Connecting Middle Grades to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 01/7/11 Unit Dog Objective: Students will investigate the resulting effects on surface area and volume when dimensions of a shape are changed proportionally. Connections to Previous Learning: Students should be able to calculate surface area and volume of a rectangular prism and draw nets for 3-dimensional figures. Connections to AP*: AP Calculus Topic: Areas and Volumes Materials: Student activity pages, 13 snap cubes for each student, grid paper copied on cardstock, tape, scissors, a sample unit dog made from snap cubes, a sample unit dog made from grid paper, and a large copy of the chart provided in step 6 of the activity Teacher Notes: The Unit Dog lesson is teacher-led. The activity pages can be given to students who finish early or as an additional assignment. Organize the students into small groups and lead the students through the following steps. Step 1: Step : Step 3: Step 4: Demonstrate how to make a unit dog from snap cubes then have each student build their own dog, using their 13 snap cubes. Note: Larger dogs should not be built with snap cubes. They are heavy and very unstable. Have each student determine the surface area and volume of their unit dog. Generally students do not correctly determine surface area if they try to count. Give them time to think about a good procedure. Once students begin to have the correct answer, have them share their process with the rest of the class. Have each student draw front, side and top views of the unit dog. Show the students how to draw the nets and construct a unit dog from the grid paper. (Before class begins, construct a unit dog to use as a model.) Help the students realize they are to construct one net for the dog s body and that the other 5 nets (head and legs) are the same net. Do not ask the students to make one net for the entire body. *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 1
Teacher Notes Step 5: In order to create a variety of different-sized dogs, assign groups to build particular dogs from grid paper nets. A group of two should construct a dog with dimensions that are double those of the unit dog. A group of three students should construct a dog with dimensions triple those of the unit dog. Four students can do quadruple and five students can do quintuple. Each group should calculate surface area and volume for their particular dog. If there are two groups for each size dog, then the groups can verify the answers of the other group. Step 6: Once all of the dogs are built, have students post their measurements on the large copy of the chart shown below. Display all of the dogs. Dimensions scaled by a factor of Surface area of the dog Area scaled by a factor of Volume of dog Volume scaled by a factor of 1 (the unit dog) 1 1 3 4 5 Step 7: As the students fill in the factors by which surface area and volume have increased, ask them how they determined the factors and how they used the factors in the calculations. Step 8: Have the students predict the surface areas and volumes for dogs with dimensions six, seven and ten times those of the unit dog. Then ask them how they determined the values. Step 9: When the unit dog activity is completed, the practice problems and/or the state assessment problems on the next two pages may be assigned for extra practice. Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Student Activity Unit Dog Practice Problems: 1. The flower pots shown below are similar. If the smaller pot has a volume of 100 cubic inches, what is the volume of the larger pot?. Myrna wants to construct a box that will hold 64 times as much as a similar box. By what factor should she multiply the dimensions of the smaller box in order to determine the dimensions of her new, larger box? 3. Quincy works in the customer service department of a store and is responsible for ordering supplies for the gift wrap department. The smallest box requires 88 square inches of wrapping paper. How much wrapping paper would be needed for a box having dimensions that are three times those of the smallest box? 4. A delivery service charges a fee based upon the volume of the box to be delivered. If they charge $3 for a 6 inch by 6 inch by 6 inch box, then how much would you expect them to charge for a box that is 1 foot by 1 foot by 1 foot? 5. An architect is working on a scale model home for a client. The linear dimensions of the scaled model will be 1 the size of the linear dimensions of the actual house. If he uses square feet 0 of wallpaper for the kitchen in the scale model, how much would he need for the kitchen in the actual house? Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 3
Student Activity State Assessment Practice: 1. A shipping company sells two types of cartons that are shaped like rectangular prisms. The larger carton has a volume of 70 cubic inches. The smaller carton has dimensions that are half the size of the larger carton. What is the volume, in cubic inches, of the smaller carton? A. 90 in. 3 B. 10 in. 3 C. 40 in. 3 D. 360 in. 3. An ice-cream carton has a volume of 64 fluid ounces. A second ice-cream carton has dimensions that are ¾ the size of the larger carton. Which is closest to the volume of the smaller carton? A. 0 fl oz B. 7 fl oz C. 36 fl oz D. 48 fl oz 3. The radius of the larger sphere shown below was multiplied by a factor of 1 to produce the smaller sphere. Radius = 1 r Radius = r How does the surface area of the smaller sphere compare to the surface area of the larger sphere? A. The surface area of the smaller sphere is 1 as large. B. The surface area of the smaller sphere is 1 as large. C. The surface area of the smaller sphere is 1 4 D. The surface area of the smaller sphere is 1 8 as large. as large. * Items taken from TAKS (Texas Assessment of Knowledge and Skills) Information Booklet, Mathematics Grades 8 11, Texas Education Agency, Student Assessment Division, 00 Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 4
Connecting Middle Grades to Advanced Placement* Mathematics A Resource and Strategy Guide Answers: 6. Dimensions scaled by a factor of Surface area of the dog Unit Dog Area scaled by a factor of Volume of dog Volume scaled by a factor of 1 (the unit dog) 54 1 13 1 16 4 104 8 3 486 9 351 7 4 864 16 83 64 5 1350 5 165 15 7. The area measures increase by squaring the linear factor. The linear factor is always multiplied by itself in the area calculation. The volume measures increase by cubing the linear factor. The linear factor is always used as a factor three times in the volume calculation. 8. A dog having dimensions six times those of the unit dog will have a surface area of 1944 square units and a volume of 808 cubic units. A dog with dimensions seven times those of the unit dog will have a surface area of 646 square units and a volume of 4459 cubic units. A dog with dimensions ten times as big would have a surface area of 5400 square units and a volume of 13,000 cubic units. 9. See answers below. Practice Problems: 0in. 1. Since the flower pots are similar the ratio between similar dimensions is the same. 10 in.. 3 3 3 3 The volume of the larger flower is (100 in )( ) (100 in )() 800in 3. Since 64 4 each dimension must be multiplied by a factor of 4. 3. The wrapping paper is covering the surface of the box and each dimension of the surface of the box is multiplied by a factor of 3, the surface area of the box is multiplied by a factor of 3 9. Quincy will need 88 square inches 9 70 square inches of wrapping paper. 4. The dimensions of the larger box are multiplied by a factor of. The delivery service would 3 charge $3 () $4 for the larger box. 5. The dimensions of the house are 0 times as large as that of the model; therefore, the surface area of the wall will be (0) time the surface area of the walls of the model. The architect will need (0) square feet 800square feet of wallpaper for the kitchen in the actual house. State Assessment Practice: 1. A. B 3. C Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 5
Answers Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 6