Teachers Guide for AP Book 7.1 Unit 6 Geometry G-1
Unit 6 Geometry: Constructing Triangles and Scale Drawings
Introduction In this unit, students will construct triangles from three measures of sides and/or angles, and will decide whether given conditions make exactly one triangle (up to congruence), more than one triangle, or no triangle. Students will also solve problems involving scale diagrams. Materials. Students will use protractors frequently in this unit. If commercial protractors are unavailable, photocopy BLM Protractors (p. G-58) onto a transparency and cut it into eight separate protractors. Do this as needed for each student to have one. In Lesson G7-5, students are asked to use a triangle-building machine. We recommend preparing enough of them ahead of time so each student and you can have one, instead of using class time to build them. See BLM Triangle-Building Machine (p. G-59) for instructions on how to make them. Alternatively, if you have class time available, you could have students make their own machine. You will need a pack of 100 paper fasteners to do this. These should be available at your local office supplies store. Grid paper. We recommend that students have grid paper and that you have a background grid on your board. If students do not have grid paper, you will need to have lots of grid paper available (e.g., from BLM 1 cm Grid Paper on p. J-1). If you do not have a background grid on your board, you will need to project a transparency of a grid onto the board so you can write over the grid and erase the board without erasing the grid. Technology: dynamic geometry software. Students are expected to use dynamic geometry software to draw geometric shapes. Some of the activities in this unit use a program called The Geometers Sketchpad, and some are instructionalthey help you teach students how to use the program. If you are not familiar with The Geometers Sketchpad, the built-in Help Centre provides explicit instructions for many constructions. Use phrases such as How to construct a line segment of given length to search the Index. NOTE: If you use a different dynamic geometry program to complete these activities, the instructions provided may need to be adjusted. Fraction notation. We show fractions in two ways in our lesson plans:
Not stacked: 1/2
If you show your students the non-stacked form, remember to introduce it as new notation.
G-2 Teachers Guide for AP Book 7.1 Unit 6 Geometry
G7-1 Angles Pages 152153 Standards: preparation for 7.G.A.2 Goals: Students will recognize lines, rays, points, and line segments. Students will compare angles and will name angles as right, obtuse, or acute. Vocabulary: acute, angle, arc, arm, degree (), endpoint, line, line segment, obtuse, point, ray, right angle, rotation, vertex Materials: large part of a small circle and small part of a large circle, made from bristol board transparency of BLM Quarter-Circle Protractors (p. G-56) overhead projector 2 thin rays cut from bristol board with arrows at one end Introduce points, lines, line segments, and rays. Draw the pictures below on the board, making sure the line segment is 70 cm long and the line is drawn shorter than the line segment: Always be sure to model using a ruler to make lines, line segments, and rays straight, since students will need to do this themselves. Point to the point, and SAY: This is called a point. Point to the line segment and SAY: A line segment is a straight path between two points, called endpoints. You can measure the length of a line segment. Have a volunteer do so. Then SAY: This line segment has lots of points. Show some of them on the board: Now point to the line. SAY: A line extends in a straight path forever in two directions. It has no endpoints. The arrows at both ends show that you can extend the line in both directions. Point out that it looks shorter than the line segment, but it is actually much longer because you can extend it as much as you want in both directions. You cannot measure the length of a line because it goes on in both directions. Point to the ray and SAY: A ray has one starting point and
line segment line
Teachers Guide for AP Book 7.1 Unit 6 Geometry G-3
goes on forever in one direction, like a ray of sunlight starting at the sun goes on forever. You cannot measure a ray either. Exercises: Name each object as a point, a line, a line segment, or a ray. a) b) c) d) e) Answers: a) point, b) line, c) ray, d) ray, e) line segment Draw on the board: Point to the picture on the left and ASK: Do these objects meet when extended as much as possible? (yes) Show the meeting by extending the line. Then point to the picture on the right and ask the same question. (no) SAY: Only the line can be extended. Show the extending and how, this time, there is no meeting point. In the Exercises below, students can signal thumbs up for yes and thumbs down for no when you take up the answers. Exercises: 1. Do the rays, lines, or line segments meet when extended where possible? a) b) c) d) e) f) g) h) Answers: a) yes, b) no, c) yes, d) no, e) no, f) yes, g) no, h) yes 2. Is the given point on the object (the line, the line segment, or the ray)? a) b) c) d) Answers: a) yes, b) yes, c) no, d) no Introduce angles. Tell students that an angle is the space between two rays that have the same endpoint. Draw on the board: Tell students that the endpoint is called the vertex and the two rays are called the arms of the angle. The vertex is easy to see, so they dont need to draw the dot to show it.
G-4 Teachers Guide for AP Book 7.1 Unit 6 Geometry
The size of an angle is the amount of rotation between the arms. Tell students that the size of an angle is the amount of rotation between the two arms. Cut out two thin rays from bristol board and tape one of them to the board. Show how you can make a small or large angle using the other ray by rotating it away from the first ray either a small or large amount. Have students stand up facing the front and rotate in place until they see various objects in the classroom, such as a clock, a bookshelf, a computer, etc. After each rotation, ask students if the rotation was a greater or lesser amount of rotation than the previous object. Draw the following pictures on the board without the arcs to illustrate what you mean by smaller and larger angles:
Explain that angles are drawn with an arc (a part of a circle around the vertex of an angle) to show how much one arm turns to get to the other. Add the arcs to the picture. Using the amount of space between the arms to compare sizes. Draw on the board: Tell students that you want to use the definition of an angle as the space between the arms as a way to compare angles to say which one is larger. Color the space between the arms in both pictures and ASK: Which angle has more space between the arms? (the one on the right) SAY: So the one on the right is a larger angle. But you need to be careful because sometimes the same angle can be drawn with shorter arms, which make it look as though there is less space between the arms. Draw the same angles as above on the board, but this time draw the one on the right with shorter arms, as shown below: Tell students that if you want to compare angles by coloring the space between the arms, you need to make sure you draw the arms the same length. Instead of coloring the space, another way to compare the space between the arms is to check which angle fits onto the other. Show students two angles: a large angle cut from a small circle and a small angle cut from a large circle. Demonstrate how the smaller angle fits onto the larger angle when you place their vertices together as shown below:
Teachers Guide for AP Book 7.1 Unit 6 Geometry G-5
Exercises: 1. Which angle is larger? a) b) c) A B A B A B Answers: a) A, b) B, c) A 2. Which picture in Exercise 1 is deceptive? Answer: The picture for part b) is deceptive because the smaller angle has longer arms drawn. The size of an angle is measured in degrees. Tell students that they can measure the size of an angle. SAY: The unit of measurement for an angle is a degree. Write on the board:
90 degrees = 90 Point at the degree symbol and SAY: We write this symbol for degrees. Show students a piece of paper and point out the horizontal and vertical sides. SAY: The amount of rotation needed to go from a horizontal line to a vertical line is 90 degrees. The angle between a horizontal line and a vertical line is called a right angle. Project BLM Quarter-Circle Protractors on the board. Draw several angles starting at the 0 mark (10, 70, 60, Bonus: 35, 75) and have students individually record the angles shown. Then continue with more angles that do not start at the 0 mark. (50 to 90, 20 to 30, 60 to 80, Bonus: 55 to 90, 75 to 85) Introduce acute and obtuse angles. Tell students that the angles they have seen so far were all less than 90 because they all fit into a right angle. Smaller angles have smaller degree measures. SAY: Angles that measure less than 90 are called acute angles and angles that measure more than 90 are called obtuse angles. You can compare an angle to a square corner to decide if it is acute or obtuse. Draw several angles on the board (see examples below): Have volunteers compare the angles to a square corner (e.g., from a sheet of paper) to decide whether the angle will measure more or less than 90. Then have the volunteers say whether each angle is acute or obtuse. (From left to right, the angles are obtuse