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3.6: Quadrilaterals

Created by: CK-12

Interior Angles

I. Section Objectives

  • Identify the interior angles of convex polygons.
  • Find the sums of interior angles in convex polygons.
  • Identify the special properties of interior angles in convex quadrilaterals.

II. Cross- curricular-Mandalas

  • Use the following image to discuss interior angles of quadrilaterals.
  • This is Figure 06.01.01
  • www.isibrno.cz/~gott/mandala/sriclr2.gif
  • This is an image of a mandala that is composed of triangles that can also be interpreted to be quadrilaterals.
  • You can use this image to discuss the measure of the interior angles of the quadrilateral with students.
  • Show them how two triangles can be combined together to become a quadrilateral.
  • Then remind students that the interior angles of a triangle add up to be 180^\circ according to the Triangle Sum Theorem.
  • Then ask the students to look at how many degrees are in a quadrilateral based on the fact that it is made up of two triangles.
  • The students will conclude that it is equal to 360^\circ.

III. Technology Integration

  • Have students complete some research on mandalas.
  • Where do they come from?
  • When were they first used?
  • What is the purpose of a mandala?
  • Have students keep a record of the websites that they visit.
  • Allow time for students to share their findings.

IV. Notes on Assessment

  • Assessment is completed through student discussion.
  • Listen to the students as they share their thoughts and ideas.
  • Be sure that they understand how the interior angles of a quadrilateral are equal to 360^\circ.

Exterior Angles

I. Section Objectives

  • Identify the exterior angles of convex polygons.
  • Find the sums of exterior angles in convex polygons.

II. Cross- curricular- Mandalas

  • Use the information that you developed in the last lesson to work on this project.
  • Use the image of the mandala on this website.
  • This is Figure 06.02.01
  • www.isibrno.cz/~gott/mandala/sriclr2.gif
  • Now tell students that today they are going to be working to design their own mandalas.
  • The students need to include triangles and quadrilaterals in their design.
  • Then they also need to identify the interior and exterior angles of the quadrilaterals.
  • Be sure to tell students to use color and creativity in their designs.
  • Then have the students write a paragraph describing the mandala and explaining the connections between the interior angles and the exterior angles and the measurements of 180^\circ and 360^\circ.
  • Allow time for students to share their work when finished.
  • Display the mandalas in the classroom.

III. Technology Integration

  • Students can go to the following website and investigate the interior and exterior angles of quadrilaterals.
  • This site is fun, interactive and colorful.
  • www.slideshare.net/guest4210b1/quadrilaterals
  • Students can participate in this as an independent study activity.

IV. Notes on Assessment

  • Assess student mandalas.
  • Did the students incorporate the triangles and quadrilaterals?
  • Are the interior angles labeled or identified in some way?
  • Are the exterior angles labeled or identified in some way?
  • Is the writing piece clearly written?
  • Does it explain the angle measures?
  • Provide students with feedback on their work.

Classifying Quadrilaterals

I. Section Objectives

  • Identify and classify a parallelogram.
  • Identify and classify a rhombus.
  • Identify and classify a rectangle.
  • Identify and classify a square.
  • Identify and classify a kite.
  • Identify and classify a trapezoid.
  • Identify and classify an isosceles trapezoid.
  • Collect the classifications in a Venn diagram.
  • Identify how to classify shapes on a coordinate grid.

II. Cross- curricular-Art

  • Use the following website and image of a geometric pattern for the following activity.
  • This is Figure06.03.01
  • www.tfaoi.com/cm/2cm/2cm511.jpg
  • Ask the students to work in teams and identify all of the different quadrilaterals in the drawing.
  • Students need to identify the quadrilateral and then write the characteristics of that quadrilateral.
  • Next, have the students work to complete their own quadrilateral art piece.
  • All of the different types of quadrilaterals should be included in the piece.
  • Also, the art design should only consist of quadrilaterals.
  • Design and color need to be included in this work.
  • Once they have done the design, have the students write a description of their work identifying each quadrilateral in the design and its characteristics.

III. Technology Integration

  • Students can go to the following website to see a video on how to identify and classify quadrilaterals.
  • www.onlinemathlearning.com/quadrilaterals.html
  • Then the students can use this information in the first part of this lesson, or this can be used to help students to solidify the information that they have already learned.

IV. Notes on Assessment

  • Assess student designs.
  • Have the students created a design composed only of quadrilaterals?
  • Is it creative?
  • Does the writing piece identify and describe each quadrilateral according to its characteristics?
  • Provide students with feedback on their work.

Using Parallelograms

I. Section Objectives

  • Describe the relationships between opposite sides in a parallelogram.
  • Describe the relationship between opposite angles in a parallelogram.
  • Describe the relationship between consecutive angles in a parallelogram.
  • Describe the relationship between the two diagonals in a parallelogram.

II. Cross- curricular-Architecture

  • Select several of the images on the following website.
  • Print these images and distribute them to the students or have the students use computers to look at the images.
  • www.trendir.com/house-design/
  • All of the homes here are constructed using many different parallelograms.
  • There are large parallelograms, small ones, all kinds of different ones.
  • Ask the students to select one of the houses and work with it to identify the elements of the parallelograms in the designs.
  • Students need to be looking for the relationship between the opposite sides of a parallelogram.
  • The consecutive angles and the diagonals- how can the relationship be determined?
  • Ask students to take notes on their home and then to share their findings in small groups.

III. Technology Integration

  • One possible integration is to have the students explore this website and the house designs further.
  • www.trendir.com/house-design/
  • Then they can actually work to design their own home using parallelograms.
  • Students can draw this out on grid paper or on plain paper.
  • The key element or focus of the home must be the parallelogram.
  • Have students share their work with their peers.

IV. Notes on Assessment

  • Assessment is completed through observation.
  • Walk around and participate in student groups as they look at the different elements of the house that they have been given.
  • Examine each house design.
  • Is the key element a parallelogram?
  • Provide students with feedback on their work.

Proving Quadrilaterals are Parallelograms

I. Section Objectives

  • Prove a quadrilateral is a parallelogram given congruent opposite sides.
  • Prove a quadrilateral is a parallelogram given congruent opposite angles.
  • Prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
  • Prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.

II. Cross- curricular- Drama

  • Assign students the task of creating a skit to prove that a quadrilateral is a parallelogram.
  • Begin this lesson by reviewing the different characteristics that make a quadrilateral a parallelogram.
  • The synopsis of the skit is “Mr./Ms. Quadrilateral needs to prove that he/she is a parallelogram. He/she has selected Geo Geometry to prove the case.”
  • Then let the students go to work.
  • They can use props, scenery (simple) and costumes.
  • Allow time for them to write a skit and rehearse it.
  • Then the students need to be given time to perform their skit.

III. Technology Integration

  • Students can explore the properties of a parallelograms with the following website.
  • www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php
  • On this website, there is a place where it lists the criteria for proving that a quadrilateral is a parallelogram.
  • Then it also has an interactive part where you can click and drag the vertices of the parallelogram to alter the side lengths, angle measures, etc.
  • The numbers change instantly on the screen as you move the vertices around.
  • A great interactive site to work with.

IV. Notes on Assessment

  • Assess each skit.
  • Did the students get the job done?
  • Did they prove that the quadrilateral is a parallelogram?
  • If so, what did they do well?
  • If not, what was missing?
  • Provide students with feedback on their work.

Rhombi, Rectangles, and Squares

I. Section Objectives

  • Identify the relationship between the diagonals in a rectangle.
  • Identify the relationship between the diagonals in a rhombus.
  • Identify the relationship between the diagonals and opposite angles in a rhombus.
  • Identify and explain biconditional statements.

II. Cross- curricular-Design Collage

  • In this activity, have the students work in groups of three.
  • Each student is going to select one of the three quadrilaterals.
  • Then they are going to create a collage about the quadrilateral that they have chosen.
  • Included in the collage should be pictures of their shape out in the world.
  • Have each group pick a theme for their collage.
  • For example, sports or nature or furniture.
  • Students can hunt through magazines for these pictures.
  • On one of the pictures, they need to draw in the angles and diagonals of the figure.
  • This is a way to demonstrate the characteristics of that figure.
  • When finished, each group should have a complete description of all three types of quadrilaterals.

III. Technology Integration

  • Students can explore the properties of rectangles, squares and rhombi on the following website.
  • www.mathsisfun.com/quadrilaterals.html
  • This is a basic site, but since these are basic figures, the information should be review.
  • There is also a place where they have interactive quadrilaterals and students can manipulate the size and configuration of the quadrilateral.

IV. Notes on Assessment

  • Assess the continuity of each design trio.
  • Is there a consistent theme?
  • Does each design show the characteristics of each quadrilateral?
  • Provide students with feedback on their work.

Trapezoids

I. Section Objectives

  • Understand and prove that the base angles of isosceles trapezoids are congruent.
  • Understand and prove that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
  • Understand and prove that the diagonals in an isosceles trapezoid are congruent.
  • Understand and prove that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
  • Identify the median of a trapezoid and use its properties.

II. Cross- curricular-Buildings/Construction

  • Use the following image of the Flatiron Building in NYC from Wikipedia.
  • This is Figure 06.07.01
  • www.en.wikipedia.org/wiki/File:Flatiron_crop_20040522_114306_1.jpg
  • Print the image so that the students can look at the image in their seats.
  • Use this as a feature of discussion.
  • Show the students how the sides of the building are the trapezoids.
  • This building is formed by three trapezoids and then the top is a triangle.
  • Are the students able to identify the type of triangle?
  • What would the shape of the building change to if the sides were composed of four trapezoids instead of three?
  • You can either have this part as a discussion or as an exploration.
  • If you want students to explore this, be sure that they have cardboard, scissors, rulers, tape and a copy of the image to work with.
  • Have the student begin by building a rough model of the Flatiron Building as it is now.
  • Then they can look at altering it by adding another trapezoid.
  • This gives students a discussion point about the building.
  • Allow time to discuss and share findings and conclusions following the exploration.

III. Technology Integration

  • Ask students to complete a websearch.
  • Students are going to google “trapezoid images” for this websearch.
  • There will be tons of different images that use trapezoids.
  • Ask the students to select five different ones to share.
  • If possible, have them print these images, if not they should make notes about the images and where the trapezoid is in the image.
  • Also be sure that students write down any websites where the images are located.
  • Allow time for students to share their work.

IV. Notes on Assessment

  • Assess student understanding of trapezoids through class discussions and sharing.
  • Provide students with feedback on their work.

Kites

I. Section Objectives

  • Identify the relationship between diagonals in kites.
  • Identify the relationship between opposite angles in kites.

II. Cross- curricular-Kites

  • Have students work to design their own kite.
  • They can look at the different kinds of kites that are possible by looking at the Wikipedia website.
  • www.en.wikipedia.org/wiki/Kite
  • In the kite, the students should show the diagonals and the opposite angles of the kite.
  • Students should be very creative with their kite and have it say something about them.
  • Use this in connection with the technology integration section.

III. Technology Integration

  • Have students do some research on kites.
  • They can do a web search on kites and see many different designs.
  • Students should be able to report on the origin of kites.
  • Three cultural elements of kites
  • Three countries known for kites
  • How and why kites fly
  • How kites are used in science and technology
  • Ask students to write a report on this topic.

IV. Notes on Assessment

  • Assess student reports and kite designs.
  • Is the design creative?
  • Does it say something about the student?
  • Is the report complete?
  • Are all of the components in it?
  • Provide students with feedback/correction as needed.

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