In my last post, I promised my solution to the fraction problem: 3 1/2 divided by 3/8. The following is what I share with teachers in an effort to explain a real-world application for this problem.
Kathy bought 3 1/2 yards of ribbon to make bows. Each bow took 3/8 yards of the ribbon. How many bows could Kathy make?
The pictorial representation, a crucial step for children and adults alike, is often overlooked. This is unfortunate because if a person can be helped to envision the problem, they can probably estimate a reasonable solution in their head. The illustration for the problem would look something like this.
When we then examine the mathematics behind this illustration, we realize the first step we took in determining the number of bows was to recognize we had seven, 1/2 yards of fabric. In the mathematical algorithm we have just changed the mixed number into an improper fraction. Next, we divide each of the yards into eight pieces because we need 3/8 of a yard to create a bow. This means that each 1/2 yard has four equal sized pieces. This is the mathematical equivalent of finding common denominators, which we don’t normally do when dividing fraction. We find that in 3 1/2 yards we would have 28/8 yards. Next we would begin to break the 28/8 into 3/8 portions. We are in effect dividing the numerators 28 by 3. The result is we could make nine whole bows with one piece left over. A common misconception, even amongst adults, is that the one leftover piece would be represented by the fraction 1/8. However, each bow takes three pieces of the ribbon so with only one piece left over we have 1/3 of a tenth bow. The mathematical algorithm would look like the following:
When one examines division of fractions conceptually, it becomes easier to estimate an answer and know if your answer is reasonable. It also proves the point, there is no need to invert and multiply. One can merely find a common denominator and divide the numerators.
So why do we invert and multiply? If you look at another problem like 6 ÷ 1/2 you are trying to determine how many 1/2′s are in 6 wholes. The answer is 12. To understand how the procedure of invert and multiply works, the problem needs to be written in fractional form, like this:
To solve the problem written in this form we would multiply both the numerator and the denominator by the reciprocal of 1/2 or 2/1. In doing so we have changed the denominator to one. When we multiply the the numerator 6/1 by 2/1, we arrive at an answer of 12/1 or 12.
Most of the time, invert and multiply is not explained this thoroughly, it is merely a “trick” we use to solve the problem. When we show all the steps, invert and multiply, is a sound mathematical procedure but it still does nothing to help students envision the division or determine the reasonableness of their answer. That is why I think division of fractions should be taught conceptually prior to introducing any other method.
Just “Sum”thing to think about.
“Don’t ask why, just invert and multiply”, talk about an arcane procedure. Maybe it’s just because I’m old, but I really can remember this being said in the math classroom. I’m not sure teachers actually use the phrase anymore, and I hope not, but I know they use the procedure.
The saying itself insinuates that one has no need to understand the rationale for dividing a fraction or dividing with fractions in the first place; one merely has to perform a trick to find the correct answer. The problem is, with no understanding, a student has no idea if the answer they compute is reasonable. Some students are even confused by the procedure itself and invert the wrong number in the problem. Again, they have no way to know whether their answer makes sense. The other misconception this creates for students is that they can change the operation in a problem for no apparent reason.
This is another one of the things I think needs to change in the math classroom. I have made it a mission to help teachers understand the concept of dividing fractions in order to help their students do the same. The first step in the process is to help teachers understand a real-world application for division of fractions. I do this by giving them the “naked number” problem 3 1/2 divided by 3/8 and asking them to quickly estimate an answer, write it on a sticky note, and put it in their pocket (where it will no doubt be washed and disintegrate in their washing machine and no one will ever see what they wrote). I never allow much time for this because most adults have the natural inclination to compute the answer using the invert-and-multiply procedure rather than to estimate. This signals a lack of benchmarking or estimation skill when it comes to fractions.
I then ask teachers to think of a real-world reason that could be solved with this fraction problem. The next step in the process if to have them illustrate the problem and its solution. Upon examining the steps taken to solve the problem, one discovers the fact that it was never necessary to invert anything.
So, if you’re still reading this post and you’re up for a challenge, try it for yourself. Take less than ten seconds (really, no more than ten seconds) and estimate an answer to the problem 3 1/2 divided by 3/8. Then think of a reason you would ever need to divide 3 1/2 by 3/8 in the first place? And what would it look like if you did? In my next post, I’ll reveal my problem, the illustration, and my rationale for banishing inverting and multiplying from the math class. I might even explain why inverting and multiplying works in the first place.
Just “Sum”thing to do in your spare time…..
I was just having a conversation with two sixth grade teachers this morning and the topic of long division came up. We had been talking about multiplication and how it would help children if we said, “groups of” instead of “times” when discussing multiplication. I made the comment that the same could be said for division, since multiplication and division are the inverse of each other. One of the teachers told me that she found it difficult to know exactly how to phrase a division problem when talking to her class. She wondered, in the problem 108 divided by 8 if she should say, “108 divided into 8 groups” or “108 divided into groups of 8″.
I think the confusion about division comes from the fact that there truly are two types of division but you don’t realize it in the “naked number” problem 108 divided by 8. It becomes apparent however, when we put the division into context. For example: We could be saying that 108 people will be divided onto 8 teams, which is a division partitive problem. When given this problem a child who is a direct modeler will make 8 groups and fair share out the 456 people onto those teams. In that instance we have divided the 108 people into 8 groups.
However, if the problem was 108 cookies were being distributed to people in packages containing 8 cookies and we were trying to determine how many people we could feed, then we have a division measurement problem. In that case, a child who is a direct modeler would take the 108 cookies and break them into groups of 8 and count the groups. Either way, you would arrive at the same answer, but the actions would be different.
The type of division problem that can be used to help students with base ten understanding is the division measurement problem. When the divisor is ten, or a power of ten, students can remove groups of ten from the dividend.
Just “Sum”thing to think about…
Some of the things that have been long accepted in the mathematics classroom, need to GO! They do nothing to help a student understand mathematics from a conceptual viewpoint. In this series, I will be talking about some of my pet peeves and things I would have outlawed if I was the “Queen of the Elementary/Middle School Math Universe”. (This is an imaginary title. Only a twisted math teacher would come up with or desire said title!)
In my opinion, the word “times” should no longer be used when teaching multiplication. It is eloquent when used in this opening sentence, “It was the best of times, it was the worst of times….”. It works well as the title for one of the world’s largest daily publications, “The New York Times”. It even has a place in the math classroom when we are teaching about its passage or how to read a clock. However, the word, “times” when it relates to multiplication, does not reflect any specific action and needs to be replaced with the phrase “groups of“. Students, especially those in the primary grades, are perfectly capable of multiplying, but are usually direct modelers. It helps them learn if they have an action to model. If we tell them they are putting candy into five groups of four, they can tell you that there would be 20 pieces of candy. If we tell them to take five times four, they have no idea what we are asking them to do.
The same can be said for division. If we ask students to take the 20 pieces of candy and break them into groups of four to share with their friends and we want to know how many children will receive candy, young children can model that action. However, if we merely ask them to do the problem 20 divided by four, we may or may not get the correct result.
Just “Sum”thing to think about!
We shouldn’t just teach rules or procedures in math. Students may learn them long enough to pass a test but without conceptual understanding to link back to, few will retain the information for any length of time. This is why teachers feel the need to re-instruct yearly on topics like math facts, fractions, and formulas. We are notorious for asking, “Why didn’t they teach you this last year?” when in fact the previous teacher did teach the topic. Students need concrete and pictorial experiences before they learn procedures and it is up to us as educators to provide them with quality problems that encourage true understanding.
Just “sum”thing to think about!