At a glance. One lesson of about 60 minutes. Students expand and simplify from lessons 1 to 4 and know area and perimeter from Years 7 and 8. This lesson writes areas and perimeters with brackets and solves problems with them.
Calculator: no calculator. Nothing here needs one. The numbers keep the arithmetic short so the modelling is the work.
What students will learn
Know
- The area of an (x + a) by (x + b) rectangle is (x + a)(x + b).
- A path or border is the outer area take away the inner area.
- A perimeter has no x² term.
Understand
- A fixed-width path round a rectangle has an area that grows by the same amount for each extra metre, because the x² terms cancel.
- Setting two area expressions equal can leave a linear equation once the x² terms cancel.
- Area and perimeter answer different questions.
Do
- Write and simplify area and perimeter expressions for rectangles, borders and paths.
- Solve a problem from an area or perimeter, and check the answer in the shape.
- Explain how an area changes as x changes.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: writing and simplifying area and perimeter expressions (questions 1 to 8).
- Criterion B. Investigating patterns: on the figure, record the path's area for x = 2 to 6 and w = 1, 2, 3. What does w do to the expression?
- Criterion C. Communicating: a labelled sketch with the sides in terms of x, then the working, then a sentence answer with units.
- Criterion D. Applying mathematics in real-life contexts: the pool, the photo border, the garden path and the frame; ask whether each answer makes sense.
The one thing to do with the figure
Collect predictions on the hook first. Then keep w = 1 and step x from 2 to 6, recording the path and the pool in two columns. Ask what the differences are in each column.
Then set w = 3 and ask for the path's expression before reading it from the panel.
A possible order
| 5 min | Warm-up from the student page, then answers. |
|---|---|
| 10 min | Hook with a vote, then the figure with a two-column table of path and pool. |
| 10 min | Key idea and the warning; worked example 2 pays off the hook. |
| 10 min | Worked example 1 and the faded example in pairs. |
| 5 min | Question 1, the check. If many choose A, ask which of area or perimeter is a length. |
| 15 min | Your turn, Levels 1 to 3, with a sketch for every context. |
| 5 min | Exit ticket, question 8. |
The answers
| Warm-up | 5x + 6; 28 m² and 22 m; x = 6. |
|---|---|
| Worked examples | x² + 13x + 30 and 4x + 26; 8x + 28. |
| Faded example | 7x + 10 square metres. |
| 1. Perimeter or area | D, 4x + 16. A is the area; B adds two sides; C doubles only the x terms. |
| 2. Area at a value of x | 64. 8²; also 9 + 30 + 25. |
| 3. The x term of an area | 4. x² + 4x − 12. |
| 4. Area of a border | 48. 4x + 8; 168 − 120. |
| 5. From a perimeter to x | 5. 4x + 20 = 40. |
| 6. Two areas equal | 3. 6x + 9 = 7x + 6. |
| 7. An L-shaped path | 9. 4x + 4 = 40. |
| 8. Area of a border | B, 12x + 66. A leaves out the corners; C is the outer rectangle; D adds the border once each way. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1 | Gave the area for a perimeter question. | Ask whether the answer is a length or a space. |
| B or C in question 1; 15 or 7.5 in question 5 | Added two sides only, or doubled only the x terms. | Sketch the rectangle and label all four sides. |
| 34, 32 or 16 in question 2 | Squared each term, gave the perimeter, or doubled. | Find the side first, then square it. |
| 8 or −12 in question 3 | Ignored the sign of −2, or gave the number term. | The grid with signed cells from lesson 2. |
| 168, 120 or 44 in question 4 | Gave the outer area or the photo, or left out the corners. | Shade the border on a sketch; count the corner squares. |
| −3 or 15 in question 6 | Sign slips in solving. | Write the operation done to both sides. |
| 10 or 18 in question 7 | Left out the corner, or counted one strip. | Expand (x + 2)² − x² in full. |
| A, C or D in question 8 | No corners, the outer rectangle, or the border counted once. | Write the outer sides first: x + 6 and x + 11. |
Questions worth asking
- Why does the path's area have no x²? Outer and inner both have x², and the subtraction cancels them.
- What does the 8 in 8x + 28 mean for the pool? Each extra metre of x adds 8 square metres of path.
- How do you know your answer makes sense? Put it back into the shape: are the lengths positive, is the border smaller than the outer area?
- Could the 2 m path ever be bigger than the pool? Yes for small pools; the what-if finds where they swap.
Differentiation
A way in for anyone stuck
- Always start from a sketch with every side labelled in x.
- Use the figure to check a value before writing the general expression.
- For borders, write outer sides first: add the border twice to each side.
Stretch
- The what-if: where the 2 m path and the pool swap over.
- A path w m wide round an x by x square: show that its area is 4wx + 4w².
Students learning in English as an additional language
“Border” and “path” here both mean a strip of the same width all round. “Area” is the space inside; “perimeter” is the distance round.
| Word | Plain meaning | Example |
|---|---|---|
| area | the space inside a flat shape, in square units | a 3 m by 5 m rug has an area of 15 m² |
| perimeter | the distance all the way round a shape | a 3 m by 5 m rug has a perimeter of 16 m |
| border | a strip of the same width all round a shape | the frame round a picture |
| square metre | the area of a square 1 m by 1 m | written m² |
| inner, outer | the shape inside, and the whole shape with its border | a pool is inner; pool and path together are outer |
Sentence starters for explaining aloud or in writing:
- The outer rectangle is … by …
- The border is the outer area take away …
- This makes sense because …
Two things not to say
“Area is always bigger than perimeter.” They are different kinds of thing and cannot be compared.
“Just take the inner bit off.” Say “subtract the bracketed inner area”.
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.
- AreaExpand the area of a rectangle (x + 9) m by (x + 4) m.
x² + 13x + 36 square metres. - Earlier workA rectangle is 12 m by 5 m. Find its perimeter.
34 m.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- BorderA tablecloth x m by (x + 1) m has a hem 1 m wide all round. Find the hem's area in x.
4x + 6: (x + 2)(x + 3) − x(x + 1). - From lesson 4Simplify (x + 3)(x + 4) − (x + 2)(x + 5).
2.
3Challenge
Reasoning, generalizing and working backwards.
- General pathShow that a path w m wide round an x by x square has area 4wx + 4w².
(x + 2w)² − x² = 4wx + 4w². - Equal areasFor which x do a square of side (x + 4) and a rectangle (x + 2) by (x + 7) have equal areas?
x = 2: x² + 8x + 16 = x² + 9x + 14.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard and by touch; the figure is drawn to scale, and every number in it is also in the panel and the caption. “Show me the steps” gives the states in words.
No device? Squared paper: draw the pool and path for x = 2, 3, 4 and count the squares.