I puzzled over what to bring to today’s meeting for days. I have a couple unfinished problems that I’ve been thinking to bring to a CAMI meeting, but in the end I chose to go with a few activities on factors, mostly from __Fostering Algebraic Thinking__, by Mark Driscoll. A group of us read the book last summer and loved the problems. There were so many good ones that we weren’t able to solve them all while reading the book. I went into this meeting hoping that the surprise of the central problem wouldn’t be ruined.

Here’s a framing question I posed: *Which of these puzzles and problems do you think could work (with modification possibly) in an adult education math class?*

I started with a concrete area activity inspired by Connie Rivera and Amy Vickers’ webinar on area models. I gave out 36 tiles to each table and asked the following question:

We came up with the following rectangles along with a question about whether orientation matters when deciding if a rectangle is unique. For example, is 3 x 12 a different rectangle than 12 x 3?

I then asked the group to look at the following and write down what they noticed.

After a few minutes, I asked them to share at their tables and then as a group.

The bottom left “stacked division” or “cake division” was new to everyone. The group noticed that it worked best when primes were used as the divisor.

The group noticed that the tree diagrams and cake division produced the same primes, though different non-prime factors showed up in the different methods.

Next, I introduced the area model puzzle:

- Keep this next part secret. Pick a pair of two digit whole numbers and create an area model for multiplying them. Label the four partial products but not the sides.

- Exchange your creation with a classmate and see if you each can figure out the two-digit numbers the other person multiplied to get those partial products.

We briefly considered the following questions, which might be good for classroom discussion:

- What strategies are useful for solving an area model puzzle?
- What strategies are useful for making hard area model puzzles?

Benjamin Dickman, who thought of the puzzle, shared some extensions on Twitter:

& find all sets of part’l prods with 3 two digit solutions (the max). Solved 1st by @MathFireworks! See Fig 5 in https://t.co/vgXwWZCPVd 2/2

— Benjamin (@benjamindickman) October 14, 2017

We then moved on to the main problem I brought:

*Consider the operation of counting the factors of a whole number (including 1 and the number itself). You might think of this as a function that counts factors. For example, the number 6 has the factors 1, 2, 3, and 6. If 6 is the input, 4 is the output. The function *d* of 6 might be written as *d*(6) = 4.*

I asked the group to start by thinking of questions and/or conjectures individually, then asked them to share at their tables. Here are some of the early conjectures and questions:

- Primes always have two factors.
- Squares have an odd number of factors.
- Which numbers have the most factors? What are the properties of these numbers?
- Is there a function? What is it?
- Can we use combinations or permutations to find the number of factors from a prime factor tree?
- Which numbers have the most factors? Multiples of 12 have lots of factors.
- What does the graph of
*d*look like? What patterns do you see in the graph? - What is
*d*(192)? Is it 12?

Tables worked on different questions for about 30 minutes, then I started to share the following table from a handout I prepared (available for download at the link above). My hope was that the table would point teachers in the right direction to derive the function, but wouldn’t tell them what it was.

*Complete the prime factorization of at least 4 integers. Add the exponents of the prime factors to the table below. (Add more columns if necessary.) What do you notice?*

integer | prime factorization |
exponent of 1st prime factor |
exponent of 2nd prime factor (if there is one) |
exponent of 3rd prime factor (if there is one) |
number of factors |

36 |
2^{2}3^{2} |
2 |
2 |
9 |

Here are few entries from Mark’s table, with additions from the group:

integer |
prime factorization |
exponent of 1st prime factor |
exponent of 2nd prime factor (if there is one) |
exponent of 3rd prime factor (if there is one) |
number of factors |

36 |
2^{2}3^{2} |
2 |
2 |
– | 9 |

24 | 233 |
3 | 1 | – | 8 |

18 | 3^{2}2 |
2 | 1 | – | 6 |

12 | 2^{2}3 |
2 | 1 | – | 6 |

We noticed that 18 and 12 both have 6 factors. They also both have 2 and 1 as exponents in their prime factorization: *3 ^{2}*

*2*and

*2*Mark explained that he could tell us another number with 6 factors by using the prime factor exponents in the table for integers 18 and 12. His next step was to use those exponents with his example, 147 (7

^{2}3.

^{2}3)*.*

One big outstanding question: What is rule for function d? Can it be derived from the table?

In attendance: Ramon, Mauricio, Linda, Maggie, Solange, Maritza, Jeremy, Mark, Lionel, Eric

Programs represented: BMCC’s Adult Basic Education Program, NYC College of Technology’s Adult Learning Center, LaGuardia Community College’s Adult Basic Skills Program and CCPI, York College’s Learning Center, Literacy Partners, and the CUNY Adult Literacy/HSE PD team

After thinking more about the d function which counts the number of factors of whole number, I had an additional thought to share.

Namely, what if you looked specifically at numbers whose prime factorization only makes use of a single prime number? For example, if a student is strategic with the values they use and compares a list of the prime factorizations and amount of factors of the powers of 2 to a list of the prime factorizations and amount of factors of the powers of 3, it might help them to gain some additional insights into the problem.

After doing this you could go on to looking specifically at numbers that make use of the same two prime factors, same three prime factors, same four, etc.

These next few thoughts I had were only tangentially related to finding the d function.

In thinking about strategically choosing numbers to factor I thought about factoring pairs of numbers like; -25 and 25 or -36 and 36 or -45 and 45 (whole numbers and their opposites). Remembering that you can’t find a prime factorization for negative values, I still forged ahead with this idea, hoping to gain some undiscovered key to how the d function works. Along the way I found myself wondering a couple of new and interesting questions; why don’t negative values have a prime factorization?

And my new favorite…

Why isn’t negative one prime?