At a glance. One lesson of about 60 minutes. Students list whole-number rectangles for a fixed perimeter or a fixed area, find the best, explain why the square wins, and use two conditions to find sides.
Calculator: no calculator. Nothing here needs one. The products are small and the exploring is the point.
What students will learn
Know
- Rectangles with the same perimeter can have different areas.
- Rectangles with the same area can have different perimeters.
- Of the rectangles with a given perimeter, the square has the greatest area; of those with a given area, the square has the least perimeter.
Understand
- Perimeter and area measure different things, so one does not decide the other.
- The loss from the square is a d by d square when the sides are d from equal.
- Half the perimeter is length plus width, which turns a perimeter problem into pairs of numbers.
Do
- List all the whole-number rectangles for a perimeter or an area.
- Choose the pair that gives the greatest area or the least perimeter.
- Find sides from two conditions, perimeter and area.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding the sides of rectangles from a perimeter or an area (questions 2 to 4).
- Criterion B. Investigating patterns: the fixed-fence figure and an organized list of pairs. Ask students to record width, length and area in a table, spot the symmetry and the maximum, and say why the pattern is a rise and a fall (questions 1, 4 and 7).
- Criterion C. Communicating: an organized table and a sentence that says why the pair nearest a square is best (the worked-example table, and question 8, the exit ticket, with a written reason).
- Criterion D. Applying mathematics in real-life contexts: fence, pens and patches, where the best answer is a recommendation, not just a number (questions 2 to 7).
The one thing to do with the figure
Make students predict, then split the room. Before the slider moves, ask half of them to guess the area at width 6 and half at width 2. Compare, then reveal 36 and 20. The surprise is that the same fence can give almost double the ground.
In the second figure ask what is the same about all five rows (the area) and what is different (the edge). Then slide to the last row and let a student say what that rectangle looks like.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The fence roll and the prediction. First figure, with the three fence lengths. |
| 10 min | Key idea and the why. Students make three rectangles of perimeter 20 on squared paper and write the area of each, then compare tables. |
| 10 min | Second figure, then the worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose the longest side, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 8, on paper or on screen. |
The answers
| Warm-up | Perimeter 26 m, area 36 m²; 1, 2, 3, 6, 9, 18; 49. |
|---|---|
| Worked example | 7 m by 8 m, area 56 m². |
| Faded example | 5 cm by 6 cm, perimeter 22 cm. |
| 1. The best rectangle for a fixed fence | D. Areas 18, 24, 28 and 30 m². A, B and C are all valid pens that are further from a square. A student who picks A has taken the longest side as the biggest pen. |
| 2. Area from a perimeter and one side | 45 m². 28 ÷ 2 = 14, 14 − 9 = 5, 9 × 5 = 45. |
| 3. Square from a perimeter | 100 m². Side 40 ÷ 4 = 10. |
| 4. Greatest area, whole-number sides | 42 m². 13 = 6 + 7, 6 × 7 = 42. |
| 5. Least perimeter for a fixed area | 32 m. Factors 6 and 10, 2 × 16 = 32. |
| 6. Two conditions on a rectangle | 14 m. 6 + 14 = 20 and 6 × 14 = 84. |
| 7. How many different rectangles | 8. Widths 1 to 8 (lengths 16 to 9). |
| 8. Larger perimeter is not larger area | A. B assumes the longer perimeter must be the larger shape. If many choose B, return to fig 1 and slide to width 1, then width 6. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Picks a pen with a long side, as if longer meant bigger. | Have them compute all four areas before choosing, then ask what the winning pair has in common. |
| 19 or 126 in question 2 | Subtracts 9 from the whole perimeter, or multiplies by half the perimeter. | Ask what half of 28 stands for (length + width), and what the other side must be. |
| 10 or 20 or 1600 in question 3 | Stops at the side, takes half the perimeter as a side, or squares the perimeter. | Ask how many sides a square has and what 40 is measuring. |
| 49 or 36 or 169 in question 4 | Rounds the square up or down, or squares half the perimeter. | Ask for a rectangle whose perimeter is exactly 26 and a table of its pairs. |
| 16 or 34 in question 5 | Gives half the perimeter, or stops at the second-closest pair. | Make a table of all factor pairs and ask which two numbers are closest. |
| 6 or 20 or 10 in question 6 | Gives the short side, the sum, or the square's side. | Check each against both conditions: add and multiply. |
| 16 or 17 in question 7 | Counts each rectangle twice, or gives half the perimeter. | Ask whether 2 by 15 and 15 by 2 are the same rectangle, and where to stop. |
| B in question 8 | Treats a larger perimeter as a larger shape. | Draw both on squared paper and count squares. |
Questions worth asking
- Why does the area go down on both sides of the square? The sides are the same pair in the other order. Listen for symmetry: width 3 and width 9 are the same rectangle.
- What is the same about all the rectangles in the second figure? The area. The perimeter is what changes: the thinner the rectangle, the more edge.
- Is there ever a time to choose a long thin pen? Yes: a field beside a river or wall, or a corridor. The square is best for the fence, not always for the job.
- Does a square always exist with whole sides? No: 30 m of fence gives 7.5 by 7.5, not whole numbers. The nearest pair is 7 by 8.
Differentiation
A way in for anyone stuck
- Give a table with columns for width, length and area, and fill in the first row together.
- Provide squared paper and let students draw each rectangle before they compute its area.
- Use perimeter 12 first (1 by 5, 2 by 4, 3 by 3), then move on to the bigger perimeters.
Stretch
- Show that for a fence of 2s metres the area of a rectangle with width w is w(s − w), and that it falls short of the square’s (s ÷ 2)² by exactly (s ÷ 2 − w)². (Check with s = 12 and w = 4: 32 and 36 − 4.)
- A rectangle has area 100 m². Is a perimeter of 38 m possible? (No: the least is the square, 10 by 10, perimeter 40 m.)
Students learning in English as an additional language
“Fixed” and “enclose” are used precisely. “Fixed” means it does not change; say “the fence is fixed at 24 m” with a hand held still. “Enclose” means to have something all the way round.
| Word | Plain meaning | Example |
|---|---|---|
| fixed | not allowed to change | The fence is fixed at 24 m while the width changes |
| enclose | to be all the way round something | A fence encloses a pen |
| whole number | a number with no fraction or decimal part | 5 is a whole number; 5.5 is not |
| square | a rectangle with four equal sides | 6 m by 6 m |
| factor pair | two whole numbers that multiply to make a number | 4 and 9 are a factor pair of 36 |
| compare | to look at two things and say how they are alike and different | Compare the areas of two pens |
Sentence starters for explaining aloud or in writing:
- The fence is …, so length plus width is …
- The area is biggest when … because …
- The nearer to a square, the … the area and the … the perimeter.
Two things not to say
“The square is always the best shape.” It is the best of the rectangles for a given fence, which is a claim about rectangles with the same perimeter.
“Bigger perimeter means bigger area.” The exit ticket exists to break this.
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.
- PairsList all the whole-number rectangles with perimeter 12 m and give each area.
1 by 5 (5), 2 by 4 (8), 3 by 3 (9). The most is 9 m². - Fixed areaWhich whole-number rectangle with area 16 m² has the least perimeter?
4 by 4, perimeter 16 m.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA school garden group has 36 m of edging. What sides give the largest rectangular bed with whole numbers, and what is its area?
9 m by 9 m, area 81 m². - SpacingFrom Unit 7, lesson 1: a rectangle has area 54 m² and length 9 m. What is its perimeter?
Width 6 m, perimeter 2 × (9 + 6) = 30 m.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA rectangle has perimeter 48 m. Show that no rectangle with that perimeter has an area of 150 m².
The most is the 12 by 12 square, 144 m². 150 is more than that, so it is impossible. - ReasoningTwo rectangles have the same area. Must the one with the longer perimeter be the longer, thinner one? Explain.
Yes: for a fixed area, the nearer to a square the smaller the perimeter, so the longer perimeter belongs to the one further from a square.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Every figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper, scissors and a few sheets of card do the same job: cut, slide and fold.