Relationships Between Area and Perimeter
Warm-up
Two rectangles: 2x10 (perimeter 24, area 20) and 4x6 (perimeter 20, area 24). Neither has both the larger area AND the smaller perimeter. The shape that minimized the perimeter also maximized the area. Is this always true? Today we investigate.
Explore
Systematic investigation: each group fixes one measurement (perimeter OR area) and varies the rectangle dimensions. Record all results in a table. Graph: one variable on each axis. Share findings: which shape gives the most area for a fixed perimeter? Which gives the least perimeter for a fixed area? (Both: the square.)
Consolidate
Practice
Students complete one fixed-perimeter and one fixed-area investigation, both recorded in tables and graphed. Solve 2 real garden-design problems. Exit ticket: for perimeter 32 cm, what rectangle dimensions give the greatest area?
Exit ticket
Students complete one fixed-perimeter and one fixed-area investigation, both recorded in tables and graphed. Solve 2 real garden-design problems. Exit ticket: for perimeter 32 cm, what rectangle dimensions give the greatest area?
1x9: area 9. 2x8: area 16. 3x7: area 21. 4x6: area 24. 5x5: area 25. Pattern: as the rectangle approaches a square, area increases. Maximum area at the square (5x5).
1x24: P=50. 2x12: P=28. 3x8: P=22. 4x6: P=20. Closest to a square wins. 4x6 has smallest perimeter among these.