columns and let the student write the headings.
- Suggest starting with small numbers, then trying a set with 0 in it.
- Prompt with a question, never the rule: “What happens to your results when you start somewhere else?”
Stretch
- What happens with the first and third numbers multiplied, against the second and fourth? (For a step of d: n(n + 2d) − (n + d)(n + 3d) = −2dn − 3d², which depends on n.)
- Try four numbers in a row and add instead of multiply. What do you find, and why? (Middle sum equals outer sum: both are 2n + 3.)
Students learning in English as an additional language
| Word in task B1 | Plain meaning |
|---|---|
| in a row | one after another, such as 7, 8, 9, 10 |
| outer | on the outside: the first and the last |
| step | how much is added each time |
| predict | say what will happen before you check |
| prove | show with algebra that something is true every time |
Sentence starters for task B1:
- I noticed that …
- My rule is: …
- I tested my rule on … and it worked / did not work because …
Task C1: Explain it to someone who missed it
Why set it. Criterion C on its own: notation, a labelled area diagram and a line of reasoning a classmate can follow, built around three slips from lessons 1 to 4.
When: After lesson 4. Time: 35 minutes. Format: Individual, written, in class or as homework. One page of words and diagrams.
The student sheet, with its task-specific clarification
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. |
Markscheme
| Task C1 question | Answer and marking |
|---|---|
| 1 (a) | Nia squared each term. (x + 5)² = (x + 5)(x + 5) = x² + 5x + 5x + 25 = x² + 10x + 25. The explanation should name the two strips she left out. |
| 1 (b) | Nia lost the sign of the number term. (−4) × 7 = −28, so the answer is x² + 3x − 28. The x term, −4x + 7x = 3x, is right. |
| 1 (c) | Nia took away only the x². −x(x + 4) = −x² − 4x, so x² + 5x + 6 − x² − 4x = x + 6. |
| 2 | A rectangle split into x², 4x, 3x and 12, with sides labelled x + 4 and x + 3. Total x² + 7x + 12. Credit a sentence linking each cell to a product. |
| 3 | A check with a value that is not 0 or 1, for example x = 2: (5)(4) − 2 × 6 = 20 − 12 = 8 and x + 6 = 8, while Nia’s 9x + 6 gives 24. |
| 4 | Any underlined word explained correctly, for example “expand: multiply out the brackets”. |
| Reading for C | Judge communication, not correctness: lines in a logical order, correct use of =, brackets and signs, a labelled diagram, and reasoning in sentences that a reader can follow. An answer slip that a reader can see and correct does not lower the C level. |
Exemplar answers
| Task C1 response | What it looks like |
|---|---|
| Level 3 to 4 extract | “(a) is wrong it should be x² + 10x + 25. (b) 28 should be minus. (c) x + 6.” Each correction is right, but there is no reason for any of them, and the reader cannot see which step Nia got wrong. |
| Level 7 to 8 extract | “(c) Nia expanded x(x + 4) correctly to x² + 4x, but the minus sign in front belongs to the whole bracket, so both terms change sign: −x² − 4x. Then x² − x² = 0 and 5x − 4x = x, giving x + 6. Check with x = 2: the first line is 5 × 4 − 2 × 6 = 8 and x + 6 = 8, but 9x + 6 = 24, so Nia’s answer fails.” Each step says what was done and why. |
Differentiation
A way in for anyone stuck
- Give a sentence frame for each mistake: Nia wrote …, but it should be … because ….
- Provide a blank 2 by 2 grid for the area diagram.
- Let the student explain one mistake aloud to a partner first, then write it.
Stretch
- Add a fourth line to Tomi’s page: explain why (x + 5)(x − 5) has no x term, with a diagram. (The strips are 5x and −5x, which cancel: x² − 25.)
- Explain to Tomi why testing with x = 0 would not have caught Nia’s mistake in (c). (At x = 0, 6 − 0 = 6 and 9 × 0 + 6 = 6.)
Students learning in English as an additional language
| Word in task C1 | Plain meaning |
|---|---|
| mistake | something that is wrong in the working |
| area diagram | a rectangle split into parts, one for each product |
| sign | plus or minus |
| check | test an answer, for example with a number |
| explain | say what you did and why |
Sentence starters for task C1:
- Nia wrote …, but …
- The minus sign in front of the bracket means …
- When x = 2, the first line gives …
Task D1: Tiling a courtyard round a fountain
Why set it. Area expressions, a budget and a decision that has to allow for waste, in one real plan. An alternative to D2: set one of them.
When: After lesson 5. Time: 50 minutes. Format: Individual, in class. A written solution with a sketch and a short note to the planning group.
The student sheet, with its task-specific clarification
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. |
Markscheme
| Task D1 question | Answer and marking |
|---|---|
| 1 | A rectangle labelled 2x + 3 by x + 4 with a square labelled x by x inside it. |
| 2 | x² + 11x + 12 square metres. (2x + 3)(x + 4) − x² = 2x² + 11x + 12 − x². 1 mark for the expansion, 1 for subtracting the fountain. |
| 3 | 6x + 14 metres. 2(2x + 3) + 2(x + 4). |
| 4 | 72 m², 3600 shillings. 16 + 44 + 12 = 72, and 72 × 50 = 3600. Check: 11 × 8 − 16 = 72. |
| 5 | A 5 m fountain: 92 m² of tiles, 4600 shillings. The budget allows 5000 ÷ 50 = 100 m². x = 5 gives 25 + 55 + 12 = 92; x = 6 gives 36 + 66 + 12 = 114, which is too much. A table or trial is a valid method; the reason x = 6 fails should be shown. |
| 6 | No. With 10% extra, 92 m² needs 92 + 9.2 = 101.2 m² of tiles, which costs 5060 shillings, 60 over budget. A 4 m fountain needs 72 + 7.2 = 79.2 m², which costs 3960 shillings. Recommend the 4 m fountain, or a slightly bigger budget for the 5 m one. |
| 7 (criterion C) | The note gives the recommended fountain size, the tile cost with units and the reason, including the extra 10%, in a form that makes sense without the working. |
Exemplar answers
| Task D1 response | What it looks like |
|---|---|
| Level 3 to 4 (D) | “Tiled area x² + 11x + 12. For 4 m it is 72, so 3600 shillings. The biggest is 5 m, 4600 shillings.” Correct, but there is no reason why 6 m fails and nothing on the extra 10%. |
| Level 7 to 8 (D and C) | “The budget buys 100 square metres. A 5 m fountain leaves 92 m² to tile and a 6 m one 114 m², so 5 m is the biggest on paper. But with 10% extra for cut tiles, 92 m² becomes 101.2 m² and 5060 shillings, which is over. I recommend a 4 m fountain: 79.2 m² with the extra, 3960 shillings, leaving 1040 for anything unexpected.” The note to the group is three sentences with units. |
Differentiation
A way in for anyone stuck
- Give the sketch with the labels to fill in.
- Give a table of x = 1 to 7 with columns for the courtyard, the fountain and the tiled area.
- For Part 3, ask: “How many square metres of tiles does the budget buy?”
Stretch
- Find the fountain size, to the nearest tenth of a metre, that uses exactly 100 m² of tiles without the extra 10%. (Between 5.3 and 5.4 m: x = 5.3 gives 98.39 m² and x = 5.4 gives 100.56 m².)
- A low wall runs round the outside edge at 30 shillings a metre. How much is it for the 4 m fountain plan? (6 × 4 + 14 = 38 m, 1140 shillings.)
Students learning in English as an additional language
| Word in task D1 | Plain meaning |
|---|---|
| courtyard | an open space with walls or buildings round it |
| fountain | a structure that sends water up into the air |
| budget | the most money that can be spent |
| tile | a flat square piece that covers a floor |
| recommend | say what someone should do, and why |
Sentence starters for task D1:
- The tiled area is … take away …
- The budget buys … square metres.
- I recommend … because …
Task D2: Picture frames for a market stall
Why set it. A border area in x, a price and a question whose exact answer is not a whole number. An alternative to D1: set one of them.
When: After lesson 5. Time: 45 minutes. Format: Individual, in class. A written solution with explanations and a short note to the frame maker.
The student sheet, with its task-specific clarification
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. |
Markscheme
| Task D2 question | Answer and marking |
|---|---|
| 1 | 12x + 36 square centimetres. (x + 6)² − x² = x² + 12x + 36 − x². 1 mark for x + 6, 1 for the expansion, 1 for the subtraction. |
| 2 | Wood 12x + 36 pesos; glass x² ÷ 4 pesos. |
| 3 | Wood 276 pesos, glass 100 pesos. 12 × 20 + 36 = 276 (check: 26² − 20² = 676 − 400 = 276) and 400 ÷ 4 = 100. |
| 4 | The claim is wrong for big pictures. 10 cm: wood 156, glass 25. 30 cm: wood 396, glass 225. 50 cm: wood 636, glass 625. The border costs more at these three sizes, but by only 11 pesos at 50 cm, because the glass grows faster (x²) than the wood (12x), so for big enough pictures the glass costs more. Credit a test at a bigger size, for example 60 cm: wood 756, glass 900. |
| 5 | Between 14 and 15 cm. The areas are equal when x² = 12x + 36. At 14: 196 against 204; at 15: 225 against 216. So the size is between 14 and 15 cm (about 14.5). A frame maker cannot measure it exactly, and it does not need to be exact. |
| 6 (criterion C) | The note says when the border costs more and when it does not, with costs in pesos for at least two sizes, in sentences that make sense without the working. |
Exemplar answers
| Task D2 response | What it looks like |
|---|---|
| Level 3 to 4 (D) | “Border 12x + 36. For 20 cm the wood is 276 pesos and the glass 100. At 10, 30 and 50 the wood costs more, so she is right.” Correct numbers, but the claim is accepted from three cases with no thought about bigger pictures. |
| Level 7 to 8 (D and C) | “The border is 12x + 36 and the picture x². At 10, 30 and 50 cm the wood costs more, by 131, 171 and then only 11 pesos, because x² grows faster than 12x. At 60 cm the glass costs more: 900 against 756, so her claim fails for big pictures. The two areas are equal between 14 and 15 cm, at about 14.5 cm, which nobody could cut exactly.” The note to the maker gives the sizes where her claim holds, with units. |
Differentiation
A way in for anyone stuck
- Give a sketch of the frame with the outside length to fill in.
- Give a table with columns: picture width, border area, wood cost, glass cost.
- For Part 3, ask: “Which grows faster as x gets bigger, x² or 12x + 36?”
Stretch
- Find the picture width, in whole centimetres, from which the glass costs more than the wood. (51 cm: wood 648, glass 650.25; at 50 cm wood 636, glass 625.)
- The maker changes to a 2 cm border. Write its area in x. (8x + 16.)
Students learning in English as an additional language
| Word in task D2 | Plain meaning |
|---|---|
| frame | the border that goes round a picture |
| border | a strip of the same width all the way round |
| claim | something a person says is true |
| grows faster | gets bigger by more each time |
| exactly | with no rounding at all |
Sentence starters for task D2:
- The border's area is … take away …
- Her claim is true for … but not for …
- The two areas are equal between … and …
Practicalities
Every task is original and written for this unit; none is copied or adapted from a published book.
The student sheets, with their task-specific clarifications, are public on the unit page so students know what each level looks like before they start. Markschemes and exemplars are only on this page.
To print a sheet, open it on the unit page, then print the page. No task needs a calculator.
The sheets are public, so a student may have seen A1 and A2. For an unseen A2, use this parallel version (same marks, same markscheme notes): Q1 (x + 7)(x − 6) (x² + x − 42). Q2 (x + 11)² (x² + 22x + 121). Q3 (x + 6)(x + 2) − (x + 3)(x + 1) (4x + 9). Q4 (x + 3)(x + 5) = (x + 1)(x + 2) + 33 (x = 4). Q5 a rug (2x + 3) m by (x + 1) m: area 2x² + 5x + 3, perimeter 6x + 8, perimeter 32 m gives x = 4 and area 55 m². Q6 72² − 28² (4400). Q7 shorter sides x and 12, longest x + 6 (9, 12, 15). Q8 two numbers that differ by 4 multiply to 4 less than the square of the number halfway between them.
The Year 3 descriptor bands on this page are band summaries written for this site; check them against the current IB MYP Mathematics guide before relying on them for reporting.