Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Are We Adding Ratios (rates?) or Fractions?

Posted by mark schwartz on May 5, 2016

Introduction

This writing is a discussion and speculation on why I think fractions are so hard. I say speculating, yet when this was presented in class, the speculation was transformed into reality based on what students said. The ideas in this writing were sometimes presented to a class at this level of detail but even if the discussion never reached this depth, the core idea was discussed with the class. After such a discussion, and when assignments were given, there was always at least one student who would ask “What kind of problem are you asking us to do?” You’ll understand why this question was asked after you’ve looked at this article.

The Story

One of the elements of this story is embedded in the title. For the most part, we tend to not pay attention to the difference between ratio and rate; they tend to blend together. Yet, texts will carefully distinguish between them. I bring this up now because some of my colleagues may be distressed by my using “ratio” where they would see what’s being described, technically and formally, as a ‘rate”. For purposes of this discussion, ratio will be the term used although it’s a fine distinction in the examples I use.

What seems to be somewhat confusing is the concept of adding ratios as opposed to the concept of adding fractions. But you contest, ratios are fractions. Yes, they are but adding ratios is a different procedure from adding fractions. At least it is as we present addition of fractions in the classroom.

I present this because I believe it carries the kernel of misconception about adding fractions. For example, for students who have struggled with addition of fractions, a question like 1/2 + 1/3 will commonly get the erroneous result 2/5. In essence, add the numerators and add the denominators. Most math instructors would say this is wrong. However, it is a correct answer depending on what question was asked!

For example, if there are 5 people in the room, 2 men and 3 women, and 1 of the men is wearing glasses and 1 of the women is wearing glasses, the ratio of men who wear glasses to all the men in the room is 1/2, while for women, the ratio of women who wear glasses to all the women in the room is 1/3. If asked “what is the ratio of those wearing glasses to all those in the room?” the answer is 2/5. This comes from simply doing a count or from adding 1/2 + 1/3.

So, the answer to the abstract question, “what is 1/2 + 1/3?” depends on what the 1/2 and 1/3 represent. If it represents 1/2 (1 of 2 equal parts of something) and 1/3 (1 of 3 equal parts of the same thing), then it represents a fraction and the fraction addition algorithm is to be used. If it represents 2 ratios — how many of the total has a certain characteristic and there are two of these ratios – then the ratio addition algorithm is to be used.

This difference between adding fractions and adding ratios is a critical and typically unvoiced point. It typically is unvoiced because the concept of “lowest common denominator” (LCD) pushes us past this consideration. Most LCD algorithms involve students’ first knowing prime numbers, prime factors, equivalent fractions and how to build the LCD. After this, the students then return to “addition”.

I would contend that students should ask (or be told) what circumstance is the context for the question. In most texts, the addition of ratios may be noted but not dwelled upon because the focus is on adding fractions. This, however isn’t explicitly stated, but it is the assumption on which “adding like fractions” is based. Yes, ratios are fractions but as noted, adding two different ratios is allowable but doesn’t use the same addition algorithm as the traditional algorithm for addition of fractions.

I propose that this issue be discussed with students as part of the introduction to fractions. I believe it clarifies the issue for students who erroneously want to add the numerators and add the denominators when adding fractions. In a sense, this is a more “natural” act than adding fractions as we teach students to do, and because it is more common in everyday situations, it seems the correct procedure to follow. For example, in a classroom of 20 students (9 females and 11 males), I could ask for 3 female and 4 male volunteers. The unvoiced part – as happens in many “selection” situations – is that 3/9 and 4/11, or 7/20 were selected. We simply don’t pay attention to the denominator!

If given the abstract question 1/2 + 1/3, I would expect students to ask “Am I adding ratios or fractions?” In my experience of allowing a discussion and demonstration of this point, students have come away with a better understanding of adding fractions. The traditional addition algorithm seemed to make more sense after discussing the difference between adding ratios vs. adding fractions. The discussion actually clarified both algorithms, as the students asked very incisive questions about the difference and how they can determine which algorithm to apply.

The reason I believe it is important to allow this as a legitimate question (an instructor can always state “from now on, all abstract statements are to be understood to be adding fractions, not ratios … unless otherwise stated”) stems from the way most texts define ratio or ratio notation. Collectively, they tend to state something like “the ratio of a to b is given by the fraction notation a/b, where a is the numerator and b is the denominator, or by the colon notation a:b”. Most texts only cite the fraction notation. But again, most texts discuss and have examples and exercises for students showing how to determine and write a ratio. But, I have never seen a text present a problem of adding ratios. I contend this is done intentionally to avoid the need to identify the difference between adding ratios and adding fractions. Yet, it seems that this identification of the difference should be included, for the reasons previously stated.

And this, I propose, is one reason why students stumble through fractions. Addition of ratios is not addressed. Further, when addition of fractions is presented it is usually preceded by a lengthy discussion of factors, prime factors, greatest common factor, least common factor, like fractions, least common denominator, and finally, equivalent fractions. Wow! … all that just to add 1/2 and 1/3. I can see why students take the easy way out and apply the ratio addition algorithm!

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