Organize Shape Families in a Hierarchy
Use the category definitions to choose a true inclusion statement.
About 20 minutes, plus any extended practice
Supplied materials
- Paper and a pencil; drawing and spoken explanations are welcome. An adult may read directions and options aloud.
- Supplied worked-case record: Every square is a rectangle because it has four right angles.
- All quantities and shape descriptions are supplied in the worked example and questions; no purchase or personal information is needed.
Learn the idea
A shape subcategory inherits all the attributes of its broader category. Squares are rectangles, rectangles are parallelograms, and parallelograms are quadrilaterals. Squares also lie in the rhombus branch because all sides are equal. A hierarchy helps explain inclusion without falsely reversing every statement.
Use the category definitions to choose a true inclusion statement. Explain which broader attributes the narrower category inherits. Test a reversed statement with a counterexample, such as a nonsquare rectangle, rather than assuming all inclusions work both ways.
Check the direction of the inclusion claim. A true narrower-to-broader inclusion does not make its reverse true. Draw broader category circles containing narrower family labels, with square in both qualifying branches.
- Draw broader category circles containing narrower family labels, with square in both qualifying branches.
- Follow the worked example. Point to each number or shape feature and say what it tells you.
- Use the category definitions to choose a true inclusion statement.
- Try the question and then the three practice questions. Check your answers and explain any change you make.
See the method in action
Every square is a rectangle because it has four right angles. Every rectangle is not a square: a 5-by-3 rectangle lacks four equal sides. The hierarchy goes from a stronger requirement to a weaker one; the reverse direction needs extra conditions. Check the direction of the inclusion claim. A true narrower-to-broader inclusion does not make its reverse true. Draw broader category circles containing narrower family labels, with square in both qualifying branches.
Check your reasoning
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Helpful tips
- Check the direction of the inclusion claim.
- A true narrower-to-broader inclusion does not make its reverse true.
See the answer and explanation
Every square is a rectangle
Squares satisfy all rectangle conditions.
Take it further
Give a counterexample to every parallelogram is a rectangle. Expected: a non-right-angle parallelogram.
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More practice: try new cases
Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.
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Check the subject sources
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