from Connect Vol.9 No.3, January/February, 1996
Focus on: Mass, Volume and Density
Grade Level: 6-8 Grade Level: K-8
Cubes, Surface Area, and Volume
A surprisingly rich invitation to mathematics for Middle School students
by Annette Raphel
In the process of exploring variables and making connections to algebra, I presented the class with the familiar painted cube problem. There are numerous versions of this problem. I built a cube made of 27 one-inch cubes and placed a sticky dot on the outside of all the faces (hmm! how many dots did I need?) Then I took the whole thing apart. I asked the students what clues I could look for in order to rebuild it so that every outside surface had a colored sticky dot and so that no internal surface had one. We talked about edges, corners, faces, interior and exterior surfaces. We analyzed how many cubes would have no dots, exactly one dot, two dots, or three dots. Then, I walked around with a tray inviting one student at a time to take a piece, any piece, and place it in a spot that made sense or hold onto it if a good spot was not yet available. We did that until we reconstructed the cube. For homework, I gave the students the painted cube problem: Using different sized cubes (ie. 4, 27, 64, 125) predict how many of the component one-inch cubes would have zero, one, two, or three dots on them. Together as a class we generalized to find algebraic formulas to predict each of those things.
The rectangular prism question
The next day I had several hundred one-inch cubes available. First we examined how many different sized rectangular prisms could be made out of fifteen of the cubes. Then we found out how many different rectangular prisms could be made from twenty of them. Students then chose any number of cubes they wished to work with and listed the dimensions of all the rectangular solids they could build. Our central question became: If you know how many cubes (volume) you have, can you predict how many different rectangular solids are possible? What do you think is the greatest number of rectangular prisms (different sizes) that can be built for volumes of less than a hundred cubic inches?
In the beginning there was a feeling that this was much too easy, until the entire class got caught up in the problem of predicting possible rectangular prisms given the volume. I had no idea what the answer was (I still dont though I am getting closer), but I suspected it had to do with prime factorization, or pairs of factors and how many of them are primes. The bell rang and no one made a move to leave as we watched the chart that we were constructing on the board. We listed volume, all the factors, prime factorization, and dimensions of all the prisms we could make. In doing so, we teased out important vocabulary and made hypotheses that were theories until someone found a counter-example. Not surprisingly, volumes that were square numbers were problematic (is 4 counted twice as a factor for 16, or only once?) and cube numbers such as 23 or 33 didnt fit too easily into our generalizations. For homework, students had to come up with summary statements of everything they knew about surface area and volume. I compiled the statements into a list.
Student statements about surface area and volume: 1. Volume can be measured in cubic measures. 2. Surface area is expressed in square units. 3. Surface area is not always larger than volume. 4. The formula for surface area is (HxWx2) + (LxHx2) + (LxWx2). 5. Volume is the amount of cubic units in a figure. 6. Surface area is the number of square units on the outer side of a figure. 7. The surface area is the measurement for the six sides of a rectangular solid. 8. The surface area cannot directly be used to find the area of a rectangular solid. 9. When you are trying to find the surface area of a cube, you have to find the area of one side and multiply by 6. 10. A figure that has the dimensions 1 x 1 x X always has the most surface area out of all the possible rectangular solids that can be made with a certain number of square units. 11. The bigger the number of square units, the bigger the surface area is. 12. If the surface area is a prime number, there is only one way to form a rectangular solid (1 x 1 x X). 13. A cube is a rectangular solid, but a rectangular solid is not necessarily a cube. 14. In order to find all the rectangular solids for a given number of cubic units, knowing the factors will help. 15. Each side interacts with the other sides only by right angles. 16. The smallest surface area is where the sum of all the dimensions is the smallest for any given group of rectangular solids. For example, 5x5x3 is the smallest surface area for a rectangular solid having a volume of 75 cubic units because 5+5+3 = 13, and the other possibilities are 1x1x75 (sum of 77), 1x5x15 (sum of 21), and 1x3x25 (sum of 29). 17. Volume is expressed by three numbers, one for each dimension. 18. Only a solid figure can have volume. 19. Rectangular prisms always have 8 corners. 20. There are different formulas for different kinds of solids. 21. To find the largest possible rectangle for any number of cubic units, take the number minus two, multiply by four and add ten. 22. To create rectangular shapes from a given number, you can find any three numbers which, when multiplied together, equal that number. 23. One cubic centimeter holds one milliliter of liquid. 24. When you form different rectangular solids out of a certain number of cubic inches, the volume will always be the same but the surface area changes. 25. Surface area is always an even number of units. 26. If the figure is a cube, the volume is a cube number (such as 23 or 33). 27. If a surface area is 4n + 2 (say n is 15, then 4n + 2 = 62) then you can always find a rectangular solid which will fit inside by knowing that it will be n units long and 1 unit wide and 1 unit high. 28. Not all even numbers of square units of surface area will contain a rectangular prism with whole number dimensions. This last statement addresses the rectangular prism problem the class is grappling with: 29. We think that if you know the number of cubes you have and you want to know how many rectangular solids you can make, you write down all the factors and count each factor which is not a prime number and thats how many rectangular solids youll have. We are still having trouble with square numbers so we need to work on our generalization further - maybe squares of certain kinds of numbers or powers of two are a little different. Authors response: This almost works in many cases, but lets look at 60. The factors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60... 1, 4, 6. 10, 12, 15, 20, 30 and 60 are not prime so there should be 9 rectangular prisms that can be built. 1x1x60, 1x2x30, 1x3x20, 1x4x15, 1x5x12, 1x6x10, 2x2x15, 2x3x10, 2x5x6, 3x4x5 are all possible; thats 10 rectangular prisms so the theory does not hold true. Finding examples and counter-examples may trigger pattern recognition which can be useful in solving this problem.
Investigations with cubes do a wonderful job of allowing for the integration of surface area, volume, multiplication, factors, generalizing algebraic formulas, and patterns. Inventing an additional statement to add to the list for the classs consideration is an irresistible next step. Figuring out how to predict the number of rectangular solids which are possible for any number of cubes or for a given surface area is engaging and mathematically rich. .
- Annette is principal of Milton Academy Lower School, Milton Massachusetts. She has taught from preschool through grade eight, is the author of Math Homework That Counts and is a frequent workshop presenter.
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