The purpose of this activity is for students to notice that a square is a special kind of rhombus and that a rectangle is a special kind of parallelogram. Working with toothpicks, students construct quadrilaterals with the same side lengths but different angles. When the side lengths are all the same, the shapes are rhombuses since the defining property of rhombuses is having 4 equal sides. The only way to make a square with the toothpicks, however, is to make 4 right angles. Hence building quadrilaterals with 4 toothpicks helps students visualize why a square is always a rhombus but a rhombus is only sometimes a square. In the same way, they see that a rectangle is always a parallelogram but a parallelogram is only sometimes a rectangle.

• Ask previously selected students to share their thinking.
• “Which of the shapes you made are parallelograms? How do you know?” (They are all parallelograms. The opposite sides are parallel in all of these shapes.)
• “Which of the shapes you built have 4 equal sides?” (The first two, the ones using 4 toothpicks.)
• Display: rhombus

• “A quadrilateral with 4 equal sides is a rhombus.”

The purpose of this activity is for students to determine if quadrilaterals are squares, rhombuses, rectangles, or parallelograms. Then they begin to outline the relationships between these different types of quadrilaterals, leading to the overall hierarchy of quadrilateral types which students investigate more fully in the next lesson.

When students draw quadrilaterals belonging or not belonging to different categories they reason abstractly and quantitatively (MP2), using the definitions of the shapes to inform their drawings. 

This activity uses MLR3 Critique, Correct, Clarify. Advances: reading, writing, representing

MLR3 Critique, Correct, Clarify

• Display the response for students to consider:
• “Mai says: All squares are rhombuses. If a shape is a square, it is also a rhombus. Rhombuses are not squares.”
• Read the explanation aloud.
• “What parts of this response are unclear, incorrect, or incomplete?” (He says a square is also a rhombus, but that rhombuses are not squares. I'm not sure what he means by rhombuses are not squares.)
• 1 minute: quiet think time
• 2 minutes: partner discussion
• Invite 2–3 groups share what they discussed. Record for all to see.
• “With your partner, work together to write a revised explanation.”
• Display and review the following criteria:
  • Use the words “all” and “some.”
  • Include a labeled diagram or other examples.
• 3–5 minutes: partner work time
• Select 1–2 groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each group shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
• “What is the same and different about the explanations?” (They all explain why a rhombus does not have to be a square.)
• Display or draw a diagram like this:

  

• “How can we use this diagram to help us revise Mai’s thinking?” (The diagram shows that all squares are rhombuses but not all rhombuses are squares.)
• “This diagram shows a hierarchy, or system for ordering shapes. It shows that squares are a type of rhombus and have the same attributes. It also shows that there are some rhombuses that are not squares.“
• “Diagrams like this can help make sense of how shapes share attributes and what makes some shapes unique. It helps us make sense of categories and subcategories of shapes.“