Frogs

CMP II ACE Investigation 1 #4, p. 13

4. The graph below shows lengths (in meters) and areas (in square meters) for rectangular cages with a fixed perimeter that are going to be built for the Zoo of Montarek.

a.Describe the shape of the graph and any special features you observe.

  1. What is the greatest area possible for a rectangular cage with this perimeter? [Hint: Refer to the graph above.]
  1. What are the dimensions of this rectangle? [Hint: what is the length? What is the width?]
  1. What is the area of the rectangular cage with a side of length 10meters?
  1. What is the area of the rectangular cage with a side of length 40 meters?
  1. Explain how these two rectangles are related.
  1. What are the dimensions of the rectangle with an area of 600 square meters? [Hint: Refer to the graph]
  1. Length:
  1. Width:
  1. What is the fixed perimeter for the rectangles represented by the graph?

HINT: To answer this question, you will need to determine the length and width of the rectangle. To do this:

  1. Pick one are of the cage (i.e. 600). Using the graph above, what it its length? What it its width? From the graph it can have one length of 20 and one length of 30 (i.e. width). What is the perimeter of a rectangle 20 by 30? Pick another area and see if you get the same perimeter.
  1. Explain how you found the perimeter.

Frogs

CMP II ACE Investigation 2 #1 p. 34

  1. A square has sides of length x centimeters. A new rectangle is created by increasing one dimension by 4 centimeters and decreasing the other dimension by 4 centimeters.

<Graphics – please remove all lines, except for the rectangle.>

  1. Make a table showing the area of the square and the area of the new rectangle for whole-number x values from 4 to 16.

x / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16
Area of square
Area of new rectangle
  1. On the same set of axes, graph the (x, area) data for both the square and the rectangle.

x-values

  1. For which representation—the table or the graph—is it easier to compare the area of a square with the area corresponding to the new rectangle?
  1. Write an equation:
  2. For the area of the original square:
  1. For the area of the new rectangle.
  1. Use your calculator to graph both equations for x values from –10 to 10.
  1. Copy the graphs onto your paper.
  1. Describe the relationship between the two graphs.

Frogs

CMP II ACE Investigation 3 #9 p. 51

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.

  1. How many handshakes does one person make with members of the other team (student members plus coaches)?

HINT: Drawing a picture may help you answer all the questions.

  1. If 8 people (6 student members and 2 coaches) all shake the same amount of hands as 1 person, how many handshakes would all 8 members of 1 team make?

b.At the end of the match, the members and coaches of the winning team exchange

high fives to celebrate their victory. How many high fives will this involve?

  1. How many people can a player exchange high fives with on his/her own team?

HINT: Drawing a picture may help you answer these questions.

  1. If 8 people all high five the same number of people, how many high fives will there be in total?

c.Before the members and coaches of one team go home, the members exchange

handshakes with the coaches. How many handshakes will this involve?

  1. 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.

  1. How many handshakes would 2 coaches then make?

Frogs

CMP II ACE Investigation 4 #4 p. 73

4.The highest dive in the Olympic Games is from a 10-meter platform. The height in meters (h) of a diver t seconds after leaving the platform can be estimated by the equation h = 10 + 4.9t – 4.9t2.

  1. Make a table of the relationship between time (in seconds) and height (of the diver from the water).

t

h / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
  1. Sketch a graph of the relationship between time (t) and height (h).

  1. When will the diver hit the water’s surface (when will h = 0)?
  1. How can you find this answer by using your graph?
  1. How can you find this answer by using your table?

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?

  1. How is this shown in the table?

ii. How is this shown in the graph?