Boxed In! Annotated Teaching Guide

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1 Boxed In! Annotated Teaching Guide Date: Period: Name: Area Review Warm-Up TC-1 Area: The space that a two-dimensional shape occupies measured in square units. For a rectangle, the area is found by multiplying the length by the width. DIRECTIONS: Find the area of each rectangle below. Show all of your work m 16 x 7 = 112 square meters 16 m 2. 8 cm 8 x 8 = 64 square cm 8 cm 3. 11mm 11 x 3 = 33 square mm 3 mm Boxed In! Teacher Materials Page 1 of 22

2 Date: Period: Name: Area Review Warm-Up Area: The space that a two-dimensional shape occupies measured in square units. For a rectangle, the area is found by multiplying the length by the width. DIRECTIONS: Find the area of each rectangle below. Show all of your work m 16 m 2. 8 cm 8 cm 3. 11mm 3 mm Boxed In! Teacher Materials Page 2 of 22

3 Boxed In! Introduction Contortionists are able to bend and flex in order to squeeze themselves into small spaces and often perform as part of a circus or theatrical act. Most contortionists have unusual natural flexibility enhanced through gymnastic conditioning and training. The practice of squeezing one s body into a small box appearing to be much too small for a person to fit in is called enterology. (TC-2) Examine these pictures of several contortionists. What do you think? Can you fold yourself into a small box? (TC-3) and (TC-4) Try rolling yourself into a ball. Was it easy? difficult? Boxed In! Teacher Materials Page 3 of 22

4 The contortionist in the above pictures is named Emma Tunbridge. She was a national gymnast from England who now lives in Australia, studies yoga and performs in various festivals and shows alongside great performers such as David Blaine, a magician and illusionist. She is most commonly known for her ability to fold herself into a 17 inch cube in less than a minute. Remember that a cube is a _three_ dimensional shape which has six equal faces and three equal dimensions of length, width, and height_. A cube is also an example of a rectangular prism. (TC-5) VOCABULARY (TC-6) Rectangular prism: a three-dimensional shape with six rectangular faces and the dimensions of length, width, and height. Example: height length width Label the length, width, and height on the rectangular prism above. Rectangular Prism Search Find two examples of rectangular prisms in the classroom. Name and describe why your examples fit the definition above. (TC-7) 1. Answers will vary. 2. Boxed In! Teacher Materials Page 4 of 22

5 Teaching Notes for Warm-Up and Introduction Materials for Warm-Up and Introduction : Warm-up sheets and introduction sheets one set per student, projector connected to a computer to show additional examples of contortionists (optional), cube examples (dice, Post-it Note cube, etc.), examples of rectangular prisms (cereal boxes, staple boxes, etc.). TC-1 The area review could be used as a warm-up to review and assess students understanding of area. TC-2 For additional background information on contortionists or more pictures to share with your class, the following websites may help: TC-3 This is a great kinesthetic activity, however, it is important to ensure that the classroom environment is supportive and safe for students to try rolling into a ball. TC-4 An optional activity would be: For practice/exploration, students will then explore (kinesthetically) the scenario of being contortionists themselves and figuring out what their volume would be and what dimensions and surface area of a box would be needed if they were lying out straight and if they could roll up into a ball and fit into a cube-like box. This will help students understand that the same volume can result in different surface areas. NOTE: For this activity it is critical that students feel comfortable and safe from ridicule about their self-image and related self-esteem. Some students may be sensitive about measuring themselves and feel like they are being compared to other students in the classroom based on body size. Therefore, while this activity would be a very concrete way for students to explore volume and surface area, it should be used with a great caution and withpre-teaching of expectations for respect and positive interactions among students. TC-5 As a whole class go through the pictures with the students showing Emma standing on the cube, getting into the cube, and lastly, inside the cube. Have a brief class discussion about what they observe and think about the pictures. Review what a cube is with the class. Have examples of cubes available to show to the class (dice, Post-it Note cube, etc.) Also, review/define dimensions and faces. These two words should be review from 5 th grade. TC-6 Define rectangular prism with the class. Show some examples such as cereal or oatmeal boxes (or other food items), staple box, a book, etc. Also, model turning the boxes to show that length, width, and height are interchangeable. It may also be helpful to have students redraw the example prism in the Vocabulary box to show the different ways it can be turned and relabeled. TC-7 Have the students find two examples of rectangular prisms (boxes) in the classroom. Give them about 5 minutes to do this. Then have the students share their findings with their small groups and then with the whole class. Boxed In! Teacher Materials Page 5 of 22

6 EXPLORATION - Volume Focus Question: What is volume? In your small group, you will explore how much space is inside the 17 inch cube Emma fits into. This space is called the volume of the figure. Then you will calculate the volume of Emma to see how she is able to fit into the space in the cube. STEP 1: Use the sheets of large inch grid paper to construct a full-size net (pattern) for a 17-inch cube. Remember that a cube will have 6 equal sides. An example of a net for a one-inch cube is shown below. The net should include all six sides and be able to be folded to form a cube or box. (TC-8) STEP 2: Fold your net into the cube. Tape all sides except the lid! Not a very big box, huh? So, how does Emma fit into that small space? Well, as a class, let s see how many inch cubes would fit by filling the bottom of the paper cube with one layer of inch cubes. A. How many cubes does it take to fill the bottom of the paper cube? 289 cubes_ B. How many layers tall is the paper cube? 17 layers C. Check your answer by stacking inch cubes the height of the cube in one corner of the paper cube. (TC-9) D. So, if 289 cubes fill the bottom layer and there are _17 layers, how many total inch cubes would fit inside the paper cube? Show your work. (TC-10) 289 cubes x 17 layers = 4,913 cubes Boxed In! Teacher Materials Page 6 of 22

7 VOCABULARY (TC-11) You just found the volume of the paper cube. Based on your work in part D, define volume in your own words: Answers will vary but should include that volume is the amount of space inside a 3D shape. Connect this to counting. E. What are the dimensions of the cube you built? Length 17 cubes Width 17 cubes Height 17 cubes F. Compare the dimension numbers with the number of inch cubes used in the layers in part D. Can you determine a general rule for finding the volume of rectangular prisms or cubes? Explain and support with numbers and/or pictures. (TC-12) Answers will vary but should include that volume is: In Words: the area of the base times the height With Symbols: length x width x height OR lwh STEP 3: Look at the first picture of Emma on page 2. If Emma was to stand straight and we drew her as a rectangular prism with the three dimensions of length, width, and height, her dimensions would be: length 12 inches, width 6.5 inches, and height 63 inches. (TC-13) A. Draw a rectangular prism below with these dimensions. 63 inches 6.5 inches 12 inches B. Now use your drawing and general rule from Step 2 to calculate the volume of Emma. Show all work. 12 inches x 6.5 inches x 63 inches = 4,914 cubic inches OR 12 inches x 6.5 inches = 78 square inches (base) 78 square inches x 63 inches = 4,914 cubic inches Boxed In! Teacher Materials Page 7 of 22

8 C. Compare Emma s volume to the volume of the 17-inch cube. How do you account for the differences? Be specific! Think carefully. Are humans really shaped like rectangular prisms? (TC-14) Emma s volume of 4,914 cubic inches is one cubic inch greater than the volume of the 17-inch cube which was 4,913 cubic inches The difference is due to the fact that Emma s dimensions are not exactly 12 x 6.5 x 63. Her head is not 12 inches wide. It is smaller. So, actually Emma s volume is a little less than our estimated calculation of 4,914 cubic inches. Therefore, Emma can fit into the 17-inch cube with a little space to spare. D. Did the amount of space or volume of Emma change from when she was standing to when she was squeezed into the cube box? Explain your reasoning. (TC-15) Volume did not change STEP 4: Time to practice what you now know about volume. You have a choice between two problems. Try problem #1 if you are beginning to understand volume but could use a little more practice. Try problem #2 if you really understand volume and need a challenge. Or, you can try them both!! (TC-16) #1 Using the picture of the rectangular prism below, find the volume. 8 feet 8 x 2 x 4 = 64 cubic feet 4 feet 2 feet #2 If the volume of the rectangular prism below is 40 units cubed, find the missing length. 4 feet 40 4 = = 5 feet 2 feet? Boxed In! Teacher Materials Page 8 of 22

9 Teaching Notes for Exploration - Volume Materials for Exploration Volume: large inch grid paper, inch cubes, one net of a cube and/or one copy of a net of a cube per student, activity pages for Exploration-Volume. TC-8 For the Exploration-Volume and Surface Area it is important to group students into small groups so they can problem solve together and so they can share materials. This activity uses quite a bit of large grid paper and requires about 305 inch cubes, so sharing materials is necessary. Pattern or net: Point out that the example given is only for a unit cube that is 1x1x1. Show students where the faces are on the pattern. If needed, have a copy cut out and demonstrate for students how to fold the pattern/net into a cube. Or, if possible have copies for students to practice folding to understand how the net works before they design the same type of pattern for the contortionist s cube. TC-9 Due to the number of cubes required (305), yet the importance for the students to see the cubes in the layer, it is necessary to either: rotate the 305 cubes between groups (groups will take differing amounts of time to construct the paper cube so this is possible because groups will be in different places at the same time) or to do this portion of the activity as a whole class (not as powerful as having the students do this in their small group). TC-10 Students should see that volume is the area of the base (length x width) times the height. Also, discuss with students why the units in volume are cubed. This is easy to demonstrate with the 3D model of the paper cube and using inch cubes. When students are finding the number of cubes to fill the bottom of the cube, link this to counting to find the area (like a postage stamp). When students are finding the number of layers, link this to counting length (like a string). TC-11 Check for understanding here. Have students share their definitions in small groups and then with the whole class. Address any misconceptions through questioning and modeling using other examples. TC-12 Students will either write the general rule in words (the area of the base x height OR length x width x height) or as an expression lwh. Check for understanding here. Do students understand how the rule counts the cubes? Have students share their rules with the class. TC-13 Students may question why we are drawing Emma as a rectangular prism. Address the difficulties and complexities of measuring Emma s dimensions exactly due to the shape of the human body. Address that using a rectangular prism to represent Emma, we are approximating her volume and this is sufficient for what we are doing in this activity. In part B, students should be applying the rule they developed in Step 2 part F. Boxed In! Teacher Materials Page 9 of 22

10 Check for general understanding, accuracy of calculations, and use of correct units. TC-14 Students will notice that Emma s estimated volume is one cubic inch greater than the volume of the box she fits into. Address that we did not measure Emma exactly, but used a rectangular prism around her general shape to approximate. If we subtract some space to account for her head which is smaller than the width used, students should understand that Emma s true volume is actually a little smaller allowing her to fit into the box. TC-15 Students should realize that Emma s volume did not change. She still takes up the same amount of space, no matter if she is standing or scrunched up into a ball. TC-16 Step 4 is an opportunity for students to practice and for the teacher to informally assess their understanding of volume. It also provides students with a choice based on their comfort and understanding volume. Boxed In! Teacher Materials Page 10 of 22

11 EXPLORATION Surface Area Focus Question: What is surface area? (TC-17) STEP 1: Paper, Paper, Paper! So, how many square inches of paper did your group use to build the cube like the one Emma fits into? Sure, to figure this out you could count all of the inch squares on the paper. But, as mathematicians, we like to calculate things as quickly as possible. So, find the area of each face (all 6 of them!) and then add all six areas together. Show your work. Each face has 289 inch squares (17x17). So, = 1,724 square inches OR 289 cubes x 6 faces = 1,724 square inches And there you have it! The surface area of the cube! Surface area is simply the total area of the faces of a three-dimensional shape. Is there a faster way to get the surface area other than counting the inch squares on each face? What if the cube wasn t made of grid paper? Explain and show work. (TC-18) Answers may vary but should include that you find the area of each face and then add them together. You use the dimensions to find the area of each face. Area of each face is 289 square inches (17 x 17) Multiply that area by 6 because all of the faces are equal. So, 289 x 6 = 1,724 square inches Boxed In! Teacher Materials Page 11 of 22

12 STEP 2: Now use your ideas for calculating surface area to find the surface area of Emma as a rectangular prism. Again the dimensions would be: length 12 inches, width 6.5 inches, and height 63 inches. Show all work! (TC-19) 2 faces x 12 inches x 6.5 inches = 156 square inches 2 faces x 12 inches x 63 inches = 1,512 square inches 2 faces x 6.5 inches x 63 inches = 819 square inches Sum of the area of the faces = , = 2,487 sq. inches STEP 3: Time to practice what you now know about surface area. You have a choice between two problems. Try problem #1 if you are beginning to understand surface area but could use a little more practice. Try problem #2 if you really understand surface area and need a challenge. Or, you can try them both!! (TC-20) #1 Using the picture of the rectangular prism below, find the surface area. 2 faces x 4cm x 2cm = 16 square cm 2 faces x 4cm x 8cm = 64 square cm 8 cm 2 faces x 2cm x 8cm = 32 square cm = 112 square cm 2 cm 4 cm #2 If the surface area of the rectangular prism below is 108 square cm, find the missing length. 2 faces x 4cm x 1cm = 8 square cm 4 cm 108 sq. cm 8 sq. cm = 100 sq. cm left 1 cm Using guess and check,? 2 numbers divisible by 2 and add to 100 and one is divisible by or x 10 x 1 = 20 2 x 10 x 4 = 40 2 x 10 x 4 = 80 2 x 10 x 3 = 60 Works Does not work The missing length is 10 cm Boxed In! Teacher Materials Page 12 of 22

13 Teaching Notes for Exploration Surface Area Materials for Exploration Surface Area: Paper cubes students built in Exploration- Volume, activity pages for Exploration-Surface Area. TC-17 Students may want to count all the inch squares. They should realize that since all the faces of a cube have an equal area, they only have to count one side and multiply by 6 faces. Discuss or model why the units for surface area are squared. Use their paper cubes to model. For students having difficulty organizing their work in finding the surface area of the faces, suggest they organize their information in a table or use a list. TC-18 The key understanding here is that surface area is the sum of the area of the faces of a figure. Check for understanding before students move on the Step 2. Have students share their ideas in groups and/or with the whole class. TC-19 Students should show all work and demonstrate an understanding of finding surface area. Also, check to see that calculations are correct and that students have the correct units. Be sure to emphasize to students that this is really an estimate since humans are not exact rectangles. TC-20 Step 3 is an opportunity for students to practice and for the teacher to informally assess their understanding of surface area. It also provides students with a choice based on their comfort and understanding surface area. Boxed In! Teacher Materials Page 13 of 22

14 BOXED IN! REFLECTIONS Volume and Surface Area (TC-21) 1. Compare the volumes calculated and the surface areas calculated for Emma. How did they differ? How were they the same? Use specific examples to support your comparisons. Answers will vary but should summarize that the volumes stayed the same while the surface areas varied with changes in the dimensions length, width, and height. 2. Describe what volume is and how you find it. Use words, numbers, and/or pictures to support your answer. Goal: For students to understand that volume is the amount of space inside a shape and that you find volume by multiplying the area of the base by the height. 3. Describe what surface area is and how you find it for a rectangular prism. Use words, numbers, and/or pictures to support your answer. Goal: For students to understand that surface area is the area of the faces or the outside of a shape and that you find surface area of a rectangular prism by using the dimensions (length, width, and height) to find the area of each face and add them together. Boxed In! Teacher Materials Page 14 of 22

15 Teaching Notes for Reflections Materials for Reflections: Reflections pages for students, color pencils. TC-21 These reflections provide students an opportunity to analyze and communicate their understanding about volume and surface area. They give the teacher an opportunity to assess those understandings. Have students complete these reflections individually. Then have them share their ideas with their small group and using a color pencil, add additional ideas shared by the students in their groups. Last, have groups share with the whole class, and have students use a second color pencil to record any new information shared with the whole class. Collect these and use to check for understanding. Boxed In! Teacher Materials Page 15 of 22

16 Stretch Out and Curl Up! (TC-22) The surface area of an animal impacts their ability to conserve heat or cool themselves in different climates/temperatures. On a hot summer day, a dog or cat will stretch out on their bellies to cool off in a shady spot, but on a cold winter day the same dog or cat will curl into a ball to stay warm. Focus Question: How does surface area and/or volume relate to an animal s ability to radiate or conserve heat? radiate means to emit rays of light or heat conserve means to save something from loss to preserve 1. Your group was given 27 unit cubes. Place the cubes in a straight length (row)to represent a dog or cat stretched out on a warm day. a. Calculate the volume of the dog/cat. Show your work! If students count the unit cubes, there are 27 cubes. So, the volume is 27 cubic units b. Calculate the surface area of the dog/cat. Show your work! 4 faces x 27 x 1 = 108 square units 2 faces x 1 x 1 = 2 square units = 110 square units 2. Now rearrange (stack) the unit cubes into a cube to represent the same dog or cat curled up on a cold winter day. a. Calculate the volume of the dog/cat. Show your work! This would make a 3 x 3 x 3 cube. So, 3 x 3 x 3 = 27 cubic units b. Calculate the surface area of the dog/cat. Show your work! Each face is 3 x 3 = 9 cubes. 6 faces x 9 cubes = 54 square units Boxed In! Teacher Materials Page 16 of 22

17 3. Compare the volumes and surface areas of the stretched out and curled up dog/cats. How are they they same? Different? Answers will vary but should include: the volumes stayed the same, but the surface areas changed. The surface area for the cube was less than the surface area for the long narrow arrangement of cubes. 4. Using your work on numbers 1-3, why do you think that surface area impacts an animal s ability to radiate heat on a warm day and to conserve heat (or avoid radiating heat) on a cold day? Be specific! Answers will vary but should include: animals that are stretched out have more surface area exposed to the elements, so their bodies lose or radiate heat away. However, when they are curled up, parts of their bodies are compacted overlapping so there is less surface area to lose heat. Using the cubes, the long narrow arrangement has all four sides of most of the cubes showing on the outside. However, when they are arranged in a cube, some of the cubes are hidden entirely in the middle of the bigger cube, so there are fewer cubes showing on the outside. Anectodate to make it stick: Ants are not harmed in a microwave because their surface area to volume is high enough to disipate any heat generated inside quickly enough not to harm them. (Don t try this at home!) Boxed In! Teacher Materials Page 17 of 22

18 Teaching Notes for Stretch Out and Curl Up! Materials Needed for Stretch Out and Curl Up!: Activity sheets for students, unit cubes, real photos of cats and dogs stretched out and curled up. TC-22 This activity provides students with an opportunity to practice their skills and understanding of volume and surface area using a real-world example. You could either have students work in small groups or individually (as a means of assessing individual understanding) (will need more unit cubes if this activity is completed individually). Access prior knowledge by having a brief class discussion about cats and dogs and how they cool off or stay warm. You could provide photos of real cats and/or dogs stretched out on a hot day or curled up in a ball. Boxed In! Teacher Materials Page 18 of 22

19 EXTENSION PROJECT RESEARCH OPPORTUNITY! (TC-23) 1. Use the Internet and/or library resources to research other animals and their unique ability to cool/stay warm based on their surface area and other characteristics. A. In your research you must include: The dimensions of the animal The climate where the animal lives Unique physical characteristics which also impact the animal s heating/cooling ability B. Write a summary of your findings. Include calculations for surface area and any other math work your conclusions based on your research and mathematical work C. Use the attached Notes and Summary Page 2. Additional challenge: Explore the relationship between an animal s volume (mass) and their surface area and how that relationship impacts their cooling/heating ability. Boxed In! Teacher Materials Page 19 of 22

20 Notes and Summary for Extension Project (TC-24) NOTES/MATH WORK Dimensions of Animal Surface Area/Other Math Work Climate Unique Characteristics SUMMARY OF FINDINGS Write in complete sentences and give specific examples and support for your conclusions. Boxed In! Teacher Materials Page 20 of 22

21 Teaching Notes Extension Project Materials Needed for Extension Project: Access to computers for students, extension project activity sheets for students, resources such as tradebooks, encyclopedias, etc. TC-23 This is an optional extension project It could be used for students who finish the other activities early and need an opportunity to extend and deepen their understanding of volume and surface area that also incorporates the use of technology. Check with your library resource specialist for search engines or research resources available at your school to give students a menu of resources available. Students could present their findings to the class verbally. They may create a poster to synthesize their findings and to enrich their presentation. Boxed In! Teacher Materials Page 21 of 22

22 Net/Pattern for a Cube Boxed In! Teacher Materials Page 22 of 22