6th Grade. Geometry.

Transcription

1 1

2 6th Grade Geometry

3 Table of Contents Area of Rectangles Area of Parallelograms Area of Right Triangles Area of Acute and Obtuse Triangles Area of Trapezoids Mixed Review Area of Irregular Figures Area of Shaded Regions 3 Dimensional Solids Nets Surface Area Volume Surface Area and Volume Application Problems More Polygons in the Coordinate Plane Glossary & Standards Click on a topic to go to that section Teacher Notes 3

4 Area of Rectangles Return to Table of Contents 4

5 Area is: Area the number of square units (units 2 ) it takes to cover the surface of a figure. ALWAYS label units 2!!! 10 ft 5 ft 5

6 Area Practice How many 1 ft 2 tiles does it take to cover the rectangle? Use the squares to find out! Look for a faster way than covering the whole figure. 10 ft 5 ft 6

7 Area The Area (A) of a rectangle is found by using the formula: A = length(width) A = lw The Area (A) of a square is found by using the formula: A = side(side) A = s 2 7

8 1 What is the Area (A) of the figure? 13 ft 7 ft 8

9 2 Find the area of the figure below. 8 9

10 3 Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. Does Michelle need to find the area or perimeter of her bedroom in order to figure out how much carpet to order? A Area B Perimeter 10

11 4 Now solve the problem... Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. How many square feet of carpet does Michelle need to order? 11

12 5 A rectangle measures 3 in by 4 in. If the lengths of each side double, what is the effect on the area? A The area doubles B The area quadruples C The area is cut in half D There is no effect 12

13 6 The area of a desktop is 24 sq. units. The length of the desktop is 6 units. What is the width of the desktop? 13

14 7 The 6th grade class at Immersion Middle School is building a giant I for their school. The I will be 10 ft. tall and 2 ft. wide. How large will the I be if measured in square inches? 14

15 8 The lumber that will be used to make the Immersion School I is 6 in by 1 ft. How many pieces of wood are needed to complete the project? 15

16 Area of Parallelograms Return to Table of Contents 16

17 Area of a Parallelogram How can we find the area of this parallelogram? Cut out your parallelogram and work with your table to come up with a way to determine the area. click 10 units 11 units Teacher Notes & Math Practice 15 units 17

18 9 What is the area of the parallelogram? click 10 units 11 units 15 units 18

19 Area of a Parallelogram Let's use the same process as we did for the rectangle. How many 1 ft 2 tiles fit across the bottom of the parallelogram? 19

20 Area of a Parallelogram Let's use the same process as we did for the rectangle. If we build the parallelogram with rows of ten 1 ft 2 tiles, what happens? How tall is the parallelogram? How can you tell? 10 ft 20

21 Area of a Parallelogram How does this help us find the area of the parallelogram? 4 ft 10 ft How do you find the area of a parallelogram? Teacher Notes 21

22 Area of a Parallelogram The Area (A) of a parallelogram is found by using the formula: A = base(height) A = bh Note: The base & height always form a right angle! 22

23 Parallelogram Area Practice Example. Find the area of the figure. 6 cm 2 cm 2 cm 1.7 cm 6 cm 23

24 Parallelogram Area Practice Try These. Find the area of the figures. 13 m 10.4 in 6.2 in 8.7 in 16 m 15 m 13 m 24

25 10 Find the area. 10 ft 9 ft 11 ft 25

26 11 Find the area. 15 in 11 in 10 in 11 in 15 in 26

27 12 Find the area. 8.4 m 13.1 m 12.2 m 8.4 m 27

28 13 Find the area. 13 cm 12 cm 7 cm 28

29 14 A box with a square opening is squashed into the rhombus shown below. What is the area of the opening? 7 in. 14 in 29

30 Solving for Missing Information A parallelogram has an area of cm 2 and a base of 9 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height. Step 1: Plug in known information. A = bh = ( ( information cm 2 9 cm A b h & Math Practice Step 2: Use inverse operations to solve 30

31 15 The height of a parallelogram is 12.6 feet and the area is 88.2 square feet. Write an equation that relates the area to the height and the base, b. Solve the equation to determine the length of the base. 31

32 16 The height of a parallelogram is 54 inches and the area is 972 square inches. Write an equation that relates the area to the height and the base, b. Solve the equation to determine the length of the base. 32

33 Area of Right Triangles Teacher Notes Return to Table of Contents 33

34 Area of a Triangle Let's use the same process as we did for the rectangle & parallelogram. How many 1 ft 2 tiles fit across the bottom of the triangle? 34

35 Area of a Triangle If we continue to build the triangle with rows of thirteen 1 ft 2 tiles what happens? 13 ft How tall is the triangle? How can you tell? 35

36 Area of a Triangle How does this help us find the area of the triangle? 5 ft 13 ft See that the rectangle we built is twice as large as the triangle. How do you find the area of a triangle? Find the area of the rectangle, then divide by ft 2 36

37 Area of a Triangle The Area (A) of a triangle is found by using the formula: 37

38 Area of a Triangle Practice Try this. What is the area of the right triangle below? 4 units 14.7 units 14 units 38

39 17 Calculate the area. 7 cm. 8 cm cm 39

40 18 Calculate the area. 9 m 9.9 m 4.1 m 40

41 19 Calculate the area. 7 in 9.9 in 7 in 41

42 ( Solving for Missing Information A triangle has an area of 70.8 cm 2 and a base of 6 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height. Step 1: Plug in known information. A = bh information 70.8 cm 2 6 cm = ( ( ( A b h Step 2: Use inverse operations to solve 42

43 20 If the area of a triangle is 117 square cm and its base is 20 cm, write an equation that relates the area to the height, h, and the base. Solve the equation to determine the height. 43

44 21 Fran is surveying a plot of land in the shape of a right triangle. The area of the land is 45,000 sq. meters. If the base of the triangular plot is 180 m long, what is the height, in meters, of the triangle? Write and solve an equation. 44

45 Area of Acute and Obtuse Triangles Return to Table of Contents 45

46 Triangles What is the difference between these three triangles? Teacher Notes 46

47 Triangle Altitudes The height of the right triangle is easy to find, it is a side. It does not always need to be a side of the triangle. The height of a triangle is also called the altitude, which is a line segment from a vertex of the triangle and perpendicular to the opposite side. h b b h Teacher Notes h b 47

48 Triangle Area Is the formula for the area of a right triangle true for all triangles? Let's see! 48

49 Triangle Area Example. Find the area of the figure. 8 cm 11 cm 11 cm 11 cm 49

50 Triangle Area Practice Try These. Find the area of the figures. 13 ft 10 ft 12 ft ft 16 50

51 22 Find the area. 10 in 6 in 8 in 9 in 51

52 23 Find the area. 10 m 12 m 9 m 14 m 52

53 24 Find the area. 10 in. 14 in. 6 in. 5 in. 53

54 25 A solar cover is needed for this triangular pool. If the area of the solar cover is 84 square meters, what does the length of the base need to be? 12 m x 54

55 (Problem derived from ) 26 Chauncey is building a storage bench for his son's playroom The storage bench will fit into the corner and then go along the wall to form a triangle. Chancey wants to buy a cover for the bench. If the storage bench is ft. along one wall and ft. along the other wall, how big will the cover have to be to cover the entire bench? 55

56 Triangle Area Use what you know to try and figure out how can we calculate the area of this triangle. Hint Teacher Notes 56

57 Step 1: Calculate the area of the square Triangle a = 1/2bh a = 1/2(2)(7) a = 7 u 2 Square a = lw a = 7(7) a = 49 u 2 Step 2: Calculate the area of the triangles. Step 3: Find the sum of the areas of the triangles. Step 4: Subtract the sum of the triangle areas from the rectangle area. Triangle a = 1/2bh a = 1/2(3)(7) a = 11.5 u 2 Triangle a = 1/2bh a = 1/2(5)(4) a = 10 u 2 Triangle Sum Difference The shaded triangle is 20.5 u 2 57

58 27 What is the area of the shaded figure? 58

59 28 What is the area of the shaded figure? 59

60 29 What is the area of the shaded figure? 60

61 Area of Any Shape This method can be used with any shape, as long as you can find the base and height of the triangles that form the surrounding rectangle. 61

62 30 What is the area of the shaded figure? 62

63 31 What is the area of the shaded figure? 63

64 Area of Trapezoids Return to Table of Contents 64

65 Area of a Trapezoid Draw a diagonal line to break the trapezoid into two triangles. Find the area of each triangle Add the area of each triangle together See the diagram below. 10 in 5 in 12 in 65

66 Area of a Trapezoid The Area (A) of a trapezoid is also found by using the formula: Note: The base & height always form a right angle! 10 in 5 in 12 in 66

67 Trapezoid Area Practice Example. Find the area of the figure by drawing a diagonal and splitting it into two triangles. 12 cm 10 cm 11 cm 9 cm 67

68 Trapezoid Area Practice Try These. Find the area of the figures using the formula. 12 ft 8 ft 7 ft 8 ft ft 13 68

69 32 Find the area of the trapezoid by drawing a diagonal. 9 m 8.5 m 11 m 69

70 33 Find the area of the trapezoid using the formula. 20 cm 12 cm 13 cm 70

71 34 The shape of the state of Arkansas resembles a trapezoid. The population density of Arkansas is 54.8 people per square mi. What is the approximate total population of this state? 280 mi & Math Practice 235 mi 210 mi 71

72 35 Each of the four sides of this tent are congruent. How much fabric was used to make all four sides of this tent? 23 in. 32 in in. 72

73 Mixed Review: Area Return to Table of Contents 73

74 36 Find the area of the figure. 5 cm 4 cm 3 cm 4 cm 11 cm 74

75 37 Find the area of the figure yd 10.5 yd 8 yd 10.5 yd 75

76 38 Find the area of the figure. 4.7 m 7.2 m 76

77 39 Find the area of the figure. 9 in 7 in 15 in 77

78 40 Find the area of the figure by drawing a diagonal and creating triangles. 17 cm 16 cm 15 cm 16 cm 22 cm 78

79 41 Find the area of the figure. 7 in 5.2 in 12.4 in 79

80 42 Find the area of the figure. 12 yd 11 yd 13 yd 12 yd 80

81 43 Find the area of the figure. 4.6 m 8.7 m 81

82 44 The Andersons were going on a long sailing trip during the summer. However, one of the sails on their sailboat ripped, and they have to replace it. The sail is pictured below. If the sailboat sails are on sale for $2 a square foot, how much will the new sail cost? Derived from 82

83 45 A wall is 56" wide. You want to center a picture frame that is 20" wide on the wall. How much space will there be between the edge of the wall and the frame? 83

84 46 Daniel decided to walk the perimeter of his triangular backyard. He walked 26.2 feet north and 19.5 feet west and back to his starting point. What is the area of Daniel's backyard? 84

85 47 If the area of a parallelogram is sq. km. and the base is km., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the height. 85

86 48 If the area of a right triangle is sq. ft. and the height is ft., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the base. 86

87 49 Below is a drawing of a wall that is to be covered with either wallpaper or paint. It is 8 ft. high and 16 ft. long. The window, mirror and fireplace will not be painted or papered. The window measures 18 in. by 14 ft. The fireplace is 5 ft. wide and 3 ft. high, while the mirror above the fireplace is 4 ft. by 2 ft. Part A: How many square feet of wallpaper are needed to cover the wall? Derived from continued 87

88 Continued from previous page. 50 Part B: The wallpaper is sold in rolls that are 18 in. wide and 33 ft. long. Rolls of solid color wallpaper will be used so patterns do not have to match up. What is the area of one roll of wallpaper? 88

89 51 Part C: How many rolls would be needed to cover the wall? 89

90 52 Part D: This week the rolls of wallpaper are on sale for $11.99/ roll. Find the cost of covering the wall with wallpaper. 90

91 53 Part E: A gallon of special textured paint covers 200 ft 2 and is on sale for $22.99/ gallon. The wall needs to be painted twice (the wall needs two coats of paint). Find the cost of using paint to cover the wall. 91

92 54 The area of a rectangular patio is square yards, and its length is yards. What is the patio's width in yards? A B C D From PARCC PBA sample test non calculator #3 92

93 55 Joanne buys a rectangular rug with an area of 35/4 square meters. The length of the rug is 7/2 meters. What is the width, in meters, of the rug? From PARCC EOY sample test non calculator #1 93

94 Area of Irregular Figures Return to Table of Contents 94

95 Area of Irregular Figures Method #1 1. Divide the irregular figure into smaller figures (that you know how to find the area of) 2. Label each small figure and label the new lengths and widths of each shape 3. Find the area of each shape Math Practice 4. Add the areas 5. Label your answer 95

96 Irregular Figure Area Example: Find the area of the figure. 2 m 12 m 4 m 8 m 4 m 2 m #1 #2 12 m 2 m 6 m 96

97 Area of Irregular Figures Method #2 1. Create one large, closed figure 2. Label the small added figure and label the new lengths and widths of each shape 3. Find the area of the new, large figure 4. Subtract the areas 5. Label your answer 97

98 Example: Find the area of the figure. Irregular Figure Area 2 m 12 m 4 m 8 m 8 m 4 m Whole 2 m 8 m Rectangle Extra Rectangle 12 m 98

99 Irregular Figure Area Practice Try This: Find the area of the figure. 3m 5m 8m 3m 99

100 Irregular Figure Area Practice Try This: Find the area of the figure. 6 ft 10 ft 18 ft 12 ft 100

101 56 Find the area. 4' 3' 5' 1' 10' 2' 8' 101

102 57 Find the area

103 58 Find the area. 8 cm 18 cm 9 cm 103

104 59 Find the area. 6 ft 4 ft 7 ft 9 ft 104

105 60 Find the area. 8 mm 8 mm 14 mm 10 mm 14 mm 8 mm 6 mm 105

106 61 Cara wants to put new carpet in both of her bedrooms. How much carpet will she need? 106

107 62 How many rectangular tiles are needed to cover this floor? Tiles 2 m 1 m (Drag and drop to check.) 107

108 Area of Shaded Regions Return to Table of Contents 108

109 Area of a Shaded Region 1. Find area of whole figure. 2. Find area of unshaded figure(s). 3. Subtract unshaded area from whole figure. 4. Label answer with units

110 Example Shaded Region Area Find the area of the shaded region. Area Whole Rectangle 10 ft 3 ft 3 ft 8 ft Area Unshaded Square Area Shaded Region 110

111 Try This Find the area of the shaded region. Shaded Region Area 12 cm 14 cm 111

112 Shaded Region Area Try This Find the area of the shaded region. 6 m 16 m 12 m 8 m 2 m 112

113 63 Find the area of the shaded region. 11' 3' 4' 8' 113

114 64 Find the area of the shaded region. 16" 15" 7" 5" 17" 114

115 65 Find the area of the shaded region. 8" 14" 9" 4" 5" 13" 115

116 66 Find the area of the shaded region. 4 yd 4 yd 4 yd 4 yd 8 yd 3 yd 116

117 67 A cement path 2 feet wide is poured around a rectangular pool. If the pool is 13 feet by 9 feet, how much cement was needed to create the path? 117

118 68 Logan wants to paint his trapezoid shaped wall shown below. He of course will not be painting over his window. One gallon of paint will cover 50 sq. feet. How many gallons of paint will he need? 4 ft 18 ft 5 ft 13 ft 23 ft 118

119 69 An advertising company is designing a new logo that consists of a shaded triangle inside a parallelogram. Part A What is the area, in square units, of parallelogram ABCD? square units From PARCC EOY sample test calculator #7 119

120 70 Part B In the new logo, what fraction of the parallelogram is shaded? square units From PARCC EOY sample test calculator #7 120

121 3 Dimensional Solids Return to Table of Contents 121

122 3 Dimensional Solids Categories & Characteristics of 3 D Solids: Prisms 1. click Have to reveal2 congruent, polygon bases which are parallel to one another 2. Sides are rectangular (parallelograms) 3. Named by the shape of their base Pyramids 1. click Have to reveal1 polygon base with a vertex opposite it 2. Sides are triangular 3. Named by the shape of their base 122

123 3 Dimensional Figures Sort the figures. If you are incorrect, the figure will be sent back. 123

124 3 Dimensional Solids Categories & Characteristics of 3 D Solids: Cylinders 1. click Have to reveal 2 congruent, circular bases which are parallel to one another 2. Sides are curved Cones 1. click Have to reveal1 circular bases with a vertex opposite it 2. Sides are curved 124

125 Face 3 Dimensional Solids Vocabulary Words for 3 D Solids: Flat surface of a Polyhedron Face Edge Line segment formed where 2 faces meet edge Vertex (Vertices) Point where 3 or more faces/edges meet 125

126 Polyhedron A polyhedron is a 3 D figure whose faces are all polygons. Sort the figures to the appropriate side. Polyhedron Not Polyhedron 126

127 71 Name the figure. A rectangular prism B triangular prism C triangular pyramid D cylinder E square cone F square pyramid 127

128 72 Name the figure. A rectangular prism B triangular prism C triangular pyramid D cylinder E cone F square pyramid 128

129 73 Name the figure. A rectangular prism B triangular prism C triangular pyramid D pentagonal prism E cone F square pyramid 129

130 74 Name the figure. A rectangular prism B triangular prism C triangular pyramid D pentagonal prism E cone F square pyramid 130

131 75 Name the figure. A rectangular prism B cylinder C triangular pyramid D pentagonal prism E cone F square pyramid 131

132 Nets Return to Table of Contents 132

133 Nets Nets are two dimensional drawings that represent the surface area of three dimensional shapes. There is more than one way to draw a net for a cube, however not all nets can be folded into a cube

134 Nets Exploratory Challenge Lab Click for Link to Lab There are some six square arranglements on your page. Sort each of the six arrangements into one of two piles, those that are nets of a cube and those that are not. Teacher Notes & Math Practice Derived from click to reveal answers 134

135 Interactive 3 D Figures and Nets Click for a web site with interactive 3 D figures and nets. 135

136 Nets for prisms will have rectangular faces and two bases for which the shape is named. Notice the two triangles are opposite from one another (bases). Prism Nets 136

137 Pyramid Nets Nets for pyramids will have triangular faces and ONE face (base) for which the shape is named. Square pyramid 137

138 76 Name the figure represented by the net. A rectangular prism B cylinder C triangular prism D pentagonal prism E cone F square pyramid 138

139 77 Name the figure represented by the net. A rectangular prism B cylinder C triangular prism D pentagonal prism E cone F square pyramid 139

140 Interactive Nets Use the packaging explorer to view more examples of nets. 140

141 For each figure, find the number of faces, vertices and edges. What is the relationship between the number of faces, vertices and edges of 3D Figures? Name Faces Vertices Edges Cube 3D Figure Patterns Rectangular Prism Triangular Prism Triangular Pyramid Square Pyramid Pentagonal Pyramid Octagonal Prism Math Practice 141

142 Euler's Formula click to reveal F + V 2 = E The number of edges is 2 less than the sum of the faces and vertices. 142

143 78 How many faces does a cube have? 143

144 79 How many vertices does a triangular prism have? 144

145 80 How many edges does a square pyramid have? 145

146 81 Paige has a figure whose faces are all congruent, and it has 4 vertices. Which shape does Paige have? A triangular pyramid B triangular prism C cube D square 146

147 82 Jonathan has 2 cubes. Henry has a square pyramid. How many edges do they have all together? 147

148 83 Which of these nets can be folded to form a cube? A C B D 148

149 Surface Area Return to Table of Contents 149

150 Surface Area Surface area is the sum of the areas of all outside faces of a 3 D figure. To find surface area, you must find the area of each face of the figure then add them together. What type of figure is pictured? How many surfaces are there? How do you find the area of each surface? 7 in 6 in 2 in 150

151 Surface Area 7 in 2 in 6 in A net is helpful in calculating surface area. Simply label each section and find the area of each. #1 6 in #2 #3 #4 6 in #5 #6 7 in 2 in 2 in 151

152 Example Surface Area Example #1 6 in #2 #3 #4 2 in 6 in #5 #6 2 in 7 in #1 #2 #3 #4 #5 #6 152

153 Surface Area Practice Try This Find the surface area of figure using the given net. #1 #2 #3 #4 12 cm #5 153

154 84 Find the surface area of the figure given its net. 7 yd 7 yd 7 yd 7 yd Since all of the faces are the What same, pattern you can did find you the notice area of while one face and multiply by it 6 to calculate finding the surface area of a area cube? of a cube. 154

155 85 Find the surface area of the figure given its net. 12 cm 9 cm 25 cm 155

156 86 The figure below represents a present you want to wrap for your friend's birthday. How many square centimeters of wrapping paper will you need? On the grid on the next slide, the distance between grid lines represents one centimeter. Use the grid to draw the net for the given figure. Then, calculate its surface area. 4 cm 8 cm 4 cm 156

157 157

158 87 Draw the net for the given figure, and calculate its surface area. 12 ft 11 ft 7 ft 4 ft 7 ft 158

159 88 This is a net of a right rectangular prism. Part A Which prism can be made using the net? A C B D 159

160 89 Part B What is the surface area, in square feet, of the prism? From PARCC EOY sample test calculator #12 160

161 Volume Return to Table of Contents 161

162 Volume Activity Take unit cubes and create a rectangular prism with dimensions of 4 x 2 x 1. What happens to the volume if you add another layer and make it 4 x 2 x 2? What happens to the volume is you add another layer and make it 4 x 2 x 3? Teacher Notes & Math Practice 162

163 Volume Volume is the amount of space occupied by or inside a 3 D Figure. The number of cubic units needed to fill a 3 D Figure (layering). Label: Units 3 or cubic units 163

164 Volume Formulas Formula 1 V= lwh, where l = length, w = width, h = height Multiply the length, width, and height of the rectangular prism. Formula 2 V=Bh, where B = area of base, h = height Find the area of the rectangular prism's base and multiply it by the height. 164

165 Find the Volume. Volume Practice 8 m 2 m 5 m 165

166 Example Volume Example Each of the small cubes in the prism shown have a length, width and height of 1/4 inch. The formula for volume is lwh. Therefore the volume of one of the small cubes is: Multiply the numerators together, Forget then how to multiply multiply the denominators. fractions? In other words, multiply across. 166

167 Volume Example Example To calculate the volume of the whole prism, count the number of cubes, and multiply it by the volume of one cube. The top layer of this prism has 4 rows of 4 cubes, making a total of 16 cubes per layer. The prism has 4 layers, 16 cubes per layer, so has 64 small cubes total. Therefore the total volume of the prism is: 167

168 Volume Example Example You can also use the formula to find the volume of the same prism. The length, width, and the height of this prism is four small cubes. Remember each small cube has a length, width, and height of 1/4 inch. Therefore, you can find the total volume finding the total length, width, and height of the prism and multiplying them together. 168

169 Example Volume Practice How would you find the volume of the rectangular prism with side lengths of 1/2 cm, 1/8 cm, and 1/4 cm? 169

170 Volume Example Try This Every cube in the rectangular prism has a length, width and height of 1/5 inch. Find the total volume of the rectangular prism. Method 1: Find volume of one small cube and multiply it by the number of cubes. One cube: Total Volume: Method 2: Find the length, width, and height of the rectangular prism and use the formula. Click to Reveal Click to Reveal 170

171 90 Find the volume of the given figure. 171

172 91 Find the volume of the given figure. 172

173 92 Find the volume of the given figure. 173

174 93 Find the volume of the given figure.the length, width, and height of one small cube is. 174

175 94 Find the volume of the given figure. The length, width, and height of one cube is. 175

176 95 A student filled a right rectangular prism shaped box with one inch cubes to find the volume, in cubic inches. The student's work is shown. & Math Practice Part A Explain why the student's reasoning is incorrect. Provide the correct volume, in cubic inches, of the box. From PARCC PBA sample test calculator #9 176

177 96 Part B A second box also has a base of 63 square inches, but it has a volume of 756 cubic inches. What is the height, in inches, of the second box? Explain or show how you determined the height. From PARCC PBA sample test calculator #9 177

178 Surface Area and Volume Application Problems Return to Table of Contents 178

179 97 A rectangular storage box is 12 1/4 in wide, 15 3/5 in long and 9 in high. How many square inches of colored paper are needed to cover the surface area of the box? 179

180 98 A teacher made 2 pair of foam dice to use in math games. Each cube measured 10 2/3 in on each side. How many square inches of fabric were needed to cover the 2 cubes? 180

181 99 A company is packaging their cereal in two rectangularshaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will hold more cereal? Input your answer, then explain your answer in a sentence on your paper. A Container A B Container B 181

182 100 A company is packaging their cereal in two rectangularshaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will require more cardboard to make the box? Input your answer, then explain your answer in a sentence on your paper. A Container A B Container B 182

183 (Problem derived from ) 101 A toy company manufactured a new set of toy blocks. The packaging manager insists that the cubes be arranged to form a rectangular prism and that the package be designed to hold the blocks exactly, with no leftover packaging. Each block measures 1 in. x 1 in. x 1 in. There are 24 toy blocks to be sold in a box. What are all of the possible box dimensions in inches? (Select all that apply.) A 1 x 1 x 24 B 1 x 2 x 12 C 1 x 3 x 8 D 2 x 2 x 8 E 2 x 3 x 6 F 1 x 3 x 6 G 1 x 4 x 6 H 2 x 2 x 6 I 2 x 4 x 8 J 2 x 3 x 4 183

184 (Problem derived from ) 102 (Cont. from previous slide) Which toy block box design will use the least amount of cardboard for packaging? Select one measurement (in inches) for each dimension of the box. Height Width Length A 1 C 1 G 4 B 2 D 2 E 3 F 4 H 6 I 8 J

185 103 A 250 in 3 box needs to be packaged for shipment. One shipping container has a length of 7 inches, a height of 5 inches, and a width of 6 inches. The other container has a length of 8 in, a height of 4 inches, and a width of 9 inches. Which container can the package be shipped in? Explain. A Container A: 7 in x 6 in x 5 in B Container B: 8 in x 4 in x 9 in 185

186 104 Small cubes with edge lengths of 1/4 inch will be packed into the right rectangular prism shown. How many small cubes are needed to completely fill the right rectangular prism? cubes From PARCC EOY sample test non calculator #6 186

187 105 The right rectangular prism is built with small cubes. Part A What is the volume, in cubic inch(es), of the right rectangular prism? Enter your fraction. From PARCC EOY sample test calculator #10 187

188 106 Part B What is the volume, in cubic inch(es), of 1 of the small cubes? Enter your fraction. From PARCC EOY sample test calculator #10 188

189 More Polygons in the Coordinate Plane Return to Table of Contents 189

190 107 Draw a polygon in the coordinate plane using the given coordinates. (4, 4) (6, 2) (8, 6) What is the area of the polygon? (Problem from ) 190

191 108 A surveyor is mapping a city block on a coordinate grid. The square shaped block has vertices at ( 4,1), ( 4, 4), and (1, 4). What are the coordinates of the remaining vertex? 191

192 109 What is the area of the square block described in the previous problem? 192

193 Glossary & Standards Teacher Notes Return to Table of Contents 193

194 3 D Figures An object with three different dimensions: length, width (or depth or breadth), and height. Also called a solid figure. One Dimensional Two Dimensional Three Dimensional length length height width width length Back to Instruction 194

195 Altitude A line segment from a vertex of the triangle and perpendicular to the opposite side. The height. h h h b b b This is not the height. It is not perpendicular to the base. Back to Instruction 195

196 Area The size of a surface. The number of square units it takes to cover a surface. 3 units 3 units 3 units x 3 units = 9 units units 2 units 1 units 1 units 30u x 10u = 300u2 20u x 10u = 200u 2 500u 2 Back to Instruction 196

197 Base & Height Base the surface that a solid object stands on Height the distance from the base to the top of a solid object. height base height base The base and height always form a right angle. Back to Instruction 197

198 Cone A 3 dimensional figure with one circular base, a vertex at the top, and one curved surface connecting the two. 1 Circular Base 1 Curved Surface 1 Vertex Back to Instruction 198

199 Cube A 3 dimensional figure with 3 pairs of parallel, congruent, square bases. 8 vertices 6 faces 12 Edges Back to Instruction 199

200 Cylinder A 3 dimensional figure with twocongruent, circular bases, and one curved surface connecting them. 2 congruent, parallel, circular bases 1 curved surface No vertices Back to Instruction 200

201 Diagonal A line that goes from one nonadjacent vertex to another. Cannot draw a diagonal, because all vertices are adjacent. Back to Instruction 201

202 Dimensions A measurement of length in one direction. 1 dimension 2 dimensions 3 dimensions length Back to Instruction 202

203 Edge The line segment where two faces meet. edge 10 edges Back to Instruction 203

204 Euler's Formula F + V 2 = E For any polyhedron that doesn't intersect itself, the number of edges is 2 less than the sum of the faces and vertices. Faces: 6 Vertices: =12 Edges: 12 Back to Instruction 204

205 Face A flat surface of a 3 d figure. Face 6 faces There is still debate over whether curved surfaces are faces. Back to Instruction 205

206 Formula An equation that describes a certain relationship between variables. l A = lw Area=length width w A = lw A = 5 8 A = 40u d=rt distance=rate time C=d circumference=diameter E=mc 2 energy=mass speed of light 2 Back to Instruction 206

207 Irregular Figure Not regular; a regular figure is a polygon with all congruent sides and angles, or a polyhedron with all regular faces. regular irregular scalene equilateral equiangular isosceles right irregular regular equilateral equiangular not equilangular not equilateral regular irregular Back to Instruction 207

208 Net A 2 dimensional pattern which can be folded into a 3 dimensional figure. = = Back to Instruction 208

209 Parallelogram A quadrilateral with opposite sides that are both congruent and parallel Four sides 2 Opposite sides A rhombus is a parallelogram with both sets of opposite sides. Opposite sides // Back to Instruction 209

210 Perimeter The distance around an object. side 3 side 2 side 1 P= side 1 + side 2 + side 3 To fence in this rectangular yard, you would measure the perimeter. l P=2l+2w w Back to Instruction 210

211 Polyhedron A three dimensional figure with all flat faces. Polyhedra "Polyhedra" is the singular form of polyhedron non polyhedra Back to Instruction 211

212 Prism A 3 dimensional figure with two congruent, parallel bases, and all other faces are rectangles. Prisms are named by the shape of their bases. Pentagonal Prism 2 triangular bases Triangular Prism 3 rectangular faces Back to Instruction 212

213 Pyramid A 3 dimensional figure with one base, a vertex at the top, and all other faces are triangles. 1 base All other faces are triangles A vertex at the top Back to Instruction 213

214 Square Units A measurement in the shape of a square with side lengths that are one unit long. 1 unit 1 unit 1 unit x 1 unit = 1 square unit Notation: sq unit unit 2 u 2 3 units 3 units 3 units x 3 units = 9 units 2 Back to Instruction 214

215 Surface Area The total area of the surface of a 3 dimensional figure. SA = u 2 6 u 2 12 u 2 6 u 2 12 u 2 8 u 2 Surface Area= 2lw+2lh+2wh SA= SA=52u 2 Back to Instruction 215

216 Trapezoid A quadrilateral with one pair of parallel sides. There are no // sides. Back to Instruction 216

217 Vertex A point where two or more straight lines meet. A Point A or vertex A The plural of vertex is "vertices" Back to Instruction 217

218 Volume The amount of space within a 3 dimensional object. Measured in cubic units V=1 1 1 V= 1 cubic unit v=lwh v= v= 36 u 3 Back to Instruction 218

219 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Math Practice Additional questions are included on the slides using the "Math Practice" Pull tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull tab. 219