A Prequel on Divisibility as Inquiry

Reaping a Landslide of Student Excitement

By Fred Harwood when I was Mathematics/IT Coordinator Hugh McRoberts Secondary School in Richmond, BC and students/Math Club members:Mark Funston, Suzanne Lee, Philip Gao, Steve Welsh, Thomas Mickie & Ekaterina Daviel

Introduction:

Many exciting student discoveries have transformed my classes into learning communities. These past discoveries provide a valuable set of “anchor papers” or standards that current students can strive to surpass. This search for meaning and for extending the science of patterns, create a vibrant sense of the dynamics of mathematics that motivate students (and teachers!) to strive for understanding and excellence as mathematical researchers.

This fall (2004) has been my most exciting of the past fifteen years with Mark Funston’s breakthrough unifying theory of factor finding creating connections to other uses at a tremendous pace.

This discovery-based learning is an invigorating process that I highly recommend for teachers. I’m employing this approach in my regular Mathematics 9 classes and in my accelerated and enriched classes in grades 9, 10, and 11. I’ve also employed it with grade 8 classes in the past and know of elementary teachers doing work like this as well.

This article will deal with the Funston Factor Finder written up my my Math 11 Enriched student, Mark Funston. Suzanne Lee interviewed him on the creative process and then Philip Gao and I extended it and applied it to new processes. Two weeks later, the math club was working on a separate concept and we saw the application of Mark’s theory’s underlying concept to exponentially increase our understanding of fraction conversions.

Setting:

I had given the students in my Enriched Math 11 Principles class an overview of divisibility tests as a mathematical tool. I taught it to them in an organized big picture approach.

Divisibility Tests

Test for divisibility by 10 (and 2 & 5 since they are factors of 10)

Last digit is divisible by 10 ( or 2 or 5)

 Test for any power of 10 (or power of 2 or 5)       

Check the same number of last digits as the power for divisibility by the number

Eg. Test for 8: check last 3 digits for ÷ by 8 because 8 = 23.

 Test for 9 note: it is almost our base value of 10 (and 3 since it is a factor of 9)      

       Add up the digits to see if divisible by 9 (or by 3)

 Test for 11 note: it is almost our base value of 10 (one of many tests but this one is similar to 9’s)

       Alternate the digits + – + – + – + etc until you are out of digits

                   and then add up the digits to see if ÷ by 11. Remember that 0 can be ÷ by 11.

Test for 7 (a student discovery that works for 3 digit and larger numbers)

       Starting on the right, break the number up into groups of 2 digits, the first digit is normal (x1), the second group is doubled (x2), the third group is x4, 4th is x8 etc until you are out of groups. Add up the products of the groups to see if divisible by 7. If the total is still too large to tell, repeat the process until it is a two digit total. (Note: 91 & 98 are ÷ by 7)

       Ex: 1572354: breaks up into 1 57 23 54: The total = 8×1 + 57×4 + 23×2 + 54×1 = 8+228+46+54 = 336 and then 336 breaks into 3   36 so 3×2+36×1 = 42 which is ÷ by 7 so 1572354 is also ÷ by 7. [Note: this is especially useful for 3 & 4 digit numbers.]

Test for composites: break into mutually exclusive factors (they must not have a factor in common other than 1)

6 = 2 x 3 so the tests for 2 and 3 must work. Eg. 2 must ÷ last digit & digits add up to ÷ by 3

12 = 3 x 4 so tests for 3 and 4 must work. Eg. 4 must ÷ last two digit s & digits add up to ÷ by 3

15 = 3 x 5 so the tests for 5 and 3 must work. Eg. 5 must ÷ last digit & digits add up to ÷ by 3

18 = 2 x 9 (not 3 x 6)                                 etc.

Some of the pieces were not in place by the end of Thursday’s period so I assigned the students to discover their own tests for 13 and for 17. Half the class was working on 13 and the rest on 17. On Monday, Mark Funston greeted me dancing and with a huge smile and a couple of pages of much more than a test for 13. The class itself had 5 of their own discoveries but Mark’s dwarfed them all and, in fact, generated many of the students other discoveries. I originally called it the Fun Key Funston but Mark has renamed it the “Funston Factor Finder”.

Here is Mark’s writeup

The Funston Factor Finder by Mark Funston

            Ever since the beginning of civilized math, we humans have tried to find little tricks to help us solve problems more quickly and efficiently. One of the major problems that we have is finding whether a number is a factor of another one, like whether 7 is a factor of 140. We have developed methods that work for different numbers, but there are so many that it’s hard to remember them all. What if there is a universal method for finding the factors of numbers?

            There is one that works for all numbers and it is called the “Funston Factor Finder” or “FCubed” for short. Let’s use the example of finding whether 7 is a factor of 140. You will be able to execute this method in 3 easy steps.

  1. Find the multiple of 7 that is closest to 100 without surpassing it

(7 x 14 = 98). Then find the difference between 100 and that number (100 – 98 = 2 = d).

  1. Separate 140 into groups of 2 digits starting from the right and continuing to the left.

II     I

1 | 40

The numbers on top are the group numbers. This will become more important with more digits.

  1. Multiply the number left of the first separator (1) by 2 (the difference between 98 and 100) and add it to the numbers on the right of the first separator.

1 x 2 + 40 = 42

Notice that 42 is a multiple of 7. This tells us that our number is also a multiple of 7. We’re finished this simple example, but this method works for all numbers so let’s try it again.

Once the number in question surpasses 4 digits (greater than 10,000), there will be more groups of 2 digits. Let’s try another example:

 Eg. Is 12 a factor of 10404?

  1. Closest multiple of 12 to 100 without surpassing it.

12 x 8 = 96 (difference from 100 is 4)

  1. Separation

III   II     I

   1 | 04 | 04

  1. We multiply group II by 4 and group III by 42 because it is in a different group. (If the number got big enough, group IV would be multiplied by 43 and so on for each of the consecutive groups. Note that Group I is multiplied by 40or 1)

(1 x 42) + (4 x 4) + 4

1(16) + 4(4) + 4

= 36 (which is a multiple of 12)

Again, this works for ALL numbers.

However complicated this method may look at first, it is really quite simple and efficient. I believe it will revolutionize the way we think about factors and it may replace the current methods of determining whether one number is the factor of another. Try it out, it really does work!

————&&&&————–&&&&————–&&&&—————-

Extensions:

I demonstrated this method to the class for 7 and it generated the “Attridge Attributed divisibility test” I’ve included in the Workshop materials. Allan Attridge was a Grade 9 whose original discovery fifteen years ago excited me about divisibility test discoveries. We then tried it for 13 and 17 and several of the students’ own discoveries were generated. A test for 11 could be that it goes into 100 nine times with a remainder of 1. So we can check for 11 by adding up pairs of digits starting from the right and checking the sum for divisibility by 11 (or by repeating the process until it is small enough to tell.) For example, 1286472 can be split into 1+28+64+72 = 165 and 165 can be split into 1+65 = 66, a multiple of 11 so 1286472 is a multiple of 11 as well. Try it on other numbers.

I demonstrated it to some of the math club members at lunch that day and Philip Gao and I thought that it could work for a factor of 10 as well. Take a number like 7. How many times does it go into 10 without going over? It goes once with 3 left over. The Funston Factor Finder extension would then say that the units digit of a two digit number would be multiplied by 30 or 1 and the ten’s digit by 31. Think of the multiples of 7 that we have stared at for years. 14 à 1×3+4=7; 21à 2×3+1=7; 28 à 2×3+8=14 and we could repeat the process to 14 to get 7. How could such an obvious pattern escape me for so long? Now the practicality of this test is not high since we’ve learned our multiplication table for 7 already but look at some of the other single digit numbers. Trying 9 (or 3) we see it goes into 10 once (or three times) with a remainder of 1. This means that each digit of a number gets multiplied by a power of one thus generating the presently practiced “add up the digits to see if the sum is divisible by 9”! Applying the theory for 10 to a factor of 10 yields the test of checking the last digit for divisibility by 2 or 5 if we loosely claim that 00 is 1 then only the last digit counts as the rest are multiplied by a power of 0.

We tried the test with the concept of 1000. For example, 11 divides into 1000 ninety times with 10 as a remainder. Progressive groups of three digits would be multiplied by a power of 10 before summing. 1286472 would split into 1×102 + 286×101 + 472×100 =100+2860+472 = 3432 which can be split into 3×10 + 432 = 462. Then if 462 is a multiple of 11 then 1286472 is too.

In writing this last paragraph I thought to try the concept of 11 into 10 with a remainder of (-1). This would generate the test of “alternating digits plus, minus, plus, minus etc and then adding to see if 11 is a factor of the sum.”

Obviously, there is no limit to the applications for factor testing but some will be less helpful than others. The important thing is that they do exist and this method can generate a wide variety of relationships between the digits of a number.

 

Crossover Extension:

Our math club met two weeks later and I mentioned again the excitement generated by Mark’s discovery and how a magazine article was being written. I thought they might discover something to form the foundation of their own fame by developing some new patterns for the conversion of fractions to repeating decimals. I had shown Steve, Thomas and Ekaterina the conversion of one seventh to a decimal by doubling 7 to 14, 28, 56, 112, 224, 448, 896, 1792 etc. By placing these into two digit registers with carryovers going into previous registers, the sum of the columns generate the six-digit repeating pattern for 1/7 as 0.142857 repeating.

These are examples of where inquiry, discovery-based learning, thinking classroom communities make a tremendous difference in students learning, being engaged and motivated to further investigations and to generate new ideas.

 

 

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