Mathematics Success Grade 8
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- Mark Morrison
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1 Mathematics Success Grade 8 S19 Warm Up Directions: Answer Questions 1 and 2 about ratios and proportionality. 1. Determine whether the ratios 9 and 8 12 your answer. form a proportion and explain 2. Determine whether the ratios 1 your answer. and 1 5 form a proportion and explain
2 S150 Mathematics Success Grade 8 Directions: Complete the following SOLVE problem with your teacher. You will only complete the S Step. Jason was graphing the relationship between the distance he could walk in a certain number of minutes. He wanted to see if the relationship between the minutes walked and the distance was proportional. He chose the times of 3 minutes and 7 minutes to use to compare the relationship. Using what you know about the relationship between similar right triangles, what is the ratio of or change in vertical over change in horizontal? y Jason s Walking Speed Distance (feet) Origin x Minutes S Underline the question. This problem is asking me to find the. Directions: Complete this page with your teacher and partner. Triangle 1 Triangle 2 A E D B 8 C F Triangle 1 and Triangle 2 are both right triangles. Explain how you know this. When we are trying to determine whether two triangles are similar, we can look at the measures of the corresponding sides.
3 Mathematics Success Grade 8 S151 Directions: Complete this page with your teacher and partner. Triangle 1 Triangle 2 E A B 8 C F 20 D 2. Explain the meaning of corresponding sides. 3. List the corresponding sides for Triangle 1 and Triangle 2. AB corresponds to BC corresponds to CA corresponds to. Once we have determined the corresponding sides, we are going to create a ratio to represent the relationship between the corresponding side lengths. AB = BC = CA = 5. What do you notice about the relationship between the numerator and the denominator in each of the ratios?. Simplify each of the ratios. AB DE = 12 = BC EF = 8 1 = CA FD = = 7. What is the ratio between each of the corresponding side lengths of Triangle 1 and Triangle 2? 8. If each ratio can be simplified to _, what can we say about the relationship between the three ratios?. They are. Because these three ratios are equivalent, we can say that the relationship between them is. 9. We can now say that Triangle 1 and Triangle 2 are similar triangles. Based on what you have discovered about the two triangles and the corresponding sides, create a definition of similar triangles with your partner.
4 S152 Mathematics Success Grade 8 Directions: Complete this page with your teacher and partner. Determine if Triangle QRS ~ Triangle PRT. P Q 7.2 R T 5 S 9 1. What is different about these two triangles from the previous example? 2. List the corresponding sides for Triangle QRS and Triangle PRT. RS corresponds to RQ corresponds to QS corresponds to 3. Once we have determined the corresponding sides we are going to create a ratio to represent the relationship between the corresponding leg lengths.. How is this relationship different from the previous example? 5. Explain how we determined if the sides were proportional for Triangle 1 and Triangle 2.. Look at the two ratios for Triangle QRS and Triangle PRT. Can they be simplified? Why? 7. What other way can we determine if the two ratios are proportional? Are the cross products equal? 8. If the cross products between the two corresponding legs are equal, then the two triangles are. Explain why. 9. We can now say that Triangle QRS and Triangle PRT are. Why?
5 Mathematics Success Grade 8 S153 Directions: Complete this page with your teacher and partner. 1. Graph the coordinates: A (2, 3), B (2, 0), and C (0, 0). 2. Graph the coordinates: D (0, 0), E (0, - ), and F ( -, - ). 3. Connect points A, B, and C. Connect points D, E, and F. What do you notice about the figure that is formed by connecting each set of points?
6 S15 Mathematics Success Grade 8 Directions: Complete this page with your teacher and partner. Use the triangles on the coordinate plane on S153 to complete the following chart and answer the questions about the relationship between the two triangles. Triangle ABC Triangle DEF Length of Vertical Leg Length of Horizontal Leg Ratio of. What are the corresponding sides of the two triangles? 5. Explain what you know about similar triangles.. Write the relationships between the corresponding sides as a proportion. Now substitute in the values for the leg lengths to determine if the triangles are similar. Is this statement true?. What does this mean? 7. There is another relationship to explore with similar triangles. That relationship is the ratio between the leg and the leg of each triangle. What is the ratio of the in Triangle ABC? What is the ratio of the in Triangle DEF? 8. What do you notice about the two ratios? 9. Is there a way to draw a line that will pass through the hypotenuse of both triangles? Draw the line to pass through the hypotenuse of both triangles.
7 Mathematics Success Grade 8 S155 Directions: Complete this page with your teacher and partner. 10. Let s choose two points on that line that we have not used in our triangles. ( - 2, - 3) and (, ). Mark them with stars. Point 1 Point 2 vertical distance horizontal distance Ratio of ( - 2, - 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. With your partner, make a prediction about the relationship between the vertical and horizontal distance of any two points on that line? Was your prediction correct? Explain. What can you conclude about the ratio of the measure of the over the no matter which two points you choose on the line? Another name for the change in vertical over the change in horizontal is.
8 S15 Mathematics Success Grade 8 Directions: Complete this page with your partner. Plot the following points: Triangle 1: (,0) (0,0) (,2) Triangle 2: (,0) (0,0) (,3) Fill out the chart using the two given points and then choose any pair of points on the line that passes through the hypotenuse. Point 1 Point 2 vertical distance horizontal distance Ratio of (, 2) (, 3) What can you conclude about the ratio of the measure of the over the no matter which two points you choose on the line?. This ratio is also known as.
9 Mathematics Success Grade 8 S157 Directions: Complete the following SOLVE problem with your teacher. Jason was graphing the relationship between the distance he could walk in a certain number of minutes. He wanted to see if the relationship between the minutes walked and the distance was proportional. He chose the times of 3 minutes and 7 minutes to use to compare the relationship. Using what you know about the relationship between similar right triangles, what is the ratio of or change in vertical over change in horizontal? y Jason s Walking Speed Distance (feet) Origin S Underline the question. This problem is asking me to find. O Identify the facts. Eliminate the unnecessary facts. List the necessary facts. L Write in words what your plan of action will be Minutes 10 x Choose an operation or operations. V Estimate your answer. Carry out your plan. E Does your answer make sense? (Compare your answer to the question.) Is your answer reasonable? (Compare your answer to the estimate.) Is your answer accurate? (Check your work.) Write your answer in a complete sentence.
10 S158 Mathematics Success Grade 8 Directions: Complete this page with your partner. Plot the coordinates for two similar triangles where the hypotenuse can be connected with a straight line Use the chart below to select two sets of points on the line through the hypotenuse. Complete the chart for two different sets of points. Point 1 Point 2 vertical distance horizontal distance Ratio of What can you conclude about the ratio of the measure of the over the no matter which two points you choose on the line?. This ratio is also known as.
11 Mathematics Success Grade 8 S159 Homework Name Date Directions: For Questions 1 and 2, find the value of x in each pair of similar triangles x 2. 9 x These triangles are similar. A R B T S C Name all corresponding sides: 3. Side AB corresponds to side.. Side AC corresponds to side. 5. Side BC corresponds to side.
12 S10 Mathematics Success Grade 8 Homework Name Date Directions: Use the coordinate graph and the information below to complete Questions 10.. Graph this set of similar triangles. 7. Graph this set of similar triangles Triangle GHI (0, 0), (0, - 8), and (, - 8) Triangle KLM ( -, 8), ( -, ), and ( - 2, ) Triangle ABC ( - 10, 0), ( -, 0), and ( -, 2) Triangle MNO ( - 2, ), (, ), and (, 8) 8. Extend the line connecting the hypotenuses for each set of similar triangles. y x Choose two sets of points that are on the line for one of the sets of triangles and complete the chart. Point 1 Point 2 vertical distance horizontal distance Ratio of
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