Activity: TEKS: Teddy Bear (G.2) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to: (A) use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships; and (G.3) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to: (D) use inductive reasoning to formulate a conjecture; and (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties; (B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles; (G.6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to: (B) use nets to represent and construct three-dimensional geometric figures; and (G.11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to: (A) use and extend similarity properties and transformations to explore and justify conjectures about geometric figures; (B) use ratios to solve problems involving similar figures; (D) describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems. Overview: This activity encourages students to explore patterns in perimeter, surface area, and volume of similar figures when the dimensions are changed. As Teddy Bear Page 1
a result, students should discover the relationship between linear ratios, area ratios, and volume ratios that compare similar three-dimensional objects. Materials: Snap cubes Centimeter grid paper Scissors Tape Calculator Teddy Bear Parts I, II, III, and IV handouts Vocabulary: Ratios Perimeter Surface area Volume Similarity Nets Grouping: Time: 2 or 4 students Two 55 minute class periods Lesson: Procedures 1. Distribute PART I of the activity to each student. Tell them that they are to read and follow the instructions, construct the ten parts with snap cubes, and assemble the parts to make BEAR 1. 2. Next have students find and record the measures listed in the table for BEAR 1. Students are then asked to predict the effect that changes in the dimensions of the bear parts will have on these measures 3. Have students build the remaining bears (at least the next two) with the centimeter graph paper. Using the graph paper, they make nets for each of the 10 parts for the enlarged teddy bears, cut them out, and tape them together. 4. Have students complete the measures in the first table for BEARS 2, 3, 4, and N. Notes Assist groups with the instructions as needed. Check to see that each group correctly lists the measures for BEAR 1. Students may need help as they begin to draw the rectangular prism nets for the bear parts. Students will need to find the pattern for each measure to complete the row for BEAR N. Teddy Bear Page 2
Procedures 5. In PART II, PART III and PART IV students use the measures (including the Nth term formulas) from the table to compare heights, perimeters, surface areas and volumes of various pairs of BEARS. They also use the patterns to find measures for BEARS not in the table and answer questions. Then students develop a rule that describes the proportional relationships they have been using. Notes This table will be used to complete the rest of the activity. Groups should complete the tables together in class. The problems from each part of the lesson could be assigned as homework. The next day groups could compare solutions, reach consensus, and write the rule. Close the lesson by discussing the proportional relationships with the whole class. You may need to discuss with students the definition of proportional. Homework: Extensions: Have students apply the rule to situations when given a drawing of the figures. Include cylinders, polygonal prisms and pyramids. For G11(D), students need to find measures when 1 or 2 dimensions are changed. The resulting figure is not similar to the original. Teddy Bear Page 3
TEDDY BEAR PART I: BUILDING THE BEAR Use snap cubes with the edge of one cube representing 1 unit of length or use centimeter graph paper to build the first teddy bear that is 5 units tall and 3 units wide. The first teddy bear has the following parts: 1 Body: 1 x 3 x 3 units 2 Front Legs: 1 x 1 x 1 units each 1 Neck: 1 x 1 x 1 units 2 Ears: 1 x 1 x 1 units each 1 Head: 1 x 1 x 2 units 2 Back Legs: 1 x 1 x 2 units each 1 Tail: 1 x 1 x 1 units The front legs attach to the top corners of the body so that the legs are even with the top of the body. The back legs attach to the bottom corners of the body so that the legs are even with the bottom of the body. The tail fits on the back of the body, in the center, even with the bottom of the body. The neck fits on the top of the body, in the center. The head fits on the top of the neck so that the back of the head is even with the back of the body. The ears fit on each side of the head so that they are even with the top of the head. The bold pieces are closest to the eye and the dashed pieces are farthest from the eye. FRONT SIDE TOP Use the teddy bear to complete the table below: N = BEAR NUMBER HEIGHT OF TEDDY BEAR PERIMETER OF BASE TOTAL SURFACE AREA VOLUME BEAR 1 1. What do you think will happen to the height of the teddy bear when the dimensions of each body part are doubled? tripled? Teddy Bear Page 4
2. What do you think will happen to the perimeter of the base of the teddy bear when the dimensions of each body part are doubled? tripled? 3. What do you think will happen to the surface area of the teddy bear when the dimensions of each body part are doubled? tripled? 4. What do you think will happen to the volume of the teddy bear when the dimensions of each body part are doubled? tripled? Using centimeter graph paper, build at least the next two teddy bears. To build each enlarged bear, multiply each dimension of Bear 1 by the BEAR number you are building. Once you have the dimensions, make nets of the teddy bear parts, cut them out and tape them together. A few of Bear 2 s parts and the dimension change process are explained below: Body: (2x1) X (2x3) X (2x3) = 2 x 6 x 6 Front Leg: (2x1) X (2x1) X (2x1) = 2 x 2 x 2 Head: (2x1) X (2x1) X (2x2) = 2 x 2 x 4 BEAR NUMBER HEIGHT OF TEDDY BEAR PERIMETER OF BASE TOTAL SURFACE AREA VOLUME BEAR 2 BEAR 3 BEAR 4 BEAR N Teddy Bear Page 5
TEDDY BEAR PART II: COMPARING HEIGHT AND PERIMETER IN SIMILAR FIGURES Using the completed tables from Part I, fill in the ratios for the following table. Simplify fractions to lowest terms and give the decimal equivalent. Use the table data to answer the questions below. RATIO OF FIGURES RATIO OF HEIGHT RATIO OF PERIMETER OF BASE FRACTION DECIMAL FRACTION DECIMAL BEAR 1 : BEAR 2 BEAR 2 : BEAR 3 BEAR 1 : BEAR 3 BEAR 3 : BEAR 4 BEAR 1 : BEAR 4 BEAR 2 : BEAR 5 1. Was your prediction in Part I about the height of the bears accurate? Explain your answer. 2. Was your prediction in Part I about the perimeter of the base of the bears accurate? Explain your answer 3. What pattern do you observe in the simplified ratios of the heights / perimeters? 4. What pattern do you observe in the decimal answers? Teddy Bear Page 6
5. Using the patterns that you observed, determine A. the ratio of the height of Bear 20 to the height of Bear 11 B. the ratio of the perimeter of the base of Bear 20 to the perimeter of the base of Bear 11. 6. If the heights of two bears were in the ratio of 100, which bears would you be 40 comparing? Explain your answer. 7. Determine which bear base has a perimeter of 192 units? Explain your answer. 8. What is the ratio of the perimeter of the base of Bear 50 to Bear 15? 9. If you know the height of Bear 1, how can you find the height of Bear N? 10. Why do you think that changing the dimensions has the same effect on height and perimeter? 11. Based upon your results, if the ratio of the heights of two similar figures is 4 5, the ratio of the perimeters would be. Teddy Bear Page 7
12. Write a rule based on your results: If the ratio of the heights of two similar figures is m/n, then the ratio of their perimeters would be. Teddy Bear Page 8
TEDDY BEAR PART III: COMPARING HEIGHT AND SURFACE AREA IN SIMILAR FIGURES Using the completed table from Part I, fill in the ratios for the following table. Simplify fractions to lowest terms and give the decimal equivalent. Use the table data to answer the questions below. RATIO OF FIGURES RATIO OF HEIGHT RATIO OF SURFACE AREA FRACTION DECIMAL FRACTION DECIMAL BEAR 1 : BEAR 2 BEAR 2 : BEAR 3 BEAR 1 : BEAR 3 BEAR 3 : BEAR 4 BEAR 1 : BEAR 4 BEAR 2 : BEAR 5 1. Was your prediction in Part I about the surface areas of the bears accurate? Explain your answer. 2. What pattern do you observe in the simplified ratios of heights and surface areas? 3. What pattern do you observe in the decimal answers? 4. Using the pattern that you observed, determine the ratio of the surface area of Bear 20 to the surface area of Bear 12? Teddy Bear Page 9
5. If the surface area of the bears were in the ratio of 256, which bears would you be 49 comparing? Explain your answer. 6. Determine which bear has a surface area of 11,232 square units? Explain your answer. 7. What is the ratio of the surface area of Bear 50 to Bear 15? 8. If you know the surface area of Bear 1, how can you find the surface area of Bear N? 9. Based upon your results, if the ratio of the heights of two similar figures is 4 5, the ratio of the perimeters would be and the ratio of the surface area would be. 10. Write a rule based on your results: If the ratio of the heights of two similar figures is m/n, then the ratio of their perimeters would be and the ratio of their surface area would be. Teddy Bear Page 10
TEDDY BEAR PART IV: COMPARING TOTAL SURFACE AREA AND VOLUME OF SIMILAR FIGURES Using the completed table from Part I, fill in the ratios for the following table. Simplify fractions to lowest terms and give the decimal equivalent. Use the table data to answer the questions below. RATIO OF FIGURES RATIO OF HEIGHT RATIO OF SURFACE AREA RATIO OF VOLUME FRACTION DECIMAL FRACTION DECIMAL FRACTION DECIMAL BEAR 1 : BEAR 2 BEAR 2 : BEAR 3 BEAR 1 : BEAR 3 BEAR 3 : BEAR 4 BEAR 1 : BEAR 4 BEAR 2 : BEAR 5 1. Was your prediction in Part I about volumes of the bears accurate? Explain your answer. 2. What pattern do you observe in the simplified ratios of heights, surface areas, and volumes? 3. What pattern do you observe in the decimal answers? Teddy Bear Page 11
4. Using the pattern that you observed, determine the ratio of the volume of Bear 20 to the volume of Bear 12? 5. If the volume of the bears were in the ratio of 1000, which bears would you be 512 comparing? Explain your answer. 6. Determine which bear has a volume of 122,472 cubic units? Explain your answer. 7. What is the ratio of the volume of Bear 50 to Bear 15? 8. If you know the volume of Bear 1, how can you find the volume of Bear N? 9. What happens to the volume if all the dimensions of the teddy bear are reduced by one-half? Explain your answer. 10. Based upon your results, if the ratio of the heights of two similar figures is 4 5, the ratio of the perimeters would be, the ratio of the surface area would be, and the ratio of the volume would be. Teddy Bear Page 12
11. Write a rule based on your results: If the ratio of the heights of two similar figures is m/n, then the ratio of their perimeters would be, the ratio of their surface area would be and the ratio of their volume would be. Teddy Bear Page 13
TEDDY BEAR ACTIVITY ANSWERS PART I: BUILDING THE BEAR Use the teddy bears you built to complete the tables below: BEAR NUMBER HEIGHT OF TEDDY BEAR PERIMETER OF BASE TOTAL SURFACE AREA VOLUME BEAR 1 5 16 78 21 BEAR NUMBER HEIGHT OF TEDDY BEAR PERIMETER OF BASE TOTAL SURFACE AREA VOLUME BEAR 2 10 32 312 (78*2 2 ) 168 (21*2 3 ) BEAR 3 15 48 702 (78*3 2 ) 567 (21*3 3 ) BEAR 4 20 64 1248 (78*4 2 ) 1344 (21*4 3 ) BEAR N 5N 16N 78N 2 21N 3 PART II: COMPARING HEIGHT AND PERIMETER IN SIMILAR FIGURES RATIO OF FIGURES BEAR 1 : BEAR 2 BEAR 2 : BEAR 3 BEAR 1 : BEAR 3 BEAR 3 : BEAR 4 BEAR 1 : BEAR 4 BEAR 2 : BEAR 5 RATIO OF HEIGHT RATIO OF PERIMETER OF BASE FRACTION DECIMAL FRACTION DECIMAL 5 10 = 1 2 10 15 = 2 3 5 15 = 1 3 15 20 = 3 4 5 20 = 1 4 10 25 = 2 5.5.67.33.75.25.40 16 32 = 1 2 32 48 = 2 3 16 48 = 1 3 48 64 = 3 4 16 64 = 1 4 32 80 = 2 5.5.67.33.75.25.40 1. Was your prediction in PART I about the height of the bears accurate? Explain your answer. An accurate prediction would be that the height doubles when the dimensions are doubled and triples when the dimensions are tripled. However students could make many different predictions. Teddy Bear Page 14
2. Was your prediction in Part I about the perimeter of the base of the bears accurate? Explain your answer. An accurate prediction would be that the perimeter doubles when the dimensions are doubled and triples when the dimensions are tripled. However students could make many different predictions. 3. What pattern do you observe in the simplified ratios of the heights / perimeters? The simplified ratio is the same as the ratio of the number of the bears being compared. For example with Bear 2 : Bear 3, the ratio of heights is 10 15 which simplifies to 2 3. The ratio of the perimeters also simplifies to 2 3. 4. What pattern do you observe in the decimal answers? The decimal answer is the same as the ratio of the bears. For example, Bear 2 : Bear 3 = 2 3 =.66 5. Using the patterns that you observed, determine: A. the ratio of the height of Bear 20 to the height of Bear 11. The ratio is 100 55 which simplifies to 20 11. B. the ratio of the perimeter of the base of Bear 20 to the perimeter of the base of Bear 11. The ratio is 320 20 which simplifies to 176 11. 6. If the heights of two bears were in the ratio of 100, which bears would you be 40 comparing? Explain your answer. You would be comparing Bear 20 to Bear 8 since 5 times 20 is 100 and 5 times 8 is 40. 7. Determine which bear base has a perimeter of 192 units. Explain your answer. Bear 12 has a base of 192 since 16 times 12 = 192. 8. What is the ratio of the perimeters of the base of Bear 50 to Bear 15? The ratio is 50 15 which simplifies to 10 3. 9. If you know the height of Bear 1, how can you find the height of Bear N? Multiply the BEAR number by 5 or 5N 10. Why do you think that changing the dimensions has the same effect on height and perimeter? Explain your answer. Students may need to refer to the figures of the teddy bears. The height and perimeter are both linear; therefore, they will change by the same factor. Teddy Bear Page 15
11. Based upon your results, if the ratio of the heights of two similar figures is 4 5, then the ratio of the perimeters would be 4 5. 12. Write a rule based on your results: If the ratio of the heights in two similar figures is m, then the ratio of their n perimeters would be m n. PART III: COMPARING HEIGHT AND SURFACE AREA IN SIMILAR FIGURES RATIO OF FIGURES BEAR 1 : BEAR 2 BEAR 2 : BEAR 3 BEAR 1 : BEAR 3 BEAR 3 : BEAR 4 BEAR 1 : BEAR 4 BEAR 2 : BEAR 5 RATIO OF HEIGHT RATIO OF SURFACE AREA FRACTION DECIMAL FRACTION DECIMAL 5 10 = 1 2 10 15 = 2 3 5 15 = 1 3 15 20 = 3 4 5 20 = 1 4 10 25 = 2 5.50.66.33.75.25.40 78 312 = 1 4 312 702 = 4 9 78 702 = 1 9 702 1248 = 9 16 78 1248 = 1 16 312 1950 = 4 25.25.44.11.5625.0625.16 1. Was your prediction about the surface areas of the bears accurate? Explain your answer. An accurate prediction would be that the surface area will increase by the original surface area times the square of the dimensional change (height). However students could make many different predictions. 2. What pattern do you observe in the simplified ratios of the heights and surface areas? The ratio of the surface area is equal to the ratio of the heights squared. Teddy Bear Page 16
3. What pattern do you observe in the decimal answers? The decimal answer for the surface area is equal to the square of the decimal answer for the heights. 4. Using the patterns that you observed, determine the ratio of the surface area of Bear 20 to the surface area of Bear 12? 400 144 = 25 9 5. If the surface area of the bears were in the ratio of 256, which bears would 49 you be comparing? Explain your answer. You would be comparing Bear 16 to Bear 7. You take the square root of 256 and 49 to determine the answer. 6. Determine which bear has a surface area of 11,232 square units. Explain your answer. 11,232 = 78 N 2 Divide by 78 and take the square root. Bear 12 has a surface are of 11,232. 7. What is the ratio of the surface area of Bear 50 to Bear 15? 2500 225 = 100 9 8. If you know the surface area of Bear 1, how can you find the surface area of Bear N? 78N 2 9. Based upon your results, if the ratio of the heights of two similar figures is 4 5, then the ratio of the perimeters would be 4 5 would be 16 25. and the ratio of the surface area 10. Write a rule based on your results: If the ratio of the heights in two similar figures is m/n, then the ratio of their perimeters would be m n and the ratio of their surface areas would be m2 n 2. Teddy Bear Page 17
PART IV: COMPARING TOTAL SURFACE AREA AND VOLUME OF SIMILAR FIGURES RATIO OF FIGURES RATIO OF HEIGHT RATIO OF SURFACE AREA RATIO OF VOLUME FRACTION DECIMAL FRACTION DECIMAL FRACTION DECIMAL BEAR 1 : BEAR 2 5 10 = 1 2 BEAR 2 : BEAR 3 10 15 = 2 3 BEAR 1 : BEAR 3 5 15 = 1 3 BEAR 3 : BEAR 4 15 20 = 3 4 BEAR 1 : BEAR 4 5 20 = 1 4 BEAR 2 : BEAR 5 10 25 = 2 5.50.66.33.75.25.40 78 312 = 1 4 312 702 = 4 9 78 702 = 1 9 702 1248 = 9 16 78 1248 = 1 16 312 1950 = 4 25.25.44.11.5625.0625.16 21 168 = 1 8 168 567 = 8 27 21 567 = 1 27 567 1344 = 27 64 21 1344 = 1 64 168 2625 = 8 125.125.296.037.421875.015625.064 1. Was your prediction in Part I about the volumes of the bears accurate? Explain your answer. An accurate prediction would be that the volume will increase by the original volume times the cube of the dimensional change (height). However students could make many different predictions. 2. What pattern do you observe in the simplified ratios of the heights, surface areas and volumes? The ratio of the surface area is equal to the ratio of the heights squared. The ratio of the volume is equal to the ratio of the heights cubed. 3. What pattern do you observe in the decimal answers? The decimal answer for the surface area is equal to the square of the decimal answer for the heights. The decimal answer for the volume is equal to the cube of the decimal answer for the heights. 4. Using the pattern that you observed, determine the ratio of the volume of 8000 Bear 20 to the volume of Bear 12? 1728 = 125 27 5. If the volume of the bears were in the ratio of 1000, which bears would you be 512 comparing? Explain your answer. You would be comparing Bear 10 to Bear 8. You take the cube root of 1000 and 512 to determine the answer. Teddy Bear Page 18
6. Determine which bear has a volume of 122,472 cubic units. Explain your answer. 122,472 = 21N 3 Divide by 21 and take the cube root. Bear 18 has a volume of 122,472. 7. What is the ratio of the volume of Bear 50 to Bear 15? 125,000 3,375 = 1000 27 8. If you know the volume of Bear 1, how can you find the volume of Bear N? 3 21N 9. What happens to the volume if all the dimensions of the teddy bear are reduced by one-half? Explain your answer. The volume will be reduced by 1 8 of the original volume. 1 2 3 = 1 8 10. Based upon your results, if the ratio of the heights of two similar figures is 4/5, the ratio of the perimeters would be 4, the ratio of the surface area 5 would be 16 25, and the ratio of the volume would be 64 125. 11. Write a rule based on your results: If the ratio of the heights in two similar figures is m/n, the ratio of their perimeters would be m m2, the ratio of their surface area would be and the ratio of their 2 n n m 3 volume would be n. 3 Teddy Bear Page 19