T616 Mathematics Success Grade 8 [OBJECTIVE] The student will explore angle relationships within triangles. [PREREQUISITE SKILLS] angles, congruency, supplementary angles, adjacent angles [MATERIALS] Student pages S307 S314 Plain Paper Ruler or Straight-edge Protractor Tape or Glue Scissors [ESSENTIAL QUESTIONS] 1. What angle relationship did we discover about the interior angles of a triangle? 2. Explain the relationship between an exterior angle of a triangle and the two nonadjacent interior angles. 3. What is the relationship between an exterior angle of a triangle and the adjacent angle? [WORDS FOR WORD WALL] interior angles, exterior angles, supplementary angles, adjacent interior angle, non-adjacent interior angles [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer [WARM-UP] (IP, I, WG) S307 (Answers on T624.) Have students turn to S307 in their books to begin the Warm-Up. Students will use knowledge of angles to complete the assignment. Monitor students to see if any of them need help during the Warm-Up. After students have completed the Warm-Up, review the solutions as a group. {Graphic Organizer, Pictorial Representation} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [1 2 Days (1 day = 80 minutes) M, GP, WG, CP, IP]
Mathematics Success Grade 8 T617 SOLVE Problem (WG, GP) S308 (Answers on T625.) Have students turn to S308 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to explore and determine the relationships between angles. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer, Pictorial Representation} Discovery Activity Interior Angles of Triangles MODELING (M, GP, CP, WG) S308 (Answers on T625.) M, CP, GP, WG: After students complete the S Step of the SOLVE problem, distribute two pieces of paper to each student to complete the next activity. Prior to beginning the activity, be sure to identify Partners A and B for activities. {Verbal Description. Concrete Representation, Graphic Organizer} Discovery Activity Interior Angles of Triangles Step 1: Have each student draw three points on the piece of paper. What shape would we create if we connected the three points on the paper? (a triangle) *Teacher Note: Have the students place the points far enough apart that they will have a large enough triangle to cut. While we want very different triangles for each student, it s important that the triangle is large enough for the activity. Next, have students use a straight-edge to connect all three points, creating a triangle. After students have finished drawing their triangles, take a moment to have the students hold their drawings up so that the rest of the class can see the triangle. Students may do this all at the same time. What do you notice about the triangles that we have drawn? (Students may have triangles that look somewhat similar to their own or they may see some that look very different. The only true similarity is that each students drawing is a triangle.) Model how to tear apart the angles of the triangle. We have two options for tearing the triangle. See the options below. 1. Tear off the points of the triangle such that you tear off three angles.
T618 Mathematics Success Grade 8 2. Tear so that three pieces of the triangle are created Either of the ways displayed will work for this activity. The point is that we will have the three angles of the triangle available to make the discovery. Next, have students place the three angles on the second piece of blank paper that was distributed. Have students line up the angles so that the sides of the triangle are adjacent, as displayed below. The solid black lines represent the sides of triangle that aren t torn and the dashed lines represent where the triangle was torn. Partner A, describe what you notice about the angles of the triangle when laid next to each other. (Together they create a straight line.) Record. Have students use a protractor or straight-edge to see that it creates a line of 180 degrees. Partner A, use a protractor to measure each of the three angles of your triangle, while Partner B does the same for his or her angles. Record them in the chart at the bottom of S308. What is the sum of the measures of the angles? (180 degrees) Record. What do you notice about your angles and your partner s angles? (They are different measurements.) What do you notice about the sum of the angles for you and your partner? (They are both 180 degrees.) Partner B, what can we conclude about the sum of the (interior), or inside, angles of a triangle? The sum is always (180) degrees.
Mathematics Success Grade 8 T619 Finding Missing Angles (M, GP, CP, IP, WG) S309 (Answers on T626.) M, GP, CP, WG: Using the knowledge that the sum of the interior angles of a triangle is always equal to 180 degrees, students will solve for the missing angles in triangles that are provided. Be sure that students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer, Pictorial Representation} MODELING Finding Missing Angles Step 1: Direct students attention to Question 1. Partner A, what do we know about the angles in the given triangle? (One angle measures 140 degrees and a second angle measures 27 degrees, while the third angle is labeled as Angle a.) Have students discuss how they may be able to use the information given to determine the third angle measurement. Partner B, what do we know about the sum of the interior angles of a triangle? (The sum of the interior angles of all triangles is 180 degrees.) Partner A, explain how we can find the missing angle and justify your answer. (Find the value that when added to 140 and 27 will equal 180 degrees.) Partner B, why do we want the total to be 180 degrees? (The sum of the interior angles of all triangles is 180 degrees.) Partner A, what operation can we use? (subtraction) Partner B, explain how we can find the missing angle using subtraction. (180 140 27) Record. Partner A, what is the difference after the subtraction? (13 degrees.) Partner B, what is the measure of Angle a? (13 degrees) Record. Step 2: Direct students attention to Problem 3. Partner B, explain how this problem is different from Problem 1. (We are only given one angle measurement.) Partner A, what do the markings on the other two angles mean? (The arcs tell us that those two angles are congruent.) Have student pairs discuss how they can use the information they are given to find the measure of the missing angles. Partner B, explain the first step to find the measurements of these two angles. (Subtract the 98 degrees from 180 to find the measurement of the sum of the other two angles.) Partner A, what is the sum of the other two angles? Defend your answer. (180 98 = 82 degrees.) Record.
T620 Mathematics Success Grade 8 Partner A, if the two angles left over are congruent, how can we find what each of their measures are? Justify your answer. (Divide 82 by 2 to get an angle measure of 41 degrees.) Record. Partner B, what are the measures of each of the remaining angles? (Each angle has a measure of 41 degrees.) Record. Step 3: Direct students attention to Problem 7. Partner A, how is this problem different than the others? (It does not have a drawing, but only given angle measures.) Partner B, explain how we can solve this problem? (Use the same strategy of subtracting the given angles from 180 degrees to find the remaining angle.) Partner A, what are the remaining degrees after we subtract the known angles? (180 75 25 = 80 degrees) Record. Partner B, what is the measure of Angle C? Justify your answer. (80 degrees) IP, CP, WG: Have students work in student pairs to complete the rest of page by completing Problems 2 6 and 8 10. After students have completed the problems, review the answers as a whole group. {Verbal Description, Graphic Organizer, Pictorial Representation} Introduction to Exterior Angles (M, GP, CP, IP, WG) S310, S311 (Answers on T627, T628.) GP, CP, WG, M: Students will explore three of the same triangles with three different exterior angles and explore the characteristics and relationship between the three. Be sure students know their roles as Partner A or Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer} MODELING Introduction to Exterior Angles Step 1: On pages S308 and S309 we worked with angles on the inside of the triangle or (interior) angles. Record. Now we are going to look at the relationships of the angles on the outside of the triangle. Partner A, can you identify the term for these angles? (exterior) Record. Take a look at the first triangle. Start your pencil at Point A and trace until you get to Point B. Now, with a straight-edge or ruler, continue the line segment by extending AB past Point B.
Mathematics Success Grade 8 T621 Partner A, explain what we created when we extended the line segment. (We created a new angle.) Partner B, describe what type of angle is on the outside of the figure. Defend your thinking. (an exterior angle, because it is outside the triangle) Partner A, what do you notice about Angle B and the new angle that is created with the extension? (The two angles create a straight line.) Partner B, what is the total measure of Angle B and the new angle measure when added together? Justify your answer. (180 because it is a straight line.) Partner B, identify and explain the relationship between the adjacent interior angle and the exterior angle. (The adjacent interior angle and the exterior angle are supplementary. If one of the two is known, the other one can be found by subtracting the known angle from 180.) Step 2: Partner B, how can we find the measure of the new angle? (Subtract the measure of Angle B from 180.) Partner A, what is the measure of the exterior angle that is created? (154 ) Record. Let s study the triangle s interior angles and the new exterior angle. Do you see any relationship between the interior and exterior angles? (Students may begin to identify the connection between the two nonadjacent angles and the exterior angles.) Partner B, What is the sum of the interior angles that are not adjacent to the exterior angle? (23 + 131 = 154 ) Record. What do you notice about the sum of the non-adjacent interior angles and the exterior angle? (They have the same measure of 154.) Record. Step 3: Direct students to the second triangle on S310. Take a look at the second triangle. Start your pencil at Point B and trace until you get to Point C. Now, with a straight-edge or ruler, continue the line segment by extending past Point C. Partner A, what do you notice about Angle C and the new angle that was created with the extension? (The two angles form a straight line.) Partner B, what is the total measure of Angle C and the new angle measure when added together? Justify your answer. (180 because it is a straight line.) Step 4: Partner A, how can we find the measure of the new angle? (Subtract the measure of Angle C from 180.) Partner B, what is the measure of the exterior angle that is created? (157 ) Record. Partner A, identify the sum of the interior angles that are not adjacent to the exterior angle? Explain your thinking (26 + 131 = 157 ) Record.
T622 Mathematics Success Grade 8 What do you notice about the sum of the non-adjacent interior angles and the exterior angle? (They have the same measure of 157.) Record. Step 5: Have students complete the process of determining the measure of the exterior angle for Triangle 3 with their partners and then review the answers as a whole group. Step 6: Direct students to the top of S311. Partner A, from our discoveries on the previous page, what do we know about the angles that are not adjacent to the exterior angle? (The sum of the non-adjacent interior angles is equal to the exterior angle.) Record. Partner B, explain the relationship between the adjacent angle and the exterior angle. (The adjacent angle and the exterior angle are supplementary. If one of the two is known, the other one can be found by subtracting the known angle from 180.) Record. Step 7: Direct students to Question 3. Partner A, what is the problem asking us to find? (the missing angle measures) Partner B, using the strategy from the previous page, explain how we can find the measure of Angle d. (The exterior angle is equal to the sum of the non-adjacent angles, so we can add the non-adjacent angles). Partner A, what will be the sum that equals Angle d? (91 + 31 = 122 ) Record. Partner B, how can we find the measure of Angle a? Defend your answer. (Angle d and Angle a are supplementary angles, therefore we can subtract the measure of Angle d from 180.) Partner A, what is the measure of Angle a? (180 122 = 58 ) Record. Partner B, is there another way we could have found the measure of Angle a if we wouldn t have known the value of the Angle d? (We could subtract the other two interior angles from 180 because the sum of interior angles has to be 180.) IP, CP, WG: Students will complete Problems 4 6 in student pairs to practice identifying the missing angle measures. They will use their knowledge that the sum of the interior angles of a triangle will total to 180 as well as the two connections regarding interior and exterior angles that they concluded at the top of S311. After students have completed the problems, review the answers as a whole group. {Verbal Description, Graphic Organizer, Pictorial Representation}