These lessons ware authored by Debra Kopcso, LSU. They are used in M1202, a geometry course for pre-service elementary teachers.
In this lesson students will investigate some special quadrilaterals: trapezoid, parallelogram, rhombus, rectangle, square, and kite. By exploring these figures on Geometer's Sketchpad, the students will first develop a mathematical definition of each of these and then investigate to discover all the properties of these figures.
Students will be able to identify and define quadrilaterals and be able to discuss the properties of these quadrilaterals. Students should also be able to determine which shapes are subsets of others and to make a tree diagram for these quadrilaterals.
Materials Needed: Computer lab with Geometer's Sketchpad program; Some quadrilaterals on Geometer's Sketchpad for students to explore.
First the teacher will construct several of the six quadrilaterals being investigated on Geometer's Sketchpad. Have students in groups of two in computer lab with Geometer's Sketchpad.
Ask the students to determine a definition of each of these figures ( they will probably know some of these definitions) by measuring sides, determining if sides are parallel or perpendicular. Have them do them in the order of the trapezoid, the parallelogram, the rhombus, the rectangle, the square, and the kite. Walk around the room to observe what you hear and answer any questions they may have.
Ask the groups to give their definition of each one and decide if all the information is necessary to define it. For example, when defining a parallelogram is it necessary to say opposite sides parallel and congruent? Is one sufficient to define a parallelogram?
Yes, it is. A parallelogram is a quadrilateral with opposite sides parallel.
They should observe that when opposite sides of a quadrilateral are parallel, they are also congruent.
Once they have defined a figure they may use that word in a definition of another figure. For example: they may use parallelogram in their definition of a rhombus or rectangle.
Have them record their definitions on the sheet provided.
After they have defined each figure, tell the students to investigate the figures further by:
determining if any sides are congruent; any angles are: congruent, right, or supplementary; then construct the diagonals for each figure and list any properties you observe: Do diagonals bisect each other? Do diagonals bisect the angles? Are the diagonals perpendicular? Are the diagonals congruent?
Have the students discuss with other groups to compare their results. Then have them record their results on the provided sheet.
Ask the students if it is possible to have a parallelogram with only one right angle?
They should try to construct one on Geometer's Sketchpad if they are not sure.
Ask them what properties of a parallelogram say if one angle is right, the others are also.
They should be able to tell you that since opposite angles in a parallelogram are congruent, if one angle is right the opposite angle is also right. Also they should tell you in a parallelogram, adjacent angles are supplementary, so if one angle is right the adjacent one is also.
Now ask the students: Is a parallelogram a trapezoid?
Is a rhombus a parallelogram? a trapezoid?
Is a rectangle a rhombus? a square? a parallelogram? a trapezoid?
Is a square a rectangle? a rhombus? a parallelogram? a trapezoid?
Is a kite a square? a rhombus? a parallelogram? a trapezoid? (Note: Some books define a trapezoid with exactly one pair of opposite sides parallel, some say it has at least one pair of opposite sides parallel. We will use the latter here. Also our definition of a kite is a quadrilateral with at least two distinct pairs of congruent adjacent sides.)
Now have the students make a tree diagram for these quadrilaterals.
Superficially, this lesson is about vocabulary. But mathematical vocabulary is not arbitrary. Good mathematical vocabulary is related to the structure of the subject it is about.
In this lesson, students' attention is drawn to the structural features of 4-sided shapes that permit classification. Different terms are related to the presence of features that permit classification.
Why is such classification important? Consider the problem of determining area. It is hard to find the area of a quadrilateral that has no special features---in general, several measurements are required. We cannot find the area of a quadrilateral if all we know abot it is the length of the sides, because the side engths don't determine the area. But if we know that the figure we have is a rectangle, then we cn find the area by measuring the lengths of two adjacent sides.
This is just one example. Depending on the presence or absence of features that are captured by words such as "square," "rectangle," "trapeziod," etc., there are numerous other properties that may or may not be present. Subsequent lessons concern these.
Students will construct some special quadrilaterals (trapezoid, parallelogram, rhombus, rectangle, square, kite) for which they have already defined, and have discovered all the properties of them.
Students will construct these quadrilaterals on geoboards, patty paper, and on Geometer's Sketchpad.
Learning Objectives:
Students will be able to apply their knowledge of these shapes to construct them.
Geoboards (one for each student); patty paper (my students have their own); computer lab with Geometer's Sketchpad.
Have the students get out their sheets on quadrilaterals with the definitions and properties.
Geoboards (20 minutes)
Pass out geoboards to each student.
Ask the students to individually construct a trapezoid. Have them compare with their group members. Have the students talk about what they did to construct their trapezoid and if they observed shapes that were not trapezoids. How do they know it is a trapezoid?
Repeat having them construct a parallelogram (that is NOT a rectangle), a rhombus (that is NOT a square), a rectangle, a square, and a kite.
For the parallelogram, ask how they know opposite sides are parallel?
You should hear them talk about slope or distance or length.
They could make a rectangle and then move the rubber bands on opposite ends the same direction and distance.
For the rhombus, ask how they know all sides are congruent?
Students will not have any trouble constructing the rectangle and square, but they should all do it and compare their figures with their group. Have them discuss how they know it is a rectangle or a square.
For the kite, have them construct one with exactly two distinct congruent adjacent sides ( in other words, NOT a rhombus). They may construct it by starting with the sides or the diagonals. Again, have them share with their group.
You may want to have some students come to the overhead and show their figures.
Patty Paper (40 minutes)
Ask the students to fold a trapezoid on patty paper. Ask the students how they know it is a trapezoid. Have them discuss their answers.
Ask the students to get out a piece of patty paper and fold a parallelogram that is NOT a rectangle. This is a good review exercise on how to fold parallel lines on patty paper(they must fold a perpendicular to a line to get a parallel line). I always suggest students number their folds on patty paper so they can remember in which order they did it later. Tell them to label this patty paper, Parallelogram, and save it. Ask the students how they did their construction. If they started with the sides first, ask them to now try to construct it by starting with the diagonals. Is this easier or harder? Ask how many first constructed by starting with the diagonals. Ask them to also construct by starting with the sides.
Have a student go to the overhead and demonstrate how to construct a parallelogram by starting with the sides first and by starting with the diagonals first (wax paper works well on the overhead because you can see the folds).
Ask the students to now fold a square. Remind them to number their folds and label their patty paper Square. They may not use the edge of the patty paper as a side. Ask the students how they did their construction. If they started with the sides first, ask them to now try to construct it by starting with the diagonals. Is this easier or harder? Ask how many first constructed by starting with the diagonals. Ask them to also construct by starting with the sides. Again have a student go to the overhead and demonstrate how they constructed their square.
I assign the students to fold the remaining shapes for homework.
Geometer's Sketchpad: (1 hour)
Have the students in a computer lab with Geometer's Sketchpad.
Have the students save their work on a disk for future reference.
Tell the students to construct a trapezoid using Geometer' Sketchpad. Tell them to click on a vertice and drag it to change the trapezoid. Does it still remain a trapezoid?
Now have the students construct a parallelogram (that is not a rectangle). Walk around the room and observe their constructions. Ask the students how they constructed their parallelogram. Can they think of another way to construct a parallelogram? They should be able to construct a parallelogram by starting with the sides and by starting with the diagonals. They should click on a vertice and drag it to see if it still remains a parallelogram.
Now have the students construct a rhombus (that is not a square). Walk around the room and observe the different ways they are constructing it. Ask them if they can think of another way to construct a rhombus. Have them record the different ways they construct the rhombus. They should be able to construct a rhombus by starting with the sides and by starting with the diagonals.
Have them construct the remaining shapes on Geometer's Sketchpad for homework. (My students have access to a computer lab during the day so they can do this. If students have a computer at home and access to the internet, they can download a demo version of Geometer's Sketchpad by getting to the homepage of Key Curriculum Press: www.keypress.com and finding products and they should see "Download demo version of Geometer's Sketchpad.)
If your students do not have a computer lab or computer at home, have them construct the remaining shapes during class.
This lesson uses a circular geoboard to investigate measures of central and inscribed angles of a circle. By making a triangle on the geoboard with one vertex of the triangle the center of the circle and the other two vertices lie on the circle, the students will first determine the measure of the central angle, and then determine the measures of the other two angles in the triangle.
Students should already know that there are 360° in a circle, that the sum of the angle measures of a triangle is 180°, and that the angles opposite equal sides of a triangle are congruent.
Students will be able to construct and determine the measure of central angles and inscribed angles in a circle on a geoboard. Students will also discover the measure of an angle inscribed in a semicircle. Students will also be able to compare and determine inscribed angles. Students will then investigate the relationship of a central angle and an inscribed angle that intercept the same arc (or arc of equal length).
geoboards, handouts with blank geoboards on them
Pass out geoboards to each student. Ask them the measure of a central angle between two adjacent pegs. (The pegs are equally spaced around the circle.) Using an overhead geoboard, make the following central angles and ask the students to determine their measures.
Diagram omitted
Now ask the students to construct the following central angles on their geoboards: 45°, 165°, 105°, 60°, 180°, and 195°. Tell them to show their angles to their group members. The students usually have a little trouble constructing the 165, 105, and 195 angles on a geoboard that has 12 pegs. Given a little time someone in the group usually figures it out.
Now on an overhead geoboard, I make two triangles with rubberbands with one vertex at the center of the circle and the other two vertices on the circle (see figure below).
Diagram omitted
I ask the students what they can say about the degree measures of angle 1 and angle 2. Angle 3 and angle 4? I have them discuss this with their group members. I walk around the room listening to their observations. I make sure they all determine that these two angles must be congruent since they are both opposite radii. In other words the triangle is isosceles since two sides are radii so the two base angles are congruent.
Now I ask the students if they can find the degree measure of angle 1, 2, 3 and 4. The students should say yes because they can first determine the measure of the central angle in the triangle, and since the other two angles are congruent they can subtract the measure of the central angle from 180 degrees and then divide by two to get the measure of each base angle.
I ask the students to construct a triangle on their geoboard that has one vertex at the center of the circle and its other two vertices on the circle. Exchange geoboards with your group members. Find the degree measure of each angle of his triangle. I encourage them to discuss their results with their group members. This process of determining these angle measures in the triangle is the foundation of the remaining work so it is essential that each student understand the process.
Now I make a triangle on the geoboard on the overhead (I have them copy this picture on their blank geoboards for their notes) but this time none of the vertices are at the center, they are all on the circle. I make it so that one side of the triangle is a diameter of the circle, and the other two sides are chords of the circle. See figure below.
Diagram omitted
I ask them to discuss with their group ways to find the measure of angles 1, 2 and 3 in the above figure. I walk around the room listening to their thinking and giving a hint that they will need to make a central angle since that is the only angle they can determine so far on the geoboard. I also tell them that they will use the previous process we did before. After I have gone to each group and seen that they are all on the correct path I go over with the class on the overhead geoboard. I add a rubberband to the triangle from the center of the circle to the vertex that is not connected to the diameter to divide the original triangle into two triangles, both having a central angle in them. Now they can clearly see how to determine the measure of each angle in the original triangle by repeating the process they did in the beginning (finding the measure of the central angle and then determining the measure of the other two congruent angles in the triangle). They must do it twice since there are two triangles that make up the original triangle. Again I go around the room until I am convinced that each student understands the process of finding the measure of each angle in the original triangle.
Now I have them refer to the following figure and determine the degree measures of angle 1, 2, and 3 as point P moves from peg A to peg K. I tell the students to draw each case on their blank geoboard handout that I give them. I tell the students to divide the cases among their group so they are not all doing each case. Have them record their results(see table below).
Diagram omitted
Ask them what happens to the angle measures when you move the third vertex of the triangle. They should notice that angles 1 and 2 change, one increasing and the other decreasing by the same amount. They should also discover that angle 3 is always 90°. They have discovered that an angle inscribed in a semicircle always has a measure of 90 degrees.
I ask them if they notice any relationship between the inscribed angles and the central angle that shares the same arc. On the overhead I construct one of the cases they have investigated (see figure below). I ask them to look back at each of their cases and see if they can find a relationship between the inscribed angle (angle 1) and the central angle (angle 4) that share the same arc. They should notice that in each case the inscribed angle is half the measure of the central angle.
Diagram omitted
I construct several insribed angles on the overhead geoboard and each time also show the central angle that subtends the same arc and ask them to find the measure of the inscribed angle. Based on the previous discovery they should determine the central angle first and then say that the inscribed angle will be half the measure of the central angle.
Now I make two inscribed angles on the overhead geoboard which subtend different arc lengths. I ask them what they can say about the measures of these two angles. Which angle is larger? Why? They should be able to tell me that the angle that subtends the larger are has a larger measure by comparing the central angles that subtend the same are as the inscribed angles.
I repeat this process with inscribed angles whose arc length is the same. Again they should be able to tell me that their measures are equal.