Unit 6 Geometry: Constructing Triangles and Scale Drawings

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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 Geometer s Sketchpad, and some are instructional they help you teach students how to use the program. If you are not familiar with The Geometer s 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: Stacked: 1 Not stacked: 1/2 2 If you show your students the non-stacked form, remember to introduce it as new notation. Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-1

2 G7-1 Angles Pages 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: point line segment line ray 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 G-2 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

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 don t need to draw the dot to show it. Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-3

4 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: G-4 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

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, acute, acute, and obtuse.) Ask a volunteer to look up the word acute in the dictionary: it means sharp. Point out that a small angle is sharper than a large angle; it would hurt more if you walked into an edge that had an acute angle than if you walked into an edge that had an obtuse angle. Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-5

6 Have students draw a mix of several acute and obtuse angles in their notebook. Emphasize the importance of using a ruler to make the rays straight, so that their partner can easily compare their angle to a square corner. Have them switch notebooks with a partner and label their partner s angles as obtuse or acute. Exercises: Does the degree measure represent an acute angle, an obtuse angle, or a right angle? a) 75 b) 83 c) 100 d) 90 e) 12 f) 94 Answers: a) acute, b) acute, c) obtuse, d) right, e) acute, f) obtuse Activity Teach students to draw lines, line segments, and rays using the line tool of The Geometer s Sketchpad. Then ask them to draw a line and an independent point. Ask them to move the point so it looks as though it is on the line. Then have them modify the line. Does the point stay on the line? (no) Now show students how to construct a line through two given points and a point on the line, so that modifying the points keep the line and the points together. (end of activity) Extensions 1. What is the angle between the hands of a clock? a) at 3:00 b) at 1:00 c) at 2:00 d) at 4:00 Answers: a) 90, b) 30, c) 60, d) 120 (MP.1) 2. Do the objects below intersect (meet)? Check all possibilities of the objects being lines, line segments, and rays in both directions. G-6 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

7 G7-2 Measuring and Drawing Angles Pages Standards: preparation for 7.G.A.2 Goals: Students will use protractors to measure and draw angles. Prior Knowledge Required: Can recognize qualitatively when one angle is larger than another Vocabulary: acute, angle, arc, arm, base line, degree ( ), endpoint, line, line segment, obtuse, origin, point, protractor, ray, vertex Materials: protractors transparency of BLM Simple Protractors (p. G-57) transparency of BLM Protractors (p. G-58) BLM Protractors (p. G-58) overhead projector Introduce protractors. Give each student a protractor and SAY: This is a protractor. Have students examine their protractors and compare them to rulers. Remind students that when they measure with a ruler, they have to line up one end of the object with the zero mark. Have students find the zero mark on their protractors (There are two of them, one at each end!) Point out that a protractor has two scales, both with the same unit, but in opposite directions. A ruler can have two scales, too, but the two scales would use different units. Explain that having two identical scales going in different directions allows you to measure the angles from both sides, but this also means that you need to decide which scale you will use each time. Project BLM Simple Protractors onto the board. Explain that the protractors on the BLM are simplified pictures of a protractor, without all the tiny markings in between the larger angles. Draw the angles below on the board: Pointing at the first picture, ASK: Is this angle an acute angle or an obtuse angle? (acute) Circle the numbers 30 and 150 that the arm of the angle passes through. ASK: Which one is the Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-7

8 answer? (30 ) How do you know? (the angle is acute, so the angle measure has to be less than 90 ) Repeat for the second picture. (this time, the angle is obtuse, so the correct choice is 150, not 30 ) Point out that there is another way to check that you are using the correct scale. The correct scale starts with zero on the arm of the angle. Exercises: Measure the angle. a) b) c) d) Answers: a) 60, 120, 120, 60 Now project BLM Protractors on the board and ask similar questions. Exercises: Measure the angle. a) b) c) d) Answers: a) 130, b) 55, c) 155, d) 45 G-8 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

9 Placing protractors on angles. Point out the base line and the origin on a large protractor or on a picture of a protractor on the board, as shown below: Have students find the base line and the origin on their protractors. Demonstrate how to place a protractor correctly, so that the base line lines up with one arm of the angle and the origin is at the vertex. Point out that this is similar to placing a ruler with the zero at the beginning of the object you are measuring. Provide students with BLM Protractors for the following Exercises. Exercises: Draw the angle on BLM Protractors two ways, starting at either zero. a) 50 b) 70 c) 15 d) 165 Selected answer: d) Have students draw an acute angle in their notebooks, then ask them to place their protractors correctly. Circulate in the classroom to check that all students have done so. Then have students measure the angle they drew. Repeat with an obtuse angle. Have students exchange notebooks with a partner and measure each other s angles to check their work. (MP.3) Draw an angle on the overhead projector and demonstrate measuring incorrectly. Have students tell you what you are doing wrong. For example, place the vertex at several different incorrect locations, such as exactly on the bottom of the protractor or along the small central arc of the protractor. You could also place the vertex correctly, but have one arm outside the range of the protractor while the other is aligned with 0 or have neither arm aligned with the zero. Extending the arms of an angle to meet the angle measures on a protractor. Draw a small angle on the overhead projector and use a protractor to demonstrate how it is difficult to read the measurements because the arms do not reach the scale. SAY: Sometimes you have to extend the arms so that you can read where they meet the scale. Do so, then have a volunteer read the angle measure. Exercises: Copy the angle onto grid paper. Then extend the arms and measure the angle. a) b) c) Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-9

10 Answers: a) 53, b) 45, c) 34 Drawing angles. Model drawing angles step by step. To draw a 60 angle: Step 1: Use a ruler to draw a ray with the arm long enough to pass through the zero mark. Place the protractor so that the origin is at the vertex of the angle, and the zero mark lies along the arm you drew. Step 2: Make a mark at 60. Be sure to use the same scale that shows zero on the arm of the angle. Step 3: Using a ruler, join the endpoint of the ray to the mark. Add an arrow to complete the second ray. Exercises: Draw the angle. a) One arm is horizontal, pointing right, and the angle opens upward and measures i) 40 ii) 72 iii) 154 b) One arm is horizontal, pointing left, and the angle opens upward and measures 120. c) One arm is vertical, pointing down, and the angle opens right and measures i) 70 ii) 96 Answers: a) i) ii) ii) b) c) i) ii) G-10 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

11 Activities Teach students to draw and measure angles using The Geometer s Sketchpad. Then ask them to try moving different points (on the arms of the angle or its vertex) so that the size of the angle becomes, say, 50. ASK: Is it easy or hard to do? (hard) When you move the line segments, does the angle change? (yes) When you move the vertex or other point on the arms, does the angle change? (yes) Show students how to draw an angle of fixed measure by using menu options. ASK: Will moving the endpoints change the size of the angle now? (no) Show students how to draw angles equal to a given angle. 2. Have students draw polygons in The Geometer s Sketchpad and measure the size of the angles and the length of the sides of these polygons. Have students check that the angle measures they obtain make sense. For example, if they click on three vertices of a quadrilateral then use menu options to measure the angle, it might produce different angles depending on the order in which the vertices were selected. Also, the software sometimes measures angles in the wrong direction, producing an answer more than 180. (end of activities) Extensions (MP.1) 1. Have students find the angle between the clock hands at a) 3:30 b) 2:30 c) 4:30 d) 12:12 e) 12:24 f) 1:36 g) 3:48 Hints: An hour is 60 minutes and a whole circle is 360. What angle does the minute hand cover every minute? (6 ) How long does it take the hour hand to cover that many degrees? (12 minutes) How do you know? (because the hour hand covers only one twelfth of the full circle in an hour, moving 12 times slower than the minute hand) If the time is 12:12, where do the hour hand and the minute hand point? (the hour hand points one fifth of the way from 12 to 1, the minute hand points 2/5 of the way from 2 to 3) What angle does each hand make with a vertical line? (hour hand: 6 ; minute hand: 72 ) What is the angle between the hands? (66 ) Answers: a) 75, b) 105, c) 45, d) 66, e) 132, f) 168, g) Some scientists think that moths travel at a 30 angle to the Sun when they leave home at sunrise. Note that the Sun is far away, so all the rays it sends to us seem parallel. a) What angle do the moths need to travel at to find their way back at sunset? Hint: Where is the Sun in the evening? b) A moth sees the light from a candle flame and thinks it s the Sun. The candle is very near to us and the rays it sends to us go out in all directions. Where does the moth end up? Draw the moth s path. Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-11

12 Answers: a) 30, b) The moth spirals toward the candle flame. G-12 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

13 G7-3 Constructing Triangles from Angles and Sides Pages Standards: 7.G.A.2 Goals: Students will learn how to recognize congruent shapes. Students will understand that two given angles with a side length between them make exactly one triangle (up to congruence), when the sum of the two angles is less than 180. Prior Knowledge Required: Can use a protractor to measure angles Can draw angles Knows that parallelograms have opposite sides equal Vocabulary: angle, congruent, degree ( ), line, line segment, parallel, protractor, ray Materials: protractors grid paper or BLM 1 cm Grid Paper (p. J-1) small triangles cut from bristol board, a different one for each student blank paper scissors Some pairs of angles can be used to make a triangle and others cannot be. Draw on the board: a) b) c) d) e) For each picture, ASK: Are the rays getting closer together, farther apart, or neither? (a) closer together, b) farther apart, c) closer together, d) farther apart, e) neither) ASK: If you were to extend the rays, would they meet? (a) yes, b) no, c) yes, d) no, e) no) Have a volunteer extend the rays in part a) so they meet. Point out that the volunteer drew a triangle. Tell students that this class is about constructing triangles. If you know two of the angles you can sometimes extend the rays to make a triangle and sometimes you cannot. (MP.8) Determining a condition on the angles for the rays to make a triangle. Draw on the board: 60 Tell all students to draw a 60 angle in their notebook with one arm being a horizontal line segment and the other arm being a ray on the left side of the first line segment, pointing upward. Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-13

14 When students finish, tell one third of the class to draw a ray on the right side of the line segment that will make the rays get closer together, another third to make them get farther apart, and the rest to try to make them stay the same distance apart. Then have everyone measure the angle they drew. Write on the board: Angles that make the rays get closer Angles that make the rays get farther apart Angles that make the rays stay the same distance apart Fill in the chart as a class. ASK: What angles make the rays get closer together? (angles less than 120 ) What angles make the rays get farther apart? (angles greater than 120 ) What angles make the rays stay the same distance apart? (exactly 120 ) Tell students that if they were trying to make the rays stay the same distance apart, they should have drawn an angle very close to 120. It won t be exact because exact measurements are not possible, but they should be close. To summarize, SAY: When the second angle is 120, the rays stay the same distance apart they are parallel and always go in the same direction they do not make a triangle. When the second angle is less than 120, the rays get closer together so they make a triangle. When the second angle is larger than 120, the rays get farther apart so they do not make a triangle. When two lines are parallel, the angles add to 180. Provide students with grid paper or BLM 1 cm Grid Paper if they do not already have grid paper. Remind students that parallelograms have opposite sides equal and if there are two parallel sides that are equal in length, students can make a parallelogram. Show students how they can draw, for example, two horizontal sides of length 3 and make parallel lines to join them. Start by drawing two line segments that go right 1 square and up 3 squares, as shown below: SAY: If both line segments go right the same number of squares and up the same number of squares, then they are parallel. Have students draw in their notebooks the parallel lines with a horizontal side of length 3 between them. Tell students that they have created two angles, as shown below: Exercise: Use a protractor to measure the two angles you just drew. Answer: 72 and 108 G-14 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

15 Bonus: Create other pairs of parallel line segments with a horizontal line segment between them. Measure the angles created in this way. (MP.8) When students finish, ASK: What do the two angles add to? (180 ) Point students attention to the 60 and 120 angles that also created parallel lines. SAY: When two lines are parallel, the angles they make in this way add to 180. If the lines weren t parallel and the angles added to less than 180, the rays would be getting closer and could create a triangle if extended. If the angles added to more than 180, the rays would be getting farther apart. (MP.8) Exercises: Add the angles to predict whether you can create a triangle satisfying these conditions. Then check your prediction. a) two angles are 70 and the side between them has length 5 cm b) two angles are 90 and the side between them has length 4 cm c) a 40 angle and a 50 angle and the side between them is 3 inches long d) a 70 angle and a 120 angle and the side between them is 7 cm long Answers: a) yes, < 180; b) no, = 180; c) yes, < 180; d) no, > 180 Congruent shapes. SAY: Two shapes are congruent when they are the same size and the same shape. Tell students that you re interested in congruent shapes because if two people each draw a triangle that satisfies certain conditions, you want to know if the two triangles have to be congruent or if there are two non-congruent triangles that satisfy the conditions. SAY: Color, pattern, and orientation don t affect congruence, only size and shape do. Exercises: Are the two shapes congruent? a) b) c) d) e) f) Answers: a) no, b) yes, c) no, d) yes, e) yes, f) yes SAY: When two shapes are congruent you can fit one exactly on top of the other with no spaces or overlaps. You might need to flip it or rotate it, but you will be able to fit it onto the other one exactly. Activity Give each student a unique, fairly small, triangle cut out from bristol board. Have students trace the triangle several times on a sheet of paper, being sure to flip and/or rotate it. Ask them to also draw some non-congruent triangles and to try to make it difficult to see which ones are congruent. Then have students exchange sheets and triangles with a partner. The partner will then predict which shapes are congruent to the triangle and check their answers by rotating or flipping to see if the triangle fits exactly. (end of activity) Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-15

16 Using a grid on the board (e.g., projected from BLM 1 cm Grid Paper), draw several shapes on the grid as shown below: Ask volunteers to draw shapes congruent to the shapes on the board, but ask them to orient the shapes differently by rotating and/or flipping them. Draw the first two shapes below on the board: Ask students to explain why the shapes are congruent. Add the third shape and ask them to explain why this shape is not congruent to the other two. (sample answer: it has a row of squares that is three squares long and the other two don t) Do two given angles and the side length between them make exactly one triangle (up to congruence), more than one triangle, or no triangle? Write on the board: a 60 angle, a 70 angle, and the side between them is 2 inches long ASK: Will it be possible to create a triangle satisfying these conditions? (yes) How do you know? ( = 130, which is less than 180 ) Tell students that you know you can create at least one triangle, but now you want to know how many different triangles you can create. Show the four ways of starting the triangle below on the board: A. B. C. D in in in 2 in 70 Give each student a letter (A, B, C, or D) and have them draw the triangle according to their letter on a blank sheet of paper, then cut out the triangle. Bonus: Draw a triangle that is different from any of the four shown, but still satisfies the conditions. G-16 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

17 When students are finished, have them cut out their triangles and get into groups of four students, each with a different letter than themselves. Have them check for congruence by rotating or flipping when necessary. (all triangles are congruent) If some students made extra triangles, they can check those for congruence, too. Tell students that all triangles satisfying the given conditions are congruent. Exploring congruent shapes. Draw on the board: SAY: You can add a square to this shape in 10 different places (point to them), but some of the ways will be congruent. Show the first five ways below on the board: Point to the first two shapes and ASK: Are these two shapes congruent? (no) Point to the third shape and ASK: Is this congruent to either of the first two shapes? (yes, to the first shape) How do you know? (you can flip the first shape to get the third shape) Is the fourth shape congruent to any previous shape? (no) How do you know? (it has a row of squares that is four squares long) How about the fifth shape? (yes, it is congruent to the fourth shape) Cross out the shapes that are the same as a previous shape, as shown below: SAY: So we have three non-congruent shapes so far. Exercise: Continue going around the shape adding squares. Did you find any new shapes or are they all congruent to a previous shape? Answer: They are all congruent to a previous shape. Draw on the board: a) b) c) d) e) f) For each shape, ASK: How many places can you add a square to this shape by going around in order? (8, 10, 12, 14, 16, 18) (MP.1) Exercises: Copy the shape into your notebook as many times as you need to. How many non-congruent ways can you add a square to the shape? a) b) c) d) e) f) Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-17

18 (MP.8) Bonus: How many non-congruent ways can you add a square to a rectangle made of 3 rows of 19 squares? Sample solution: e) Answers: a) 3; b) 3; c) 2; d) 4; e) 5; f) 5; Bonus: 12, because except for the 3 by 3 array, the pattern would be 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, and so on. Extensions (MP.3) 1. The pattern in the last Bonus question of the lesson should be 3, 3, 4, 4, 5, 5,. Why does it change when there are 3 rows in each square? Answer: When there are 3 rows of 3 squares, adding a square to the side does the same thing as adding a square to the top or bottom. In any of the other cases, adding a square to the side is always different from adding a square to the top. (MP.1) 2. How many non-congruent shapes can you make by removing one square from the 3 by 3 array? Answer: 3 shapes, as shown below: (MP.1) 3. How many non-congruent shapes can you make by removing 2 squares from the 3 by 3 array? Answer: 8 shapes. Encourage students to proceed systematically in looking for the answer. For instance, they might start by finding all the shapes they can make after they have removed a corner square: Then they could try removing a middle square on the outside of the figure: G-18 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

19 (Notice the last two shapes have been crossed out because they are already on the list for the previous figure.) Finally, they could try removing the middle square (but all of the shapes that can be made after removing the middle square are already listed). WORKBOOK 6:1 PAGE Investigate to find what the relationship is between the angle measures a and b in the picture below that makes the rays to get closer together and hence makes a triangle. a Answer: b > a b Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-19

20 G7-4 Drawing Triangles on Grid Paper Pages Standards: 7.G.A.2 Goals: Students will draw triangles on grid paper to satisfy given conditions. Students will understand that some conditions are consistent with more than one non-congruent triangle. Prior Knowledge Required: Understands that congruent shapes are the same shape and size Can draw a 90 angle on grid paper Can measure angles Vocabulary: angle, congruent, degree ( ), protractor, ray, unique triangle Materials: grid paper or BLM 1 cm Grid Paper (p. J-1) transparency of BLM 1 cm Grid Paper (p. J-1) Review drawing triangles with a protractor and ruler. Remind students that in the last lesson they drew triangles from given conditions in particular, two given angles and a side between them. ASK: What do you have to know about the angles to know that it s possible to create the triangle? (that they add to less than 180 ) If the conditions do make a triangle, how many different triangles do they make? (only 1; all possible triangles are congruent) What tools did you use to draw the triangles? (a protractor and a ruler) Drawing right triangles with grid paper instead of a protractor and ruler. Tell students that they will draw triangles from given conditions, but instead of using a protractor and a ruler, they will use grid paper. Explain that some angles are really easy to draw using grid paper. ASK: How can you use grid paper to draw a 90 angle? (the horizontal and vertical lines make a 90 angle with each other) Tell students that in the Exercises below, all lengths are in units equal to the length of one grid square. Exercises: Draw a triangle according to the given conditions. a) two sides of length 3 and the angle between them is 90 b) a side of length 2, a side of length 5, and the angle between them is 90 Have students cut out their triangles from the Exercises above and check with a partner that their triangles from each part are congruent. G-20 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

21 Drawing 45 angles on grid paper. Have students draw a 90 angle on grid paper, then fold it in half. Have students predict what the degree measure will be (45 ) and explain their prediction. (45 is half of 90) Have students use a protractor to check that their prediction is correct. Then have students look at the fold marks they made and ASK: How can you use the grid lines to draw a 45 angle on grid paper? (go up or down one unit for every unit you go right) NOTE: For the Exercises below, you may need to get students started by drawing a vertical or horizontal ray for them. Exercises: On grid paper, draw as many ways of making a 45 angle as you can. Bonus: Draw 135 angles. Sample answers: Drawing triangles that have two 45 angles. Exercises: a) Draw a triangle with two 45 angles. The side between the angles is i) 2 units long ii) 5 units long iii) 7 units long b) Then cut out the triangles and make sure they are congruent to a partner s triangles. Bonus: Draw the first triangle from part a), but not on grid paper. Make sure it is congruent to the triangle you drew on grid paper. (MP.8) The third angle in a triangle with two 45 angles is always 90. Then have students use a protractor to measure the third angle. ASK: What do you notice? (the third angle is always 90 ) Tell students that when two angles in a triangle are 45, the other angle is always 90. NOTE: In the next lesson students will learn that the sum of the angles in a triangle is always 180. For now, students only need this special case. Non-congruent triangles with the same angles in both triangles and one same side length. Draw on the board: Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-21

22 Have students list some of the things these two triangles have in common. Prompt them to think about angles and side lengths. (sample answers: both have two 45 angles and one 90 angle, both have at least one side of length 4) SAY: These two triangles have a lot in common. ASK: Are they congruent? (no) How do you know? (they are not the same size) If some students say they are not the same shape, tell students that actually they are the same shape, and that would be easier to see if you could rotate one of them. ASK: Which triangle has the 90 angle between two 4 cm sides? (the second one) Which triangle has the 90 angle opposite the 4 cm side? (the first one) SAY: If you know two angles and a side length, or even three angles and a side length, you can still have two non-congruent triangles satisfying the conditions. If there is exactly one triangle (up to congruence) that satisfies the conditions, then you can say that the triangle is the unique triangle satisfying the conditions. Exercise: On grid paper, draw two non-congruent triangles with two 45 angles and a side of length 3. Answer: Both triangles will have a 90 angle. One triangle will have the sides adjacent to the 90 angle of length 3, and the other triangle will have the side opposite the right angle of length 3. Extension How many non-congruent triangles can be made to satisfy the conditions: a unique triangle, no triangle, or more than one triangle? a) a 90 angle, a 45 angle, and the side between them is 5 cm long b) two 45 angles and the side between them is 2 inches long c) three 45 angles d) two 45 angles and a 90 angle e) two 45 angles and a side 2 inches long f) two 90 angles and the side between them is 4 inches long Answers: a) a unique triangle, b) a unique triangle, c) no triangle, d) more than one triangle, e) more than one triangle, f) no triangle G-22 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

23 G7-5 Constructing Triangles from Three Sides or Three Angles Pages Standards: 7.G.A.2 Goals: Students will determine the conditions for which three side measures or three angle measures make exactly one triangle, more than one triangle, or no triangle. Prior Knowledge Required: Can determine whether two triangles are congruent by trying to fit one on top of the other Can measure angles Vocabulary: angle, congruent, degree ( ), point, protractor, unique triangle, vertex Materials: a triangle-building machine for each student (plus one for yourself), made using BLM Triangle- Building Machine (p. G-59) and two paper fasteners paper clips rulers Constructing triangles from three side lengths. Provide each student with a triangle-building machine made from BLM Triangle-Building Machine and two paper fasteners. Also give each student a paper clip. Tell students that when they use their triangle-building machine to make triangles, the longest side will always be 10 units long. They could change the machine to make any number of units up to 10, but in this class, they will always use 10 as the longest side. Show students how to make a triangle with any given side lengths. For example, to make a triangle with side lengths 5, 7, and 10, fold one of the strips under at the 5 and the other strip under at the 7, and then rotate them until the 5 dot and the 7 dot meet. Tell students to make sure they keep all the strips flat along the table. Show students how they can do it wrong if they stretch the strips by lifting one end off the table. That makes the dots meet at a point farther from the other side than they should. Students can hold the third vertex of the triangle together with a paper clip, then check with a partner to see if their triangles are congruent. Remind students that when two triangles are congruent, one will fit exactly on top of the other without gaps or overlaps, but they might need to rotate or flip the triangle to make it fit exactly. Make sure students do at least one of the following Exercises. Students who finish early can do more. Exercises: Use the triangle-building machine to construct the triangle. Check with a partner that your triangles are congruent. a) 3, 9, 10 b) 6, 8, 10 c) 8, 8, 10 Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-23

24 If some students find that their triangles are not congruent, emphasize the importance of building the machine correctly and of not stretching any sides. Point out also that if some students made the 3 and 9 sides from part a) on opposite sides of the 10, then one partner will need to flip their triangle to fit onto the other one. Be sure that students have done it correctly and have found their triangle to be congruent to those of everyone else who has made the triangle with the same side lengths. (MP.8) Not all side lengths make a triangle. Tell students that not all possible side lengths will work. Demonstrate with sides of length 2, 3, and 10 (see diagram below). SAY: The sides of length 2 and 3 are too far from each other to complete the triangle. 2 cm 3 cm 10 cm Tell students that you would like to investigate when three side lengths make a triangle. In the Exercises below, students will need their triangle-building machine. Exercises: Complete the table. Three Sides Sum of the Two Do the Sides Make Shortest Sides a Triangle? a) 4, 5, 10 b) 4, 9, 10 c) 7, 7, 10 d) 2, 4, 10 Bonus: 3, 7, 10 Answers: a) 9, no; b) 14, yes; c) 14, yes; d) 6, no; Bonus: 10, no (MP.3) SAY: When the shorter sides put together are too short compared to the longest side, they can t fit together to make the third point of the triangle. Have volunteers try to articulate the precise rule. Some students might say that the shorter sides put together can t be shorter than the longest side, others might say that the shorter sides put together have to be longer than the longest side. Point out that there is a slight difference between the two answers if they both come up. Then tell students you want to know what happens when the longest side is exactly equal to the shorter two sides put together. Demonstrate by trying to make a triangle. Point out how the sides of length 3 and 7 land onto the side of length 10 exactly (you will need to turn the triangle machine face down to see this) and so, while the two points do meet at a third point, the triangle has no height, so it s not a triangle! (MP.6) Use this to emphasize the importance of students being precise while they work. If they made their triangle incorrectly or stretched the sides by lifting one end off the table, they would make an incorrect triangle. Exercises: Without constructing a triangle, decide whether the side lengths make a triangle. a) 3 in, 5 in, 8 in b) 2 mm, 5 mm, 6 mm c) 18 mi, 23 mi, 42 mi d) 2 cm, 3 cm, 4 cm e) 2 cm, 5 cm, 8 cm f) 17 in, 3 in, 20 in (MP.2) Bonus: 4 cm, 18 mm, 23 mm (Hint: Be careful with the units.) G-24 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

25 Answers: a) no; b) yes; c) no; d) yes; e) no; f) no; Bonus: yes, because 4 cm = 40 mm, and > 40 When do three angle measures make a triangle? SAY: We know when three side lengths make a triangle. Now I want to know when three angle measures make a triangle. Have students draw triangles using a ruler to make sure that the sides are straight, but without paying attention to the side lengths. Have them measure the angles in their triangles. When students finish, point out that the sum of the angles in a triangle is always exactly 180 and it is only measurement error that will make them get something slightly different. Then remind students that, in the previous lesson, they learned that when two angles in a triangle are 45, the third angle is 90. SAY: That makes sense because = 180. SAY: Let s assume we have three angles that add to 180, so we know that they make a triangle. Write on the board: 40, 60, 80 SAY: I want to know if they will make a unique triangle or if there will be more than one noncongruent triangle. Have students make predictions, then suggest ways of checking the prediction. SAY: You don t know the length of the side between the 40 and 60 angle, so try making one up and see if it works. Write on the board: A. a 40 angle, a 60 angle, and a 5 cm side between them B. a 40 angle, a 60 angle, and a 6 cm side between them C. a 40 angle, a 60 angle, and a 7 cm side between them Point to A and ASK: Do these conditions make a unique triangle? (yes) Repeat for B and C. (both yes) Point to each in turn again, and ASK: What will the third angle be? (80 ) How do you know? (because the three angles have to add to 180 ) Have students use a ruler and protractor to draw all three triangles. SAY: So all of these triangles have angles 40, 60, and 80, so you can actually make many triangles having angles 40, 60, and 80. None of these triangles are congruent because if you put the 40 angle from A onto the 40 angle from B, the 60 angles won t match up. In B, it is farther from the 40 angle than it is in A. The triangles are all the same shape, but they are all different sizes. Extensions (MP.1) 1. a) A triangle with perimeter 8 has whole-number side lengths. What are the sides of the triangle? b) A triangle with perimeter 13 has whole number side lengths. Its shortest two sides are equal. What are the sides of the triangle? Answers: a) 2, 3, 3; b) 4, 4, 5 Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-25

26 (MP.7) 2. Can the quadrilateral below exist? Hint: Divide the shape into a triangle and square Answer: Dividing the shape into a triangle and a square makes a right triangle with side lengths 4, 3.5, and 8, but < 8, so the triangle, and hence the quadrilateral, does not exist. G-26 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

27 G7-6 Constructing Triangles from Three Measures Pages Standards: 7.G.A.2 Goals: Students will be given three measures and decide whether the three measures determine a unique triangle, no triangle, or more than one triangle. Prior Knowledge Required: Can use a protractor and a ruler to draw triangles Can determine whether two triangles are congruent Can measure angles Knows that the sum of the angles in a triangle is 180 Knows what acute, right, and obtuse angles are Vocabulary: acute, angle, congruent, degree ( ), endpoint, obtuse, point, ray, right angle, unique triangle, vertex Materials: protractors rulers Review the conditions for three given angles to make a triangle. Tell students that in this unit they have been learning what kinds of conditions on triangles make a unique triangle, more than one triangle, or no triangle. ASK: When do three angles make a triangle? (when they add to 180 ) Review the terms right angle (90 ), acute angle (less than 90 ), and obtuse angle (greater than 90 ). Then have students signal their answers to the questions below. ASK: Can there be a triangle with a) two right angles? b) two obtuse angles? c) three acute angles? d) exactly two acute angles? e) a right angle and an obtuse angle? f) an acute angle and an obtuse angle? (MP.3) Have volunteers explain their reasoning for any no answers or draw a sample for any yes answers. (a) no, two right angles already add to 180 ; b) no, they add to more than 180 ; c) yes, for example an equilateral triangle; d) yes, any triangle with a right angle or an obtuse angle will have exactly two acute angles; e) no, they add to more than 180 ; f) yes, any triangle with an obtuse angle will also have an acute angle) Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-27

28 ASK: Do three angles that add to 180 make a unique triangle or more than one triangle? (more than one) Ask a volunteer to sketch on the board two non-congruent triangles having angles 90, 45, and 45. (see sample answers below) Review the conditions for three given side lengths to make a triangle. ASK: When do three side lengths make a triangle? (when the shortest two sides add to more than the longest side) What if the shortest two sides add to the same as the longest side? Do they make a triangle? (no) Why not? Remind students that the two shorter sides will meet at a third point, but that third point is on the longest side, so the triangle has no height and is not actually a triangle. Another way to frame the side lengths rule for triangles. Remind students that the perimeter of a shape is the distance around the shape. Draw on the board: 4 cm 5 cm 3 cm ASK: What is the perimeter of this triangle? (12 cm) How did you get that? (add the three side lengths) Tell students that they can always get the perimeter of a triangle by adding its three side lengths. Write on the board: Longest Side Perimeter (cm) (cm) Sum of the Two Shortest Sides Did it Make a Triangle? (MP.3) Point to the first row and ASK: What is the sum of the two shortest sides of the triangle? (6 cm) PROMPT: The longest side is 8 cm and the sum of all three sides is 14 cm. Write 6 in the third column. ASK: If the two shortest sides add to 6, and the longest side is 8, do the sides make a triangle? (no) Complete the first row, then have volunteers help you complete the table. (7, no; 8, no; 9, yes; 10, yes) ASK: How can you explain the rule in terms of longest side and perimeter? (when the perimeter is more than double the longest side, you can make a triangle) Exercises: Can you make a triangle with a) longest side 10 cm and perimeter 18 cm? b) longest side 5 inches and perimeter 12 inches? c) longest side 7 mi and perimeter 18 mi? d) longest side 1.83 miles and perimeter 3.4 miles? Bonus: longest side 18 ft and perimeter 11 yd? Answers: a) no; b) yes; c) yes; d) no; Bonus: no, because 11 yards = 33 feet, and 33 < 18 2 G-28 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

29 Review the conditions for two angles and a side. SAY: If two angles are given and the length of the side between them is given, then you can make a unique triangle satisfying the conditions. Have students construct a triangle with a 40 angle, a 50 angle, and a 4 cm side between them. When students finish, write on the board: a 40 angle a 50 angle a 4 cm side ASK: Does the triangle you drew satisfy all these conditions? (yes) What will the other angle always be? (90 ) How do you know? (because the three angles have to add to 180 ) Add this condition to the list on the board: a 40 angle a 50 angle a 4 cm side a 90 angle SAY: The 4 cm side doesn t have to be between the 40 and 50 angles; it can also be between the 40 and 90 angles or between the 50 and 90 angles. So there are actually three different cases. Show the three different cases on the board: cm 4 cm cm Provide each student with a protractor and a ruler. Exercises: Draw all three possible triangles you can make with a 40 angle, a 60 angle, and a 4 cm side. Bonus: How many non-congruent triangles can you make with a) a 40 angle and a 70 angle and a 4 cm side? b) two 60 angles and a 4 cm side? Answers: There are three non-congruent triangles depending on where the 4 cm side is; Bonus: a) there are only two different triangles because the other angle is 70, and the triangle with the 4 cm side between one 70 angle and the 40 angle is congruent to the triangle with the 4 cm side between the other 70 angle and the 40 angle; b) there is only one possible triangle because all angles are 60, so no matter which two 60 angle the 4 cm side is between, all the triangles are congruent. When students finish the Exercises above, point out that when they are told the side length between two given angles, then the triangle is unique. But if they are just given a side length without being told where it is, there can be more than one triangle. Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-29

30 Two sides and an angle. SAY: There are four ways you can be given three pieces of information about a triangle. Write on the board: three given side lengths three given angle measures two angles and a side two sides and an angle SAY: We ve looked at the first three so far, but we haven t considered the last one yet. Write on the board: a 3 cm side a 5 cm side an 80 angle (MP.3) SAY: Let s consider all possible cases. ASK: Where can the 80 angle be relative to the sides? (between the two sides; beside the 3 cm side, but not the 5 cm side; beside the 5 cm side, but not the 3 cm side) Demonstrate beginning each case as shown below. NOTE: In the first picture, both sides have given lengths. The next two diagrams have one side without a given length, so that side requires a ray to show that the length is infinite. 5 cm 3 cm cm 5 cm Point out that in the second case, students need to try to find a point on the ray that is 5 cm from the other endpoint of the 3 cm side. They can do this by using a ruler and rotating it around the endpoint of the 3 cm side. ASK: At what point on the ray is the distance 5 cm? In the third case, they need to try to find a point on the ray that is 3 cm from the other endpoint of the 5 cm side. Tell students that some cases might not be possible and they have to decide which ones are possible. The best way to do that is to try to construct all three triangles. G-30 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

31 Exercises: 1. Try to construct all three triangles. Which ones are possible? Answer: Only the first two are possible; in the third case, the ray is always farther than 3 cm from the opposite vertex. 2. Draw two non-congruent triangles with a 3 inch side, a 5 inch side, and a 90 angle. Answers: The two possibilities are for the 90 angle to be between the 3 inch side and the 5 inch side, and for the 90 angle to be adjacent to the 3 inch side but not the 5 inch side. Extensions (MP.3) 1. Sara says a triangle can have longest side 5 cm and perimeter 16 cm. Explain why she s wrong. Answer: If the perimeter is 16 cm, then the sum of the remaining sides is 11 cm. Therefore, the 5 cm side is not the longest. (MP.3) 2. Find four different triangles that have a 3 cm side, a 5 cm side, and a 30 angle. Answer: There are three different cases to consider. Case 1: the 30 angle is between the two sides. 5 cm 30 3 cm Case 2: the 30 angle is adjacent to the 5 cm side, but not to the 3 cm side. 3 cm 3 cm 30 5 cm This case has two triangles. Case 3: the 30 angle is adjacent to the 3 cm side, but not to the 5 cm side. 5 cm 30 3 cm Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-31

32 3. How many non-congruent triangles can you make with a 60 angle and two 3 cm sides? Answer: Only one any triangle you can make with these conditions will be an equilateral triangle with all sides equal to 3 cm. NOTE: In Grade 8, students will learn that when two sides are equal in a triangle, the angles opposite those sides are equal as well. They will then be able to prove their discovery from Extension 3. G-32 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

33 G7-7 Counterexamples Pages Standards: 7.G.A.2 Goals: Students will determine whether statements are true or false. Students will use counterexamples to disprove false statements and reasoning to prove true statements. Prior Knowledge Required: Knows that congruent triangles can fit exactly on top of one another without overlaps or gaps Knows that the sum of the angles in a triangle is 180 Knows that two non-congruent triangles can have all the same angles Vocabulary: angle, area, congruent, counterexample, degree ( ), perimeter Materials: BLM Sudoku Introduction (pp. G-60 61, see Extension 1) BLM Sudoku Another Strategy (p. G-62, see Extension 1) BLM Sudoku Advanced (p. G-63, see Extension 1) Anno s Hat Tricks by Akihiro Nozaki and Mitsumasa Anno (see Extension 8) Introduce the term counterexample. Draw and write on the board: All circles are shaded. Have a volunteer identify which circle shows that the statement isn t true. (the third one) Tell students that an example that proves a statement false is called a counterexample to the statement. NOTE: For the Exercises below, draw the triangles so that A and D are isosceles, B is right scalene, and C is equilateral but rotated. Exercises: Which shape is the counterexample to each statement? a) All triangles are striped. b) All triangles have a horizontal side. Bonus: All triangles have at least two equal sides. Answers: a) D, b) C, Bonus: B Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-33

34 Recognizing when a statement does not apply to all examples. Draw and write on the board: All circles are shaded. ASK: What is this statement about? (circles) Underline all the circles. Emphasize that the statement refers only to the circles it doesn t matter whether any of the other shapes are shaded. ASK: Are all circles shaded? (no) Have a student circle the counterexample. (E) Erase the underlining and the circling and repeat with new statements (see below), underlining the relevant shapes first. Emphasize in each case that the sentence is only about the shapes you underline; the shapes that are not underlined don t matter. Write the following statements on the board, one at a time: All squares are large. All large squares are shaded. All shaded circles are small. Have volunteers name the counterexample for each statement. (D, F, and A) (MP.3) Exercises: Use the same picture to name a counterexample for the statement. a) All shaded shapes are circles. b) All unshaded shapes are small. c) All shaded shapes are large. d) All unshaded shapes are squares. e) All large shapes are squares. f ) All large shapes are shaded. g) All small shapes are unshaded. h) All small unshaded shapes are squares. Answers: a) B, b) F, c) C, d) E, e) A, f ) F, g) C, h) E As students do the Exercises above, encourage them to first write down the shapes that the statement is talking about. For example, the statement in part a) is about the shaded shapes A, B, and C. This is where students should look for a counterexample. For students who need extra help, you can draw the shapes in their notebooks for them, and they can underline the shapes each question is referring to (and erase the underlining before doing each new question). False statements about words. Write on the board: All words start with the letter b. G-34 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

35 Ask if each of the words below is a counterexample to the statement and have students explain why or why not: bat (no, because it does start with b) cat (yes, it is a word that does not start with b) boat (no, because it does start with b) bxcv (no, because it does start with b OR no, it s not a word and the statement only talks about words, so something that is not a word cannot be a counterexample) xcvb (no, because it s not a word,) (MP.3) Exercises: Circle the counterexample to the statement. a) All nouns have an e. red brown truck bike b) All even numbers have a digit c) All numbers divisible by 5 have ones digit Answers: a) truck, b) 34, c) 40 Review the word vowel if necessary. The letters a, e, i, o, u, and sometimes y are vowels. Exercises: 1. To which of these statements is Bob a counterexample? A. All names have two vowels. B. All names have three letters. C. All names have four letters. D. All boys names start with D. E. All names are boys names. F. All names read the same backward as they do forward. Answers: A, C, and D. 2. Find a counterexample to the three statements in Exercise 1 for which Bob is not a counterexample. Try to find one example that works as a counterexample to all three statements at the same time. Sample answer: Sara. (MP.3) Bonus: Explain why there cannot be a counterexample to all six statements in Exercise 1. Answer: To be a counterexample to D, the name would have to be a boy s name. On the other hand, to be a counterexample to E, the name would have to not be a boy s name. So, there cannot be a counterexample that disproves both D and E at the same time. Therefore, there cannot be a counterexample that disproves all six statements at the same time. If a hint is necessary for the Bonus above, point students attention to statements D and E. Explain that to reject D, you need a boys name that doesn t start with D and, for E, you need a name that is not a boys name. Teacher s Guide for AP Book 7.1 Unit 6 Geometry G-35

36 Proving a statement true by checking all examples. Draw and write on the board: All squares are shaded. Demonstrate checking all the squares to see if they are shaded. They are, so the statement is true. Repeat with the following statement: All triangles have a horizontal side and have volunteers check all triangles. (again, the statement is true) Repeat with All squares have a horizontal side. (this statement is false; I is a counterexample) Point out that to show a statement is true, students need to check all examples. To show a statement is false, they need to identify just one any counterexample. Have students use the same shapes to complete the following Exercises. Exercises: Decide whether the statement is true or false. a) All striped shapes are large. b) All triangles are large. c) All large circles are shaded. d) All small squares have a horizontal side. e) All small shapes have a horizontal side. Bonus: All large shaded triangles are equilateral. Answers: a) true; b) false; D; c) false, H or K; d) true; e) false; J; Bonus: false, E Using reasoning to prove a statement true. Write on the board: Whenever it is raining, there are clouds. ASK: Is this statement true or false? (true) Do you have to check for clouds every time it rains to know that the statement is true? (no) How do you know without checking that it is true? (rain can only come from clouds) Tell students that there is often a reason why a statement is true. When there is, students don t have to check all examples to prove it. Review the words even and odd as they apply to numbers. (even numbers are multiples of 2; odd numbers are not even) Then write the following two statements on the board: All even numbers have an even digit. All even numbers have an odd digit. Tell students that one of the statements is true and the other is false. ASK: Which statement is true? (all even numbers have an even digit) How do you know it s true? (because the ones digit G-36 Teacher s Guide for AP Book 7.1 Unit 6 Geometry

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