Demystifying Surface Area and Volume Teachers Edition

Transcription

1 Demystifying Surface and Volume Teachers Edition These constructions and worksheets can be done in pairs, small groups or individually. Also, may use as guided notes and done together with teacher. CYLINDER 1. Use the net of the cylinder provided. Measure in centimeters and record the radius of the circle, and the length and width of the rectangle. TC-1 Be sure to check that students measure in centimeters and round to the nearest tenth. Discuss with students which side is length--suggest use the longest side. Do not include flaps in measurements! Answer: radius should be approximately 4 cm, length 5 cm, width 10 cm. radius = length = width =. Cut out the circles and rectangle. Use tape and construct the cylinder. Sketch a picture of your cylinder in the space below: TC- Length and width are used instead of base and height to eliminate confusion later (seetc-). Help students tape cylinder together, tabs may go on outside. Check student s sketches as they may have difficulty drawing three-dimensional shapes.. The rolled up rectangle is called the lateral face of the cylinder. What shape is the base of the cylinder? How do you know what is the base? Can you have more than one shape as a base? Explain. Answer: The base is a circle. Bases are the shapes that are in parallel planes. Either circle is the base; it is the only shape that is possible for a base of a cylinder. 4. Calculate the circumference of the cylinder. How does this value compare to the dimensions of the rectangle? Explain. Answer: The length of the rectangle should be the same as circumference of circle. 5. The amount of paper used to make a cylinder is the surface area. How can you calculate the surface area of your cylinder? Explain. Answer: Add together the area of the two circles and the area of the rectangle Demystifying Surface and Volume Teacher Materials Page 1 of 11

2 6. What is the area of each region and total surface area? Include the units of measurement. Circles Rectangle Total Surface Answer: The area of the two circles should be approximately cm, and the total surface area cm cm, the rectangle 7. When you fill a cylinder with something, such as plain M & M s, you are finding volume. Remember, to calculate volume, multiply the area of the base times the height of the cylinder. What measurement from part 1 above corresponds to the cylinder s height? TC- Be sure to check that students understand width of original rectangle is the height of the cylinder. 8. What is the volume of your cylinder? (Remember to include the appropriate units) Answer: 50.4 cm, x 10cm = 50.4 Volume = cm 9. If 60 plain M & M s take up 1 cm of space, approximately how many M & M s would fill your cylinder? Explain and show your work. Answer: Volume divided by 1 times 60 = number of M & M s. Approximately 51 plain M & M s Demystifying Surface and Volume Teacher Materials Page of 11

3 RIGHT TRIANGLE PRISM 1. Use the net provided for a right triangle prism. Measure in centimeters the dimensions of the right triangles and rectangles. TC-4 Have students fold rectangles instead of cutting, Also, discuss what is considered the length and width of the right triangles. Do not measure flaps! Length Width Triangles Rectangle A Rectangle B Rectangle C Answer: Triangle length and width are approximately 5. cm by 5 cm, rectangle A is 15.4cm by 5 cm, rectangle B is 15.4 cm by 5. cm, and rectangle C is 15.4 cm by 7.4 cm.. Cut out the net of triangles and rectangles. Use tape and fit together the net to make a triangular prism. Sketch below different orientations of the prism: TC-5 Be sure to allow ample time to think here or discuss with partners. A good question to ask is if there are only two orientations. As an alternative, have students draw one triangular prism, but look at a neighbor s and discuss any differences. Answer: prism could be drawn with one of the triangles on the bottom, or could be drawn with any of the three rectangles on the bottom. 4 possible orientations.. What shape is considered the base of this prism, regardless of its orientation? Explain how and why this shape makes sense. Answer: The triangle. It is the only shape in parallel planes. 4. The shapes that connect the bases are called lateral faces. What shapes are the lateral faces, and are they all the same size? TC-6 This is a good place to review edges and if any edges have the same length. Answer: the lateral faces are the rectangles. They are not the same size. Demystifying Surface and Volume Teacher Materials Page of 11

4 5. What measurement do all of the lateral faces have in common? What does this measurement mean with respect to the prism? TC-7 Students may have trouble here, check for student understanding of the measurement it is the height of the prism. Sometimes an alternative word such as thickness helps comprehension. Answer: the length of 15.4 cm. This is the height of the prism, as it connects the bases. 6. With a ruler, measure in centimeters the hypotenuse of the right triangle. What measurement does this correspond to from part 1? Answer: approximately 7.4 cm, which is the width of rectangle C. NOTE: Pythagorean Theorem is discussed in question # 1..you can discuss it now or wait until then. 7. The amount of paper used to make a triangular prism is the surface area. How can you calculate the surface area of your triangular prism? Explain. Answer: add together the areas of the triangles and the three rectangles. 8. What is the area of each piece and total surface area? What are the units of measurement? Triangles Rectangle A Rectangle B Rectangle C Total Surface Answer: The triangles have area of approximately 6 cm, rectangle A 77 cm, rectangle B cm, and rectangle C cm. Total surface = cm, 9. When you fill a prism with something, such as plain M & M s, you are finding volume. To calculate volume, multiply the area of the base times the height of the prism. Calculate the volume of your triangular prism, show work. Demystifying Surface and Volume Teacher Materials Page 4 of 11

5 TC-8 Students have trouble understanding which height to use. The area of a triangle is half base times height, whereas height of prism is a different value. Be sure to check student work! Students may also wonder why their volume value is less than surface area value. Remind students that one is in centimeters squared, the other in centimeters cubed. Answer: 1 cm x 15.4 cm = 00. cm Volume = Tom, a friend in your class, is confused. He knows that the area of a triangle is bh, and that volume involves height. He does not know which one to use when. Help him out and explain the difference between them. TC-9 Check for student understanding! Answer: the base and height of the triangle are not used in determining the height of the prism. They are used in calculating the area of the base. The height is 15.4 cm. 11. If 60 plain M & M s take up 1 cm of space, approximately how many M & M s would fill your triangular prism? Explain and show your work Answer: Volume divided by 1 x 60 = 1001 plain M & M s 1. Another friend of yours, Jane, says that if you know the two legs of the right triangle base, you can easily find the third side without measuring with a ruler. Why does she think this? Do you agree with Jane? TC-10 A good review question is how using Pythagorean Theorem for calculating. This would be a good time to check if students get the same answer by calculating and measuring. Answer: Yes, you can use the Pythagorean Theorem, as the length and width of the right triangle are the same as the legs in the Pythagorean Theorem Demystifying Surface and Volume Teacher Materials Page 5 of 11

6 RECTANGULAR PRISM 1. Use the net provided of the rectangular prism. Measure in centimeters the dimensions of all the rectangles. TC-11 Check for measurement in centimeters, and do not measure the flaps! Rectangles A Length Width Rectangles B Answer: The length and width of rectangle A is 15.cm x 5cm, and 5cm x 5cm for rectangle B. Length is the longest side.. Susan wants to use base and height instead of length and width when she measures the dimensions of the rectangles. Is this okay? Explain TC-1 Take a class vote to see what students think and discuss. Answer: Yes, either is fine.. Cut out the net of the rectangular prism. Use tape and fit together to make the prism. Sketch below different orientations of the prism: TC-1 Are there more than two? Alternative is to have students compare with neighbor. It is ok for students to tape flaps on the outside of the prism. Answer: Yes, although rectangle A is the same on four of the six sides. This is a good time to show another example, such as a shoebox, where there are three different rectangles that can make up a rectangular prism. 4. What shape will all the bases and lateral faces be? Does it matter which orientation you use to determine surface area or volume? Explain why or why not. TC-14 Check with students, discuss whether all orientations will work. Answer: All of the faces are rectangles. It does not matter which orientation students use. This is a good time to have half of the class place rectangle A as the base and the other half rectangle B. In question 6, show students that each orientation will give the same surface area. Demystifying Surface and Volume Teacher Materials Page 6 of 11

7 5. How can you determine the surface area of your rectangular prism? Answer: Add up the areas of all of the faces. 6. Calculate the area of each face of your prism; be sure to include your units. How many of each rectangle do you need? Rectangles A Rectangles B Total Surface Answer: Need four of rectangle A, 4(15. cm x 5 cm) = 04 cm, two of rectangle B, (5 cm x 5 cm) = 50 cm. Total surface area = 54 cm. 7. Meagan remembers from middle school that you can determine volume of a rectangular prism by calculating length times width times height. She is having trouble figuring out how to look at her prism and determine which side is which. How would you help her? Explain. Answer: It does not matter in a rectangular prism; any face could be considered the base. Usually the longest edge would be considered the length, and the other two edges the width and height. 8. If you calculated volume by multiplying the area of the base times the height, would you get the same answer as Meagan? Which dimension would you use as the height? Explain. Answer: The height of the prism depends on which rectangle is the base. If rectangle A is the base, then the height is 5 cm. If rectangle B is the base, then 15. cm is the height. Regardless of method, the volume is (15.cm x 5 cm x 5cm = 80 cm ) 9. If 60 plain M & M s take up 1 cm of space, approximately how many M & M s would fill your rectangular prism? Show your work. Answer: Volume divided by 1 x 60 = 1900 M & M s Demystifying Surface and Volume Teacher Materials Page 7 of 11

8 10. Look around the classroom, around your school, or outside. Are there any examples of cylinders or triangular prisms you can see? List them below: TC-15 You can include rectangular prisms if desired. It is challenging to find right triangular prisms. Answers will vary. Demystifying Surface and Volume Teacher Materials Page 8 of 11

9 Skateboard Parks and Camping: Surface and Volume in the real world One of the more popular locations for prisms is a skateboard park. A ramp is drawn in the space below. The height of the ramp is 6 feet, the skating width is 5 feet, and the entire length of the ramp along the ground is 0 feet. Mark the picture below with these dimensions. 1. The ramp is a combination of a rectangular prism and right triangle prism. Do you have enough information to determine the surface area of the entire ramp structure? Explain. TC-16 Allow time for students to mark diagram. They should notice that they are missing information. The skating width is the face of the hypotenuse of the right triangle. Answer: No. Students need the length of the rectangle.. If the top of the ramp is a rectangle with dimensions of 5 feet by 7 feet, calculate the surface area of the ramp structure (including the floor). Is there a hidden side of the rectangular prism that is not going to be used in this calculation? What about in the triangular prism? Explain. TC-17 Allow time for students to discuss with classmates. Answer: The surface area does not include the side where the triangular prism and rectangular prism are joined. Teacher could demonstrate this with a net if necessary. Surface area is ( 5 ft x 7 ft) rectangles, ( 6 ft x 7 ft) rectangles, 1(6 ft x 5 ft ) rectangle, 1 (1 ft x 5 ft) rectangle, (6 ft x 1 ft) triangles, and 1 (5 ft x 14. ) rectangle. Total surface area is approximately 98.6 ft. Calculate the volume of the skateboard ramp. Do you need to worry about the hidden faces when you calculate volume? Explain. TC-18 Students may have a hard time understanding that the hidden face does not affect volume. A net of the entire ramp may be helpful. Answer: The volume of the rectangular prism portion is (7 ft x 6 ft x 5 ft) = 10 ft, and the volume of the triangular prism is (9 ft x 5 ft) = 195 ft, for a total volume of 405 ft. Demystifying Surface and Volume Teacher Materials Page 9 of 11

10 4. Your friend Brian thought this ramp was cool and decided that he wanted to make one at home out of plywood. He already has a framework made and has to add sheets of plywood to finish it. Which calculations would he want to use, the surface area or volume? Would the hidden side(s) be necessary for construction? Explain. Answer: He would need to use the calculations for surface area. Skateboarding occurs on the surface of the ramp, whereas volume is the amount of space inside the ramp structure. The hidden side (where the rectangular and triangular prism meets) is not necessary if the framework is there. 5. If plywood is sold in sheets that are 8 feet by 4 feet, and they cost $ 1.95 a sheet, how much would Brian have to pay in order to make the ramp? TC-19 Tell students to assume that portions of one sheet can be used on the faces of the ramp. An extension could be to have students figure out how much plywood is needed if leftovers from one plywood sheet cannot be used on other faces of the ramp. NOTE: if students know that a partial sheet cannot be bought, then it is a good time to discuss use of an overestimate. Answer: Plywood sheets cover ft. Surface area divided by x $1.95 is approximately $ Most ramps at skateboard ramps are poured concrete. Concrete is sold in cubic yards, so how much concrete would be needed to make this ramp? If the cost of concrete were $1.75 a cubic yard, what would be the total cost? TC-0 Tell students to assume that the forms have already been made and paid for, and that the volume of the ramp is roughly the same. Remind students that there are feet in a yard, and each dimension has to be converted, thus dividing by 7 ( x x ) Answer: Volume needs to be converted in to cubic yards. Volume divided by 7 is approximately 15 cubic yards. 15 x $1.75 = $ Betty loves to go camping. Her family has a tent that used to be her grandfathers. It is made of olive green canvas material, and looks like an isosceles triangular prism when it is set up. Betty drew a picture of it below: Demystifying Surface and Volume Teacher Materials Page 10 of 11

11 7. What shape is the floor of the tent? Is this the base of the prism? Explain. TC-1 Check for student understanding that the base of the prism is an isosceles triangle, yet the floor of the tent is a rectangle. Answer: floor is a rectangle; base of prism is isosceles triangle. 8. Betty knows that the floor of the tent is 1 feet by 8 feet, with the front being the longest. She also knows that she cannot stand up inside the tent without hitting her head. She estimates that the peak of the tent is 5 feet. Calculate the length of the legs of the isosceles triangle in feet, show work. TC- Allow students time to use Pythagorean Theorem to find the legs. Remind them which sides of the triangle are congruent. Answer: Divide isosceles triangle in half, using a median. This length is 5 ft and divides the long side of the triangle in half, 6 ft. Using Pythagorean Theorem, = 61, ft. The length of each leg is 7.8 ft. 9. How much olive green canvas material (including the floor) does it take to make Betty s tent? Answer: 1 (1 ft x 8 ft) rectangle, (8 ft x 7.8 ft) rectangles, and (1 ft x 5 ft) triangles. Surface area is approximately 80.8 ft. 10. Betty hates mosquitoes. She has a bug spray that claims to kill mosquitoes in the immediate area, 10 in per pump of spray. Ignoring the volume of Betty, how many pumps of spray should she use inside the tent to ward off the mosquitoes? TC- Check for student understanding of need for volume of the tent. Also, discuss the reasonableness of this answer! Answer: Volume is 0 ft x 8 ft = 40 volume by 7, which is 6480 in. Divide 6480 in by 10 pumps. She probably does not have enough pumps in her bug spray! ft. Need to convert to cubic inches. Multiply in, to get approximately 648 Demystifying Surface and Volume Teacher Materials Page 11 of 11