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 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 ‘Fibonacci’ Category

Fibonacci: Surprise and Pattern in Mathematics

Posted by mark schwartz on June 18, 2016


Teaching math involves a lot of things, first of which is “do I have their attention?” and secondly, “how do I keep their attention?” It took me a while to realize that these were the wrong questions. The real questions are “does math have their attention and does it keep their attention?” There is a difference. We’ve all developed presentation modes, perhaps several of them, tailored to the content and the class … we do things that feel comfortable but also things that seem to keep students’ attention. Let me show you one that I’ve used in a developmental class that gets conversation going beyond the actual exercise and leads to two things important to math that I like to emphasize as the semester continues: surprise and pattern.

The Story.

An interesting way to demonstrate surprise and pattern is the use the Fibonacci series. Don’t teach it, just use it. In fact, in some instances it’s never mentioned but in most instances, students want to know how the exercise came out the way it did.

The first thing is to have students work in groups of two or three. This group behavior engages each student in the process of deciding who is going to do what part of the activity, and as they work together they realize that working together is actually a fun math experience. I typically randomly assign students to groups in an effort to get them to work with someone new and this involves their moving around to sit with their new partner. This activity alone is typically something that they’ve never experienced in a math class. I give them time to introduce themselves to each other.

Once all the hubbub of moving and introductions is over, I ask them to get out their calculators or phones that have calculators because they’re going to need them for this exercise.

I then ask them to designate a writer and a “calculator” and ask that the writer get out a piece of paper and make a list down the page from 1 to 25. When everyone is ready, I then ask each group, one group at a time, to give me two numbers from 1 to 9, inclusive. Sometimes the word “inclusive” needs a little explanation. The writer is then to put their pair of numbers as the first and second number in the list. If the question is asked about the order of the numbers, tell them that it’s their choice. I then write their choice on the board. I do this because I want each group to have its’ unique set of two numbers and also if they see what’s been selected it will help avoid duplicates. Once every group has done it, I ask them if they’re ready to go on.

The instruction is to add the first and second numbers to get the third number. Then take the 2nd and 3rd numbers to get the fourth and continue this way until they’ve filled in their list to 25 numbers. I take one of the pairs from the board and show them how it works. I again ask if there are any questions before they continue. I point out that as they move down the list the numbers may get big and the job of the calculator is to get the numbers correct, so work slowly. As an aside, this sometimes leads to the group members deciding that they will all do the calculations to check on each other and make sure they’re right. If no more questions, get to work. I roam the room and watch them work and comment occasionally.

After all the groups have their list of 25 numbers, I ask them to slowly and carefully do these things: divide the 25th number by the 24th number, ignore the decimal point, and write down only the first 4 digits from the left (sometimes I have to show them what this means).

After they’re all done, I announce that I will now tell each group what their 4 numbers are. Using the list of their pairs on the board, I start with announcing 1618 for the first two groups and then I quit and announce that they all have the same number.

This is the first surprise. They typically verify with each other that this is true. Usually someone will ask how that happened and this begins the conversation; how can each group start with two different random selected numbers and yet come out with the same number? Enjoy the conversation and let them roam around for a while; they might hit on what happened. If they ask you for the answer, don’t give it but you can provide a hint that the answer is hidden in the process; can they see any patterns?

This first day exercise lays the foundation for students to realize that a lot of the math they’ll be doing will have surprise and pattern and that as we go through the semester, I’ll be referring to this a lot (which I do; after all, rules, formulae and algorithms are “frozen” patterns, aren’t they)?

And finally, when the class is over and they’re gathering their stuff to leave, there are animated conversations about what they just did and that’s very satisfactory to me. You might want to experience it too.


Posted in basic math operations, Fibonacci, math instruction, mathematics, remedial/developmental math | 2 Comments »

A Toddler, Pascal and Fibonacci Climb Steps

Posted by mark schwartz on April 14, 2016


I once taught a Contemporary Math class, designed as a survey course for those whose major required one math course but nothing heavy like a full Algebra, although some students had Pre-Algebra or Algebra in high school. Among the topics were binomial expansion, Pascal and Fibonacci. A student asked if these guys knew each other and when I asked him what he meant he said “I thought most of these guys stuff was connected somehow”. It wasn’t an elegant question but it turned out to be insightful. One of the exercises I had the class do was “a toddler climbs the steps”. As I thought about his question and the exercise and played with it some more I came to see a connection between these guys stuff. As a class we discussed our way through the story that follows and learned considerably more because his question led us to it.

The Story

Let’s start with a set of steps and a toddler. The toddler is about to climb a set of steps but is limited to two ways; either 1 step at a time or 2 steps at a time, or a combination of 1-step and 2-step climbing.

The question is how many different ways the toddler might climb a fixed number of steps. What this means, for example, is … suppose there are only 2 steps. The toddler could climb 1 step at a time, which can be represented as 1.1, for a total of 2. The toddler could also simply climb the 2 steps once.

Continuing this, if the toddler were to climb 3 or 4 or 5 etc., the number of ways to climb can be determined by simply mapping the possibilities and counting them. To clarify the entries in the “ways-to-climb” column in the table below, take a look at the ways-to-climb for 3 steps. “1.1.1” means that the baby climbed them 1 at a time. “1.2” means that the baby climbed one step followed by a 2-step climb and “2.1” means the baby first climbed 2 at a time, followed by climbing one. These are the only 3 ways that the baby could climb 3 steps.

# steps           Ways to climb                                                                                                       Total

1                        1                                                                                                                        1

2                        1.1 2                                                                                                                   2

3                        1.1.1     1. 2     2.1                                                                                              3

4                  1.1.2   1.2.1   2.1.1   2.2                                                                         5

5                 2.2.1    2.1.2    1.2.2                         8

6                 2.2.2                                                                      13

An interesting and familiar pattern shows up in the “Total” column. It’s the Fibonacci series. But the question is “why’? This question can be answered by looking first at the relationship between Pascal’s triangle and the Fibonacci series.

Pascal’s triangle is a series of rows showing the coefficients of the terms that result from the expansion of the binomial

(x + y)n using n = 0 through 6. which will generate 7 rows. Pascal’s triangle, left aligned, is:


1    1

1     2     1

1     3     3     1

1     4     6     4       1

1     5     10   10     5     1

1     6       15   20   15   6     1

The Fibonacci series is found in the triangle by starting at a “1” in the left-hand column, adding the numbers as you move diagonally up left to right at about 45 degrees. For example;

In the fifth row, moving diagonally up, there is a 1, 3, 1, totaling 5.

In the sixth row, moving diagonally up, there is a 1, 4, 3, totaling 8.

In the seventh row, moving diagonally up, there is a 1, 5, 6, 1, totaling 13.

These are a sequence in the Fibonacci series.

But the pattern goes even deeper. Pascal’s triangle only shows the coefficients of the terms in the expansion. For example, the expansion of ( x + y )3 is   x3 + 3x2y + 3xy2 + y3, which are the coefficients in the fourth row of Pascal’s triangle. What the baby climbing the steps does is show the pattern, somewhat hidden, of the expansion and further, shows how the Fibonacci series can be seen in the triangle.

Let’s begin with the number of steps = 2. In this case, and in all subsequent cases, let 1 step = x and 2 steps = y. For 2 steps then the outcomes in the ways-to-climb column would be x.x and y, not 1.1 and 2. Here we will use exponential notation for x.x giving x2 and this will be done for all subsequent cases.

So, for 2 steps that has 2 outcomes, they are x2 and y. Now look at Pascal’s triangle and the Fibonacci series that can be generated from it.

An expansion of ( x + y )1 gives           x + y

An expansion of ( x + y )2   gives        x2   +   2xy +   y2

An expansion of ( x + y )3 gives          x3 +  3x2y +  3xy2 +  y3

An expansion of ( x + y )4 gives          x4 +   4x3y + 6x2y2 + 4xy3 +  y4

The expansions have been “spaced” so that you can see that if you add the coefficients of x2 in the second expansion and the coefficient of y in the first expansion you get 2. If you add the coefficients x3 and 2xy you get 3. If you add the coefficients of x4, 3x2y and y2 you get 5 and 2, 3, 5 are a sequence in the Fibonacci series.

Below in rows 1 and 2 is an example showing the generation of the relationship between “ways to climb” when the toddler climbing 5 steps. Row two is the first term of (x + y)5 , the second term of (x + y)4 and the third term of (x + y)3. This is the same as what you get moving diagonally up in Pascal’s triangle.

Row 1. Ways to climb                2.2.1   2.1.2    1.2.2

Row 2. xy form                   x5                                         4x3y                                                     3xy2

If you add the coefficients of the terms in row 2 above you get 8, which is the same as finding the Fibonacci series from Pascal’s triangle beginning in the sixth row of the above Pascal’s triangle. What’s more, row 2 demonstrates the relationship between the coefficients in Pascal’s triangle, the number of ways the toddler can climb steps, the terms in a binomial expansion and finally, the Fibonacci series. You can see that the entries in row 1- each term having 1s and 2s – are, in essence, the exponents of x and y in row 2, where 1 step = x and 2 steps = y. At the beginning of this piece, I noted that you could find the number of ways by mapping it, but given climbing steps is connected to the binomial expansion, use the binomial expansion. For example, if there were 10 steps, in how many ways could the toddler climb these 10 steps?

Here’s how get the number of steps; generate a series of combinations using nCr because that’s how you can get the coefficients of the terms in a binomial expansion. For example given x2y2, 2C2 gives 6. But it’s not a single calculation.

Knowing that the Fibonacci series is the “Total” outcome as seen in Table 1 and also that the Fibonacci series is the outcome by moving diagonally up at 450 in Pascal’s triangle, the nCr calculation is a series of calculations using the following pattern:

Start with 10C0 and generate the series by decreasing “n” by 1 and increasing “r” by 1 each time. In this case, the next two terms would be 9C1 followed by 8C2, etc..

The question then is “how do you know when to stop”? At some point in generating this series, r will be greater than n and since this is illegal (your calculator will tell you!), you’re done. Just add up the values that have been generated. Given 10 steps, there are 89 different ways for the toddler to climb them. This may seem awkward, but as I indicated earlier, you could map the outcomes but in this case and with larger numbers it would be very time consuming and very trying to map it. Stick with nCr.

As I noted in the introduction we learned a lot as we worked through this. More importantly though, in my opinion, was the engagement of the class in doing the work and seeing all the connections. It didn’t exactly follow the curriculum, but it did follow a student’s imaginative question.

You can leave a comment below if you’d like to … it would be appreciated.










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