Westland Middle School. Math 7. Unit 3 Topic 1 Student Packet

Transcription

1 1 1 Westland Middle School Math 7 Unit 3 Topic 1 Student Packet

2 A Brief History of Pi ( ) Date: Period: 2 Pi has been known for almost 4000 years but even if we calculated the number of seconds in those 4000 years and calculated pi to that number of places, we would still only be approximating its actual value. Here s a brief history of finding pi: The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca BC) indicates a value of for pi, which is a closer approximation. The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of for pi. The first calculation of pi was done by Archimedes of Syracuse ( BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, Archimedes showed that pi is between 3 1/7 and 3 10/71. A similar approach was used by Zu Chongzhi ( ), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes method but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for pi, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places. Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in Page 1 of 1 2

3 Circle Vocabulary Summarizer Date: Period: 3 In your own words, define the terms and show an example of each term on the circle below. Be sure to label each example in your definition. Center Diameter Radius Radii Briefly describe the circumference of a circle: Page 1 of 1 3

4 Circumference & Diameter Problem Solving Date: Period: 4 Before completing each prompt, read the question and determine which form of pi would be best used to accurately answer your question. Circle the best form of pi. Then, determine a solution for each problem. Form of pi Prompt 1) The distance around the wheel of a truck is 9.42 feet. What is the diameter of the wheel? ) A dog is tied to a wooden stake in a backyard. His leash is 3 meters long and he runs around in circles pulling the leash as far as it can go. What is the circumference of the space the dog has to run around? Page 1 of 2 4

5 Circumference & Diameter Problem Solving Date: Period: 5 Form of pi Prompt 3) An asteroid hit the earth and created a huge round crater. Scientists measured the distance around the crater as 78.5 miles. What is the diameter of the crater? ) Barthemele is biking around a circular track and jamming to some tunes. His bike tires are 2 feet in diameter, and the track is 1000 feet in diameter. How many times do Barthemele s tires go round in one lap around the track? Page 2 of 2 5

6 6 Historic Bicycle The circumference of a circle, C =!d, where d is the diameter. Basil saw a strange old bicycle at the museum. It had one very big wheel and one very small one. It was called an Ordinary or a Penny Farthing. At home Basil looked it up on the internet and found that the big wheel could have a 52 inch diameter and the small wheel could have an 18 inch diameter. 1. What is the circumference of the big wheel? Show how you figured it out. inches 2. How far would you travel in one turn of the big wheel? Give your answer in feet and inches. Show how you figured it out. feet inches 3. How many times must the cyclist turn the big wheel to travel 1 mile? A mile is 1760 yards. Give your answer to the nearest 10 turns. Show how you figured it out. 4. How many times does the small wheel turn when the cycle travels 1 mile? Show how you figured it out. Copyright 2009 by Mathematics Assessment Resource Service. All rights reserved. Historic Bicycle 6

7 Lab: Circles 7 A Circle A ( A ) Page 1 of 12 7

8 Lab: Circles 8 B Circle B ( B) Page 3 of 12 8

9 Lab: Circles 9 C Circle C ( C) Page 5 of 12 9

10 Lab: Circles 10 D Circle D ( D) Page 7 of 12 10

11 Lab: Circles 11 This page intentionally left blank. Page 8 of 12 11

12 Lab: Circles 12 E Circle E ( E) Page 9 of 12 12

13 Lab: Circles 13 F Circle F ( F) Page 11 of 12 13

14 Lab: The Relationship between Diameter & Circumference Date: Period: 1) Measure the circumference of each circle to the nearest cm or in (using string and a meter stick). 2) Measure and record the diameter of each circle. 3) Using a calculator, divide the circumference by the diameter. Round to the nearest hundredth. 4) Repeat the steps for each circle. Station Circumference in cm or in. (C) Diameter in cm or in.(d) C d A B C D E F a) Compare the quotients you obtained when you divided each circle s circumference by its diameter (last column). What do you notice? 14 b) What generalizations can you make about the relationship between diameter of a circle and its circumference? Page 1 of 1 14

15 More Circumference & Diameter Problem Solving Date: Period: 15 Before completing each prompt, read the question and determine which form of pi would be best used to accurately answer your question. Circle the best form of pi. Then, determine a solution for each problem. Form of pi Prompt 1) What is the circumference of a 12-inch pizza (A 12-inch pizza is a pizza who s diameter is 12 inches.)? ) The distance around a carousel is yards. What is the radius? ) Miguel has ordered 100 dinner plates. Each plate is 12 inches in diameter, with gold trim along the edge. What is the total number of inches of gold trim for all 100 plates? Page 1 of 2 15

16 More Circumference & Diameter Problem Solving Date: Period: 16 Form of pi Prompt 4) In modern cataract surgery (eye surgery), the ophthalmologist (eye doctor) makes an incision that is 6% of the circumference of the cornea. If the diameter of a patient s cornea is 8 mm, what is the length of the incision? ) Ariel bought a new bike. The wheels have a diameter of 20 inches. How far will the bike travel if the wheel rotates one time around? Page 2 of 2 16

17 Area of a Circle Problem Solving Date: Period: 1) A semi-circle shaped rug has a diameter of 2 feet. What is the area of the rug? 17 2) A spinner has 6 sectors, half of which are red and half of which are black. If the radius of the spinner is 3 inches, what is the area of the red sectors? 3) Consider a circle that has a circumference of 28π centimeters (cm). a) What is the area, in cm 2, of this circle? Show all work necessary to justify your response. b) What would be the measure of the radius, in cm, of a circle with an area that is 20% greater than the circle in part (a)? Show all work necessary to justify your response. 4) Nicki is opening a pizzeria. He has decided that the small pizza should be $6. If Nicki charges a constant amount per square inch, what are appropriate prices for the medium and large pizzas? Explain how you determine each price. Adapted from Page 1 of 2 17

18 Area of a Circle Problem Solving Date: Period: 18 5) A storm is expected to impact 7 miles in every direction from a small town. What is the area that the storm will affect? 6) Sukai is looking for a new sprinkler. Emma posted a used sprinkler online that she claims will water a circle 500 square feet in area. Sukai s yard is 40 ft x 10 ft. Is it possible that Emma s sprinkler will cover every inch of grass in Sukai s yard? 7) Lamar is making cookies. He rolls the dough into the square below, and is cutting circular cookies from the square. a) If Lamar cuts circular cookies that are 2 inches in diameter. How many cookies can he cut without gathering the scraps? How much dough will be left over? b) How much dough would be left over if Lamar cut one giant cookie 12 inches in diameter? c) How many cookies could Lamar make from the leftovers? Adapted from Page 2 of 2 18

19 Area of Circular Regions Date: Period: 1) Determine the circumference and area of each circle with the following dimension. a) radius = 4.2 in b) diameter = 5.4 cm Sketch and label the circle: Sketch and label the circle: 19 Circumference Circumference Area Area Page 1 of 2 19

20 Area of Circular Regions Date: Period: 2) The circle to the right has a diameter of 12 cm. Calculate the area of the shaded region. How do you know? 20 3) The circle to the right has a diameter of 10 ft. Calculate the area of the shaded region. How do you know? 4) The circle to the right has a radius of 4 in. Calculate the area of the shaded region. How do you know? Page 2 of 2 20

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22 Pizza Crusts Date: Period: 22 Adapted from MARS Page 1 of 2 22

23 Pizza Crusts Date: Period: 23 Adapted from MARS Page 2 of 2 23

24 The Circumference of a Circle & the Area of the Date: Period: Region it Encloses 1) How would you describe a circle? 24 2) If a circle has center A and radius AB, is point A on the circle? Is point B on the circle? Explain. 3) Imagine that a circle with center is drawn on 1 4 inch grid paper as shown below. a) What is the radius of the circle? b) Determine the circumference of the circle. c) Use the grid to estimate the area of the region enclosed by the circle. Figure not drawn to scale. d) What are you measuring when you find the circumference of a circle? What are you measuring when you find the area of a circle? Adapted from Illustrative Mathematics Page 1 of 1 24

25 Attribute Cards Date: Period: 25 SHAPE A A 4-sided polygon with 4 right angles. The base 1 measures 1 inches and the height is 4 inches. 2 SHAPE E A geometric figure with a radius that measures 2.5 inches. SHAPE B A 4-sided polygon with 4 right angles. Each side measures 4 inches. SHAPE F A geometric figure with a diameter measuring 4 inches. SHAPE C A 4-sided polygon with 4 right angles. 1 Each side measures 2 inches. 4 SHAPE D A 3-sided polygon with one right angle. The base measures 3 inches and the height is 4 inches. SHAPE G A 4-sided polygon with bases that are parallel. The two bases measure 3 and 6 inches. The height measures 4 inches. There are two right angles. SHAPE H A 4-sided polygon whose base measures 4 inches 1 and height is 2. 2 Page 1 of 1 25

26 Drawing Geometric Shapes Date: Period: Using only a ruler, set square, small string, and a pencil, draw the following shapes and determine their area and perimeter/circumference. 1) Draw a square PQRS with side length equal to 5 cm. Label side and angle measurements. Determine the area and perimeter. 26 2) Draw a segment AB 6 cm. in length. Draw a circle whose diameter is segment AB. Determine the area and circumference. Adapted from: Page 1 of 2 26

27 Drawing Geometric Shapes Date: Period: 3) Draw a parallelogram with a base of 3.75 in. and a height of 2 in. Determine the area and perimeter. 27 4) Draw a right trapezoid with bases that are 3 cm and 6 cm and a height of 4 cm. Determine the area and perimeter. Adapted from: Page 2 of 2 27

28 How Does It Measure Up? Date: Period: Using only a ruler, protractor, set square, string, and a pencil, draw the following shapes. 1) Line segment AB with a length of inches 28 2) Line segment MN with a length of 7.2 cm. 3) Circle P with radius PR whose length is 4 cm. Page 1 of 2 28

29 How Does It Measure Up? Date: Period: 4) A rectangle with the dimensions 5.4 cm. and 7.2 cm. 29 5) A parallelogram, with a height of 1 inch and a base of 4.5 inches, with angles that are NOT 90. Page 2 of 2 29

30 30 30 use measurement to find the area and perimeter of shapes 1. This parallelogram is drawn accurately. Make any measurements you need, in centimeters, and calculate: a. The area of the parallelogram. Show your calculations. The area of a parallelogram = base x height b. The perimeter of the parallelogram. Show your calculations. 2. The diagram below shows the same parallelogram again. a. Find the area of Triangle A. b. Find the area of Triangle B. c. Explain how you found your answers. Triangle A Triangle B Copyright 2007 by Mathematics Assessment Page 57 Journey Test 7 Resource Service. All rights reserved.

31 Which triangle has a larger perimeter, Triangle A or Triangle B? Explain how you can tell without measuring. 4. Sketch a right triangle with the same area as Triangle A. Your diagram does not need to be accurate. Show how you figured it out. 9 Copyright 2007 by Mathematics Assessment Page 58 Journey Test 7 Resource Service. All rights reserved.

32 Smarter Balanced Assessment Item (7.G.4) Date: Period: 32 Page 1 of 1 32

33 Beyond Basic Area Date: Period: 33 For #1 and #2, one can of red paint can cover 200 in 2. 1) A red and white target is being painted. The radius of the smallest circle is in. and each successive, larger circle is inches larger in radius than the circle before it. One can of paint can cover 200 in 2. How many cans of paint will be needed to paint the target? 2) Max is making a stop sign for his clubhouse. It is an octagon with eight equal sides and eight equal angles, with the dimensions given in the image. He wants to paint one side of the sign red. How many cans of will he need? 3) The diameter of the circle is 12 in. Write and explain a numerical expression that represents the area. Adapted from Page 1 of 3 33

34 Beyond Basic Area Date: Period: 34 4) The circle below has a diameter of 12 cm. Calculate the area of the shaded region. Sasha, Barry, and Kyra wrote three different expressions for the area of the shaded region. Describe what each student was thinking about the problem based on their expression. Sasha s expression: 6 Barry s expression: 6 6 Kyra s expression: 6 5) Find the area of the shaded region of the circle below and explain your thinking. 12. Adapted from Page 2 of 3 34

35 Beyond Basic Area Date: Period: 35 6) Calculate the area of the figure below that consists of a rectangle and two quarter circles, each with the same radius. Leave your answer in terms of pi. 7) The vertices and of rectangle are centers of circles each with a radius of 5 inches. Find the exact area of the shaded region. Adapted from Page 3 of 3 35

36 Keeping Up with the Joneses Date: Period: The Jones family is renovating their house. 36 1a) The diagram below shows the new countertop purchased by the Jones family. Determine the square footage of counter space the Joneses will have, not including the sink. b) The Jones family would also like to tile their kitchen floor, but they will not tile beneath the counter or sink. If the kitchen is 14 feet long and 10 feet wide, calculate the amount of square footage they will need to cover with tile. Describe how you determined your solution. 2) Each floor tile features a star created by four quarter circles within a square. If each quarter circle has a radius of 7 inches, what is the area of the star? Adapted from Page 1 of 2 36

37 Keeping Up with the Joneses Date: Period: 37 3) In addition to the kitchen, the Jones family is purchasing new wall-to-wall carpet for the house. They will carpet everything except their bathrooms, kitchen, and closets (shaded on the floor plan below). How many square feet of carpeting must be purchased for their home? 1 unit = 2 feet Bath Kitchen Bath 4) A new layer of grass sod will be installed in the Jones family home front, back, and side yards. They will not need sod in any of the shaded areas in the diagram below. The house has the dimensions shown in #3. The house sits on a trapezoidal lot that is 72 feet deep. The shed in the backyard is 8 feet long and 3 feet deep. The garage and driveway are 12 feet wide. The driveway extends 14 feet beyond the house. The front flower bed is a semicircle and the walkway is a quarter circle with a radius of 3 feet. Determine the number of square feet of sod that will need to be purchased. Adapted from Page 2 of 2 37

38 Strategy Analysis Date: Period: 38 As a group, analyze the strategies you used to determine area solutions in the resource, Keeping Up with the Joneses. Every scenario required additional reasoning beyond basic area formulas. In each box below, provide a detailed example of when you used each of the following operations in your reasoning, above and beyond the area formulas. You must refer to each scenario at least once. Addition Subtraction Multiplication Division Page 1 of 1 38

39 Track and Field Date: Period: A scale drawing of an Olympic track and the field within the track are shown below. a) What shapes make up the figure? 39 b) What is the radius length of each half-circle? How do you know? c) What are the dimensions of the rectangle? How did you determine them? d) Find the total area of the track and field. e) An athlete runs along the perimeter of the track. How far have they run? meters meters Page 1 of 1 39

40 Build a Box Date: Period: Challenge: Create a prism from the given 8x 8 square using the fewest number of cuts. 1) What type of prism have you created? 40 2) What is the area of the base of your prism? 3) What is the surface area of your prism? 4) What is the volume of your prism? 5) Predict: Could a classmate have created a different prism using the same 8x8square? How might the surface areas of those prisms compare? How might their volumes compare? Page 1 of 1 40

41 41 Candle Box Task description Pupils design a 2D net for a box, given a 3D picture of the box. Resources Ruler and pencil; it may be useful to have spare copies of the worksheet available; scissors available but provide only on request Key Processes involved Representing: Represent a 3D object in 2D. Analysing: Visualise how the box will look when opened flat; draw their net accurately, joining faces appropriately and keeping the correct orientation of the pictures on the faces. Interpreting: Imagine the net folded up again (after drawing it) and locate a sufficient number of flaps so that faces may be joined together; ensure that glue flaps do not interfere with one another. Communicating and reflecting: Draw the box design clearly and labelled. Teacher guidance Check that pupils fully understand the task context before they begin, with points such as: You are asked to draw a net, in one piece, for a candle box. What will the box look like when it is assembled? What types of flaps will you need to include? Some are to be glued and some not; how will you tell which ones are which? What is important about the candle designs on the sides and top of the box? Pupils may tackle this task in different ways, but they might be expected to: make 3D mathematical models by linking given faces or edges, and draw common 2D shapes in different orientations on grids recognise and use common 2D representations of 3D objects 2010 Bowland Charitable Trust 1 41

42 Candle Box 42 Candle Box Hinge Top View Thumb hole for opening Tom is making a little gift box to hold a big candle. I want the top and the base to be regular hexagons. The sides will be rectangles. A little candle design will go on each side. There will be a thumb hole to help you open the box. Please help Tom by drawing an accurate plan for making his box using the dotted paper. It should be drawn so that when it is cut out it will all be in one piece. One of the sides has been drawn to start you off. Remember: flaps are needed for gluing the box together; shade these in flaps are needed for fastening the lid, but these will not be glued! so don t shade them draw the candle on the lid and the thumb hole draw a picture of the Birthday Candle on each side get it right way up! 2010 Bowland Charitable Trust 2 42

43 Candle Box 43 Draw your design below: 2010 Bowland Charitable Trust 3 43

44 Candle Box 44 Assessment Guidance Progression in Key Processes Representing Analysing Interpreting and evaluating Representation of a 3D object in 2D Geometric accuracy and completeness of net Interpretation of net in relation to the problem; placing of glue flaps and drawings of candle Communicating and reflecting Accuracy and completeness of final design P R O G R E S S I O N Represents part of the box on the isometric paper. Pupil A Represents most of the box on the isometric paper. Pupil B Represents all faces of the box on the isometric paper. Pupil C Represents all faces and glue flaps of the box on the isometric paper. Pupil D Draws some faces correctly, but does not join them up appropriately. Pupil A Draws most faces correctly; joins some up correctly, but with several omissions and inaccuracies. Pupil B Draws a net accurately, including the correct number of faces, joined appropriately. Pupil C Draws a net accurately, including the correct number of faces, joined appropriately. Pupil D Cannot interpret own drawing in terms of 3D box. The picture on the box is not considered or is incorrectly placed. Pupil A After drawing the net, tries to imagine it folded up again. Draws candles and glue flaps but in wrong orientations/ positions. Pupil B Locates some glue flaps and lid flaps, but may not distinguish them. Candle picture omitted or in wrong orientation on some faces. Pupils C and D Locates glue flaps and lid flaps, and distinguishes between them. Ensures that these do not interfere with one another. Candle picture in correct orientation on all faces. Clearly draws some faces of the box, with inaccuracies and/or omissions. Clearly draws most faces of the box, but with inaccuracies and/or omissions. Pupils A and B Clearly draws and labels the box design. Pupils C and D 2010 Bowland Charitable Trust 4 44

45 Composite and Surface Area Date: Period: Is it possible to determine the area of the composite figure below? Explain your thinking units 15 units 32 units Page 1 of 1 45

46 46 Fruit Boxes A grocer wants to sell fruit in boxes. He wants to make the boxes from square card 36 inches long and 36 inches wide as shown. 4 inches inches F F F F 3036 inches (Diagrams not drawn to scale.) L L 3630 inches The shaded areas are cut away and the rest is folded along the dashed lines. The sides are folded up and stuck together using the four flaps marked F. The lid has two flaps, marked L, which are not glued. 1. Calculate the volume of the finished box. Show your work. Please continue your work on the page opposite _ Copyright 2011 by Mathematics Assessment Page 1 Fruit Boxes Resource Service. All rights reserved. 46

47 47 47 Fruit Boxes (continued) 2. Suppose he starts with the same square of card, but changes the 4 inches to a different measurement. What is the largest volume he can make the box? Show your calculations. Copyright 2011 by Mathematics Assessment Page 2 Fruit Boxes Resource Service. All rights reserved.

48 It Figures Date: Period: 48 Page 1 of 4 48

49 It Figures Date: Period: 49 Page 2 of 4 49

50 It Figures Date: Period: 50 Page 3 of 4 50

51 It Figures Date: Period: 51 Page 4 of 4 51

52 It Figures Capture Sheet Date: Period: 52 Figure: Surface Area Sketch: Figure: Surface Area Sketch: Describe the net of the figure: Describe the net of the figure: Figure: Surface Area Sketch: Figure: Surface Area Sketch: Describe the net of the figure: Describe the net of the figure: Page 1 of 1 52

53 Surface Area Date: Period: Calculate the surface area of the square pyramid. 2. A rectangular prism is removed from a larger rectangular prism. Determine the surface area of both pieces. 3. The house below is a prism with a square base. It has a roof that is a right pyramid. Determine the surface area of the house. Adapted from Page 1 of 3 53

54 Surface Area Date: Period: Determine the surface area. a. b. c. d. Adapted from Page 2 of 3 54

55 Surface Area Date: Period: 5a. Determine the surface area of the cube. 55 5b. A square hole with a side length of 4 inches is drilled through the cube. Determine the new surface area. 6. The base rectangle of a right rectangular prism is 4 ft. 6 ft. The surface area is 288 ft 2. Find the height. Let be the height in feet. Adapted from Page 3 of 3 55

56 Surface Area and Volume Date: Period: 56 1) A triangular prism has a volume of 3240 cubic feet. If the height of the prism is 30 feet, what is the area of the base? 2) The surface area of a cube is 864 square meters. a. What is the length of each side of the cube? How do you know? b. How would you determine the volume of the cube? What is the volume? 3) Diamond was worried that her nephew s birthday gift would open in the mail, so she duct taped it shut with fun Superman tape. She covered the box below with duct tape and did not overlap. The Superman tape costs 10.6 cents per square inch. How much money will Diamond have to spend on Superman duct tape? Page 1 of 1 56

57 Volume Right Rectangular Prisms Date: Period: Determine the volume of each prism. Be sure to correctly identify the base, B. a. b. c. d. Adapted from Page 1 of 2 57

58 Volume Right Rectangular Prisms Date: Period: 2. Calculate the volume of the composite prism: 58 3a. Show that the following figures have equal volumes. cm cm cm cm cm. cm 3b. How can it be shown that the prisms will have equal volumes without completing the entire calculation? Adapted from Page 2 of 2 58

59 Volume Scenarios Date: Period: What do you notice about the prisms below? What do you wonder? 59 Page 1 of 2 59

60 Volume Scenarios Date: Period: The Jefts family wants to install a pool like the one below. 1) What is the maximum volume of water the pool can hold? 60 2) The largest water delivery truck holds 1200 cubic feet of water. The Jefts have one truckload of water placed into the pool. If Lynne is 5 feet tall, will the water be over her head? Explain. Page 2 of 2 60

61 61 Cubes & Rectangular Prisms Date: Period: 61 Page 1 of 2

62 62 Cubes & Rectangular Prisms Date: Period: 62 Page 2 of 2

63 Drawing Slicing Planes Date: Period: 63 1) In the following exercises, the points at which a slicing plane meets the edges of the right rectangular prism have been marked. Each slice is either parallel or perpendicular to a face of the prism. Use a straightedge to join the points to outline the rectangular region defined by the slice and shade in the rectangular slice. Adapted from Page 1 of 2 63

64 Drawing Slicing Planes Date: Period: 64 2) Use the dimensions to sketch the slice from each prism below and provide the dimensions of each slice. Adapted from Page 2 of 2 64

65 Slicing Rectangular Prisms Capture Sheet Date: Period: 65 Rectangular Prism Model Directions: Slice the cube parallel to the shaded base. Describe the two-dimensional shapes that result when a plane slices a rectangular prism parallel to its base. Rectangular Prism Model Directions: Slice the cube perpendicular to the shaded base. Describe the two-dimensional shapes that result when a plane slices a rectangular prism perpendicular to its base. Summary: What have you notice about two-dimensional shapes created by slicing three-dimensional figures? Explain your findings. Page 1 of 4 65

66 Slicing Rectangular Prisms Capture Sheet Date: Period: 66 Rectangular Prism Model Directions: Slice the rectangular prism parallel to the shaded base. Describe the two-dimensional shapes that result when a plane slices a rectangular prism parallel to its base. Rectangular Prism Model Directions: Slice the rectangular prism perpendicular to the shaded base. Describe the two-dimensional shapes that result when a plane slices a rectangular prism perpendicular to its base. Summary: How are these two-dimensional shapes similar or different from the two-dimensional shapes created when slicing a cube? Page 2 of 4 66

67 Slicing Rectangular Prisms Capture Sheet Date: Period: Rectangular Prism Model Directions: Slice the rectangular prism parallel to the shaded base. 67 Describe the two-dimensional shapes that result when a plane slices a rectangular prism parallel to its base. Rectangular Prism Model Directions: Slice the rectangular prism perpendicular to the shaded base. Describe the two-dimensional shapes that result when a plane slices a rectangular prism perpendicular to its base. Summary: What have you notice about two-dimensional shapes created by slicing three-dimensional figures? Explain your findings. Page 3 of 4 67

68 Slicing Rectangular Prisms Capture Sheet Date: Period: 68 Rectangular Prism Model Directions: Rectangular Prism Model Directions: Can you slice the rectangular prism that result in a triangle crosssection? Can you slice the rectangular prism that result in a trapezoidal cross-section? If possible, describe how you sliced your figure. If possible, describe how you sliced your figure. Rectangular Prism Model Directions: Can you slice the rectangular prism that result in a pentagonal crosssection? Can you slice the rectangular prism that result in a hexagonal cross-section? If possible, describe how you sliced your figure. If possible, describe how you sliced your figure. Summary: What are you noticing or discovering about two-dimensional shapes created by slicing three-dimensional figures? Explain your findings. Page 4 of 4 68

69 What s the Result? Date: Period: Directions: Name the two-dimensional plane sections that result in the slice made in the threedimensional figure Adapted from Howard County Public Schools Page 1 of 1 69

70 Sometimes, Always, Never Date: Period: 70 Read and think about each statement. Determine whether the statement is sometimes true, always true, or never true. If the statement is only sometimes true, give a counterexample. 1. If you cut/slice a square pyramid perpendicular to its base, then the cross section will be a square. 2. If you cut a prism parallel to its base, then the cross section will be the same shape as the base. 3. If you cut a prism parallel to its base, then the cross section will be congruent to the base. 4. If you cut a pyramid parallel to its base, then the cross section will be congruent to the base. 5. If you make a cut perpendicular to the base of a pyramid, then the cross section will be a triangle. 6. If you make a cut parallel to the base of a pyramid, then the cross section will be a triangle. 7. If you make a cut to any right rectangular prism, parallel or perpendicular to its base, then the cross section will be a rectangle. 8. If you cut a sphere, then the cross section is a circle. Adapted from MSDE Page 1 of 3 70

71 Sometimes, Always, Never Date: Period: 71 ANSWER KEY Directions: Read and think about each statement. Determine whether the statement is sometimes true, always true, or never true. If the statement is only sometimes true, give a counterexample. Never 1. If you cut/slice a square pyramid perpendicular to its base, then the cross section will be a square. Always 2. If you cut a prism parallel to its base, then the cross section will be the same shape as the base. Always 3. If you cut a prism parallel to its base, then the cross section will be congruent to the base. Never 4. If you cut a pyramid parallel to its base, then the cross section will be congruent to the base. Sometimes 5. If you make a cut perpendicular to the base of a pyramid, then the cross section will be a triangle. (only a triangle if cut is from apex of pyramid) Sometimes 6. If you make a cut parallel to the base of a pyramid, then the cross section will be a triangle. (only if the base of the pyramid is a triangle) Always 7. If you make a cut to any right rectangular prism, parallel or perpendicular to its base, then the cross section will be a rectangle. Always 8. If you cut a sphere, then the cross section is a circle. Adapted from MSDE Page 3 of 3 71

72 72 Right Rectangular Pyramids Date: Period: 72 Page 1 of 1

73 Slicing Game Capture Sheet Date: Period: 73 Shape with slicing example Parallel Slice Generalizations about slice(s) performed Perpendicular Slice Skew Slice Page 1 of 1 73

74 Slicing Prisms and Pyramids Date: Period: 74 Directions: Examine the three-dimensional figures below. Use your pencil to draw the different ways you could create a parallel slice to each of the figure s base. Examine the diagrams of another student. 1) How are your diagrams the same? How are they different? 2) What does your new surface look like? Page 1 of 3 74

75 Slicing Prisms and Pyramids Date: Period: 75 Directions: Examine the three-dimensional figures below. Use your pencil to draw the different ways you could create a skew slice to each of the figure s base. Examine the diagrams of another student. 3) How are your diagrams the same? How are they different? 4) What does your new surface look like? Page 2 of 3 75

76 Slicing Prisms and Pyramids Date: Period: 76 5) Examine the three-dimensional figures below. Use your pencil to draw a slice through each figure that creates the same cross-section. Explain what you created. Page 3 of 3 76

77 The Most Sides! Date: Period: 77 1) Directions: Draw a slice that creates the maximum possible number of sides for each planesection. Explain how you got your answer. Adapted from Page 1 of 1 77

78 Applications with Composite Volume Date: Period: 78 1) Find the volume of the prism below. 2) Find the volume of the prism below. Adapted from: Page 1 of 1 78

79 Capture Sheet: Composite Figures Date: Period: Challenge Task A: Keepin it Cool 79 Challenge Task B: Play On Adapted from: Page 1 of 2 79

80 Capture Sheet: Composite Figures Date: Period: Challenge Task C: Tiny House 80 Challenge Task D: Log Splitter Wedge Adapted from: Page 2 of 2 80

81 Challenge Tasks: Composite Figures Date: Period: Challenge Task A: Keepin it Cool 81 A Styrofoam cooler does a good job of keeping things cold because the material is a poor conductor of heat. A closed container of Styrofoam creates a cold zone into which heat from the outside enters at a very slow rate. Styrofoam has good insulating properties because it has millions of tiny air bubbles that slow the progress of heat through the material. Nina constructed an insulated ice cooler. Her cooler shown below was made from a large cube of Styrofoam with a hollow inside that is a right rectangular prism with a square base. The figure on the right is what the box looks like from above. Discuss with your group how you would find the volume of the insulation. Determine the volume of the Styrofoam insulation in cubic centimeters. Determine the amount of ice the cooler can hold in cubic centimeters. Adapted from: Page 1 of 4 81

82 Challenge Tasks: Composite Figures Date: Period: Challenge Task B: Play On 82 Toys can be for entertainment and for education. Shape toys have always been a popular way to introduce small children to geometric shapes. Below is a child s wooden shape sorting toy constructed by cutting a right triangular prism out of a right rectangular prism. Discuss with your group how you would find the volume of the wooden triangular prism and the wooden block into which it fits. Determine the volume of the material in each of the two parts of the wooden shape sorting toy. Adapted from: Page 2 of 4 82

83 Challenge Tasks: Composite Figures Date: Period: Challenge Task C: Tiny House 83 The tiny house movement is a social movement where people are reducing the space they live in. Tiny houses come in all shapes and sizes but they focus on smaller spaces and the belief that less is more. Discuss with your group what prisms make up the tiny house shown. Determine the volume of the structure. Adapted from: Page 3 of 4 83

84 Challenge Tasks: Composite Figures Date: Period: Challenge Task D: Log Splitter Wedge 84 Wood harvested from trees must be split into smaller sections to be burned in a fireplace or a campfire. A log splitter wedge is a hand tool that helps split the wood in multiple directions. Wedges are typically constructed from high carbon steel. Discuss with your group what prisms can be combined to create the log splitter wedge. Determine how many cubic centimeters of steel is needed to construct the wedge. Adapted from: Page 4 of 4 84

85 85 The base of the right prism is a hexagon composed of a rectangle and two triangles. Find the volume of the right hexagonal prism using the formula =. Lesson 23: The Volume of a Right Prism Date: 9/14/ Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution NonCommercial ShareAlike 3.0 Unported 85 License.