Slide 1 / 219 Slide 2 / 219 6th Grade Geometry 2015-12-01 www.njctl.org Table of Contents Area of Rectangles Click on a topic to Area of Parallelograms go to that section 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 Slide 3 / 219
Table of Contents Area of Rectangles Click on a topic to Area of Parallelograms go to that section Area of Right Triangles Area of Acute and Obtuse Triangles Area of Trapezoids Vocabulary Words are bolded Mixed Review in the presentation. The text Area of Irregular Figures box the word is in is then Area of Shaded Regions linked to the page at the end 3-Dimensional Solidsof the presentation with the Nets word defined on it. Surface Area Volume Surface Area and Volume Application Problems Teacher Notes More Polygons in the Coordinate Plane Glossary & Standards Slide 3 () / 219 Slide 4 / 219 Area of Rectangles Return to Table of Contents Area is: Area the number of square units (units 2 ) it takes to cover the surface of a figure. Slide 5 / 219 ALWAYS label units 2!!! 10 ft 5 ft
Area Practice How many 1 ft 2 tiles does it take to cover the rectangle? Slide 6 / 219 Use the squares to find out! Look for a faster way than covering the whole figure. 10 ft 5 ft Area Slide 7 / 219 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 1 What is the Area (A) of the figure? Slide 8 / 219 13 ft 7 ft
1 What is the Area (A) of the figure? Slide 8 () / 219 13 ft 7 ft 2 Find the area of the figure below. Slide 9 / 219 8 2 Find the area of the figure below. Slide 9 () / 219 8
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? Slide 10 / 219 A B Area Perimeter 3 Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. Does Michelle need *Note to find - perimeter the area is or a perimeter of her bedroom linked order vocabulary to figure word. out how much carpet to order? Click on the text box to go to the vocab page. A B Since the carpeting will Area cover the floor in her room, it is area, A. Perimeter Slide 10 () / 219 4 Now solve the problem... Slide 11 / 219 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?
4 Now solve the problem... Slide 11 () / 219 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? 5 A rectangle measures 3 in by 4 in. If the lengths of each side double, what is the effect on the area? Slide 12 / 219 A The area doubles B The area quadruples C The area is cut in half D There is no effect 5 A rectangle measures 3 in by 4 in. If the lengths of each side double, what is the effect on the area? Slide 12 () / 219 A The area doubles B The area quadruples C The area is cut in half D There is no effect B The area quadruples
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? Slide 13 / 219 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? Slide 13 () / 219 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? Slide 14 / 219
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? Slide 14 () / 219 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? Slide 15 / 219 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? Slide 15 () / 219
Slide 16 / 219 Area of Parallelograms Return to Table of Contents Area of a Parallelogram Slide 17 / 219 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 15 units Area of a Parallelogram Print this rectangle out full page so that each student has one. After reviewing their How can we find the predictions, area of this ask how parallelogram? can we change Cut the out your parallelogram and work parallelogram with your into table a rectangle? to come Click up on with the a way "click box" to reveal the height drawn and to determine the area. labeled. Have students cut this triangle off Teacher Notes & Math Practice click their parallelogram. You can also trace over the triangle yourself on the board and then move it to the other side. This activity addresses MP1. Additional questions to ask: - How should you start the problem? - Why would you choose 10 do units perform a certain step? - What plan can [This you object make is a pull to tab] solve the problem? 11 units Slide 17 () / 219 15 units
9 What is the area of the parallelogram? Slide 18 / 219 click 10 units 11 units 15 units Area of a Parallelogram Slide 19 / 219 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? Area of a Parallelogram Slide 20 / 219 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
Area of a Parallelogram How does this help us find the area of the parallelogram? Slide 21 / 219 4 ft 10 ft How do you find the area of a parallelogram? Area of a Parallelogram How does this help us find the area of the parallelogram? Slide 21 () / 219 4 ft Teacher Notes 10 ft How do you find the area of a parallelogram? Green triangle can be moved. Area of a Parallelogram Slide 22 / 219 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!
Example. Parallelogram Area Practice Slide 23 / 219 Find the area of the figure. 6 cm 2 cm 2 cm 1.7 cm 6 cm Example. Parallelogram Area Practice Slide 23 () / 219 Find the area of the figure. 6 cm 2 cm 2 cm 1.7 cm 6 cm Parallelogram Area Practice Slide 24 / 219 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
Parallelogram Area Practice Slide 24 () / 219 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 10 Find the area. Slide 25 / 219 10 ft 9 ft 11 ft 10 Find the area. Slide 25 () / 219 10 ft 9 ft 11 ft
11 Find the area. Slide 26 / 219 15 in 11 in 10 in 11 in 15 in 11 Find the area. Slide 26 () / 219 15 in 11 in 10 in 11 in 15 in 12 Find the area. Slide 27 / 219 8.4 m 13.1 m 12.2 m 8.4 m
12 Find the area. Slide 27 () / 219 8.4 m 13.1 m 12.2 m 8.4 m 13 Find the area. Slide 28 / 219 13 cm 12 cm 7 cm 13 Find the area. Slide 28 () / 219 13 cm 12 cm 7 cm
( 14 A box with a square opening is squashed into the rhombus shown below. What is the area of the opening? Slide 29 / 219 7 in. 14 in 14 A box with a square opening is squashed into the rhombus shown below. What is the area of the opening? Slide 29 () / 219 7 in. 14 in Solving for Missing Information Slide 30 / 219 A parallelogram has an area of 137.7 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 = Step 2: Use inverse operations to solve ( information 137.7 cm 2 9 cm A b h
Solving for Missing Information Slide 30 () / 219 A parallelogram has an area of 137.7 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. & Math Practice This question addresses MP2. Step 1: Plug in known Additional information. Questions to Ask: - What variables are used to represent the A = bh variables? - Why are those variables chosen? - = Are there different variables we could have chosen? [This object ( is a pull tab] - Why is it important to have variables instead of the original values? Step 2: Use inverse operations to solve ( information 137.7 cm 2 9 cm A b h 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. Slide 31 / 219 Slide 31 () / 219
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. Slide 32 / 219 Slide 32 () / 219 Slide 33 / 219 Area of Right Triangles Return to Table of Contents
Slide 33 () / 219 Teacher Notes Students will first complete the Area of Right Triangles Exploratory Challenge Lab Area of Right the Triangles lab section at: prior to starting the slides in this section. This is found in https://njctl.org/courses/ math/6th-grade-math/ geometry/ Return to Table of Contents Area of a Triangle Slide 34 / 219 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? Area of a Triangle Slide 35 / 219 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?
Area of a Triangle Slide 36 / 219 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 2 32.5 ft 2 Area of a Triangle Slide 37 / 219 The Area (A) of a triangle is found by using the formula: Area of a Triangle Practice Slide 38 / 219 Try this. What is the area of the right triangle below? 4 units 14.7 units 14 units
Area of a Triangle Practice Slide 38 () / 219 Try this. What is the area of the right triangle below? 4 units 14.7 units 14 units 17 Calculate the area. Slide 39 / 219 7 cm. 8 cm. 10.5 cm 17 Calculate the area. Slide 39 () / 219 7 cm. 8 cm. 10.5 cm
18 Calculate the area. Slide 40 / 219 9 m 9.9 m 4.1 m 18 Calculate the area. Slide 40 () / 219 9 m 9.9 m 4.1 m 19 Calculate the area. Slide 41 / 219 7 in 7 in 9.9 in
( 19 Calculate the area. Slide 41 () / 219 7 in 7 in 9.9 in Solving for Missing Information Slide 42 / 219 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 = ( Step 2: Use inverse operations to solve ( ( information 70.8 cm 2 6 cm A b h Solving for Missing Information Slide 42 () / 219 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
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. Slide 43 / 219 Slide 43 () / 219 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. Slide 44 / 219
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. Slide 44 () / 219 Slide 45 / 219 Area of Acute and Obtuse Triangles Return to Table of Contents Triangles Slide 46 / 219 What is the difference between these three triangles?
Triangles Slide 46 () / 219 What is the difference between these three triangles? Teacher Notes Label the triangles as right, acute and obtuse. 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. Slide 47 / 219 h h b b h b 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, Label which the is a triangles line segment as right, from a vertex of the triangle and perpendicular acute to the and opposite obtuse. side. Then students will complete the Area of Acute and Obtuse Triangles Exploratory Challenge Lab hprior h to starting the rest of the slides in this section. This is found in the lab section at: b b Teacher Notes Triangle Altitudes h https://njctl.org/courses/math/ 6th-grade-math/geometry/ Slide 47 () / 219 b
Triangle Area Slide 48 / 219 Is the formula for the area of a right triangle true for all triangles? Let's see! Triangle Area Slide 49 / 219 Example. Find the area of the figure. 8 cm 11 cm 11 cm 11 cm Triangle Area Slide 49 () / 219 Example. Find the area of the figure. 8 cm 11 cm 11 cm 11 cm
Triangle Area Practice Slide 50 / 219 Try These. Find the area of the figures. 13 ft 10 ft 12 ft 14 16 20 11 ft 16 Triangle Area Practice Slide 50 () / 219 Try These. Find the area of the figures. 13 ft 10 ft 12 ft 14 16 20 11 ft 16 22 Find the area. Slide 51 / 219 10 in 8 in 9 in 6 in
22 Find the area. Slide 51 () / 219 10 in 8 in 6 in 9 in 23 Find the area. Slide 52 / 219 10 m 12 m 9 m 14 m 23 Find the area. Slide 52 () / 219 10 m 12 m 9 m 14 m
24 Find the area. Slide 53 / 219 10 in. 14 in. 6 in. 5 in. 24 Find the area. Slide 53 () / 219 10 in. 14 in. 6 in. 5 in. Slide 54 / 219
Slide 54 () / 219 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? (Problem derived from ) Slide 55 / 219 (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? Slide 55 () / 219
Triangle Area Use what you know to try and figure out how can we calculate the area of this triangle. Slide 56 / 219 Hint Triangle Area Use what you know to try and figure out how can we calculate the area of this triangle. Slide 56 () / 219 Teacher Notes Calculate and subtract the area of the surrounding triangles from the area of the whole rectangle. Hint Students should complete each step on the next slide, and click on the reveal box to check answers as they go. 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 Slide 57 / 219 Triangle Sum 11.5 7 + 10 28.5 Difference 49-28.5 20.5 The shaded triangle is 20.5 u 2
27 What is the area of the shaded figure? Students type their answers here Slide 58 / 219 27 What is the area of the shaded figure? Rectangle Students type their A=lw answers here A=5(4) A=20 u 2 Slide 58 () / 219 1 A=1/2bh A=1/2(1)(4) A=2 u 2 2 A=1/2bh A=1/2(4)(3) A=6 u 2 3 A=1/2bh A=1/2(1)(5) A=2.5 in 2 Area of rectangle 20 u 2 - Sum of triangles - 10.5 u 2 = Area [This of Shaded object is a Triangle pull tab] 9.5 u 2 28 What is the area of the shaded figure? Students type their answers here Slide 59 / 219
28 What is the area of the shaded figure? Rectangle Students type their A=lw answers here A=5(4) A=20 u 2 Slide 59 () / 219 1 A=1/2bh A=1/2(1)(4) A=2 u 2 2 A=1/2bh A=1/2(4)(3) A=6 u 2 3 A=1/2bh A=1/2(1)(5) A=2.5 in 2 Area of rectangle 20 u 2 - Sum of triangles - 10.5 u 2 = Area [This of Shaded object is a Triangle pull tab] 9.5 u 2 29 What is the area of the shaded figure? Students type their answers here Slide 60 / 219 29 What is the area of the shaded figure? Students type their answers here Rectangle A=lw A=4(7) A=28 in 2 1 Slide 60 () / 219 1 A=1/2bh A=1/2(4)(4) 2 A=8 in 2 2 A=1/2bh 3 A=1/2(2)(7) A=14 in 2 3 A=1/2bh A=1/2(2)(3) A=3 in 2 Area of rectangle 28 in 2 - Sum [This object of triangles is a pull tab] - 25 in 2 = Area of Shaded Triangle 3 in 2
Area of Any Shape Slide 61 / 219 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. Area of Any Shape Square A=s 2 A=5 2 This method A=25 can be inused 2 with any shape, as long as you can find the base and height of the 1 A=1/2(3)(3) triangles that form the surrounding rectangle. A=4.5 u 2 2 A=1/2(2)(3) A=3 u 2 3 A=1/2(2)(4) A=4 u 2 4 A=1/2(2)(1) A=1.5 u 2 Area of square 25 u 2 - Sum of triangles - 13 u 2 = Area [This of Shaded object is Quadrilateral a pull tab] 12 u 2 Slide 61 () / 219 30 What is the area of the shaded figure? Students type their answers here Slide 62 / 219
30 What is the area of the shaded figure? Students type their answers here Rectangle A=lw A=4(5) A=20 in 2 1 A=1/2(2)(3) A=3 u 2 2 A=1/2(1)(3) A=1.5 u 2 3 A=1/2(3)(2) A=3 u 2 4 A=1/2(1)(3) A=1.5 u 2 Slide 62 () / 219 Area of rectangle 20 in 2 - Sum of triangles - 9 in 2 = Area of Shaded Rhombus 11 in 2 31 What is the area of the shaded figure? Students type their answers here Slide 63 / 219 31 What is the area of the shaded figure? Square A=s Students type their answers here 2 A=5 2 A=25 in 2 1 A=1/2(1)(3) A=1.5 u 2 2 A=1/2(1)(2) A=1 u 2 3 A=1/2(1)(3) A=1.5 u 2 4 A=1/2(4)(2) A=4 u 2 Area of rectangle 25 in 2 - Sum of triangles - 8 in 2 = Area [This of object Shaded is a pull Rhombus tab] 17 in 2 Slide 63 () / 219
Slide 64 / 219 Area of Trapezoids Return to Table of Contents Area of a Trapezoid Slide 65 / 219 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 Area of a Trapezoid Slide 66 / 219 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
Trapezoid Area Practice Slide 67 / 219 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 Trapezoid Area Practice 12 cm Slide 67 () / 219 Example. 10 cm 11 cm Find the area of the figure by drawing a diagonal 9 cm and splitting it into two triangles. 12 cm 10 cm 11 cm 9 cm Trapezoid Area Practice Slide 68 / 219 Try These. Find the area of the figures using the formula. 12 ft 10 8 ft 7 ft 8 ft 7 6 8 9 ft 13
Trapezoid Area Practice Slide 68 () / 219 Try These. Find the area of the figures using the formula. 12 ft 8 ft 7 ft 8 ft 7 6 8 10 9 ft 13 32 Find the area of the trapezoid by drawing a diagonal. Slide 69 / 219 9 m 8.5 m 11 m Slide 69 () / 219
33 Find the area of the trapezoid using the formula. Slide 70 / 219 20 cm 12 cm 13 cm 33 Find the area of the trapezoid using the formula. Slide 70 () / 219 20 cm 12 cm 13 cm 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? Slide 71 / 219 280 mi 235 mi 210 mi
34 The shape of the state of Arkansas resembles a trapezoid. The population density This question of addresses Arkansas is 54.8 MP4. people per square mi. What is the approximate total Additional questions to ask: population of this state? & Math Practice 280 mi - Are there any other states that we can measure with our area formulas? - What are some of the benefits/disadvantages of using an area model in real life? Slide 71 () / 219 235 mi 210 mi 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. Slide 72 / 219 32 in. 36.5 in. 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. Slide 72 () / 219 32 in. 36.5 in.
Slide 73 / 219 Mixed Review: Area Return to Table of Contents 36 Find the area of the figure. 5 cm Slide 74 / 219 4 cm 3 cm 4 cm 11 cm 36 Find the area of the figure. 5 cm Slide 74 () / 219 4 cm 3 cm 4 cm 11 cm
37 Find the area of the figure. Slide 75 / 219 10.5 yd 10.5 yd 8 yd 10.5 yd 37 Find the area of the figure. Slide 75 () / 219 10.5 yd 10.5 yd 8 yd 10.5 yd 38 Find the area of the figure. Slide 76 / 219 4.7 m 7.2 m
38 Find the area of the figure. Slide 76 () / 219 4.7 m 7.2 m 39 Find the area of the figure. Slide 77 / 219 9 in 7 in 15 in 39 Find the area of the figure. Slide 77 () / 219 9 in 7 in 15 in
40 Find the area of the figure by drawing a diagonal and creating triangles. Slide 78 / 219 17 cm 16 cm 15 cm 16 cm 22 cm 40 Find the area of the figure by drawing a diagonal and creating triangles. Slide 78 () / 219 17 cm 16 cm 15 cm 16 cm 22 cm 41 Find the area of the figure. Slide 79 / 219 7 in 5.2 in 12.4 in
41 Find the area of the figure. Slide 79 () / 219 7 in 5.2 in 12.4 in 42 Find the area of the figure. Slide 80 / 219 12 yd 11 yd 13 yd 12 yd 42 Find the area of the figure. Slide 80 () / 219 12 yd 12 yd 11 yd 13 yd
43 Find the area of the figure. Slide 81 / 219 4.6 m 8.7 m 43 Find the area of the figure. Slide 81 () / 219 4.6 m 8.7 m 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? Slide 82 / 219 Derived from
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? Slide 82 () / 219 $96 Derived from 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? Slide 83 / 219 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? Slide 83 () / 219 18 inches on each side
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? Slide 84 / 219 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? Slide 84 () / 219 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. Slide 85 / 219
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. Slide 85 () / 219 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. Slide 86 / 219 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. Slide 86 () / 219
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? Slide 87 / 219 Derived from continued Slide 87 () / 219 Continued from previous page. Slide 88 / 219 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?
Slide 88 () / 219 51 Part C: How many rolls would be needed to cover the wall? Slide 89 / 219 Slide 89 () / 219
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. Slide 90 / 219 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. Slide 90 () / 219 $11.99 x 2 = $23.98 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. Slide 91 / 219
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. If the wall needs to be painted twice, we need to paint a total area of 84 ft 2 x 2 = 168 ft 2. One gallon is enough paint for this wall, so the cost will be $22.99. Slide 91 () / 219 54 The area of a rectangular patio is square yards, and its length is yards. What is the patio's width in yards? Slide 92 / 219 A B C D From PARCC PBA sample test non-calculator #3 54 The area of a rectangular patio is square yards, and its length is yards. What is the patio's width in yards? Slide 92 () / 219 A B C D A From PARCC PBA sample test non-calculator #3
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? Slide 93 / 219 From PARCC EOY sample test non-calculator #1 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? Slide 93 () / 219 From PARCC EOY sample test non-calculator #1 Slide 94 / 219 Area of Irregular Figures Return to Table of Contents
Area of Irregular Figures Method #1 Slide 95 / 219 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 4. Add the areas 5. Label your answer Area of Irregular Figures Method #1 Slide 95 () / 219 1. Divide the irregular figure into smaller figures (that This you slide know addresses how to find MP8. the area of) Additional Questions to Ask: 2. Label each small figure and label the new lengths and - How can you chose which method to widths of each shape use? Math Practice 3. Find - the How area the of each methods shape similar? different? - Why is it possible to use previous 4. Add formulas the areas in this new situation? 5. Label your answer Irregular Figure Area Slide 96 / 219 Example: Find the area of the figure. 4 m 2 m 8 m 4 m 2 m #1 #2 12 m 2 m 6 m 12 m
Area of Irregular Figures Method #2 Slide 97 / 219 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 Example: Find the area of the figure. Irregular Figure Area 4 m 2 m 12 m 8 m Slide 98 / 219 8 m 4 m Whole 2 m 8 m Rectangle Extra Rectangle 12 m Irregular Figure Area Practice Slide 99 / 219 Try This: Find the area of the figure. 3m 5m 8m 3m
Irregular Figure Area Practice Slide 99 () / 219 Try This: Find the area of the figure. 3m 5m 8m 3m Whole Square 3m 5m 5m 8m 3m Extra Square Irregular Figure Area Practice Try This: Find the area of the figure. 6 ft Slide 100 / 219 18 ft 10 ft 12 ft Irregular Figure Area Practice Try This: Find the area of the figure. 6 ft 18 ft 6 ft Slide 100 () / 219 10 ft 18 ft 10 ft Whole Triangle 12 ft Bottom Trapezoid 12 ft
56 Find the area. Slide 101 / 219 4' 3' 5' 1' 10' 2' 8' 56 Find the area. Slide 101 () / 219 4' 3' Top Rectangle Vertical Rectangle 2' 5' 2' 4' 1' 3' 3' 10' 1' 5' 5' 2' 8' Bottom Rectangle Total Area 8' 57 Find the area. Slide 102 / 219 12 20 25 10 13 10
57 Find the area. Slide 102 () / 219 12 12 Total Area 20 10 13 10 10 20 13 10 25 Whole New Figure New Rectangle 25 58 Find the area. Slide 103 / 219 8 cm 18 cm 9 cm 58 Find the area. Slide 103 () / 219 8 cm 18 cm 9 cm 8 cm 18 cm 9 cm Triangle 9 cm Rectangle Total Area
59 Find the area. Slide 104 / 219 6 ft 7 ft 4 ft 9 ft 59 Find the area. Slide 104 () / 219 6 ft 7 ft 6 ft Total Area 4 ft 7 ft 9 ft 4 ft Triangle 9 ft Trapezoid 60 Find the area. Slide 105 / 219 8 mm 8 mm 14 mm 10 mm 14 mm 8 mm 6 mm
60 Find the area. Slide 105 () / 219 8 mm 8 mm 14 mm 10 mm 14 mm 8 mm 6 mm 14 mm 8 mm 8 mm Total Area 10 mm 8 mm 14 mm 6 mm Area of Triangle 1 Area of Rectangle Area of Triangle 2 61 Cara wants to put new carpet in both of her bedrooms. How much carpet will she need? Slide 106 / 219 61 Cara wants to put new carpet in both of her bedrooms. How much carpet will she need? Slide 106 () / 219 Area of Bedroom 1 Area of Bedroom 2 Total Area
62 How many rectangular tiles are needed to cover this floor? Slide 107 / 219 Tiles 2 m 1 m (Drag and drop to check.) 62 How many rectangular tiles are needed to cover this floor? Area of Tile Slide 107 () / 219 Tiles 2 m 1 m (Drag and drop to check.) Total Area Slide 108 / 219 Area of Shaded Regions Return to Table of Contents
Area of a Shaded Region Slide 109 / 219 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 2. Example Shaded Region Area Slide 110 / 219 Find the area of the shaded region. Area Whole Rectangle 10 ft 3 ft 3 ft 8 ft Area Unshaded Square Area Shaded Region Shaded Region Area Slide 111 / 219 Try This Find the area of the shaded region. 12 cm 14 cm
Shaded Region Area Slide 111 () / 219 Try This Find the area of the shaded region. Area Whole Square 12 cm Area Triangle 14 cm Area Shaded Region Shaded Region Area Slide 112 / 219 Try This Find the area of the shaded region. 16 m 6 m 12 m 8 m 2 m Try This Shaded Region Area Area Trapezoid Slide 112 () / 219 Find the area of the shaded region. 6 m 16 m 12 m Area Rectangle Area Shaded Region 8 m 2 m
63 Find the area of the shaded region. Slide 113 / 219 11' 3' 4' 8' 63 Find the area of the shaded region. Area Whole Rectangle Slide 113 () / 219 11' 3' 4' Area Unshaded Area Shaded Region 8' 64 Find the area of the shaded region. Slide 114 / 219 16" 7" 15" 5" 17"
15" 7" 5" 64 Find the area of the shaded region. Slide 114 () / 219 Area Parallelogram 16" Area Triangle Area Shaded Region 17" 65 Find the area of the shaded region. Slide 115 / 219 8" 14" 9" 4" 5" 13" 65 Find the area of the shaded region. Area Whole 8" Area Rectangle Slide 115 () / 219 14" 9" 4" 14" 5" 8" 9" 4" Area Shaded Region 5" 5" 13" 13"
66 Find the area of the shaded region. 4 yd Slide 116 / 219 4 yd 4 yd 3 yd 4 yd 8 yd 66 Find the area of the shaded region. 4 yd Area Rectangle 4 yd Slide 116 () / 219 4 yd 4 yd 8 yd Area of 2 Triangles 3 yd Area Shaded Region 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? Slide 117 / 219
67 A cement path 2 feet wide is poured around a rectangular pool. If the pool is 13 Area feet Path by 9 & feet, Pool how much cement was needed to create the path? Slide 117 () / 219 Area Pool Area Path 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? Slide 118 / 219 18 ft 4 ft 5 ft 13 ft 23 ft 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 Area of Trapezoid gallons of paint will he need? Slide 118 () / 219 18 ft Area of Window 4 ft 13 ft 5 ft 23 ft
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? Slide 119 / 219 square units From PARCC EOY sample test calculator #7 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? 24 square units Slide 119 () / 219 square units From PARCC EOY sample test calculator #7 70 Part B In the new logo, what fraction of the parallelogram is shaded? Slide 120 / 219 square units From PARCC EOY sample test calculator #7
70 Part B In the new logo, what fraction of the parallelogram is shaded? Slide 120 () / 219 1/4 square units From PARCC EOY sample test calculator #7 Slide 121 / 219 3-Dimensional Solids Return to Table of Contents 3-Dimensional Solids Slide 122 / 219 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
3-Dimensional Figures Sort the figures. If you are incorrect, the figure will be sent back. Slide 123 / 219 3-Dimensional Solids Slide 124 / 219 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 Face 3-Dimensional Solids Vocabulary Words for 3-D Solids: Flat surface of a Polyhedron Slide 125 / 219 Face Edge Line segment formed where 2 faces meet edge Vertex (Vertices) Point where 3 or more faces/edges meet
Polyhedron Slide 126 / 219 A polyhedron is a 3-D figure whose faces are all polygons. Sort the figures to the appropriate side. Polyhedron Not Polyhedron 71 Name the figure. A rectangular prism Slide 127 / 219 B triangular prism C triangular pyramid D cylinder E square cone F square pyramid 71 Name the figure. A rectangular prism Slide 127 () / 219 B triangular prism C triangular pyramid D cylinder E square cone F square pyramid C
72 Name the figure. Slide 128 / 219 A rectangular prism B triangular prism C triangular pyramid D cylinder E cone F square pyramid 72 Name the figure. Slide 128 () / 219 A rectangular prism B triangular prism C triangular pyramid D cylinder E cone F square pyramid A 73 Name the figure. Slide 129 / 219 A rectangular prism B triangular prism C triangular pyramid D pentagonal prism E cone F square pyramid
73 Name the figure. Slide 129 () / 219 A rectangular prism B triangular prism C triangular pyramid D pentagonal prism E cone F square pyramid D 74 Name the figure. Slide 130 / 219 A rectangular prism B triangular prism C triangular pyramid D pentagonal prism E cone F square pyramid 74 Name the figure. Slide 130 () / 219 A rectangular prism B triangular prism C triangular pyramid D pentagonal prism E cone F square pyramid E
75 Name the figure. Slide 131 / 219 A rectangular prism B cylinder C triangular pyramid D pentagonal prism E cone F square pyramid 75 Name the figure. Slide 131 () / 219 A rectangular prism B cylinder C triangular pyramid D pentagonal prism E cone F square pyramid B Slide 132 / 219 Nets Return to Table of Contents
Nets Nets are two-dimensional drawings that represent the surface area of three-dimensional shapes. Slide 133 / 219 There is more than one way to draw a net for a cube, however not all nets can be folded into a cube... 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. Slide 134 / 219 Derived from click to reveal answers Nets Exploratory Challenge Lab Each group Click for of students Link to Lab will need a There are some set six of 20 square (nets A-T). arranglements They are sized on to your page. Sort each of the wrap six around arrangements a cube with into side one of two piles, those that are lengths nets of of a 4 cube cm, which and those can be that made are not. from 8 Unifix cubes. Each group needs one of these cubes. Teacher Notes & Math Practice The slides that cover nets cover MP6. Additional Questions to Ask: - Why are nets useful? - What are the advanatages/ disadvantages of a net? Slide 134 () / 219 Derived from click to reveal answers
Interactive 3-D Figures and Nets Click for a web site with interactive 3-D figures and nets. Slide 135 / 219 Prism Nets Slide 136 / 219 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). Slide 137 / 219
76 Name the figure represented by the net. Slide 138 / 219 A rectangular prism B cylinder C triangular prism D pentagonal prism E cone F square pyramid 76 Name the figure represented by the net. Slide 138 () / 219 A rectangular prism B cylinder C triangular prism D pentagonal prism E cone F square pyramid C 77 Name the figure represented by the net. Slide 139 / 219 A rectangular prism B cylinder C triangular prism D pentagonal prism E cone F square pyramid
77 Name the figure represented by the net. Slide 139 () / 219 A rectangular prism B cylinder C triangular prism D pentagonal prism E cone F square pyramid F Interactive Nets Slide 140 / 219 Use the packaging explorer to view more examples of nets. 3D Figure Patterns 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 6 8 12 Slide 141 / 219 Rectangular Prism Triangular Prism Triangular Pyramid Square Pyramid Pentagonal Pyramid Octagonal Prism 6 8 12 5 6 9 4 4 6 5 5 8 6 6 10 10 16 24 Math Practice
Euler's Formula Slide 142 / 219 click to reveal F + V - 2 = E The number of edges is 2 less than the sum of the faces and vertices. 78 How many faces does a cube have? Slide 143 / 219 78 How many faces does a cube have? Slide 143 () / 219 6
79 How many vertices does a triangular prism have? Slide 144 / 219 79 How many vertices does a triangular prism have? Slide 144 () / 219 6 80 How many edges does a square pyramid have? Slide 145 / 219
80 How many edges does a square pyramid have? Slide 145 () / 219 8 81 Paige has a figure whose faces are all congruent, and it has 4 vertices. Which shape does Paige have? Slide 146 / 219 A triangular pyramid B triangular prism C cube D square 81 Paige has a figure whose faces are all congruent, and it has 4 vertices. Which shape does Paige have? Slide 146 () / 219 A triangular pyramid B triangular prism C cube A D square
82 Jonathan has 2 cubes. Henry has a square pyramid. How many edges do they have all together? Slide 147 / 219 82 Jonathan has 2 cubes. Henry has a square pyramid. How many edges do they have all together? Slide 147 () / 219 32 83 Which of these nets can be folded to form a cube? A C Slide 148 / 219 B D
83 Which of these nets can be folded to form a cube? A C Slide 148 () / 219 All of them. B D Slide 149 / 219 Surface Area Return to Table of Contents Surface Area Slide 150 / 219 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
Surface Area Slide 151 / 219 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 Example Surface Area Example Slide 152 / 219 #1 6 in #2 #3 #4 2 in 6 in #5 #6 2 in 7 in #1 #2 #3 #4 #5 #6 Surface Area Practice Slide 153 / 219 Try This Find the surface area of figure using the given net. #1 12 cm #2 #3 #4 #5
Surface Area Practice Slide 153 () / 219 Try This Find the surface area of figure using the given net. #1 #2 #3 #1 #2 #3 #4 12 cm #4 #5 #5 84 Find the surface area of the figure given its net. Slide 154 / 219 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. 84 Find the surface area of the figure given its net. Slide 154 () / 219 7 yd 7 yd 7 yd #1 - #6 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.
85 Find the surface area of the figure given its net. Slide 155 / 219 12 cm 9 cm 25 cm 85 Find the surface area of the figure given its net. #1 #2 #3 12 cm 9 cm 25 cm 12 cm #1 9 cm #2 #3 #4 25 cm #5 #4 #5 Slide 155 () / 219 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. Slide 156 / 219 4 cm 8 cm 4 cm
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 #1,#3, #5 and #6 A = 4(8) represents one centimeter. Use the grid to draw the net A = 32 cm for the given figure. Then, calculate its surface 2 area. 4 cm 8 cm #1 8 cm 4 cm #2 #3 #4 #5 #6 4 cm 4 cm 4 cm #2, #4 A = 4(4) A = 16 cm 2 A = #1 + #2 + #3 + #4 + #5 + #6 A = 32 + 16 + 32 + 16 + 32 + 32 A = 160 cm 2 Slide 156 () / 219 Slide 157 / 219 87 Draw the net for the given figure, and calculate its surface area. Slide 158 / 219 12 ft 11 ft 7 ft 4 ft 7 ft
87 Draw the net for the given figure, and calculate its surface area. #1 #2 #3 Slide 158 () / 219 12 ft 7 ft 4 ft 7 ft 11 ft 11 ft #1 12 ft #4 #3 4 ft 7 ft 7 ft #2 #4 88 This is a net of a right rectangular prism. Slide 159 / 219 Part A Which prism can be made using the net? A C B D 89 Part B What is the surface area, in square feet, of the prism? Slide 160 / 219 From PARCC EOY sample test calculator #12
89 Part B What is the surface area, in square feet, of the prism? Slide 160 () / 219 1300 square feet From PARCC EOY sample test calculator #12 Slide 161 / 219 Volume Return to Table of Contents Volume Activity Slide 162 / 219 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?
Volume Activity Slide 162 () / 219 Lead students to discover that the area Take unit cubes and of the create base times a rectangular the height equals prism the with dimensions of 4 x 2 x 1. volume. Teacher Notes & Math Practice This question addresses MP5. What happens to the volume if you add another layer and make it 4 x 2 x 2? Additional Questions to Ask: - Why is it important to be precise What happens to the when volume adding cubes? is you add another layer and make it 4 x 2 x 3? - Can you provide an explanation for your formula? - Does the unit matter for the formula? Volume Slide 163 / 219 - 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 Volume Formulas Slide 164 / 219 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.
Find the Volume. Volume Practice Slide 165 / 219 8 m 2 m 5 m Find the Volume. Volume Practice Slide 165 () / 219 2 m 5 m VOLUME: 2 x 5 10 8 (Area m of Base) x 8 (Height) 80 m 3 VOLUME: V = B h V = l w h V = 5 2 8 V = 10 8 V = 80 m 3 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: Slide 166 / 219 Multiply the numerators together, Forget how then to multiply multiply the denominators. fractions? In other words, multiply across.
Volume Example Slide 167 / 219 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: Example Volume Example You can also use the formula to find the volume of the same prism. Slide 168 / 219 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. 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? Slide 169 / 219
Example Volume Practice How would you find the volume Since it already of the tells rectangular you the prism side with lengths, you side can lengths simply of plug 1/2 cm, it 1/8 cm, into the volume and 1/4 formula. cm? Slide 169 () / 219 Try This Volume Example Every cube in the rectangular prism has a length, width and height of 1/5 inch. Find the total volume of the rectangular prism. Slide 170 / 219 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 90 Find the volume of the given figure. Slide 171 / 219
90 Find the volume of the given figure. Slide 171 () / 219 91 Find the volume of the given figure. Slide 172 / 219 91 Find the volume of the given figure. Slide 172 () / 219
92 Find the volume of the given figure. Slide 173 / 219 92 Find the volume of the given figure. Slide 173 () / 219 93 Find the volume of the given figure.the length, Slide 174 / 219 width, and height of one small cube is.
93 Find the volume of the given figure.the length, Slide 174 () / 219 width, and height of one small cube is Method. 2 Method 1 94 Find the volume of the given figure. The length, width, Slide 175 / 219 and height of one cube is. 94 Find the volume of the given figure. The length, width, Slide 175 () / 219 and height of one cube Method is 1. Method 2
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. Slide 176 / 219 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 95 A student filled a right rectangular prism-shaped box with one inch cubes to find the volume, in cubic inches. The The student's reasoning is incorrect because they student's work is shown. did not count the top layer as part of the height. The calculation should have been 63 x 10, which equals a total of 630 cubes. Therefore, the volume is 630 cubic inches. Slide 176 () / 219 & Math Practice Part A This slide addresses MP3. Additional questions to Ask: - What evidence can you use to support the error analysis with? - What are other common student errors? - What advice can you tell other students to avoid making the error? 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 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. Slide 177 / 219 From PARCC PBA sample test calculator #9
96 Part B A second box also has a base of 63 square inches, but it has a volume of Volume 756 cubic is equal inches. to the What area is of the the height, in inches, of the second base times box? the Explain height. or V=bh show how you determined the height. 756 = 63 x height of the cubes 756/63 = height of the cubes 12 = height of the cubes So, the height of the box is 12 inches since there are 12 1-inch cubes stacked on top of each other. Slide 177 () / 219 From PARCC PBA sample test calculator #9 Slide 178 / 219 Surface Area and Volume Application Problems Return to Table of Contents 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? Slide 179 / 219
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? 12.25in 9in Slide 179 () / 219 15.6in 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? Slide 180 / 219 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? Slide 180 () / 219 Area of 1 cube Area of 2 cubes
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. Slide 181 / 219 A Container A B Container B 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. Volume Container A Volume Container B Slide 181 () / 219 A Container A B Container B Container A will hold more cereal becuase the container has a greater volume. 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. Slide 182 / 219 A Container A B Container B
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 A 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 5.5in 7.25in Container B 10.75in Container B will require more cardboard because it has a larger surface area. Slide 182 () / 219 12in 8.5in 3.25in 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 (Problem derived from ) F 1 x 3 x 6 Slide 183 / 219 B 1 x 2 x 12 C 1 x 3 x 8 D 2 x 2 x 8 E 2 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 4 101 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 A, B, C, in G, inches? H, J (Select all that apply.) A 1 x 1 x 24 (Problem derived from ) F 1 x 3 x 6 Slide 183 () / 219 B 1 x 2 x 12 C 1 x 3 x 8 D 2 x 2 x 8 E 2 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
(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. Slide 184 / 219 Height Width Length A 1 C 1 G 4 B 2 D 2 H 6 E 3 F 4 I 8 J 12 (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. Slide 184 () / 219 Height Width Length A 1 B 2 C 1 D 2 E 3 F 4 B, E, G G 4 H 6 I 8 J 12 103 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. Slide 185 / 219 A Container A: 7 in x 6 in x 5 in B Container B: 8 in x 4 in x 9 in
104 Small cubes with edge lengths of 1/4 inch will be packed into the right rectangular prism shown. Slide 186 / 219 How many small cubes are needed to completely fill the right rectangular prism? cubes From PARCC EOY sample test non-calculator #6 105 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. Slide 187 / 219 From PARCC EOY sample test calculator #10 105 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. Slide 187 () / 219 3/8 From PARCC EOY sample test calculator #10
106 Part B What is the volume, in cubic inch(es), of 1 of the small cubes? Enter your fraction. Slide 188 / 219 From PARCC EOY sample test calculator #10 106 Part B What is the volume, in cubic inch(es), of 1 of the small cubes? Enter your fraction. Slide 188 () / 219 1/64 From PARCC EOY sample test calculator #10 Slide 189 / 219 More Polygons in the Coordinate Plane Return to Table of Contents
107 Draw a polygon in the coordinate plane using the given Students type their answers here coordinates. Slide 190 / 219 (4, -4) (6, -2) (8, -6) What is the area of the polygon? (Problem from ) 107 Draw a polygon in the coordinate plane using the given Students type their answers here coordinates. Slide 190 () / 219 (4, -4) (6, -2) (8, -6) What is the area of the polygon? (Problem from ) 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? Slide 191 / 219
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? Slide 191 () / 219 Vertex [This object is is a pull (1,1) tab] 109 What is the area of the square block described in the previous problem? Slide 192 / 219 109 What is the area of the square block described in the previous problem? Slide 192 () / 219
Slide 193 / 219 Glossary & Standards Return to Table of Contents Slide 193 () / 219 Teacher Notes Vocabulary Words are bolded in the presentation. The text Glossary box the & word Standards is in is then linked to the page at the end of the presentation with the word defined on it. Return to Table of Contents 3-D Figures An object with three different dimensions: length, width (or depth or breadth), and height. Also called a solid figure. Slide 194 / 219 One-Dimensional Two-Dimensional Three-Dimensional length length width width length height Back to Instruction
Altitude A line segment from a vertex of the triangle and perpendicular to the opposite side. The height. Slide 195 / 219 h h h b b b This is not the height. It is not perpendicular to the base. Back to Instruction Slide 196 / 219 Base & Height Slide 197 / 219 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
Cone A 3-dimensional figure with one circular base, a vertex at the top, and one curved surface connecting the two. Slide 198 / 219 1 Circular Base 1 Curved Surface 1 Vertex Back to Instruction Cube Slide 199 / 219 A 3-dimensional figure with 3 pairs of parallel, congruent, square bases. 8 vertices 6 faces 12 Edges Back to Instruction Cylinder A 3-dimensional figure with twocongruent, circular bases, and one curved surface connecting them. Slide 200 / 219 2 congruent, parallel, circular bases 1 curved surface No vertices Back to Instruction
Diagonal Slide 201 / 219 A line that goes from one nonadjacent vertex to another. Cannot draw a diagonal, because all vertices are adjacent. Back to Instruction Dimensions Slide 202 / 219 A measurement of length in one direction. 1 dimension 2 dimensions 3 dimensions length Back to Instruction Edge Slide 203 / 219 The line segment where two faces meet. edge 10 edges Back to Instruction
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. Slide 204 / 219 Faces: 6 Vertices: 8 6+8-2=12 Edges: 12 Back to Instruction Face Slide 205 / 219 A flat surface of a 3-d figure. Face 6 faces There is still debate over whether curved surfaces are faces. Back to Instruction Slide 206 / 219
Slide 207 / 219 Net A 2-dimensional pattern which can be folded into a 3- dimensional figure. Slide 208 / 219 = = Back to Instruction Slide 209 / 219
Perimeter Slide 210 / 219 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 Polyhedron Slide 211 / 219 A three dimensional figure with all flat faces. Polyhedra "Polyhedra" is the singular form of polyhedron non-polyhedra Back to Instruction Prism A 3-dimensional figure with two congruent, parallel bases, and all other faces are rectangles. Slide 212 / 219 Prisms are named by the shape of their bases. Pentagonal Prism 2 triangular bases Triangular Prism 3 rectangular faces Back to Instruction
Pyramid Slide 213 / 219 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 Square Units A measurement in the shape of a square with side lengths that are one unit long. Slide 214 / 219 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 Surface Area Slide 215 / 219 The total area of the surface of a 3-dimensional figure. SA = + + + + + 8 u 2 6 u 2 12 u 2 6 u 2 12 u 2 8 u 2 Surface Area= 2lw+2lh+2wh 2 12+2 6+2 8 SA=24+12+16 SA=52u 2 Back to Instruction