Soft Drink Project Part 4: The Math

This is the fourth in a series of posts detailing a student-posed math project. To get the full picture, please read the previous posts beginning with:

• Surface Area
  • I would hope so. The initial lesson was designed to have students explore how changing the dimensions of a prism would affect the surface area. This was the starting point of the learning. Students designed several boxes and calculated areas to compare. Students were discussing the formulae for areas of triangles and rectangles–using arguments to justify their design. Triangles, circles, and various polygons were calculated. Even an irregular shape was estimated.

A student estimates the SA of his newly minted shape

Student uses grid shading to create an accurate estimate

• Measurement
  • The dimensions of every shape had to be labeled. This was particularly interesting for triangles. The students had to decide which dimensions were necessary/important to include to calculate the surface area. The most eluding measurement was the diameter of the can. The bevel along the top and bottom made this an interesting problem. Students recalled their skills from a previous lesson to take the measurement.

Student measures the diameter of two cans

• • Pythagorean Theorem
    • I wasn’t expecting to fit this into the project, but it usually manages to squeeze itself into places where triangles are present. Students often forgot that the height of any triangle is needed to calculate its area. (Well at least until they become familiar with the idea of semi perimeter). I would ask them if they had a way to calculate the height and most remembered the theorem. Few knew how to use it. I think students remember the theorem because it represents a particularly traumatic experience in school mathematics. Regardless, students found their height and calculated areas.

• • Error Estimation
    • Many students realized the effect that human error has on mathematics. Their centimetre rulers only contained one digit of accuracy while their tracing and cutting skills contained far less. One group found it particularly distressing that their Pythagorean Theorem calculation differed from their measurement of the height.
    • Student 1 – “Does that mean the theorem is wrong?”
    • Student 2 – “No, it means that our measurements are wrong.”
    • Students blind recognition of mathematics is troubling. Nowhere along the line did they ever question the fact that side-squared-plus-side-squared-equals-hypotenuse-squared.

• Combinatorics
  • This was pretty much exclusive to the group that created the Tetris box design. As they were drawing up possible arrangements, I asked them how many possible arrangements existed. They could then make an objective choice as to which to bring to life. Essentially this was a tiling problem with irregular tiles composed of 4 squares.

• Algebra
  • The student who designed the triangular prism cans to fit snugly inside the triangular prism box was well on his way to completion when I asked him how big each of his cans were. After a discussion, and a quick Google-ing to see the conversion between cubed centimetres and millilitres, we decided that we should shift the height of the cans so they held exactly 355mL. The area of the base of his can was fixed at 21.35 squared centimetres. (An equilateral triangle with sides of 7cm). He knew he wanted the volume to be 355 cubic centimetres. He recalled our prior tasks with prisms to set up the following equation:
  • Volume = Area of base * height = 355
  • Volume = 21.35 * h = 355
  • He then went on to experience his very first real application of algebra. He made sure to mention to everyone who would listen that his cans contained exactly the same amount of liquid as the regular cans.

The last 355mL triangular prism can is created (with help from my fabulous EA)