Lesson 7

Rectangles and Squares

Warm-up: What Do You Know About This Shape? (10 minutes)

Narrative

The purpose of this What Do You Know About...? is for students to share what they know about a square. Students revisit this same question in the lesson synthesis.

Launch

  • Display the image.
  • “What do you know about this shape?”
  • 1 minute: quiet think time

Activity

  • Record responses.

Student Facing

What do you know about this shape?

square

Student Response

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Activity Synthesis

  • “What is this shape called?” (a square)
  • “How do you know it is a square?” (It has 4 equal sides and 4 angles that are 90 degrees.)
  • “We are going to come back to this question in the lesson synthesis.”

Activity 1: Quadrilateral Clues (15 minutes)

Narrative

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.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____ , so I matched . . .” Encourage students to challenge each other when they disagree.
Advances: Conversing, Representing

Required Materials

Materials to Copy

  • Quadrilateral Clues

Required Preparation

  • Create a set of cards from the blackline master for each group of 2.
  • Gather diagram from a previous lesson.

Launch

  • Groups of 2
  • Give each group of 2 a set of cards.

Activity

  • 2 minutes: independent think time
  • 5 minutes: partner work time
  • Monitor for statements about properties of shapes and conjectures such as:
    • a rhombus has 4 equal sides
    • squares are rhombuses
    • a trapezoid can be a rectangle because it has at least one pair of parallel sides
    • a rectangle is a parallelogram because it has two pairs of parallel sides
    • it is impossible to find a square that isn’t a rectangle

Student Facing

Spread out your shape cards so you and your partner can see all of them.

Work together to find a shape that fits each clue. If you don’t think it is possible to find that shape, explain why. You can only use each shape one time.

  1. Find a quadrilateral that is not a parallelogram.
  2. Find a rhombus that is also a square.
  3. Find a rhombus that is not a square.
  4. Find a trapezoid that is not a rectangle.
  5. Find a rectangle that is not a square.
  6. Find a parallelogram that is not a rectangle.
  7. Find a square that is not a rectangle.

Student Response

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Activity Synthesis

  • 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.)
venn diagram. squares, rectangles, rhombuses all inside of parallelograms. squares inside of rhombuses. parallelograms insides of trapezoids. trapezoids inside of quadrilaterals.
  • “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.)

Activity 2: Always, Sometimes, Never (20 minutes)

Narrative

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.

Action and Expression: Develop Expression and Communication. Provide access to a variety of tools: colored pencils, markers, etc. Invite students to use the tools in order to help organize the anchor chart to aid in completing the task accurately.
Supports accessibility for: Conceptual Processing, Attention

Launch

  • Groups of 2

Activity

  • 5 minutes: independent work time
  • 5 minutes: partner work time
  • Monitor for students who:
    • make drawings of shapes to help answer each question
    • think about defining properties of the different shapes

Student Facing

Write always, sometimes, or never in each blank to make the statements true.

For each statement that is completed with “sometimes,” draw a figure for which the statement is true and another figure for which the statement is not true.

  1. A rhombus is ________________________ a square.
  2. A square is ________________________ a rhombus.
  3. A triangle is ________________________ a quadrilateral.
  4. A square is ________________________ a rectangle.
  5. A rectangle is ________________________ a parallelogram.
  6. A parallelogram is ________________________ a rhombus.
  7. A trapezoid is ________________________ a parallelogram.

Student Response

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Activity Synthesis

  • “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 synthesis of the previous activity.
venn diagram. squares, rectangles, rhombuses all inside of parallelograms. squares inside of rhombuses. parallelograms insides of trapezoids. trapezoids inside of quadrilaterals.
  • “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.)

Lesson Synthesis

Lesson Synthesis

“Today we looked at relationships between different types of quadrilaterals including trapezoids, parallelograms, rectangles, rhombuses, and squares.”

Display the image from the warm-up:

square

“During the warm-up we said the shape is a square because . . . (include statements students made earlier in the warm up). What do you now know about this shape?” (This is a square, but it also is a rectangle. It can also be called a rhombus or parallelogram.)

“Why is a square also a rhombus?” (All of its sides are the same length.)

“Why is a square also a rectangle?” (All of the angles are 90 degrees.)

“If a shape is a rectangle, is it also a square?” (Sometimes. It depends on the lengths of the sides of the rectangle. If all four sides are equal then it is a square, but if all four sides are not equal then it is not a square.)

“If a shape is a rectangle and a rhombus, is it also a square?” (Yes, because it has 4 right angles and 4 equal sides.)

Cool-down: Quadrilaterals in the Venn Diagram (5 minutes)

Cool-Down

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