When this happens, there is a delightful interplay of creative freedom and originality on the one hand, and the restrictive logic of the previously existing structure on the other hand. [2]
In the case of ratios the restriction is the intuition we have when we compare relative magnitudes.
Vector spaces of n-tuples have proven to be a valuable starting point for many interesting concepts, even on an elementary level. Z. P. Dienes, for example, suggests if addition of scalars is taught as a special case of addition of vectors (the "deep end" approach) that the more general setting will provide the framework for a more general understanding. [3] There are three essential properties of a vector space needed for what follows. We list them for n = 2.
| Picking (in thousands of bushels) | Cortland | MacIntosh | Northern Spies |
| First Yield | 2 | 6 | 4 |
| Second Yield | 3 | 6 | 0 |
I asked the students to state various ratios of the quantities display At this point I did not criticize the "reduction" of "two to four" (2: to 1:2 although we shall see later that the students were asked specifically whether they wanted this to be a property of ratios in the ultimate definition. The first difficulty they encountered was with the zero second yield of Northern Spies (to be referred to as "second N"). Some students suggested that a ratio should not be defined if one of the items to be compared is 0. If this were done, they reasoned, then 0: 8 and 0: 4 would both "reduce" to 0:1. They disliked the possibility that numbers would be the same relative size when compared with 0. Others having identified ratios with fractions from previous training, suggested that we should allow 0 only if it is the first entry, so that 0: 4 will defined but 4: 0 will not. Further class discussion convinced some students that such an alternative would not allow for the symmetry indicated by the fact that we can report first N to second N by 4: 0 as easily and meaningfully as we can report second N to first N by 0:4. At this point I added that a ratio only indicates relative size; unlike division for example, no operation is performed on the numbers. The alternatives were summarized on the blackboard as follows:
Zeros in ratios should be:
The students were then asked to vote on which alternative they thought should hold for ratios as they understood them. The third was near-unanimous choice. I suspect the decision was prompted by reasoning somewhat along the following lines: "This is the first time I've been able to choose the rules. Just to be different, I'll choose one different from that for fractions. Since I'm not sure about what to do with 'not defined' I'll choose the third." That is, given the opportunity to make a daring or a conservative choice, students will respond to the novelty of the approach with the more daring alternative. Davis found this be true as well in classrooms in which student initiative was encouraged. [4]
One side benefit to teaching mathematics democratically is the it stimulates thoughtful discussion and exposes misconceptions. One student volunteered that 0: 4 for him meant that second N is 4 times first N. I then asked him to describe the ratio 1: 4. At that, the student conceded that he no longer had any intuition about 0: 4 (which, course, is an improvement if initially his intuition was incorrect).
The following appeared elsewhere on the blackboard:
A ratio is like a fraction except:
A. 0 is treated differently: it is permitted in either entry.I then added a second way in which ratios are different from fractions:
B. Ratios of 3 or more things make sense, as 2:6:3.
Why on the other hand is "2/6/3" meaningless as a fraction? The students pointed out that (2/6)/3 2/(6/3). This illustrated the fact that division is not associative and provided an opportunity for a brief review of the fundamental laws. of algebra, noting especially what does not hold (for example, that division and subtraction are not commutative).
In order to get yet a third way in which ratios differ from fractions I asked the question, "What is the ratio of Cortlands to Macs, for the two yields taken together?" The answer was quickly given, 5:12. I then suggested that taking the two yields together is a kind of addition, and wrote
2:6
3:6 5:12
In discussion this was compared with the addition of corresponding fractions: 2/6 + 3/6 = 5/6. This suggested that we add to our list of differences:
C. Addition of ratios is given bybut we add fractions as follows:a: b + c:d = (a + c) : (b + d),
a/b + c/d = (ad + bc) /bd.
A student commented that since he and others had been reporting the ratio of 2:6 as 1: 3 without being corrected by the teacher, he wondered about the following problem from the point of view of consistency:
2:6 = 1:3
3:6 = 1:2 5:12 =/= 2:5
The reader will recognize that our addition is not well-defined. If it is not noticed spontaneously, it can be brought up as a point of discussion. After a short inquiry about how the student knew that 5:12 =/= 2:5, I focused on the obvious contradiction among the following three desiderata: (a) Ratios can be reduced. (b) Ratios are added as described in C above. (c) For ratios, if equals are added to equals, the results are equal.
Which of these assumptions should be eliminated? Again a vote was taken. The discussion preceding the vote was stimulating, including examples in which each assumption could fail (see section 4 below). I remained as neutral as possible; the conclusion reached, but only the slimmest majority, was to eliminate (a). An especially keen student, one of those who voted for elimination of (c), pointed out that now (c) becomes meaningless because we are without a definition of what means for two ratios to be equal. After some discussion, the class agreed on the definition; a: b = c:d if and only if a = c and b = d.
Note at this point that all the properties of ratios are vector space properties.
As for the second vote, an alternative to (b) is to add ratios as you add fractions. An alternative to (c) is harder to come by. I suggest using geometry to show that analogs of (a) and (b) can hold but the analog of (c), fails. Reducing ratios can be thought of as a scale change, so a ratio can be interpreted as a similarity class. To be specific we can think of 2:4 as a rectangle with dimensions 2×4. (A rectangular solid similarly provides the geometric analog of a triple ratio.) Then the analog of "2:4 × 1: 2" is "a 2 × 4 rectangle is similar to a 1 × 2 rectangle." If we now interpret addition as proposed algebraically above we get 2:6 + 3:6 = 5:12. Geometrically we have,
But 2:6 = 1:3 and 3:6 = 1:2, so 1:3 + 1:2 = 2:5. Geometrically,
However, 5:12 =/= 2:5; that is, a 5 × 12 rectangle is not similar to a 2 × 5 rectangle. Similar figures "added" do not in general give similar figures. This shows that for some meanings of = and +, addition is not well-defined. This falsifies (c).
The traditional treatment of ratios may now be reviewed with an introductory question like, "What do we mean by two fractions being equal?"