Rabbit Pen Design: Perimeter, Area, Right Angles, and Shape Names
The fourth grade has 24 meters of fencing to enclose a rectangular rabbit pen in the school garden, and all sides must be whole meters. Three groups stake out plans: 10 m by 2 m, 8 m by 4 m, and 6 m by 6 m, each arguing that its pen gives the rabbits the most room. The custodian will approve only a pen with an area of at least 30 square meters. Meanwhile, a corner-checking group reads one corner as 60° on a protractor, and a shape group claims its four-equal-sided figure is a square even though the corners are not right angles. The teacher has until 1:30 to settle all three disputes, choose one plan, and explain the decisions so the groups can revise their posters before the custodian arrives.
1. Which pen plan should the teacher approve? Compare the three plans using both perimeter and area, and address the custodian's 30 square meter rule.
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Every plan spends all 24 meters of fencing: 10 + 2 + 10 + 2 = 24, 8 + 4 + 8 + 4 = 24, and 6 + 6 + 6 + 6 = 24. So perimeter cannot decide the argument - only area can. Area: 10 × 2 = 20 m², 8 × 4 = 32 m², and 6 × 6 = 36 m². This shows that equal perimeters do not give equal areas; the closer a rectangle comes to a square, the more area it holds. The custodian's rule eliminates 10 × 2 (20 m²) but not the other two. The teacher should approve 6 × 6 for the most room and explain that 8 × 4 is legal but gives the rabbits 4 fewer square meters.
2. A group's protractor reading shows a corner of 60°. Explain how to check whether the tool was used correctly and what the reading means for the pen.
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Set the protractor up before trusting any number: the center mark goes exactly on the corner, the baseline lies along one fence side, and the reading is taken on the scale that starts at 0 on that side. Most protractors print two scales, so a ray reading 60 on one scale reads 120 on the other - both describe the same physical angle, not two angles. A true right angle reads 90 on both scales, so a 60/120 pair cannot be a right angle: the corner really is 60° and one stake must move until the reading is 90. If the readings are not supplementary (60 and 120), the setup was wrong and the measurement must be redone.
3. A group says its figure is a square because all four sides are equal, even though the corners are not right angles. Correct the claim and give the teacher a quick test students can apply.
Read the worked answer
Four equal sides alone make a rhombus, not a square. A square needs four equal sides and four right angles, so every square is also a rectangle and a rhombus, but the reverse is not true - a rhombus with slanted corners is neither a square nor a rectangle. The quick test is to check the corners with a right-angle tester, such as an index card corner or the protractor, and count: four right angles plus four equal sides gives a square; four equal sides with no right angles gives a rhombus. The group should relabel its figure as a rhombus and note that a square would also fit the rhombus description, which is the hierarchy idea the poster is missing.