1ACE Exercise 4. Name Date Class

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1 1ACE Exercise 4 Investigation 1 4. A farm wants to add a small rectangular petting zoo for the public. They have a fixed amount of fencing to use for the zoo. This graph shows the lengths and areas of the rectangles they can make. Rectangular Petting Zoos Area (m 2 ) y a. Describe the shape of the graph and any special features you observe. Shape: Special Features: Length (m) b. What is the greatest area possible for a rectangle with this perimeter? b. What are the dimensions of this rectangle? x HINT Refer to the graph above. HINT What is the length? What is the width? 167

2 1ACE Exercise 4 (continued) c. What is the area of the rectangle with a side of length 10 meters? c. What is the area of the rectangle with a side of length 40 meters? c. Explain how these two rectangles are related. d. What are the dimensions of the rectangle with an area of 600 square meters? HINT Refer to the graph. Length: Width: e. What is the fixed amount of fencing available for the petting zoo? HINT To answer this question, you will need to determine the length and width of the rectangle. To do this: 1. Pick one area of the cage (i.e., 600). Using the graph, a rectangle with an area of 600 can have one length of 20 and one length of 30 (width). What is the perimeter of a rectangle 20 by 30? 2. Pick another area and see if you get the same perimeter. 3. Explain how you found the perimeter. 168

3 2ACE Exercise 1 Investigation 2 1. A square has sides of length x centimeters. One dimension increases by 4 centimeters and the other dimension decreases by 4 centimeters, forming a new rectangle. x x + 4 x x 4 a. Make a table showing the side length and area of the square and the area of the new rectangle for whole-number x-values from 4 to 16. The side lengths have already been filled in for you. x (side length) Area of Square Area of New Rectangle 0 9 b. On the same set of axes, graph the (x, area) data for both the square and the rectangle. Area x-values 169

4 2ACE Exercise 1 (continued) Investigation 2 c. Suppose you want to compare the area of a square with the area of the corresponding new rectangle. Is it easier to use the table or the graph? c. Table or graph? c. Explain. d. Write an equation for the area of the original square in terms of x: d. the new rectangle in terms of x: d e. Use your calculator to graph both equations. Show values of x from 10 to 10.. Copy the graphs onto your paper. d. Describe the relationship between the two graphs. 170

5 3ACE Exercise 9 Investigation 3 9. In a school math league, each team has six student members and two coaches. a. At the start of a match, the coaches and student members of one team exchange handshakes with the coaches and student members of the other team. HINT Drawing a picture may help you answer all the questions. a. How many handshakes does one person make with members of the other team (student members plus coaches)? a. If 8 people (6 student members and 2 coaches) all shake the same number of hands as 1 person, how many handshakes do all 8 members of 1 team make? b. At the end of the match, the members and coaches of the winning team exchange handshakes. How many handshakes occur? How many people can a player exchange handshakes with on his/her own team? HINT Drawing a picture may help you answer these questions. If 8 people all shake hands with the same number of people, how many handshakes will there be in total? 171

6 3ACE Exercise 9 (continued) Investigation 3 c. The members of one team exchange handshakes with their coaches. How many handshakes occur? How many handshakes can 1 coach make with 2 coaches and 6 student members of the other team? HINT Drawing a picture may help you answer the questions. How many handshakes would 2 coaches then make? 172

7 4ACE Exercise 4 Investigation 4 4. The highest dive in the Olympic Games is from a 10-meter platform.the height h in meters of a diver t seconds after leaving the platform can be estimated by the equation h = t 4.9t 2. a. Make a table of the relationship between time (in seconds) and height (of the diver from the water). t h b. Sketch a graph of the relationship between time (t) and height (h). h (height) t (time) c. When will the diver hit the water s surface (when will h = 0)? c. How can you find this answer by using your graph? c. How can you find this answer by using your table? 173

8 4ACE Exercise 4 (continued) Investigation 4 d. When will the diver be 5 meters above the water (h = 5)? e. When is the diver falling at the fastest rate? HINT When is t the biggest? e. How is this shown in the table? e. How is this shown in the graph? 174

9 Unit Test 1. One of the equations below represents the height h in feet of a thrown baseball as it changes over time t in seconds. The ball starts at about shoulder height, rises to a maximum, and then falls to the ground. h = 16t t + 4 h = 16t t + 4 h = 16t h= 16 t + 4 a. Which of these equations is a reasonable model for the given situation? Explain your choice. b. How is shoulder height represented in the correct equation? c. What maximum height will the ball reach? Explain how you found your solution. 2. Write an equation for the area of this rectangle in factored form and expanded form. What are the two factors that would form the factored form of the equation? What is the length of the horizontal side? What is the length of the vertical side? Write the factored form of the equation: A = x 4 x 2 HINT Remember, factored form is when an equation is made up of products of factors, such as (x + 1)(x) =. 175

10 Unit Test (continued) How could the factored form of the equation be expanded to give you the expanded form of the equation? Write the expanded form of the equation: A = HINT Remember, expanded form is an equation that is made up of sums or differences of terms, such as x 2 + x + 1 =. 3. The table below represents a pattern. n The number, s n, in the sequence a. Describe the pattern of change between the numbers in the sequence. b. Use the pattern you have described to find (6, ) and (10, ). c. Write an equation for calculating the nth number in the sequence. 176

11 Unit Test (continued) d. Is 117 in the sequence? If it is in the sequence, explain which number in the sequence it is. HINT Is it the tenth? Is it the eighteenth? If it is not in the sequence, explain why not. e. Is 10,100 in the sequence? e. If it is in the sequence, explain which number in the sequence it is. e. If it is not in the sequence, explain why not. 177

12 Unit Test (continued) 4. a. You know that with 40 meters of fencing the greatest rectangular area that you can enclose is a square In the space at the right above draw the square you get if you increase each side by 1 unit. b. How does adding 1 to each side of the square of part (a) increase the perimeter? Increase the area? c. Draw the largest rectangular area you can enclose from fencing with length 4n meters. d. Increase the length of each side of the figure you drew in part (c) by 1 unit. How does adding 1 unit to each side increase the perimeter? Increase the area? 178

Name Date PD. Pythagorean Theorem

Name Date PD. Pythagorean Theorem Name Date PD Pythagorean Theorem Vocabulary: Hypotenuse the side across from the right angle, it will be the longest side Legs are the sides adjacent to the right angle His theorem states: a b c In any