The glossary of the math book lying about in Jackie’s room has this gem:
Jackie and I want students making sense out of this statement (without ever subjecting them to it.) What does it mean to state that two ratios are equal? How do you know they are equal? What do you mean by equal? We recently had them look at some images, determine whether or not the ratios they saw in them were equal, and explain why. It was interesting, to say the least, and they were surprisingly challenged, especially on the ‘splainin part. They’ve also been developing some conjectures about what they believe is true about equal ratios. All this has got me pondering (more than usual) about fractions and ratios, how these ideas are the same, and maybe more importantly, how they are not.
Here’s what I’m thinking. Fractions can be considered ratios, if viewed as part-to-whole comparisons: I have 3 out of 4 squares of a chocolate bar. Each number in the ratio quantifies a separate pieces of information: the 3 is an amount/number/value, an adjective-ish word/symbol that states how many parts I have, and the number 4 quantifies how many parts are were the whole candy bar before I snitched some. Conversely not all ratios are fractions because they could compare parts to parts, this to that, etc.
When we extend the ratio point of view, we find the size/value of the part depends on the size/ value of the whole unit, which can be an-y-thing. Take 3/4. For one whole hour, the part is 45 minutes; for one whole dollar, the part is 75¢; for one whole yard, the part is 27 inches; for one whole deck of cards, the part is 39 cards; for one whole 8oz candy bar, the part is 6 oz. That’s probably overkill on the examples, but the point is that the independent/dependent variables thing belongs to ratios, not fractions.
Yet the entire fraction itself is also a stand-alone quantity: I have 3/4 of a candy bar. When the same ratios from above are abstracted to a quantity, each are located in the exact same place on a single number line. One point to rule them all. So, 45/60 = .75/1.00 = 27/36 = 39/52 = 6/8 because they each represent the same idea (3/4 of one whole). This “sameness” can be demonstrated by ‘simplifying’ or ‘reducing’ each value to 3/4. Both simplifying and scaling up require you to shift back into fractions-as-ratios mode (although no one mentions it when they are showing you “how”.)
The most perplexing thing so far about is these layers of meaning– fractions as ratio and quantity. (There’s division, too, but I’m not getting into that here.) It would not surprise me that this is the source of confusion and anxiety whenever the f-word gets mentioned. (Fractions. What were you thinking?)
Ratios by contrast are not placed on a single number line. You need two lines to represent both quantities in the relationship; for every 3 parts (one axis) there is a whole of 4 (the other axis). There are an infinite number of points possible, located somewhere between the two axes. Put enough of ’em there and a pattern emerges. If you define rates as a comparison of two different units, like time and distance, they are always relational, never a single quantity/fraction. Oh, wait, they become a quantity, or are treated as such, as soon as you generalize the relationship into an algebraic equation! Drat. (Is this what is meant by ‘constant of proportionality’?) No wonder kids get confused. Is this a ratio or a fraction or division or what? Yes.
Let’s model equality. Consider the candy bar. Suppose the 3 remaining parts are shared evenly among 3 people, each person receiving one. Yet if a same-sized candy bar contains twice as many parts, each person would get two parts. They are getting exactly the same amount of candy in both cases. Therefore 3/4 and 6/8 are (still) equal because they are the same amount. This justifies the idea of equal quantities. The size of the whole did not change (it remained constant), which is why we can scale fractions (as ratios) up and down to get common denominators for comparison or addition/subtraction purposes. Notice though that to get 8 parts, each original part in the model had to be divided. However, when you write this process symbolically, it translates bizarrely to multiplying both values by 2. Is this a WTOther F-word moment for anyone else but me? I mean, if I give you one when you asked for two, and to solve the conflict I break one into two parts, wouldn’t you be kinda miffed?
Now look at another candy bar model that’s a part : whole comparison, 3 parts to 4 total. Scaling up by doubling the 3 shaded parts as well as the 4 total parts (which totally matches the symbolize version) I get…..3/4 of TWO candy bars, and all the pieces are still the same size!! WTF again!
Here’s what so significant and what we want our students to truly understand–ratios are about relationships between two quantities, not about a single quantity. Even thought the amount of candy changed, the relationship between did not. There’s still 3 parts for every 4 parts. So now 3/4 = 6/8 for an ENTIRELY DIFFERENT REASON. And you get your two pieces after all.
When we asked the 7th graders studying proportionality why the ratios 3/4 and 6/8 are equal, they said things like, “…because 3×2 = 6 and 4×2 = 8.” They’re calling on their prior experiences with fractions (nothing new to learn here!) but are they thinking about quantity or relationship? I suspect it’s quantity, and I wonder which candy bar model would they choose to represent their idea of “equal” ratios, or if they can identify the scale factor in either model. Others simply said, “….because they are the same number” so at least we know they’re thinking about. In any case, students seem pretty sure that ratios are just fractions and that equal means same value and the same-old reasoning about fractions applies. So let’s thrown them a curve ball:
Really? 3/4 and 6/8 are equal? Are you sure?