Chapter 7 · Consolidation and enhancement
Lesson 7.5 · Mathematics plaza — who encloses a larger area?
Hua's Classroom for Shanghai Maths — One Lesson One Exercise (English edition, pages 139–141).
Lesson Alignment
How this Shanghai Maths lesson connects to learning goals, Common Core, and Eureka Math.
Learning Objectives
Students will be able to:
- 01Explore how side length changes affect area. (Question 1)
- 02Conclude when P is fixed, area is greatest when closer to a square. (Question 2)
- 03Count fencing ways between two walls. (Question 3)
Common Core Standards
| Code | Standard (focus for this lesson) |
|---|---|
| 3.MD.D.8 | Perimeter. |
| 3.MD.C.7 | Area. |
Eureka Math Correspondence
| Shanghai Maths | Focus | Eureka Math |
|---|---|---|
| Isoperimetric | Max area | G3 M7 plaza |
| Fence | Wall as one side | G3 M7 |
Core Concepts
Explore which rectangle has greatest area for a fixed perimeter.
Side & Area
Double the side of a square → area becomes 4 times.
Maximum Area
For a fixed perimeter, the shape closest to a square has the largest area.
Fence Against a Wall
Use the wall as one side; enumerate whole-metre length/width pairs.
Key Terms & Definitions (Reference)
Quick glossary for this lesson.
| Term | Definition |
|---|---|
| Fixed perimeter | Total edge length stays the same. |
| Maximum area | Largest interior for that perimeter. |
Key Answers
Quick reference for this lesson’s exercise answers.
Key answers
- 4 cm²; 9 cm²; area becomes $n^2$ times when side becomes $n$ times.
- $P=20$: $9\times1$, $8\times2$, $7\times3$, $6\times4$, $5\times5$; areas 9, 16, 21, 24, 25. Same perimeters; different areas; max when $5\times5$.
- MCQ: B; C; D.
- Against wall: 8 m by 4 m; max area 32 m².
- Corner walls: 25 ways; max area 169 m².
- Original field 288 m².