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You Be the teacher

You Be the teacher. Student Responses.

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You Be the teacher

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  1. You Be the teacher Student Responses

  2. Peggy Popcorn, where every kernel is popped and buttered with real butter, makes the best gourmet popcorn in town. Peggy Popcorn is packaged in a box like the one pictured below. E. Ficiency thinks that the company can save money by building a better box. In her design, the volume stays the same but the surface area decreases, so it would cost Peggy Popcorn less to build the box.  The Task (Level A – Spatial Sense and Geometry) •Find the volume and surface area of the box above. •Give integral dimensions of a second box that has the same volume but a smaller surface area. •Explain how you arrived at your answers.

  3. Discuss various solutions to this problem with your neighbor. • Decide what the solution should look like that would be acceptable. • Decide what solution would not be acceptable.

  4. Solution Possible Solutions: Volume = lwh = 2(24)(8) = 384 in3 Surface Area = 2(lw)+2(lh)+2(wh) = (2x8x24)+(2x24x2)+(2x8x2) Since the volume must remain the same, the factors of 384 just need to be manipulated. 2x24x8 is the same as 4 x 12 x 8 The closer the dimensions are in size, or the closer the box resembles a cube, the smaller the surface area.

  5. Solution Possible correct answers include:

  6. Incorrect answers (same volume, but greater or equal surface area is most often occurring on student papers are:

  7. You Be The Teacher Scoring Rubric

  8. You Be The Teacher Scoring Authentic Papers

  9. Meghan’s Response: 2 x 24 x 8 = 384 that’s the volume cause its length times width times height for surface area add all the area front 2 x 24 = 48 and back would be the same soback = 48side 2 x 8 = 16other side = 16top 24 x 8 = 192bottom = 192 512 cause the volume has to stay the same I need other numbers that multiply to be 384384 ÷ 3 = 128 128 ÷4 = 32 so 3 x 4 x 32 = 384 front 3 x 4 = 12back = 12side 4 x 32 = 128other side = 128top 3 x 32 = 96bottom = 96 472 it works This response is a 3 because the student’s answers are correct and her explanations are easy to follow and demonstrate understanding of the concepts.

  10. Cornelia’s Response: volume = Bh so the area of the base is 192 times 2 is 384another box with volume 384 is 2 x 2 x 96 surface area = 2(area of top + side + front) 2(192 + 16 + 48) = 512 another box with smaller surface area is 2 x 3 x 4 This response is a 1. Although the volume and surface area are correct, the work and explanation does not demonstrate a connection between the two. It also appears that she misinterpreted the problem.

  11. Kesley’s Response: 1st I found the volume (lwh), then I found the surface area which means to add up the areas of all 6 sides. then to keep the volume the same I divided the biggest number by 2 and multipled the smallest number by 2, because they are opposite operations I really didn’t change anything. then I found the surface area of the new box and it worked – it should because 4 and 12 are closer to each other than 2 and 24. Surface Area 2 x 24 = 48 24 x 8 = 192 12 x 8 - 96 Volume 2 x 24 x 8 = 384 4 x 12 x 8 = 384 48 48192192 16 16512 48 48 32 32 9696352 This response is a 3. All the answers are correct. The work and explanations are easy to follow and demonstrate full understanding of concepts involved.

  12. Karen’s Response: 2 times 24 is 48 so area = 48 now use the 8 too and x 8 = 384 for volume the surface = 24 in wide and 2 in high your surface is 48? well, the depth has to change for a smaller surface, if I make the box say 4 x 20 the area and surface is 80 but that’s more surface so how about 1 inch by 24 than my area and surface is 24 and my depth is 384 ÷ 4 is 16 so the new box is 1 x 24 x 16, same area less surface. This response is a 1. The response shows some understanding of volume. The work does not show good understanding of surface area or its connection to volume.

  13. Jorge’s Response: Volume is Length x Width x Height 24 x 8 x 2 = 384 cubic inches. Surface Arfea is find the area of all the surfaces and add them up 24 x 8 = 192, 2 x 8 – 16, 8 x 2 = 16, 2 x 24 = 48, 24 x 8 = 192, 24 x 2 = 48 192 + 16 + 16 + 48 + 192 + 48 = 512 square inches The close the box is to a cube the less surface area it will have The new dimensions should be 7.268 x 7.268 x 7.268 This response is a 3. The response demonstrates full understanding of volume, surface area, and their relationship.

  14. Torrie’s Response: I took the box apart and drew lines Then I counted 192 boxes on the top, 16 on the side and 48 boxes for the other side. adding them alltogether you get 256 boxes To make the box have a smaller surface but stay the same I took 2 rows of boxes off the top and added them to the side so the big gox would have the same amount of boxes but a smaller surface, and then theres only 22 rows on the top (and the front too or its not a box anymore) but 16 on the side. I kept the two high. Answer: 22 x 16 x 2 This response is a 0. The student does not address volume in the response. She shows some understanding of surface area and attempts to calculate it but does not complete the process. The response does not demonstrate understanding of creating a new box within the parameters of the problem.

  15. Oanh’s Response: area = bh and volume is lwh area = 24 x 2 = 48 and volume = 24 x 8 x 2 = 384 to make the same volume but smaller area the base or length would have to be smaller cause the height is already small enough. If I change the 24 to a 12 that would make the area 12 x 2 = 24 and that’s smaller but to keep the same volume I would have to change the 24 to a 12 and the 8 to a 16 (if ones in half the others double) and leave the height the new box is 12 x 16 x 8 This response is a 1. The student finds the correct volume but uses area instead of surface area. Her logic for finding the new box with the same volume is valid but she wrote the wrong numbers.

  16. Gregorio’s Response: • There area a lot of answers for a better box. • To keep the volume the sameyou have to find three other numbers that multiply up to 384 besides those ones. • To find a smaller surface area you have to change the three numbers to make them more box like and not flat like. • Then you have to find the three numbers that work for moth of those things. volume=384 inches cube surface area=512 inches square This response is a 2. The student finds the correct surface area and volume; his explanation and pictures demonstrate an understanding of the relationship between the two; but he fails to find the dimensions of the second box.

  17. Alfred’s Response: Volume = lwh = 384 sq in Surface area = 2(lw) + 2(wh) + 2(lh) – 512 c in Dimensions: 2 x 8 x 24 x3 ÷3 6 x 8 x 8 check lwh = 384.sq in surface area = 2(lw) +2(wh) + 2(lh) = 320 c in (I chose to x and ÷ the biggest and smallest numbers to make the rectangular box closer to a cube) for new dimensions This response is a 3. The explanation is clear and concise and the answers (except for the units) are correct. The response demonstrates understanding of the concepts involved.

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