Mathematics Success Grade 8

Transcription

1 T308 Mathematics Success Grade 8 [OBJECTIVE] The student will analyze the constant rate of proportionality between similar triangles, construct triangles between two points on a line, and compare the ratio of the vertical change to the horizontal change between the two similar triangles. [PREREQUISITE SKILLS] ratios, graphing, ordered pairs, similarity [MATERIALS] Student pages S149 S160 Rulers Colored pencils [ESSENTIAL QUESTIONS] 1. Explain the meaning of corresponding in relation to similar triangles. 2. How do you determine similarity? 3. How is the slope of a line related to the similar slope triangles formed by the line? [WORDS FOR WORD WALL] corresponding, proportional, similar triangles, ratios, vertical, horizontal, slope, means, extremes, y-axis, x-axis, hypotenuse [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A and 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, Algebraic Formula, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer [WARM-UP] (IP, I, WG) S149 (Answers on T316.) Have students turn to S149 in their books to begin the Warm-Up. Students will determine if two fractions are proportionate. Monitor students to see if any of them need help during the Warm-Up. Have students complete the problems and then review the answers as a class. {Verbal Description} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [3 Days (1 day = 80 minutes) - (M, GP, WG, CP, IP)]

2 Mathematics Success Grade 8 T309 SOLVE Problem (WG, GP) S150 (Answers on T317.) Have students turn to S150 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 find the relationship of similar triangles on the coordinate plane. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graph, Graphic Organizer, Graph} Identifying Similar Triangles (M, GP, WG, CP) S150, S151 (Answers on T317, T318.) WG, CP, GP, M: Have students turn to S150. Students will be working on how to identify similar triangles. Assign the roles of Partner A and Partner B for specific discussions during the lesson. {Pictorial Representation, Algebraic Formula, Verbal Description} MODELING Identifying Similar Triangles Step 1: Direct students attention to Triangle 1 and Triangle 2. Have student pairs discuss observations about the two triangles and offer the opportunity for pairs to share their observations. Partner A, Triangle 1 and Triangle 2 are both what kind of triangles? (right). Explain your thinking. (The right angle is marked with a small square in each triangle representing a 90-degree angle.) Record. Step 2: Direct students attention to S151. Have student pairs talk about the meaning of corresponding sides. Partner B, explain the meaning of corresponding sides. (Corresponding sides are the sides of the triangles that are in the same position.) Record. Have student pairs work together to determine the corresponding sides for Triangle 1 and Triangle 2 and then share answers as a whole class. Partner A, what is the corresponding leg to leg AB, in Triangle DEF? (DE) Record. Partner B, what is the corresponding leg to leg BC, in Triangle DEF? (EF) Record. Partner A, what is the corresponding leg to leg CA, in Triangle DEF? (FD) Record.

3 T310 Mathematics Success Grade 8 Step 3: Direct students attention to Questions 4 and 5. Have student pairs discuss how the corresponding relationships are represented. If we look at Triangle ABC we want to find corresponding sides (matches) to Triangle DEF. Partner A, explain how the corresponding relationship is written. (as a ratio in the form of a fraction) Partner B, what is the corresponding side to leg AB, in Triangle DEF? (DE) What are their lengths? (6, 12) Record. Partner A, what is the corresponding side to leg BC, in Triangle DEF? (EF) What are their lengths? (8, 16) Record. Partner B, what is the corresponding side to leg CA, in Triangle DEF? (FD) What are their lengths? (10, 20) Record. Partner A, what do the numbers in each ratio represent? (the length of the corresponding sides) Ask your students to look at each of the numerical ratios and discuss the relationship between each numerator and denominator. Partner B, describe what you notice about each of the ratios. (The ratios can be simplified.) Record. Step 4: Direct students attention to Questions 6 through 8. Ask students to look at each of the ratios and discuss how they can be simplified. Shared answers may include: simplify like fractions, find the GCF, or divide the numerator and denominator using the GCF. Have student pairs simplify each ratio. Partner B, what did you discover about each ratio? (They all simplify to 1 2.) Record. Ask your students to discuss their observations and conclusions about the relationship between the three ratios of the corresponding sides. Partner A, explain the relationship between the three ratios. (They are equivalent.) Record. Partner B, when we have ratios that are written as equivalent fractions we know that they have what kind of relationship? (proportional) Record. Step 5: Have students pairs look at Question 9. The statement given is that Triangle 1 and Triangle 2 are similar triangles. Given that true statement, have students work together to create a definition of similar triangles with their partners. Guide students to use the information they have discovered on S151 in order to create their definition. Bring the class back together as a whole group to share possible definitions. (Answers may vary slightly, but work together to create a definition that includes the following: Similar triangles are triangles whose corresponding sides form a proportional relationship.) Record.

4 Mathematics Success Grade 8 T311 More on Identifying Similar Triangles (WG, GP, CP, M) S152 (Answers on T319.) WG, CP, GP, M: Have students turn to S152. Students will be working on how to identify similar triangles when one is inside the other. {Pictorial Representation, Algebraic Formula, Verbal Description, Graphic Organizer} MODELING More on Identifying Similar Triangles Step 7: Direct students attention to S152. Have student pairs make observations about the two triangles on S152. (One triangle is inside the other.) Record. (Also discuss that the symbol ~ means similar.) Have student pairs work together to list the corresponding sides for Triangle QRS and Triangle PRT. (RS corresponds to RT, RQ corresponds to RP, QS corresponds to PT) Record. Have student pairs answer Questions 3 and 4 together. Review the values for the corresponding leg lengths in Question 3. Have students share their observations for Question 4. (No measurements are given for RQ and RP, the hypotenuse of each triangle.) Record. Ask students to discuss the significance of not having the hypotenuse measurement and then draw a conclusion about it not being given information. Bring the class back together and share ideas. (Possible answers can include: unnecessary information, you can use two sides to determine proportionality.) Share with students that if the triangles they are comparing are right triangles, they only need the measurements of the two legs to determine proportionality. Step 8: Have students reference Question 8 on S151 and then complete Questions 5 and 6 in student pairs. Regroup as a whole class and share answers for Questions 5 and 6. Ask students to share other ways they can determine proportionality of ratios. (Multiply the cross products to see if they are equal.) Record. Partner A, share the cross product of the extremes. (36) Record. Partner B, share the cross product of the means. (36) Record. *Teacher Note: Reference and review bolded vocabulary as necessary using the word wall. Step 9: Have student pairs answer Questions 8 and 9. Regroup as a whole class to share answers. What is vital here is that the students understand that if the corresponding sides are proportional, then the two triangles are similar.

5 T312 Mathematics Success Grade 8 Similar Triangles on the Coordinate Plane (WG, GP, CP, IP, M) S153, S154, S155, S156 (Answers on T320, T321, T322, T323.) WG, CP, GP, M: Have students turn to S153. Distribute rulers and colored pencils. Students will be working with similar triangle relationships on the coordinate plane. Be sure students know their designation as Partner A or Partner B. {Pictorial Representation, Algebraic Formula, Verbal Description, Graph, Graphic Organizer} MODELING Similar Triangles on the Coordinate Plane Step 1: Have student pairs work together to plot the coordinates in Question 1 and Question 2. Review graphing of coordinate points as necessary. Partner A, identify what figures are formed. (They form right triangles.) Record. Partner B, explain how you know they are right triangles. (Some possible answers are: a right angle is formed by two of the legs, we have two perpendicular lines, etc.) Step 2: Have students turn to S154. They will need to refer to the coordinate graph on S153 to complete the questions on S154 and S155. Review the meaning of the terms vertical and horizontal, referencing the word wall as needed. Step 3: Have students look at the graphic organizer on the top of S154. Partner A, explain how to identify the vertical leg of Triangle ABC. (The vertical leg runs parallel to the y-axis.) What is the length of AB? (3 units) Record. Partner B, explain how to identify the horizontal leg of Triangle ABC. (The horizontal leg runs parallel to the x-axis.) What is the length of BC? (2 units) Record. Triangle ABC Triangle DEF Length of Vertical Leg AB: 3 units DE: 6 units Length of Horizontal Leg BC: 2 units EF: 4 units

6 Mathematics Success Grade 8 T313 Step 4: Have student pairs work together to determine the vertical leg length and horizontal leg length of Triangle DEF and then share answers as a whole group. (DE = 6 units; EF = 4 units) Record in the graphic organizer. Partner A, identify the side that corresponds to AB. (DE) Record. Partner B, identify the side that corresponds to BC. (EF) Record. Step 5: Ask student pairs to review together what they know about the corresponding sides of two similar triangles. (The corresponding sides form a proportional relationship.) Record. Have student pairs complete Question 6 and review the answers. Step 6: Have students refer to the chart on the top of S154. Ask them to discuss with their partners other relationships that might exist between the similar triangles. Bring the class back together to share answers. Step 7: There is a relationship that exists in each triangle between the leg and the leg. (vertical, horizontal.) Record. Have students look at Row 3 in the graphic organizer and describe the ratio relationship. vertical leg ( ) horizontal leg Partner A, identify the ratio that represents the length of the vertical leg over the horizontal leg for Triangle ABC. ( 3 ) Record in the graphic 2 organizer and in Question 7. Triangle ABC Triangle DEF Length of Vertical Leg AB: 3 units DE: 6 units Length of Horizontal Leg BC: 2 units EF: 4 units Ratio: vertical leg horizontal leg 3 2 Partner B, identify the ratio that represents the length of the vertical leg over the horizontal leg for Triangle DEF. ( 6 ) Record in the graphic 4 organizer and in Question 7. Step 8: Ask partners to discuss Questions 8 and 9 and then share answers as a whole class. Model how to draw the line that will pass through the hypotenuse of both triangles. Refer students to the word wall as needed for a review of the hypotenuse. Step 9: Direct students attention to the graphic organizer on S155 in Question 10. Have students identify the categories of each of the 5 columns. 6 4 Point 1 Point 2 Change in vertical distance Change in horizontal distance ( - 2, - 3) (4, 6) 9 6 Ratio of vertical leg horizontal leg 9 6 = 3 2

7 T314 Mathematics Success Grade 8 Partner A, identify Point 1 in the first row. ( - 2, - 3). Have students plot the point on the coordinate graph on S153 and make the point with a star. Partner B, identify Point 2 in the first row. (4, 6). Have students plot the point on the coordinate graph on S153 and mark the point with a star. Step 10: On the coordinate graph on T320, model for students how to draw a triangle using those two points. Partner A, identify the vertical distance that represents the vertical leg of the new triangle. (9) Record in the graphic organizer. Partner B, identify the horizontal distance that represents the horizontal leg of the new triangle. (6) Record in the graphic organizer. In Column 5, have student pairs write the ratio and simplify it. Then have them share answers as a whole group. Step 11: Have student pairs complete the following steps for Rows 2 and 3. Choose two other sets of points on the line. Mark the points you use with hearts and diamonds. Determine the change in vertical distance. Determine the change in horizontal distance. Write the change as a simplified ratio in the table above. Point 1 Point 2 Change in vertical distance Change in horizontal distance Ratio of vertical leg horizontal leg ( - 2, - 3) (4, 6) = 3 2 Answers will vary. Answer will vary. Step 12: Based on the points student pairs chose, have them make a prediction about the relationship between the vertical and horizontal distance of any two points on that line. Record the prediction. Step 13: Encourage several student pairs to share their points by graphing them on the coordinate wall chart. (Be sure to check for accuracy before allowing them to post.) Step 14: Based on the individual student graphs and the graphs on the wall chart, determine if your prediction was right or wrong and explain. (Yes) Record. Have them also include the explanation. (The ratio of vertical over horizontal change could always be simplified to 3 2.) Record.

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