In this activity, students deepen their understanding of the quadrilateral hierarchy. Students recognize quadrilaterals with specific attributes. Students consider the defining attributes of each type of quadrilateral as they decide whether or not certain shapes exist. For example, a square which is not a rectangle does not exist because a square has 4 right angles (and 4 equal sides). However, there are rectangles that are not squares because the 4 sides of a rectangle do not need to have the same length.

As students work on these problems, monitor for those who experiment and try to draw shapes with different attributes and for those who think about the defining attributes of each shape category. As needed, remind students of the convention for these materials that a trapezoid is a quadrilateral with at least one pair of parallel sides. 

• Invite previously selected students to share.
• “Clare says, ‘Some rhombuses are squares and some rectangles are squares.’ Do you agree with her?” (Yes, we saw with the toothpicks that a rhombus can be a square but it doesn't have to be. Rectangles are squares when the 4 sides are equal, but the 4 sides don’t need to be equal, so not all rectangles are squares.)
• Display or draw a diagram like the one below or use the diagram from a previous lesson and ask, “How does the diagram show the relationship between rhombuses and rectangles?” (It shows that squares are both rhombuses and rectangles.)

• “Are all squares rectangles? How does the diagram show this?” (Yes, a square has 4 right angles and the diagram shows that squares sit inside of rectangles.)

The purpose of this activity is for students to use their understanding of the hierarchy of quadrilaterals to determine if statements relating shape categories are sometimes, always, or never true. Students may draw examples of the shapes to help them answer the questions or they may think of defining attributes. The synthesis gives students an opportunity to have a discussion about these statements and apply what they have learned to make sense of the hierarchy as it is represented in a diagram. For example, as seen in the previous activity, squares are included inside rectangles on the diagram because all squares are rectangles.

• “Is it possible to have a rhombus that is also a square? How do you know?” (Yes, I drew one. Any square has 4 equal sides so it is also a rhombus.)
• Display the diagram from the Activity Synthesis of the previous activity.

• “How does the diagram show that a rhombus is sometimes a square?” (A part of the rhombus bubble includes squares.)
• “How does the diagram show how parallelograms are related to trapezoids?” (It shows that a parallelogram is always a trapezoid but there are trapezoids that are not parallelograms.)