Common Denominators

This is an essay for parents, illustrating a useful math topic which schools do a poor job of teaching. It is not a lesson for kids, but it could become the basis for one.

Common Denominator

What's a common denominator? Why is it used?

In 2003 while substitute teaching in Silicon Valley's broken schools, I encountered 7th, 8th and 9th grade students who didn't understand what a common denominator was, or why it was useful, though they did remember bits and pieces of how to find them.

Say one recipe contains 3 ounces of fat and makes 18 servings. Another contains 5 ounces of fat and makes 28 servings. How do the amounts of fat per serving compare?
Say 13 out of 31 people in a rural area vote Yes on something in one study and 20 out of 52 vote Yes in another study. Which study had a greater fraction of Yes votes? By how much? How can we compare these two fractions, 13/32 and 20/52?

Whenever we compare amounts, we need to convert the amounts to a common currency to compare them. This is true with fractions, too. We need to divide them up into same-sized slices so that we can compare the number of slices in one to the number of slices in another.

The word "common" means "standard," as in the sentence: A common problem in California classrooms is students are asked to learn a lot of concepts and processes instead of spending time understanding numbers and relationships.

Say you have $5. Do you have enough to buy a book that costs 3 euros? If the currencies are different, we don't know- first we have to convert dollars into euros or euros into dollars or both into something else. Which is longer, 4 meters or 13 feet? We need to convert them into a common measurement to compare them, whether this is a common unit, or just draw these lengths next to each other in the same (common) place. We need a common measure to compare things.

"Denominator" is a common trait or characteristic, or an average level or standard. In math, it's the numerical size of the units being used. Usually this is 1 (one) when we're just counting things. But when we're considering fractional parts, it's the number below the line. When we're dealing with thirds, the denominator is 3. When we're dealing with hundredths, the denominator is 100.

There's a similar word, "Denomination" which means one of a series of kinds, values or sizes, as in currency (paper money) or weights. For instance, we might say that the denominations of paper money that I carry are ones, fives, tens and twenties- rarely fifties. Or that stamps come in 37 cent denominations. This word is also used for the variations of a religion.

So when we compare two fractions, we need to express them in terms of the same (common) kinds of slices (denominator) in order to compare them.

How do we find this "common denominator," that is, a small slice that fits an even number of times into each fraction? In the first example, comparing 13/32 to 20/52, if we slice each 32nd into 52 parts, and each 52nd into 32 parts, we find that each fraction is now expressed in a certain number of 1664ths, so we can compare them easily.

To make it simpler, we might notice that in the first fraction, using 32nds slices one unit into 4x8 pieces and using 52nds slices the unit into 4x13 pieces. There's a common factor of 4, or to look at it another way, the first fraction slices a quarter of the votes into 8 pieces and the 52nds slice a quarter of the votes into 13 pieces. So instead of dividing each 32nd into 52 pieces, we can divide the 8ths (of the quarter) into 13 pieces and the 13ths (of a quarter) into 8 pieces. Thus we can find a common denominator of 4*13*8, or 416ths, which is smaller and easier to work with than 1664ths.

Though it is correct to call this the "smallest common denominator," usually we say "least" instead of "smallest", so it is called the "least common denominator." Even mathematicians think this is a dumb name, so it's commonly known by its initials, LCD. We find it easier to work with smaller numbers, so the LCD is the common denominator we usually look for.

What's the least useful way of choosing a name? Considering ambiguous names. In "least useful way", the word "least" modifies "useful". It means the way that is least useful, not the smallest of all the useful ways. But with "least common denominator," the word "least" modifies "common denominator", not "common". Thus, "least common denominator" is an ambiguous phrase. A better name would be "smallest common denominator."

It's poor naming like this that helps make math particularly confusing. If we taught concepts, we'd teach "smallest common denominator" and AFTER kids had the concept down, we'd warn them that it is often called "least common denominator" and even abbreviated LCD.

Even "smallest common denominator" is confusing conceptually. Halves give us a smaller common denominator (2) than eighths, but halves are bigger than eighths. Better might be "simplest common denominator."

California standards emphasize learning jargon over learning math. And little of the jargon is necessary for developing math skills- it should all be taught AFTER the math skills are taught. This is especially confusing for the myriad students for whom English is a second language.

What happens if we have 12 different denominators? Don't we really just want a standard, simpler way to name fractions?

Yes. Let's just pick a standard denominator and use that. In general, people all over the world like to work in hundredths. If we use long division, we find that
13/32 = .406250 ~= .41
20/52 = .384615 ~= .38
so using hundredths, it's pretty easy to compare them, although not precisely.

When we talk, the phrase "point four one versus point three eight" isn't as clear at 41 hundredths vs 38 hundredths, and we commonly call a hundredth a "percent", and say "forty one percent vs thirty eight percent." By using percents a lot and getting familiar with how simple fractions translate to percents, percents become a common currency, or common denominator, for expressing fractions.

Percents, and writing out fractions as decimals, means that we have to do long division and round, so it's not the perfect solution. By having several ways to work such problems, we can choose a method that fits the problem best.

In summary, fractions occur all the time. To compare two fractions, we need a common denominator, just like to compare the costs of two items, we need a common currency. Sometimes we find an exact common denominator for two fractions, and sometimes we look for the smallest of these, or "least common denominator." At other times, especially if we want to communicate the size of fractions to others, we use a common denominator that we're all familiar with, such as percents. If we need more precision, or are using calculators, we often use thousandths or a decimal representation with more digits.

This might seem like a simple lesson for an adult, but it can't be taught to most kids. Most kids need to discover these relationships for themselves, and spend time exploring each aspect of these concepts, working with models that can be broken into fractions, drawing pictures and measuring. Students need to slice up fractions to discover a common way of counting them, and measure them against a 100 cm ruler as a common measure. And they need to do this many times in many different situations to really understand it thoroughtly.

There's plenty of room for lessons from a teacher, but the lessons need to be delivered to curious students who are discovering the nature of the problems themselves, as well as much of the solutions.

In a class of 20-30 students, they will have plenty of insights to share. The teacher can help by asking questions and reframing lessons and by guiding the students toward illuminating the lessons, but the learning must be done by the students. The terms "least commonn denominator" and LCD should either only be mentioned after they have the concepts down, or during a lesson in which the students look for a good name for the concept themselves. Names should only be taught after a thorough introduction to the parts of the names.

Teaching a lesson like this should take weeks, with discussions and measurements and experiments and trying different methods. Sometimes a child will be exposed to an idea, but it'll take a week or a month or two for it to really gel so it can be a solid, unforgettable building block for the child's math education.

This method of good teaching doesn't fit with California's curriculum where each topic is taught for a day or two and several related topics are completed in a week. Though there is often review, in the meantime new topics are taught and the child forgets. Kids often get review questions wrong because they haven't spent enough time with the initial learning. Review questions ask them to dig in their memories, rather than utilize their understanding, because the understanding just isn't there.

There's always the next year- in California, most topics are repeated year after year, at a slightly more advanced level. A more advanced level is possible because the child is older. However, many children are face with completely relearning most topics because they just didn't have time the last time, except to learn that math is confusing and they're just not good at math.

And, of course, California's STAR test is geared to the same material, again testing to see if the child can regurgitate learned procedures and shallow understandings.

California students are failing the exit exams in math in record numbers, and the exit exam is easy for any student with a rudimentary understanding of math. The problem is that Califoria's math standards rob most kids of even a rudimentary math education.

-Randy Strauss