How cubey is your prism? the No Child Left Behind law has called for teachers to be highly qualified to teach their subjects.

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1 How cubey is your prism? Recent efforts at improving the quality of mathematics teaching and learning in the United States have focused on (among other things) teacher content knowledge. As an example, the No Child Left Behind law has called for teachers to be highly qualified to teach their subjects. The knee-jerk reaction would be to require more traditional mathematics courses for middle school licensure. In my professional development work in recent years, it seems that the connections between higher mathematics and middle school mathematics do not come easily to teachers. As an example, many teachers who were successful Calculus students in college do not recognize that the process of taking first and second differences (see e.g. Lappan, et al. 2001) is related to derivatives. As a result, it seems important to ask What kinds of learning experiences can preservice middle school mathematics teachers have that will deepen their content knowledge in a way that builds connections to the mathematics they will teach? I suggest that one way to deepen preservice teachers content knowledge, while keeping the mathematics relevant to the mathematics they will teach is to pose challenging extensions of middle school math problems. Middle school mathematics is rife with opportunities to ask questions beginning with words such as, What if...? that will challenge preservice and practicing secondary mathematics teachers. This article reports on one such investigation in a middle school mathematics and methods course. Setting the Stage The surface area of a rectangular prism is calculated with a simple formula: 2lw + 2lh + 2hw, where l is the length of the prism, w is the width and h is the height. Students in many modern middle-school classrooms derive this formula through experiences building prisms p. 1

2 from unit cubes and with counting the number of unit squares visible on each of the faces. They may observe that there are two rectangular faces with dimensions l units by w units, two with dimensions l by h and two with dimensions h by w. One way for teachers to give students practice with this formula, and with the concept of surface area, while also calling on students higher-order thinking skills is to ask students to find the surface areas of all prisms with a fixed volume. If we use only whole-number side lengths, and if we agree that a 2 in by 3 in by 4 in prism is the same as a 3 in by 2 in by 4 in prism, there are six rectangular prisms with a volume of 24 cubic inches. These are listed in no particular order in Table 1. Imagine for a moment that the surface area of prism F has not yet been calculated. Now line up prisms A-E in order from least surface area to greatest, as in Figure 1. Where should prism F go in the lineup? Can this be determined visually before calculating the surface area? In the lineup in Figure 1, height seems to be the determining factor. As we move to right in the figure, both the surface areas and the heights increase. But looking only at height means there are at least two places the 2 in by 2 in by 6 in prism (prism F) could fit (see Figure 2). If we orient prism F so that its height is 2 in, then we predict that it will have a smaller surface area than that of prism E (which has a height of 3 inches in the lineup). If we orient prism F so that it has height 6 in, then we predict that it will have the same surface area as does prism C. These cannot both be right, so there must be something more to consider. As a group, middle school students generally agree that prism F appears to fit between prisms C and E in the lineup, and that F s surface area is between the surface areas of C and E. When asked why, common responses include F is more compact than C, but E is more compact than F; F has more squares on the inside than C, but fewer than E; and F is more like a cube p. 2

3 than C, but E is more like a cube than F. This last observation often leads to some form of the following conjecture: Given two rectangular prisms with the same volume, the one that is closer to a cube will have the smaller surface and vice versa; the prism with the smaller surface area will be closer to a cube. In this paper, we define cubeyness informally as the property we can perceive visually that allows us to rank the six prisms with volume of 24 cubic inches with respect to surface area without having to calculate the surface areas directly. Cubeyness could also be expressed as compactness. The question we investigated in class was, How can we measure cubeyness? We can distinguish the cubeyness of prisms A F visually. If we allow fractional dimensions, there could be two prisms, X and Y that have surface areas quite close to each other. Prisms X and Y might each look equally cubey. How can we compute their cubeyness without computing their surface areas? Measuring cubeyness I An intuitively appealing way to measure cubeyness is to add the dimensions of the prism: 3 length + width + height. A cube would minimize this sum ( = 3i ), so the smaller the sum of the dimensions, the more cubey the prism is. Table 2 shows that this measure correctly sorts our six prisms by surface area. This measure has been suggested in middle school classrooms. Its intuitive appeal, together with the fact that it works on all of the prisms at hand, may lead to its acceptance as a valid measure of cubeyness. When our class took on the challenge of examining this measure closely however, the prisms in Table 3 arose. As in the conjecture, we have two prisms with a common volume (64 in 3 ). Our cubeyness measure suggests that prism L is closer to a cube, and so by our conjecture it p. 3

4 should have a smaller surface area. But it does not. Notice that neither of these prisms is particularly cubey (see Figure 3). The non-cubeyness of these two prisms is borne out by their surface areas, which are substantially greater than the surface area of the cube with a volume of 64 in 3 (which is 96 in 2 ). This is a case where the cubeyness of a prism is hard to perceive, and it demonstrates the need for an accurate measure. Together, prisms L and M refute one of two things: either the conjecture or the measure. Rather than reject the conjecture outright, the class sought to formulate a better measure. Measuring Cubeyness II Returning to the original task of ordering the prisms by visually perceiving their cubeyness, another measure of cubeyness arose: Given two prisms with a common volume, the one with a longer shortest side will be more cubey. In the event of a tie, compare the next-shortest side; the one with the longer side in this second comparison will be more cubey. Table 4 shows that this measure correctly sorts prisms A-F. Furthermore, the measure correctly predicts that prism M, with a shortest dimension of 2 inches, has a smaller surface area than prism L, with a shortest dimension of 1 inch. One student s attempt to algebraically prove the relationship between smallest dimension and surface area yielded nothing but frustration. Similarly, attempts to strategically generate counterexamples were in vain. The measure is clearly quite good. But the standard techniques failed to determine whether the measure is perfect. Computer software settled the matter. Fathom is a software package for teaching and learning statistics. In addition to a user-friendly interface, one of the most powerful aspects of Fathom is the ability to sample (see e.g. Author 2008). The graph in Figure 4 represents 1,000 p. 4

5 randomly generated prisms, each with a volume of 24 cubic inches. After setting up the formulas for generating the prisms, Fathom takes about 15 seconds to generate and graph these prisms. The graph in Figure 4 is what we should expect; as the shortest side gets longer, the surface area decreases. This is an inverse relationship. The maximum shortest side is slightly less than 3. Given that the cube root of 24 is slightly less than 3, this makes sense the cube would have the smallest surface area and would have the longest possible shortest side. At first glance, this graph appears to demonstrate the perfection of this measure? On closer inspection, there are a lot of points in the bottom-left corner of the graph. Figure 5 zooms in on these points, and demonstrates that the measure is imperfect. There exist pairs of prisms, P and Q, such that P has a longer shortest side than Q, but such that P has greater surface area than Q. It is worth noting that the differences here are quite small, so the measure is very good, but it is not perfect. Once again we need to decide whether to reject the measure or the conjecture. Rejecting the measure leads to the more interesting mathematics. Measuring Cubeyness III We discussed the cubeyness problem with many colleagues in mathematics and math education. The variety of their ideas has been astonishing. Several proposed measures, together with short explanations follow. Not all of these measures have been confirmed or refuted. They also do not represent the full range of ideas people have suggested for measuring the cubeyness of a rectangular prism. In all of what follows, let l be the length of the prism whose cubeyness is being measured, w the width, h the height, V the volume and s the cube root of the volume (this may be thought of as the length of the side of a cube with volume V.) p. 5

6 1. Let l w h. l w w h h l This measure was devised in response to the additive measure: l + w + h. The reasoning was this: l + w + h does not work because surface area is not purely an additive concept. Instead, it is a multiplicative concept. Therefore, we should make a multiplicative comparison. This measure sets up the ratios of the sides and multiplies them. In the case of a cube, l = w = h, so each of the ratios is equal to 1, so their product is 1. The idea is that, the closer this product is to 1, the more cubelike the prism. Unfortunately due to its setup, this particular product is always equal to Let l w h. l w w h l h This measure fixes the flaw of the preceding one. It simplifies to l2 h 2, which turns out to be just as good as Let l w h. l h This is the simple ratio of the longest side to the shortest one. The closer to 1 this measure is, the more cubey the prism. Prisms L and M are counterexamples, just as they are for the original additive measure. 4. An algebraic topologist friend looked at the problem this way: Each rectangular prism can be viewed as the image of a transformation of a cube. If you start with the cube, you can stretch it along one dimension, squash it along another, etc. Each transformation in three dimensions can be described by a 3 by 3 matrix. The eigenvalues of this matrix should tell us something about the cubeyness of the image. p. 6

7 Moving on, we consider Let l w h and let s be the cube root of the volume. lw hs This measure would be equal to 1 for a cube, and is greater than 1 for all other prisms. The smaller the measure, the more cubey the prism. 6. l 3 + w 3 + h 3 3 This is the average of the volumes of the cubes with side lengths l, w and h. For a cube, this would be equal to the cube s volume. The smaller this measure, the more cubey the prism. 7. ( lw) 2 + ( lh) 2 + ( wh) 2 s Larger surface areas occur when at least one of the terms in the formula for surface area is larger than the others (because for a cube, these terms would all be equal). We can simplify things by not including the factor of 2. Then by squaring each term (lw, lh and wh), the differences among the terms are accentuated. Therefore prisms with large surface areas will have very large measures. The smaller this measure, the more cubey the prism should be. 8. s l + s w + s h 3 This is the mean of three ratios. Each ratio has this form: the length of the side of the cube to a dimension of the prism. The idea is that we compare each dimension to the cube s side length. In the case of a cube, the measure comes out to 1. A bit of experimentation suggests that for non-cube prisms, the measure comes out to be less than one. Another form of this ratio is: p. 7

8 9. s 3 * 1 l + 1 w + 1 h Note that, when volume is fixed, s is a constant, so a simpler version of this measure is: l + 1 w + 1 h In all of cases 8 10, the greater the measure, the less cubey (and so the greater the surface area). Measuring Cubeyness IV Surprisingly, this last measure works. Figure 6 shows a very strong correlation. A bit of algebraic manipulation shows why this is so: 1 l + 1 w + 1 h = wh lwh + = wh + lh + lw lwh lh lwh + = 1 2 surface area volume lw lwh If two prisms have the same volume, then the denominator of this measure is equal and we are comparing the numerators, which is equivalent to comparing their surface areas. It is important to note that we can compute this measure without computing the surface area directly. For instance, Table 5 shows these measures computed for the original six prisms with volume of 24 cubic inches. Extra credit questions p. 8

9 The graph in Figure 6 suggests a linear relationship between the cubeyness measure and the surface area. What is the slope of this line? The algebraic proof that this cubeyness measure correlates exactly with surface area suggests at least one new formula for surface area. Can you find such a formula? Try it out on some familiar prisms. It would be nice if our cubeyness measure would scale. In its present form, a 2 by 3 by 4 prism has a different cubeyness measure (viz ) than a 4 by 6 by 8 prism ( 24 ). These two prisms are geometrically similar; therefore they should be equally cubey. How can we modify our cubeyness measure so that it scales? So what have we learned? We take two lessons away from this exploration: one is a general principle for guiding the mathematical development of future and practicing mathematics teachers, the other is related to a very real teaching problem all teachers face. 1. Challenging mathematics can arise by asking a new question in a familiar situation (see also Author 2009). Perhaps this is a model that might better fit the demands of being highly qualified in mathematics than simply requiring more formal mathematics. Perhaps deep investigation of middle school mathematics problems is a way to improve teachers content knowledge in ways that will be useful in their classrooms. 2. This investigation provides both a model and a concrete example of differentiated instruvtion. The model is for a teacher to ask students What if? questions and to be on the lookout for counterexamples. It is easy to accept the first seemingly good idea as correct (here, the sum of the sides as a cubeyness measure); it takes much more practice to develop a habit of asking good What if? questions that will challenge these ideas (e.g. What if we were p. 9

10 not limited to whole-number side lengths? or What if our volume had even more factors, such as 64 units 3?) Additionally, these future teachers are now equipped with a specific question they may use to challenge their strongest students and they know much more about the possible directions their students could go. Conclusion This article has used cubeyness as the specific context for a larger idea-that of deep investigation of middles school mathematics ideas. Along the way, the investigation challenged preservice teachers to think about measurement, geometry, statistics and algebraic manipulation in new ways. But there are many places in middle school mathematics where a task for middle school students can be tweaked to become a challenging task for preservice and practicing mathematics teachers. We conclude this article with an example from probability and statistics. When middle school students are listing out all of the possible outcomes for a roll of 2 dice, it is common for them to list only 21 possibilities. (1,6) and (6,1) are not both listed, for example. Sorting out the reasons for needing to list both (a,b) and (b,a) when counting equally likely outcomes is a challenging task in teaching and learning probability. The task is to imagine that you do not know whether we should list both (a,b) and (b,a), and to design an empirical test that will determine whether (a,b) and (b,a) should be counted as the same outcome or as two different outcomes. You should formulate your test completely prior to rolling any dice. You should design a test that can be performed in 5 minutes or less (although p. 10

11 you may spend substantially longer than that in formulating it), and it should be good enough that you are willing to stake your teaching license on its generating the correct result. BONUS: Sometimes middle school students overcompensate and want to count doubles twice. Does your test help to determine whether this is correct? How might you modify your test? p. 11

12 CUBEYNESS Figure 1: Prisms A-E ordered by surface area; least (left) to greatest (right). p. 12

13 CUBEYNESS Figure 2: Does prism F have height of 2 inches or 6 inches? p. 13

14 Figure 3: Which prism is more cubey, L or M? p. 14

15 Sample of Collection Scatter Plot shortest_side Figure 4: One thousand randomly generated prisms with a volume of 24 cubic inches. The length of the shortest side is on the x-axis; surface area is on the y-axis. p. 15

16 Sample of Collection Scatte shortest_side Figure 5: Consider the red point in this graph. There are prisms with longer shortest sides and greater surface areas (although not by much!) p. 16

17 Sample of prisms 2000 Scatter Plot reciprocal_sum Figure 6: There is a very strong correlation between the reciprocal sum and the surface area of a rectangular prism. p. 17

18 prism length width height surface area volume A 1 in 1 in 24 in 98 in 2 24 in 3 B 1 in 2 in 12 in 76 in 2 24 in 3 C 1 in 4 in 6 in 68 in 2 24 in 3 D 1 in 8 in 3 in 70 in 2 24 in 3 E 2 in 4 in 3 in 52 in 2 24 in 3 F 2 in 2 in 6 in 56 in 2 24 in 3 Table 1: Dimensions of rectangular prisms with whole-number side lengths and a fixed volume of 24in. p. 18

19 prism sum of dimensions surface area volume cube approx approx in 3 E 9 in 52 in 2 24 in 3 F 10 in 56 in 2 24 in 3 C 11 in 68 in 2 24 in 3 D 12 in 70 in 2 24 in 3 B 15 in 76 in 2 24 in 3 A 26 in 98 in 2 24 in 3 Table 2: The six prisms with whole-number side lengths and fixed volume of 24 in 3 are sorted correctly by the cubeyness measure length + width + height. As the sum of the dimensions increases, so does the surface area. p. 19

20 prism length width height sum of dimensions surface area volume cube in in in in in 2 in 3 L in in in in in 2 in 3 M in in in in in 2 in 3 Table 3: Dimensions of two rectangular prisms with whole-number of these prismss incorrectly predict their side lengths and a fixed volume of 64 in 3. The sums of the dimensions comparative surface ares. p. 20

21 prism shortest side next-shortest side surface area volume E 2 in 3 in 52 in 2 24 in 3 F 2 in 2 in 56 in 2 24 in 3 C 1 in 4 in 68 in 2 24 in 3 D 1 in 3 in 70 in 2 24 in 3 B 1 in 2 in 76 in 2 24 in 3 A 1 in 1 in 98 in 2 24 in 3 Table 4: The six prisms with whole-number side lengths and fixed volume of 24 in 3 are sorted correctly by the cubeyness criterion: the prism with the longer shortest side is more cu- As the length bey; in the event a tie, the prism with the longer next-shortest side is more cubey. of the shortest side decreases, the surface area increases. p. 21

22 prism cubeyness measure computation cubeyness measure surface area volume E = = in in 3 F = = in in 3 C = = in in 3 D = = in in 3 B = = in in 3 A = = in in 3 Table 5: The cubeyness measure 1 l + 1 w + 1 h co omputed for each of the original six prisms. As the measure increases, so does the surface area. While the correlation between the measure and the surface area is perfect, computing the measure does not require computing the surface area. p. 22

23 References Author. What can we do with a common numerator? Mathematics Teaching in the Middle School (May 2009): Author. Probability in practice: The case of turkey Bingo. Mathematics Teacher (November 2008): Lappan, Glenda., James T. Fey, William Fitzgerald, Susan Friel and Elizabth Phillips. Connected Mathematics: Frogs, Fleas and Painted Cubes. Student Edition. Upper Saddle River, NJ: Prentice Hall, p. 23