Designing an Energy Drink Can Teacher Materials

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1 Lesson Outline: Designing an Energy Drink Can Teacher Materials Changing Cylinder Attributes I. The opening exercise (elicitation question) is the Energy Drink Can Problem. Hand out the problem description and have students think-pair-share their initial ideas. None of the three options on the worksheet are the perfect dimensions for the given requirements, but each of the three should generate discussion. Students should initially journal their own response before discussing it briefly with one other student. Finally, the teacher will solicit responses for the whole class to hear. The teacher should use a Socratic method of questioning throughout this initial investigation being careful not to reveal relationships of attributes that are intended to be discovered later in the lesson. II. After students have been prompted to consider cylinder attributes, they will begin the Construction of Cylinders component of the lesson. During this activity, students will spend some time manipulating cylinders and solidifying the attributes through hands-on experience. Each student should be given just one cylinder net to cut out and construct. (Most of the cylinders will have to be made before the lesson begins.) In the end, there needs to be a set of eight unique cylinders created for every 4 students in class. Ideally, each set of cylinders will be copied onto a different color of paper. Each student needs to cut out and tape one cylinder together. Then measurements need to be recorded along with the computation of volume and surface area. In the end, each cylinder can be taped to a placard that identifies it. Break the lesson here if needed for a two-day lesson. radius = height = volume = SA= III. The critical part of the lesson follows with the Cylinder Sorting activities. Once the attributes of all cylinders have been recorded, then each group of students can begin the four steps found on the Cylinder Sorting worksheets. Students should take their time while carefully graphing these functions. Their findings need to be clearly explained on the bottom half of each worksheet. All graphs should be titled and both axes must be clearly labeled with appropriate units. Designing an Energy Can Teacher Materials Page 1 of 15

2 Consolidated Table of Cylinder Attributes radius height volume surface area (cm) (cm) (cu. cm.) (sq. cm.) The four fixed attribute graphs from the Cylinder Sorting Activity : Cylinders with a Fixed Radius Volume (cu. cm) Height (cm) Cylinders with a Fixed Height Volume (cu. cm) Radius (cm) Designing an Energy Can Teacher Materials Page 2 of 15

3 Cylinders with a Fixed Radius Surface Area (sq. cm) Height (cm) Cylinders with a Fixed Height Surface Area (sq. cm) Radius (cm) 6 IV. As a summary activity, students should revisit their initial journal regarding the Energy Drink Can Problem. They can see if their initial ideas were right or wrong and change them if needed. Finally, each student should try to find the correct dimensions for doubling the volume of the can with the least amount of additional surface area. Students will have to use a guess and check method to eventually discover that the doubled volume occurs with the least amount of added surface area if the new can has a diameter equal to its height of almost 9 cm. A Final Assessment is included in the lesson materials along with a grading rubric. Designing an Energy Can Teacher Materials Page 3 of 15

4 Energy Drink Can Problem: This elicitation question is intended to get the students thinking about cylinders and their potentially changing attributes. You re the project leader for a team of packaging consultants who has been given the task of creating a new drink can that will double the volume of an upstart energy drink. The dimensions of the original can are shown below along with three new options intended to double the volume with the least amount of additional aluminum needed for the new can. Which of the following three options would be best and why? Can O Caffeine The original measures just 6 cm in diameter and 10 cm tall. Option A Option B Option C TWINS Option A doubles the diameter while keeping the height the same. Option B doubles the height while keeping the diameter the same. Option C simply packages two of the original sized cans together with plastic. As the leader of the packaging team, determine which of the three options listed above fits the objective of doubling the volume with the least amount of additional aluminum. Only Option B fits the criteria. Option C uses too much aluminum because it contains an extra top & bottom. Option A cannot be considered because its volume actually quadruples the original volume. Designing an Energy Can Teacher Materials Page 4 of 15

5 Cylinder Sorting Name Key Step 1: Identify which cylinders have the same radius, then arrange those in order of least to greatest volume. Set the other three cylinders aside for now. Complete the data table below, then fill in the graph to determine the relationship between volume and radius. height 3 cm volume 37.7 cm 3 Cylinders with a Fixed Radius 4 cm 6 cm 10 cm 12 cm 50.2 cm cm cm cm 3 Volume (cm 3 ) Height (cm) (a) Is the relationship between the height & volume of a cylinder a function? Why or why not? Yes it is a function. Each height corresponds with exactly one volume. (b) Is it linear? Yes. (c) Is it a direct variation? Yes it is a direct variation because it is linear and it would pass through the origin since a cylinder with no height would have no volume Designing an Energy Can Teacher Materials Page 5 of 15

6 Step 2: This time identify which cylinders have the same height, then arrange those in order of least to greatest volume. Set the other four cylinders aside for now. Complete the data table below, then fill in the graph to determine the relationship between volume and radius. radius volume Cylinders with a Fixed Height 1 cm 18.8 cm 3 2 cm 75.4 cm 3 3 cm cm cm cm 3 Volume (cm 3 ) Radius (cm) (a) Is the relationship between the radius & volume of a cylinder a function? or why not? Why Yes it is a function. Each radius corresponds with exactly one volume (b) Is it linear? No. (c) Is it a direct variation? No, it is not linear. Designing an Energy Can Teacher Materials Page 6 of 15

7 Step 3: Once again identify which cylinders have the same radius, then arrange those in order of least to greatest surface area. Set the other three cylinders aside for now. Complete the data table below, then fill in the graph to determine the relationship between surface area and radius. height surface area Cylinders with a Fixed Radius 3 cm 62.8 cm 2 4 cm 75.4 cm 2 6 cm cm 2 10 cm cm 2 Surface Area (cm 2 ) 12 cm cm 2 Height (cm) (a) Is the relationship between the height & surface area of a cylinder a function? Why or why not? Yes it is a function. Each height corresponds with exactly one surface area. (b) Is it linear? Yes. (c) Is it a direct variation? Yes it is a direct variation because it is linear and it would pass through the origin since a cylinder with no height would have no volume. Designing an Energy Can Teacher Materials Page 7 of 15

8 Step 4: Once again identify which cylinders have the same height, then arrange those in order of least to greatest surface area. Set the other four cylinders aside for now. Complete the data table below, then fill in the graph to determine the relationship between surface area and height. radius surface area Cylinders with a Fixed Height 1 cm 44.0 cm 2 2 cm cm 2 3 cm cm cm cm 2 Surface Area (cm 2 ) (a) Radius (cm) Is the relationship between the radius & surface area of a cylinder a function? Why or why not? Yes it is a function. Each radius corresponds with exactly one volume (b) Is it linear? No. (c) Is it a direct variation? No, it is not linear (d) How does the steepness of the two graphs with the fixed height compare? (Step 2 and step 4) The graph in step 2 actually becomes steeper even though it starts off with less of a slope. This tells us that if given a fixed cylinder height and an increasing radius, then the volume will eventually increase more rapidly than the surface area. Designing an Energy Can Teacher Materials Page 8 of 15