“Eric, this may be useful for a lesson in self-advocacy if not a word problem in math. I had two metrocards, and both, super annoyingly, had $2.45 remaining on them. (How and why is that?!) No time to dither at the ticket slot machine, I asked the attendant in the booth if she could combine the fares onto one card, and to my great surprise, she did it. I thanked her about three times.

So I now had $4.90 on one card and zero on the other, and passed the gate.

I am not sure what this means for the leftover on the one card with a balance, and how to possibly make it pay down the line, but at least I’m on that train.”

http://devlinsangle.blogspot.com/2017/11/mathematics-and-supreme-court.html

]]>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?

]]>For example, can 26 be written as the sum of 4 consecutive numbers?

To use Solange’s method, we would add 2 to 26 and then divide by 4, which gives us 7. Because it goes in evenly, it passes the test and so we know 26 can in fact be written as the sum of 4 consecutive numbers. But what is that 7? Going further, the consecutive numbers that add up to 26 are 5, 6, 7, and 8. Solange’s method gave us the third number in the sequence. We did the same thing with 46 and again, the result was the 3rd consecutive number in the sequence – 10, 11, 12, 13.

Our first conjecture was that Solange’s method would give us the number in the sequence that was one less than the number of consecutive numbers we were looking for. While testing if numbers could be written as the sum of 4 consecutive numbers, the number that came out of Solange’s expression was the 3rd. So we thought maybe if we were looking for numbers that can be written as the sum of six consecutive numbers, maybe her method would give us the 5th number. So we tried it out with 27. 27 plus 3 divided by 6 is 5, which is the fourth number in the sequence (2+3+4+5+6+7 = 27). We tried again with 39, and got 7, which is the fourth consecutive number in the sequence (4+5+6+7+8+9=39.)

So what is the pattern? Will Solange’s expression always give us one of the consecutive numbers? And which number in the sequence will it give us?

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