Math Background

Lesson: Triangles and Quadrilaterals
Introducing the Concept

Having students construct and measure figures will help them internalize their understanding of the concepts.

Materials: 5 sheets of easel paper, marker, large chalkboard protractor and meterstick or transparent protractor and ruler to use with an overhead projector; ruler, protractor, scissors, pencil, and several sheets of paper for each student

Preparation: Draw each of the following figures on one sheet of easel paper, leaving room beneath the drawing to list the properties of the figure: parallelogram, rectangle, rhombus, trapezoid, square.

Prerequisite Skills and Background: Students should know the various types of angles (acute, obtuse, right, complementary, supplementary, and so on), recognize common quadrilaterals (rectangle, square, parallelogram, and so on), and be able to use a protractor and ruler.

  • Ask: Who can tell me what a triangle is? (a polygon with three sides) Good. Now I'd like everyone to take a ruler and draw a triangle on the paper.
    Draw a triangle on the board.
  • Say: Now I'd like you to measure each of the three angles in your triangle. Write down their measures and add them.
    Walk around the room to see that students are measuring their angles correctly. After everyone is done, select six students to share the angle sums they found. Write the sums on the board.
  • Ask: What do you notice about the sums of the angles for the six triangles?
    Students should say that they seem to add to about 180 degrees.
  • Say: Label the angles inside your triangle “a,” “b,” and “c.” (Do this on the triangle on the board.) Cut out the triangle and rip off the angles. Now arrange the three angles so they share a common vertex. What do you notice about the outside sides for the three angles? (They form a straight line.)
  • Say: On your paper, draw a line segment 5 cm long. At each endpoint of your segment, draw an angle measuring 60 degrees. Extend the sides of the angles until they meet to form a triangle.
    Draw a picture on the board like the one below, so students can see what to do.
    angles
  • Say: Measure the two new sides of your triangle. What do you notice about the lengths of the sides? (They are about 5 cm each.) Yes, that's right. This is a triangle with all its sides congruent. Does anyone know what we call such a triangle? (an equilateral triangle)
    Write equilateral triangle on the board and then write the definition of equilateral triangle.
  • Say: I'd like you to draw another triangle. This time draw a triangle with two sides that are 4 cm long.
    Draw a sketch on the board of what you want.

    Have the students show the class the different triangles they drew.

  • Ask: Does anyone know what we call this kind of triangle? (an isosceles triangle)
    Write isosceles triangle on the board, and write the definition of isosceles triangle.
  • Say: Note that the definition of isosceles triangle says “at least two sides are congruent.” That means that every equilateral triangle is also an isosceles triangle. However, not every isosceles triangle is an equilateral triangle.
  • Say: We now know what we call a triangle with at least two sides congruent and what we call a triangle with three sides congruent. If a triangle doesn't have any congruent sides, we call it a scalene triangle.
    Write scalene triangle and the definition on the board.
  • Say: Please draw a scalene triangle on your paper.
    Have some students show their scalene triangles to the class.
  • Say: We classify triangles by the lengths of their sides into the three classes we have just discussed−scalene, isosceles, and equilateral. (Point to those three names on the board as you say them.) We can also classify triangles by the measures of their angles. That's what we are going to do now.
  • Ask: Who can tell me what a right angle is? (an angle whose measure is 90 degrees) If we have a triangle with a right angle, we call it a right triangle. See if you can make a right triangle, using your ruler and protractor.
    Walk around the room and help students who are having difficulty. Have some students show their right triangles to the class. Have these students rotate their triangle to show different orientations.
  • Ask: What is an obtuse angle? (an angle whose measure is greater than 90 degrees and less than 180 degrees)
  • Say: Good. I'm drawing a triangle with an obtuse angle. (Draw an obtuse triangle on the board.) This is called an obtuse triangle because it has an obtuse angle.
  • Say: If a triangle has only acute angles, we call it an acute triangle. Thus, we have three ways to classify triangles based on angle measures-acute, right, and obtuse.
    Write these three names on the board next to the name for classifying triangles by the lengths of their sides.

    Now, draw a picture of any quadrilateral on the board.

  • Ask: What do we call a polygon with four sides? (a quadrilateral) We can draw a line from one vertex to an opposite vertex like this. (Draw this on the board.) This creates two triangles. What is the sum of the angle measures for each triangle? (180 degrees) That's right. Therefore, the sum of the angle measures for a quadrilateral is 360 degrees.

    On the board, tape the picture of the trapezoid.

  • Ask: Does anyone know what this figure is called? (a trapezoid) What do you know about trapezoids? (They have one pair of parallel sides.)

    Write the name trapezoid and the fact that a trapezoid has one pair of parallel sides under the drawing.

  • Say: Draw a trapezoid on your paper by drawing two line segments parallel to each other, using the two sides of your ruler. Make one segment shorter than the other and connect the endpoints of the two line segments you drew. Measure the angles in your diagram and see if you discover anything.
    Students should find that angles that share a common side are supplementary. Write that on the board as well.

    On the board, tape the picture of the parallelogram.

  • Ask: Does anyone know what we call this figure? (a parallelogram) Yes. Why do we call it a parallelogram? (Its opposite sides are parallel.)
    Write the term parallelogram and the fact that a parallelogram has opposite sides that are parallel under the drawing of the parallelogram.
  • Say: To help us find other properties that parallelograms have, I'd like you to draw a picture of one on your paper. To do this, take your ruler and draw two parallel lines by drawing lines on both sides of your ruler. Now take your ruler and hold it at an angle to these parallel lines, and draw two more lines intersecting them to form a parallelogram.
    Illustrate this on the chalkboard.
  • Say: Measure each of the angles and each of the sides and make a list of the properties you discover.
    After giving them some time, ask students for things that they discovered about the parallelogram. List them under the drawing of the parallelogram. Students should find that opposite sides are congruent, opposite angles are congruent, and any two angles sharing a common side are supplementary.

    Tape up the picture of the rectangle.

  • Ask: What do we call this figure? (a rectangle)
    Write the word rectangle and the definition−a quadrilateral with four right angles−on the sheet of paper.
  • Ask: Can you see any items in this room that are rectangular in shape? (Students may say the door, the windows, the chalkboard, and so on.) What else do you know about rectangles other than that they have four right angles?
    List the other properties of rectangles on the sheet of paper. The list should include the following. (1) It is a parallelogram (or its opposite sides are parallel), and, since it is a parallelogram, (2) its opposite sides are congruent, (3) its opposite angles are congruent, and (4) any two angles sharing a common side are supplementary.

    Tape up the picture of a rhombus.

  • Ask: Does anyone know what this figure is called? (a rhombus)
    You may have to tell students the name of this figure.

    Have someone come up front and measure the sides of the rhombus.

  • Ask: What did you find out about the lengths of the sides? (They are all the same length.)

    Write the name rhombus and the definition−a quadrilateral with all four sides congruent.
    Now have someone come up and measure the angles in the rhombus.

  • Ask: What did you find out about the angles? (The opposite angles are congruent and angles sharing a common side are supplementary.)
  • Say: Very good. In fact, this too is a parallelogram.

    Write the properties that the students discovered under the picture of the rhombus.
    Tape up the picture of a square.

  • Ask: What do we call this shape? (a square) What do you know about squares?
    (They are quadrilaterals with four congruent sides and four right angles.)
  • Say: There are different ways to define square. Our book defines it as “a rectangle with all sides congruent.” (Write the term square and its definition under the drawing of the square.) Since it has four congruent sides and is a quadrilateral, it is also is a rhombus. Since both rectangles and rhombuses are parallelograms, a square is a parallelogram, and it has all the properties of a rectangle, a rhombus, and a parallelogram.
    Write the properties of the square under the drawing.
  • Say: We can make a diagram to help us remember the relationships between the types of quadrilaterals.
    Draw the diagram below, discussing the relationships as you place the figures on the chart.
diagram

Houghton Mifflin Math Grade 6