1. a) How many equilateral triangles can you construct by joining the points in this isometric grid? b) How many parallelograms?

c) How many trapezoids?

In each case, how do you know you have found all possibilities?

2. Given the square grid shown, draw quadrilaterals having segment AB as one of the sides. All four vertices of the quadrilateral must be grid points.

a) How many parallelograms are possible?

b) How many rectangles are possible?

c) How many rhombuses are possible?

d) How many squares are possible?

In each case, be prepared to make an argument that there are no more.

A B

3. Find the following shapes in the figure. List them by giving the vertices. Be prepared to justify your answers.

N

M

K

A B

CD

H

E

F

G

I

J

L

a) 3 squares

1- 2- 3-

b) 2 rectangles that are not squares

1- 2-

c) a parallelogram that is not a rectangle

d) 7 congruent right isosceles triangles

1- 4- 7- 2- 5- 3- 6-

e) 2 isosceles triangles that are not

congruent to those in part d)

1- 2-

f) a rhombus that is not a square

g) a scalene triangle with no right angles

h) a right scalene triangle

i) a trapezoid that is not isosceles

j) an isosceles trapezoid

4. A pentomino is made by five connected squares that touch only on a complete side. There are 12 non-congruent pentominos.

Draw all 12 pentominos on a sheet of centimeter graph paper. Label the pentominos that have reflection symmetry and draw all lines of symmetry. Label the pentominos that have rotation symmetry and give the smallest angle of rotational symmetry.

This is a notpentomino

This is a pentomino

5. Use 3 pentomino shapes to form a 3 x 5 rectangle. Use 5 pentomino shapes to

form a 5 x 5 square. Use all of the pentominos to make a 6 x 10 rectangle.

6. Use your protractor to measure angles E, D, C, and F with dots on their

vertices in the following semicircle. Make a conjecture based on your findings.

What are some differences between a conjecture and a proof?

B A

CD

E

F

7. Find the measures of

!

"1,

!

"2 ,

!

"3 and

!

"4 if lines l, m, and n are parallel. Is this picture drawn to scale?

x+20°

x+50°

3x+12°

!4!3

!2

!1

8. Saima is attempting a tessellation with regular pentagons. She ends up with gaps as shown in this figure. What kind of quadrilateral is made by a gap? What are the angle measurements of that quadrilateral? Be prepared to justify your answer without using a protractor.

9. Find the measure of the following angles drawn on isometric grids without using your protractor. What are the properties of the isometric grid that allow you to make your conclusions?

10. Find the measure of angles a and b. Show your work. Is the drawing a scale drawing?

b

a

65°

80°

70°

48°

11. Calculate the measure of the indicated angles. Is the diagram drawn to scale?

Find the measure ofthese angles.

!DCA

!ECD

!F

!BKF

Given that!F"!H

4 7 °

3 4 °

1 0 3 °

5 8 °

KH

A

F

B

C

D

E

G

12. A ninth grade student drew this figure to investigate the sum of the angles in

a convex 7-gon. He said that, because there are seven triangles in the picture, the sum of its vertex angles equal to 7 times 180° = 1260°. Where has this student gone wrong? Working from his picture show how to find the correct answer.

13. If one circle represents the set of rectangles and the other circle represents the set of parallelograms, which of the diagrams best represents the relationship between rectangles and parallelograms? I II III

a) If one circle represents the set of rhombuses and the other circle represents the set of rectangles, which of the diagrams best represents the relationship between rhombuses and rectangles?

b) Find a different relationship to illustrate the remaining diagram.

14. These conversations were overheard in the classroom, in stores, and in other places where people discuss mathematics. Comment on the reasoning expressed or implied.

a) Gail draws a horizontal line through a parallelogram and says, “If I cut along this line, the two pieces fit on top of each other. So my line must be a line of symmetry!”

b) Fred chose the floor tiling pattern shown below using squares and octagons. After studying it for a few minutes, he decided that each angle of the octagon measures 135°, even it is not a regular octagon.

15. A saw blade is made by cutting 17 congruent right triangles out of a regular

17-gon as shown below. If the angle M is the right angle, what is the measure of the angle made by the sharp points of the blade? The line segment MN is coincident with a radius of the 17-gon.

L

N

M

16. Identify the types of symmetry present in the following patterns.

Create a pattern in this square that has symmetry through a vertical line but no other symmetries.

17. Determine the isometry that maps the shape on the left onto the shape on the

right. All but one can be done with one simple isometry (rotation, translation or reflection). Which one? In each case be ready to demonstrate the transformation using tracing paper. Include the center of rotation, line of reflection, arrow of translation as needed.

18. Prior to conversion to a decimal monetary system, the United Kingdom used the following coins. 1 pound = 20 shillings 1 penny = 2 half-pennies = 4 farthings 1 shilling = 12 pence a) How many pence were there in a pound? b) How many half-pennies were in a pound? c) How many farthings were equal to a shilling?

19. A restaurant chain has sold over 80 billion hamburgers. A hamburger is about one-half inch thick. If the moon is 240,000 miles away, what percent of the distance to the moon is the height of a stack of 80 billion hamburgers?

20. A teacher and her students established the following system of measurements for the Land of Names. 1 jack = 24 jills 1 james = 8 jacks 1 jennifer = 60 james 1 jessica = 12 jennifers Complete the following table. Jill(s) Jack(s) James(s) Jennifer(s) Jessica(s)

Jill = 1

Jack = 1

James = 1

Jennifer= 1

Jessica= 1

21. George is building a large model airplane in his workshop. If the door to his workshop is 3 feet wide and 6 ½ feet high and the airplane has a wingspan of 7.1 feet, will George be able to get his airplane out of the workshop?

22. Jason has an old trunk that is 16 inches wide, 30 inches long, and 12 inches high. Which of the following objects would b\he be able to store in his trunk? Explain your answers. a) a telescope measuring 40 inches b) a baseball bat measuring 34 inches c) a tennis racket measuring 32 inches

23. A rectangle whose length is 3 cm more than its width has an area of 40

square centimeters. Find the length and the width of the rectangle.

24. Given are the lengths of the sides of a triangle. Indicate whether each triangle is a right triangle, an acute triangle, or an obtuse triangle. a) 70, 54, 90 b) 63, 16, 65 c) 24, 48, 52 d) 27, 36, 45 e) 48, 46, 50 f) 9, 40, 46

25. If the price for each pie is the same, which is the better buy – a 10 centimeter diameter circular pie or a square pie 9 centimeters on each side?

26. Suppose that every week the average American eats one-fourth of a pizza. The average pizza has a diameter of 14 inches and costs $8.00. There are about 250,000,000 Americans, and there are 640 acres in a square mile. a) About how many acres of pizza do Americans eat every week? b) What is the cost per acre of pizza in America?

27. Which costs more to fence? a) A garden in the shape of a circle with a diameter of 12 feet with wrought iron that costs $121 per yard. b) A square garden with sides of length 12 feet with fencing that costs $163 per yard.

28. Larry says the area of a parallelogram can be found by multiplying length times width. So the area of the parallelogram below must be 20 x 16 = 320 sq. in. Do you agree? If not, what could you do to give Larry an intuitive feeling about its area? Is it possible to find the exact area in this case? Explain.

20 in.

16 in.

45°

29. A student has a tennis can containing three tennis balls. To the students’ surprise, the perimeter of the top of the can is longer than the height of the can. The student wants to know if this fact can be explained without performing any measurements. Can you help?

30. Find the area of triangle ABC in each of the following triangles

10 cm

3 cm

6 m

5 m

8 cm 10 cm

30 cm40 cm

50 cm

4 cm

3 m

3 cm6 cm

31. Which playground has more room? a) A circle that has a circumference of 100 meters. b) A square that has a circumference of 100 meters.

32. Find the area of each of the following quadrilaterals:

6 cm

5 cm

9 cm

10 cm

7 cm

14 cm

5 cm

5 cm

3 cm3 cm

8 cm

10 cm

27 cm

10 cm

20 cm

4 cm

6 cm

4 cm45°45°

33. Which room costs more to carpet? a) Room dimensions: 6.5 m by 4.5 m and carpet cost = $13.85/sq. m b) Room dimensions: 15 ft by 11 ft and cost = $30/sq. y

34. Find the area of each of the following. Leave your answers in terms of π. Which of these drawings drawn to scale?

20 cm

10 cm

3 cm

6 cm

36°

35. Find the area of each of the following regular polygons: a) regular triangle with side length s. b) a regular hexagon with side length 4 inches.

36. The screens of two television sets are similar rectangles. The 20-in set

costs $400, and the 27-in set with similar features costs $600. The dimensions given are the diagonal lengths of the sets. If a customer is concerned about the size of the viewing area and is willing to pay the same amount per square foot, which is a better buy?

37. Find x in each of the following:

x

4

4 m

3 m

1 m

x

x

8

10

17

38. Two cars leave a house at the same time. One car travels 60 km/hr north, while the other car travels 40 km/hr east. After 1 hr, how far apart are the cars?

39. Two airplanes depart from the same place at 2:00 p.m. One plane flies south at a speed of 376 km/hr, and the other flies west at a speed of 648 km/hr. How far apart are the airplanes at 5:30 p.m?

40. Starting from point A, a boat sails due south for 6 mi, then due east for 5 mi, and then due south for 4 mi. How far is the boat from point A?

41. A wire 10 m long is wrapped around a circular region. If the wire fits exactly, what is the area of the region?

42. Suppose a wire is stretched tightly around Earth’s equator. If the wire is cut, its length is increased by 20 m, and the wire is then placed back around the Earth so that the wire is the same distance from Earth at every point, could you walk under the wire? [The radius of Earth is approximately 6400 km.]

43. Find the surface area and volume of the following: a) b)

3.5

9 in.

9 in.

8 in.

8 in.

3 in. 3 in.

3 in. 3 in.

44. A sketch of the Surenkov home is shown in the following figure, with the

recent sidewalk addition shaded. Using the measurement indicated on the figure and the fact that the sidewalk is 4 inches thick with right angles at all corners, determine how many cubic yards of concrete were used.

14'

55'

42'

6'

5'

4'

The flower beds are both 3 feet wide.

sidewalk

Surenkov Home

flowers

Patio

45. a) Construct a model of the swimming pool picture below by making a net.

Use a scale of 1 cm for each meter. b) How many square meters of tile are needed to cover the inside of the swimming pool illustrated below?

6 m

20 m

26 m

1 m

13 m

14 m

c) How much water does the pool hold?

46. A paper cup has the shape shown in the first drawing (a

frustrum). If the cup is sliced open and flattened, the sides of the cup have the second shape (the shaded part of a sector). Use the dimensions given to calculate the number of square meters of paper used in the construction of 10,000 of these cups

23.3 cm

11.5 cm

38°

8 cm

5.2 cm

11.5 cm

47. The first three steps of a 10-step staircase are shown at the right.

80 cm

15 cm

20 cm

a) Find the amount of concrete needed to make the exposed portion of the 10-step staircase. b) Find the amount of carpet needed to cover the fronts, tops, and sides of the steps.

48. A soft-drink cup is in the shape of a right circular cone with capacity 250

ml. If the radius of the circular base is 5 cm, how deep is the cup?

49. The water level in an aquarium, measuring 2 ½ ft long by 1 ft wide, is ½ inch from the top. The owner wants to add to the aquarium 200 solid marbles, each with a radius of 1.5 cm. Will the addition of these marbles cause the water in the aquarium to overflow?

50. The bases of a prism are regular hexagons. The other sides are squares. If

the perimeter of one of the bases is 72 inches, what is the surface area of the prism? What is the volume of the prism?

51. Find the surface area and volume of a Giant’s wooden

napkin ring pictured at the right.

20 in

16 in

15 in

52. Archimedes’ tomb bore an engraving of a sphere inscribed in a right circular

cylinder to commemorate a discovery of which he was particularly proud. The discovery concerned the ratio of the volume of the sphere to the volume of the cylinder and the ratio of the surface area of the sphere to the total surface area of the cylinder. Find those two ratios in simplest form.

53. A cubical box with edge length 12 in. is filled to the top with 27 congruent

balls, arranged in 3 layers of 9 each. A second cubical box of the same size is filled to the top with 64 congruent balls arranged in 4 layers of 16 each. Compare the total volume of the balls in the two boxes.

54. Calculate the volume and surface area this bolt. The

hole through the nut has diameter 1 cm. Each side of the hexagon is 2 cm. and the depth of the bolt is 0.75.

55. Scotty just moved into a new house and the landscaper ordered 1 cubic yard

of topsoil for his 15’ by 24’ garden. If the topsoil is spread evenly, about how thick will it be – a light dusting, about 1”, or about 1’? Justify your answer.

56. If one were to double each dimension of a fish tank that is the shape of a

rectangular prism, how would the capacity of the tank also double?

57. The Great Pyramid of Khufu (also known as the Cheops) was original 481 feet

tall with a square base that was 751 feet on each side. Answer these questions:

a. How many football fields (100 yards by 160 feet) would fit in the

same area as the base of this pyramid? b. This pyramid has the same volume as about how many bedrooms that

are 15 feet by 15 feet by 12 feet?

58. If the ice cream in an ice cream cone were to melt, could the cone hold all of the ice cream. Assume that the cone is a perfect cone, with a height of 5 inches and a circular base with a diameter of 2.5 inches, and that the ice cream is a perfect sphere with a diameter equal to the diameter of the cone.

59. At a factory that produces switches, a batch of 3000 switches has just been produced. To check the quality of the switches, a random sample of 75 switches are selected to test for defects. Our of these 75 switches, 2 were found to be defective. Base on these results, what is the best estimate you can give for the number of defective switches in the batch of 3000? Explain your reasoning.

60. A researcher wants to estimate the umber of rabbits in a region. The researcher sets some traps and catches 30 rabbits. After tagging the rabbits, they are released unharmed. A few days later the researcher sets some traps again and this time catches 35 rabbits. OF the 35 rabbits trapped, 4 were tagged, indicating that they had been trapped a few days earlier. Based on these results, what is the best estimate you can give for the number of rabbits in the region? Explain your reasoning.

61. Shante caught 17 ladybugs every day for 4 days. How many ladybugs does Shante need to catch on the fifth day so that she will have caught an average of 20 ladybugs per day over the 5 days.

62. Tracy’s times swimming 200 yards were as follows:

2:45 2:47 2:44

How fast will Tracy have to swim her next 200 yards so that her average time for the 4 trials is 2:45? Explain your reasoning.

63. Explain how you can quickly calculate the average of the following list of

test scores without adding the numbers:

83, 79, 81, 76, 81

64. In Ritzy county, the average annual household income is $100,000. In neighboring Normal county, the average annual household income is $32,000. Does it follow that in the two-county area the average annual household income is $66,000. Explain.

yellow

blue

blue

green

green

red

red

red

65. Determine the probability of spinning each of the following on the spinner at the right. a) red b) either red or green c) either red or yellow d) a red followed by a yellow in two spins e) a blue followed by a green

66. At family math night at school features the following fame: There are two opaque bags, each containing red block and yellow blocks. Bag A contains 3 red blocks and 5 yellow blocks. Bag B contains 5 red blocks and 15 yellow blocks. To pay the game, you pick a bag and then you pick a block out of the bag without looking. Your win a prize if you pick a red block. Kate thinks she should pick from bag 2 because it contains more red blocks. Is this the best strategy? Explain.

67. There are 4 black marbles and 5 red marbles in a bag. If you reach in and randomly select two marbles, what is the probability that both are red? Explain.

68. Suppose you have a penny, a nickel, a dime, and a quarter. You throw them in the air and let them all land on the table. What is the probability that exactly two coins will land face up and two will land face down? Draw a tree diagram to explain your answer.

69. Suppose you have 100 light bulbs and 1 of them is defective. If you pick out two of the light bulbs at random, what is the probability that one of them is defective?

70. A game at a fundraiser: there are 20 rubber ducks floating in a pool. One of the ducks has a mark on the bottom indicating that the contestant wins a prize. Each contestant pays 25¢ to play. A contestant picks 2 of the ducks. If the contestant picks th duck with the mark on the bottom, the contestant wins a prize $1.

a) What is the probability that a contestant will win a prize? b) If 100 people play the game, about how many people would you expect to

win? c) Based on your answer to part b), how much money should the duck game

expect to earn for the fundraiser if 100 people play?

71. A children’s game has a spinner that is equally likely to land on any one of four colors: red, blue, yellow, or green. What is the probability of spinning a red followed by a green in 2 spins. Explain how to solve this problem with fraction multiplication and explain why this method makes sense.

72. There are three boxes One of the 3 boxes contains 2 envelopes, the other two

boxes contain 3 envelopes each. In the box with two envelopes, one of the 2 envelopes contains a prize. No other envelope contains a prize. If you pick a random box and a random envelope from that box, what is the probability that you will win the prize? Explain how to solve this problem with fraction multiplication and explain why this method makes sense.

73. Suppose the average number of children per family for employees at a certain university is 2.58. Could this average be a mean? a median? a mode? Answer the same question as above if the average number was reported to be 2.5.

74. If 99 people had a mean income of $27,000, how much is the mean income increased by the addition of a single income of $500,000.

75. To receive an A in a class, Beth needs at least a mean of 90 on five exams. Beth’s grades on the first four exams were 84, 95, 86, and 94. What minimum score does she need on the fifth exam to receive an A in the class?

76. The mean of five numbers is 6. If one of the five numbers is removed, the mean becomes 7. What is the value of the number that was removed?

77. Following are raw test scores from a history test:

86 85 87 96 55

90 94 82 68 77

88 80 89 85 74

90 72 80 76 88

73 64 79 73 85

a) Construct an ordered stem and leaf plot for the given data. b) Construct a grouped frequency table for these scores with intervals of 5. c) Draw a histogram of the data. d) Construct a box and whiskers plot of the data.

78. Following are three boxes containing letters:

MATH AND HISTORY A B C a) From box A, three letters are drawn one by one without replacement and

recorder in order. What is the probability that the outcome is HAT? b) From box 1 three letters are drawn one by one with replacement and recorded

in order. What is the probability that the outcome is HAT? c) One letter is drawn at random from box A, then another from box B, and then

another from box C, with the results recorded in that order. What is the probability that the outcome is HAT?

d) If a box is chosen at random and then a letter is drawn at random from the box, what is the probability that the outcome is A?

79. A dart board has the design of a tangram set. If a

dart may hit any point on the board with equal probability, what is the probability that it hits

a) A triangle b) The small square c) One of the smallest triangles

80. An electric clock is stopped by a power failure. What is the probability that

the second hand is stopped between the 3 and the 4?

81. A box contains five slips of paper. Each slip has one of the numbers 4, 6, 7, 8, or 9 written on it. There are two players for the game. The first player reaches into the box and draws two slips of paper and adds the two numbers. If the sum is even, the player wins. If the sum is odd, the player loses.

a) What is the probability that the first player wins? b) Does the probability change if the two numbers are multiplied?

82. Four blue socks, 4 white socks, and 4 gray socks are mixed in a drawer. You pull out two socks, one at a time, without looking.

a) Draw a tree diagram for this situation. b) What is the probability of getting a pair of socks the same color c) What is the probability of getting two gray socks? d) Suppose you pull out four socks, what is the probability of getting two

socks the same color?

83. The land area of the Earth is approximately 57,500,000 mi2 The water area of Earth is approximately 139,600,000 mi2. If a meteor lands at random on the planet, what is the probability, to the nearest tenth, that it will hit water?