Marveling At The Historical

Math Oldies But Goodies

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    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 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.

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Archive for the ‘place value systems – different bases’ Category

Counting Sheep

Posted by mark schwartz on May 26, 2016


This was presented in a course titled Contemporary Math. The course was designed for students whose major didn’t require math beyond the basics. It was to familiarize students with a wide array of math ideas, but none in depth. This article is how students were introduced to different base number systems and operations in bases other than ten. Students were remarkably novel in their response to the idea in this article, modifying it to fit other circumstances. Some of their ideas are included. The following story is a handout to students, only up to the point where the exercise was extended. The class worked in groups.

The Story:

Below are shown 5 holding areas each individually fenced in, labelled A through E, from right to left. They are for the farmer’s sheep, which are in the East Pasture. The areas are labeled A through E, from right to left because when the sheep come to the holding areas, they enter from right to left, starting with A. The fence around each area is low enough so that the sheep can get in and out by jumping over the fence.

E        D        C        B        A            East  Pasture

South Pasture

So, a sheep that wants to get into area A or move from area to area must jump over the fence. Sheep move only from right to left between areas – A to B to C, etc. –  and only one sheep per area is allowed. The farmer may change his mind later and allow more than one sheep per area.

But these are not ordinary sheep! They are specially bred to have amazing jumping power. If an area is empty, a sheep will simply jump in. However, if they see a sheep already in the next area, they will jump over that sheep to land in the next empty area. It gets sillier! If a sheep coming in sees that there is a sheep in area A and also in area B, the incoming sheep will jump over both of them to land in area C, the next empty area.  An incoming sheep jumps over all filled areas to the next empty area.  AND THEN, any sheep that gets jumped over leaves the area by jumping into the South Pasture.  They get a little weird when other sheep fly over their head!

Now, I wasn’t allowed to bring pictures of the sheep because they are so special, so you can use anything you want to represent your sheep (a coin, a wad of paper, a stick-figure sheep, a pen, etc.). So, see what happens when your sheep come in from the East Pasture. Let’s start with 5 of them. Although 5 sheep entered, if you do it as the sheep would, sheep would be only in areas A and C when you’re done. If this doesn’t happen, call me over. Once everyone has accomplished this, we’ll play with a different number of sheep.

Extending the Concept: After playing with 5 sheep, the class was asked to extend the idea by finding out which areas would be filled if 6 sheep came in from the East Pasture. Once each group got an answer and all groups agreed, the results for 5 and 6 sheep were “tabled” on the board and then this was extended to 7, 8, 9, and 10 sheep, which was also recorded.  There was a lot of interaction within and between groups, which is a desirable outcome of this exercise. Here’s what the class saw once all of them finished the exercise and there was agreement between groups about the correct arrangement of sheep.

# sheep     E     D     C     B     A

5            0      0      1     0      1

6            0      0      1     1      0

7            0      0      1     1      1

8            0      1      0     0      0

9            0      1      0     0      1

10          0      1      0     1      0

I then asked that everyone let their sheep sleep and then asked: “Without actually doing it can you predict what the arrangement would be for a dozen sheep?” I walked around the classroom and listened to the within-group discussions. and prompted them to let their sheep sleep. Every group eventually concluded correctly that only areas C and D would be occupied. We talked about how they figured it out.

In most classes, no one in the class was aware that what they were actually doing was counting in binary. In some classes there were tech and computer savvy students, who were aware of the binary system (base 2). I asked them not to shout it out or share it with those in their group. This activity led to a discussion about the binary system and its use in computers. At this time, the discussion was extended to seeing how the base 2 and base 10 systems functioned similarly by demonstrating the place-value system. The class was then asked to wake up their sheep and do the following:  the condition for the sheep was changed to allow three sheep to be in an area at the same time and the class was asked to find the arrangement in the areas if 19 sheep came in from the East Pasture. The result was …

# sheep     E     D     C     B     A

19          0     0      1      0      3

This was discussed in detail and I asked them to identify in what base they had counted and yes, they figured it out. Then the question asked by one of the students (it always comes up at some point in this exercise) was: do all bases work the same, even if a dozen sheep were allowed in each area?  We then had another fun discussion. Then someone asked (but we didn’t discuss it) how it would work if a different number of sheep were allowed in each area – what a great question!

Summary. This exercise was done typically over two class periods and explored the place-value system and the different base systems. Further, the classes played with basic operations within each base as well as conversions between bases. A final note. In one of the classes, there was an education major and she wanted to know if the basic sheep-counting exercise could be used in elementary school to introduce students to the ideas we covered in class. Her idea was to line up 5 chairs in a row and have students enter this row of chairs just as the sheep had. My answer was a definite “yes” but not to label the activity as an exploration of different base number systems. Rather, just let the children play and the concept will incubate and at some point later in their education if they study different base number systems, I strongly suspect they will have an “aha” moment.


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