Two things I want to share tonight. One of them has multiple parts.

One of my international students shared a lovely gift with me yesterday. It’s a food treat that her mom sent here for her to share. I have a few food allergies so I was concerned but did not want to tell her because it felt rude. Luckily, there are a number of boys in my from who can translate the ingredient list for me. Pretty cool. Oh yeah, I’m allowed to eat it – no nuts.

I blogged a couple of days ago about a problem on my calc BC final. Here is the problem

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s

Alice in Wonderland:

“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”

Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen. Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty.

Twenty-four students took this final and a number of them really did a wonderful job in explaining their reasoning. I’m going to present a handful of the best responses here.

- 4 x 5 = 12 is number base 18. 4 x 6 = 13 is number base 21. 4 x 7 = she will say 14 in number base 24. 4 x 12 = 19 number base 39, 4 x 13 = 20? number base 42. If following the pattern, as one of the numbers remains constant 4, another increasing by 1 each time, we get the product increasing by 1 each also. This is possible for number base 18 , increasing 3 each time. This 4 x 13 = 20 in number base 42 accordingly. However, 4 x 13 in number base 42 is one full round with another 10, which in base 42 as there exists another symbol for that suppose A, thus 4 x 13 = 1A and will never equal to 20 this way.
- She is using number base to calculate it. Every time 4 times initial number plus 1 and number base that is increased by 3. It will never get to twenty because the number base is always growing as number grows. Number base is growing faster than the number we multiply by. (Note – this one was accompanied by meaningful, but scribbly, calculations)
- (This answer starts with all the calculations hinted at in the first answer i presented) Because the base in continually increasing by 3 and the answer is only increasing by one, the answer will never be able to get out of the ones digit.
- The base is increasing and in order to get to 20 the result of the calculation must be EXACTLY twice as big as the base., which is not possible.

All of these were accompanied by calculations on the side. We spent about two and a half days talking about number bases and I must admit I was really impressed by the patience that my students had with this problem. Nice way to end the year!