Place value is among the most powerful notational achievements of humanity. The MathLab we describe here is admittedly contrived, but it is believed that place value is too important a concept to be overlooked as a valid Q/R/V MathLab. The activity suggested here can be performed in any base, but we will describe it using base 4. The question we pose is the simplest that can be devised so that the two best pedagogical tools for teaching and learning about place value can be utilized: snap-to cubes (bricks) and Styrofoam® Cups (clicks).
The crux of the pedagogy is to create our own odometer for base 4. To do this, we simply take a few Styrofoam® Cups and mark each one with the numbers 0, 1, 2, and 3 as seen in the photo (photo, left).
To begin operation, be sure that all four zeros are facing the operator (photo, center). To record, say, the base-4 depiction of eleven items, we rotate one “click” for each count: One, … rotate the rightmost cup to 1. Two, … rotate same cup to 2. Three, … rotate it to 3. For four, everyone understands that something has to give. Thus, the second cup (from the right) is “engaged” with the first cup, so that, when the first cup rotates to the 0, the second cup advances to its 1. For five, … second cup remains at 1, first cup rotates to 1. Six, … second cup remains at 1, first cup rotates to 2. Seven, … second cup remains at 1, first cup rotates to 3. For eight, again something must give. Thus, the two cups are again “engaged” and the second cup rotates to 2 as the first advances to 0. For nine, … second cup remains at 2, first cup rotates to 1. For ten, … second cup remains at 2, first cup rotates to 2. Finally, for eleven, the second cup remains at 2 and the first is advanced to 3. Thus, the “odometer” reading of 0023 (photo, right) now represents the count of eleven in base 4, namely 23. This should be practiced by various students until the idea becomes obvious.
Once the base-4 odometer is mastered, the odometer should then be concurrently run along with the following activity using the snap-to cubes. [Frankly, it is better to teach, say base-4, first with odometers (rather than other manipulatives) because it is much truer to the idea of place value—which, after all, is about notation. This is because of the confusion that is brought by the fact that there are four symbols, but one of them is the 0. Thus, not more than three things can be “held” since 4 itself does not exist. (Remember, in base-4, 4 is actually 10.) This causes a certain confusion for students that simply is not present in the odometer, where it is obvious when “something has to give”—namely, a second wheel has to be engaged (i.e., a “bundle” must be formed). Nonetheless, we leave the pedagogy to the instructor.] So, the crux about the snap-to cubes is that one is never permitted to hold more than three objects. Thus, instead of a few Styrofoam cups, we have a few students sitting in a right-to-left arrangement. Again, we count out eleven items. For one, the first (rightmost) student is handed one cube. For two, he is handed a second cube. For three, he is handed a third cube. Now, for four, the student is handed a fourth cube, but is not allowed to hold them (because he has no way of expressing his holdings with only one of the digits 0, 1, 2, and 3). Thus, he must “bundle” the four items (he snaps the four cubes together) to obtain a single item (a single bundle of the four cubes, a 4-bundle) that he must then “pass” to the student to the left of him (i.e., facing us; i.e., to his right). He (the first, rightmost, student) is now free to take more cubes. For the counts of five, six, and seven, the first student takes a cube for each while his colleague still holds the one larger bundle. For eight, however, he must again bundle the four items into a single larger item and then pass it to the next student who now holds two such bundles. For nine, ten, and eleven our first student gets another three cubes. Thus, the final holdings show the rightmost student having three single cubes, but the student to the left of him holds two larger bundles (four cubes in each bundle). These two 4-bundles and three unit cubes match up to the 23 display of our base-4 odometer. But now, the fact that, in base 4, 23 represents 2x(4^1)+3x(4^0) comes to light.
With a little practice, and 99 cubes, the students eventually appreciate that the base 4 representation for these many cubes is 1203 (base-4 odometer reading) and that is means there are 1 group of 64, 2 groups of 16, no groups of 4, and 3 single cubes (i.e., 1203 = 1×4^3+2×4^2+0x4^1+3×4^0).
We can now pose a MathLab challenge. For example, a team of students might secretly place some, say, 158 single snap-to cubes into a bag. These cubes are now removed one at a time and lined up in a row. Another team of students must carefully operate a base-4 Styrofoam cups odometer through the 158 clicks to arrive at the final display that is the base-4 representation of the base-10 158 (in base 4, this should be 2132). When this is completed, another team of students should empirically verify that this is the correct representation by arranging, from those 158 cubes, two 64-bundles, one 16-bundle, three 4-bundles, and 2 single cubes.
In this way, we have
Q. posed a question (“What is the base-4 representation for the number of cubes in the bag?”) that
R. requires the mathematics of bricks (bundling cubes on the basis of powers of 4) and clicks (operating a base-4 odometer) to resolve; and for which
V. the result is empirically verified by matching the odometer reading to the corresponding holdings of bundles.
It is worth mentioning two ways in which such MathLabs can be made more realistic (i.e., real world). One is to use base 2 under the auspices of the fact that machines, such as computers, ultimately perform in this base because of the no-yes, false-true, off-on, etc. states that the two symbols 0 and 1 can articulate.
The other is to use a large base, such as base-36 brought by the 36 symbols: @, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U,V, W, X, Y, Z to compress a very long string of information (such as a vehicle’s informational string of digits that might encode a manufacture date, a plant location, a color, etc.) into a much compacted size (i.e., the vehicle identification number).