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.

Walk the Clock: It’s Fractions

Posted by mark schwartz on August 3, 2016

Introduction

For some reason, or perhaps reasons, fractions don’t make sense to many students. Despite the visual representations in text and/or the use of manipulatives such as Cuisenaire rods, fractions seem to remain a mystery to students. One day I asked all my basic math classes “What makes fractions so hard?” The overwhelming response focused on remembering the steps of the 4 basic operations. For them, operations with fractions seemed nothing more than trying to remember the steps to get the answer. Somehow, math instruction throughout elementary and secondary education led students to think not about what fractions mean and what they represent but rather to think about how to “solve the problem”. So, I played with an idea which seems to have provided a way for students to “see” fractions a little differently.

The Story

DON’T TELL STUDENTS THIS IS FRACTIONS! If someone asks if this is fractions, tell them it will be discussed after the activities are done. I’ve provided an idea on how to do this in the discussion section following the demonstration of the activity.

The students preferably will work in groups of 3 (or 2, depending on the size of the class). The minimum grouping is 2. Each group gets a magic marker and 12 paper plates. The students are to number the plates 1 to 12. The plates are to be placed on the ground as a clock face. This activity is best done outdoors but if not, move the desks and chairs to allow for each group to have enough space for one person to walk inside and one person to walk outside a clock face circle. If neither of these spaces are available, the plates can be cut down in size and placed on a table top. If it’s a rainy cold day and going outside is a bad idea, and if the curriculum allows and time allows, make it a “review” day and hope for sunny and warm tomorrow. This activity works indoors and on the desk top but outside is best; it’s more fun. If done on the desk top, 2 markers per group will be needed. These markers will be the “walkers” in the activity (this will be explained below).

Here’s how it works. Lay out the plates as a clock face. One of the people in the group will walk outside the circle (call this person the outsider); one person will walk inside the circle (call this person the insider); the third person will be the reader/recorder (call this person reader). Give each group a copy of the activities (a sample is below) which states what the insider, outsider and reader are to do. Once the groups have figured out who will do what, give a demonstration of what they are to do, using the 1st activity.

Using the first activity and using one group to demonstrate, note that both walkers will walk twice. Both walkers start at “12”. In each activity, the insider walks first and then the outsider. The first walk is done when the insider reaches “12”. The second walk for both starts where their first walk ended. The reader is to watch and verify that each walker takes the right number of steps (others in that group can help verify).

1st Activity: on the first walk, the insider walks 2 units while the outsider walks 1 unit. On the second walk, the insider walks 3 units while the outsider walks 1 unit. The reader will note “outsider.insider”. In this activity, the record should show 10.12.

If there is confusion about the walking and/or the recording, just repeat the first activity. When everyone’s ready, move on to the next activities.

2nd Activity: on the first walk, the “insider” walks 4 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 6 units while the “outsider” walks 1 unit. The reader should note 5.12.

3rd Activity: on the first walk, the “insider” walks 2 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 3 units while the “outsider” walks 2 unit. The reader should note 14.12. (There should be questions on how to record this. Show students “military” time.

It’s important that if more activities are to be done, don’t allow students to do it. The reason: activities provided by students may result in a very time consuming set of walks and more critically, present a new issue to handle. For example, although subtraction of fractions can be done this way, I suggest not doing it. You could get a negative answer and you might want to avoid this. Just stick with one concept at a time; adding fractions (although they may not realize it). Given this, you might want to prepare and walk through a bunch of activities and be careful that none of them take too much time, yet enough time for the students to play and enjoy it.

Again, do not say anything about fractions at this point, but what has happened is that the problem 1/2 + 1/3 has been done. The record “outsider.insider” is 10.12 , or in reduced fractional notation is 5/6. Most likely, someone has noticed that the insider always has a value of 12. You sort of have to weasel your way around this and don’t yet call it a common denominator.

A Little Discussion. After these activities, you can transition to presenting fractions as you usually do. But, here’s one idea to consider in talking to students about how this activity demonstrates addition of fractions. What is seen and used but not referenced is the common denominator of 12. This explains why the insider’s walking the line twice isn’t counted twice. In the problem 2/3 + 3/4 , the denominator could be any multiple of 12 but in this case since it is 12 and you know it, don’t count it twice. Students may balk at this idea but it can be explained further. The insider always walks the line twice but always restarts the 2nd walk at “12”, while the outsider restarts the 2nd walk where the first walk ended so the insider’s walks aren’t added, rather they simply repeat.

Also not seen is the addition, but it occurs in the outside walk when the second walk starts where the first walk ended. The outside walker’s position at the end of the first walk is added to the beginning of the second walk. Please note that using this method for a problem such as 1/2 + 1/3 would give the answer 10/12, not 5/6, so clearly reducing fractions has to be addressed before this activity. Further, you might question how to get from this activity to the “rules” for addition and subtraction, but that’s not the point, although it can be seen because both fractions in this example, were converted to equivalent fractions with a denominator of 12, although in this case and others, it wouldn’t necessarily be the lowest common denominator. This again, could create a teaching moment, discussing the issue of common denominator versus lowest common denominator.

I suggest that different sets of students get a chance to walk the line. In fact, teams of students could do it; two walk and the others verify that their walking is accurate. Further, the point at which the transition from this activity to the traditional fraction work is to be made is a matter of how the class is collectively responding. In some instances, students caught on and realized that this was adding fractions. But even if they caught on, I still had them walk through all the activities. In several classes, students wanted more exercises. I think it was because it was a nice warm day. It’s a judgment call.

One more thing. Recall that the purpose of this activity is to give a visual and kinesthetic sense to the “rules” and it does seem to have a positive effect on students. When we got to the traditional rules and procedures, I heard students talking about how it “matched up” with what they were doing outside. Play with it.

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