Subject:
Geometry
Material Type:
Lesson Plan
Level:
Middle School
Grade:
7
Provider:
Pearson
Tags:
  • 7th Grade Mathematics
  • Angles
  • Parallelograms
  • Rectangles
  • Rhombus
  • License:
    Creative Commons Attribution Non-Commercial
    Language:
    English
    Media Formats:
    Interactive, Text/HTML

    Education Standards

    Diagonals Of A Rhombus

    Diagonals Of A Rhombus

    Overview

    Students learn how the diagonals of a rhombus are related. They use interactive sketches to learn about the properties of the angles and diagonals of squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.

    Key Concepts

    • The sum of the measures of the angles of all quadrilaterals is 360°.
    • The alternate angles (nonadjacent angles) of rhombuses and parallelograms have the same measure.
    • The measure of the angles of rectangles and squares is 90°.
    • The consecutive angles of parallelograms and rhombuses are supplementary. This applies to squares and rectangles as well.
    • The diagonals of a parallelogram bisect each other.
    • The diagonals of a rectangle are congruent and bisect each other.
    • The diagonals of a rhombus bisect each other and are perpendicular.

    Goals and Learning Objectives

    • Measure the angles formed by the intersection of the diagonals of a rhombus.
    • Explore the relationships of the angles of different squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.
    • Explore the relationships of the diagonals of different squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.

    Generalize From Repetition

    Mathematical Practices in Action

    Mathematical Practice 8: Look for and express regularity in repeated reasoning.

    Have students watch the video and listen to the dialogue between Karen and Sophie as they work together to find quadrilaterals with diagonals that are perpendicular. This video shows students engaged in Mathematical Practice 8: Look for and express regularity in repeated reasoning.

    After the students have watched the video, have them talk about how Karen and Sophie approached the problem. What might they have done differently? Why did they start by listing all the types of quadrilaterals they might explore?

    In the discussion, elicit that it is important in mathematics to notice patterns, things that regularly repeat. This can help you come up with rules or features that you can explore across figures. When you find a characteristic in one figure, ask yourself whether it applies to all figures of that type or only a subset of the figures.

    Students could come up with conclusions like:

    • All rhombuses are parallelograms and the properties of parallelograms apply to rhombuses.
    • The diagonals of a rhombus are perpendicular.

    Teacher Note: Some students may notice there are special cases in which rectangles, parallelograms, and trapezoids can have perpendicular diagonals. Acknowledge that this observation is correct, but make sure they understand which shapes always have perpendicular diagonals.

    ELL: When showing the video, make sure that ELLs can follow the explanations. “Chunk” the video by pausing at key times to allow ELLs time to process the information. Show the video again if needed. Ask questions to check for understanding before moving on to the discussion. If you notice that ELLs do not understand the important parts of the video, show it one more time.

    Opening

    Generalize From Repetition

    Watch the video where Sophie and Karen generalize about the diagonals of a quadrilateral.

    Discuss the following with your classmates.

    • How did the students decide if the diagonals are perpendicular? Could they have used a different method?
    • How did Karen summarize what they discovered?

    VIDEO: Mathematical Practice 8

    Math Mission

    Lesson Guide

    Discuss the Math Mission. Students explore angle relationships and diagonal relationships in quadrilaterals.

    Opening

    Explore angle relationships and diagonal relationships in quadrilaterals.

    Explore Angles in Quadrilaterals

    Lesson Guide

    For the Quadrilaterals Sketch interactive, students may work individually or with a partner. Make sure that students understand how to manipulate the figures.

    SWD: Students with visual-spatial challenges may struggle to independently find patterns and to observe similarities between parallelograms and rhombuses. Model how to manipulate the parallelogram to make a rhombus.

    Interventions

    Student has difficulty getting started.

    • What are you trying to do?
    • What do you think is true about angle measures in a parallelogram?
    • What angle measures do you see?

    Student has a solution.

    • Explain your conclusions.
    • Can you make a different quadrilateral with the same characteristics?

    [common error] Student sees different parallelogram types as different (i.e., a rectangle is not a parallelogram).

    • What figures are shown in the sketch about angles?

    Possible Answers

    • The sum of the angles is always 360°. The sum of the opposite angles varies. The sum of the adjacent angles is always 180°.
    • The sum of the measures of the angles of any quadrilateral is always 360°. The sum of the opposite angles will vary based on the figure; the alternate angles of a rectangle and a square always measure 90°. The adjacent angles in a parallelogram, rhombus, square, and rectangle are supplementary.

    Work Time

    Explore Angles in Quadrilaterals

    Use the Quadrilaterals Sketch interactive to explore the angles in squares, rectangles, rhombuses, parallelograms, and other free form quadrilaterals.

    • For each type of figure, find the sum of the angles, the sum of opposite angles, and the sum of adjacent angles.
    • What can you conclude?

    INTERACTIVE: Quadrilaterals Sketch

    Hint:

    How are the angles in each figure related? What do these relationships tell you about the figure?

    Make a Figure

    Mathematical Practices

    Mathematical Practice 8: Look for and express regularity in reasoning.

    • Most students will be able to generalize about the angles of quadrilaterals and the diagonals of quadrilaterals—particularly after using the sketches for all the figures.

    Interventions

    Student has difficulty getting started.

    • What are the lengths of the parts of the diagonals?
    • What is true when the diagonals are perpendicular?

    Student doesn't see that each interior angle of the quadrilateral is shown as two angles within the interactive.

    • What is the measure for angle A ?
    • What is the measure for angle B ?

    [common error] Student sees different parallelogram types as different (i.e., a rectangle is not a parallelogram).

    • Can you make each of the figures in the sketch about diagonals have perpendicular diagonals? Congruent diagonals? Perpendicular and congruent diagonals?

    Possible Answers

    • Rectangles and squares always have two adjacent angles that have the same measure. Trapezoids may have two consecutive angles with the same measure.
    • Squares always have diagonals that are congruent and perpendicular. Kites may have perpendicular and congruent diagonals.
    • Parallelograms, rhombuses, rectangles, kites, and squares have opposite angles that have the same measure. (Note: All kites and parallelograms have at least one pair of congruent alternate angles; therefore, rhombuses, rectangles, and squares must as well.)

    Work Time

    Make a Figure

    Use the Quadrilaterals Sketch interactive to manipulate figures with the following characteristics. If you were able to make a figure, explain how you did it.

    • A figure with two adjacent angles that are equal
    • A figure with congruent and perpendicular diagonals
    • A figure with opposite angles that are equal

    INTERACTIVE: Quadrilaterals Sketch

    Prepare a Presentation

    Preparing for Ways of Thinking

    Look for these types of responses to share during the Ways of Thinking discussion:

    • Students who manipulate the sketch in a unique way, such as making a “flat” parallelogram, or a concave quadrilateral
    • Students who draw the correct conclusions about quadrilaterals

    Challenge Problem

    Answer

    • Yes, a rectangle can have diagonals that are perpendicular to each other if the rectangle is a square.

    Work Time

    Prepare a Presentation

    Based on your explorations, write conjectures about the angles and the diagonals in quadrilaterals. Write as many conjectures as you can.

     

    Challenge Problem

    • Is it possible for a rectangle to have diagonals that are perpendicular to each other?

    Make Connections

    Lesson Guide

    Have students share the different things they did with each interactive sketch or anything they thought was unusual for the sketch (such as a concave quadrilateral).

    ELL: Ensure students use precise mathematical language. Allow ELLs to use diagrams along with their primary language. Provide students with a safe learning environment where they can attempt new mathematical concepts.

    Mathematics

    Lead a discussion, having students share their thinking. Ask questions such as the following:

    • When you added the angle measures, did they come out to exactly 360°? Why?
    • Describe the properties of parallelograms.
    • Describe the properties of rectangles.
    • Describe the properties of rhombuses.
    • Describe the properties of squares.
    • Describe the properties of quadrilaterals.
    • Can you define a parallelogram, rhombus, rectangle, and square by using just the diagonals?
    • Can you define a parallelogram, rhombus, rectangle, and square using just angles? (Answer: No, more information is needed; for example, a square and a rectangle have four angles that measure 90°.)
    • If you look at a quadrilateral, what information would help you define it further?

    Performance Task

    Ways of Thinking: Make Connections

    • Take notes about your classmates' conjectures concerning the angle relationships and diagonal relationships in quadrilaterals.

    Hint:

    As your classmates present, ask questions such as:

    • When you added the angle measures, when did the sum come out to exactly 360°? Explain.
    • What characteristics are true for parallelograms?
    • What is true for all quadrilaterals?
    • Can you define a parallelogram, rhombus, rectangle, and square just by the relationships between the diagonals?
    • Can you define a parallelogram, rhombus, rectangle, and square just by the angles?
    • If you wanted to define a quadrilateral, what information would you use beyond saying that it has four sides?

    Angles and Diagonals in Quadrilaterals

    A Possible Summary

    The sum of the measures of the angles of all quadrilaterals is 360°. The opposite angles of a rhombus and a parallelogram have the same measure. The measure of the angles of rectangles and squares is 90°. The adjacent angles of a parallelogram and a rhombus are supplementary. This applies to a square and a rectangle as well. The diagonals of a parallelogram bisect each other. The diagonals of a rectangle are congruent and bisect each other. The diagonals of a rhombus bisect each other and are perpendicular.

    ELL: Make sure you write what you are saying for the summary on an anchor chart so students can follow along. It will be helpful to hang this chart in a prominent location in your room to support students when they are working with properties of quadrilaterals.

    Formative Assessment

    Summary of the Math: Angles and Diagonals in Quadrilaterals

    Write a summary about the angles and diagonals in quadrilaterals.

    Hint:

    Check your summary:

    • Do you describe what you know about the angles in a quadrilateral?
    • Do you describe what you know about the diagonals in a quadrilateral?

    Reflect on Your Work

    Lesson Guide

    Have each student write a brief reflection before the end of class. Review the reflections to find out if any students are having difficulty understanding the properties of angles and diagonals of quadrilaterals.

    Work Time

    Reflection

    Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

    Something I wonder about quadrilaterals is …