Guess what! Another problem solving group -- http://www.mathlesstraveled.com/ -- also showcased the Number Bracelet problem, and they have added a cool visual showing all possible bracelets in all mods through 12: http://www.mathlesstraveled.com/?p=520. This also leads to a couple of suggestions for presentation software.

mastermind

My game suggestion from last class was Mastermind, and the interactive version available from NLVM: http://nlvm.usu.edu/en/nav/frames_asid_179_g_2_t_1.html?open=instructions. We didn't have much time to consider it, but perhaps a next step is to consider a book, MASTERMIND® Mathematics Logic, Strategies, and Proofs that teases out the higher level thinking skills.

Tootsie Roll Game (“Iterated Sharing” from Solve This)

A group of student sits in a circle, each with a pile of wrapped candies. (Wrapped candy is used because each piece will be handled by many people before being eaten.) Some people have 20 or more, others none, and the rest some number in between. The distribution is arbitrary except for the fact that everyone has been given an even number of pieces. A reserve supply is set aside.
The group now follows these instructions: Give half of your candy to the person on your left and half to the person on your right (and hence receive a supply of candy from the people on your left and right). Do this simultaneously. Now recount your candy supply. If you now have an odd number of pieces, take an extra piece of candy from the reserve supply. This boosts your pile back up to an even number of pieces and everyone is ready to perform the maneuver again.
What happens to the distribution of candy among the group if this maneuver is performed over and over again? Will people be forever taking extra pieces from the center, so everyone’s amount of candy will grow without bound? Or will the distribution stabilize or equalize in some sense? Will one person end up with all the candy? Might “clumps” of candy move around the circle with each iteration or some strange oscillatory pattern emerge? Is it possible to predict what the result will be?
Taking it further: What happens if instead of adding pieces, you eat any odd piece of candy to bring your pile back down to an even number?
Hint: Based on the investigations my class performed, and the patterns they found, I encourage you to examine what the range of distributions is. For example, if you started with 0 – 18 pieces (a range of 18), what happens? And is it different if you started with 0-20 pieces (a range of 20)?
My classes enjoyed this activity, though my advanced class looked into in much further than my lower level class. They tried several rounds to test and prove conjectures. It was a difficult game to play with a class larger than 12 students. I tried putting them into two groups, but this was difficult since I was facilitating both groups. It was also difficult to play with desks in a circle. Playing on the floor or at a large table would have made things easier.

Math Circle Documents

I added a Math Circle Documents Link ....if you have documents that you would like to share - send them to me and I'll post them.

-Kendall

Bertine started us with teaching strategies for half-life. Steve & Kendall suggest a penny shake: shake, remove heads, shake again, remove heads, etc.

Lynne gave a presentation about the Teacher Circles to the UW Math Club on Friday, Sept 25. The following day at an event for families of prospective students, the Math Club used the ropes and Conway’s Tangles. Kids said, “cool”.

Many had tried the human knot. Bertine found that his mixed gender group was reluctant, so he went to single-gender tangles. Lynne tells of using this same Tangle for UW freshman during Summit 2009, having talked about touching BEFORE doing the tangle. Dave's 5th graders held onto two opposite ends of a shoestring, not hands.

Lynne tells a little of a talk at UW’s Center for Learning, e-volution with Gardner Campbell, "Narrate, Curate, Share: Integrative Learning and Web 2.0".

Steve is shared his “lock boxes” teaching strategy. (Steve has 56 physical lock boxes, and about 130 questions organized by the answers.) He tells his 7th graders that in a problem solving situation, rather than looking at looking for a rule, that they should look all around to figure things out. Steve is working with kids who have difficulty with even basic concepts and words can go out, discover relationships, etc. His example is to find a number that is 6 less than the product of the 2nd and 6th primes. Kids don’t immediately start asking questions --- learned helplessness of 'I don’t get it' --- now they read, investigate, etc. Mary adds that last year with her SPED kids would ask to do a lock box, and helped each other.

Ariane did number bracelets with all of her HS classes. Her point was that the problem of number bracelets engaged high performing and low performing students. One class was having a test under a sub, and the class wanted permission to make number bracelets after their test. A young man said that there were only 100 combos, so he would go home over the weekend, work out all 100 and let the class know what happens.

Lynne posted an NCTM site where they claim they “combine understanding with skill”. We had a small attendance (6), but the conversations were animated as we all face the challenge of implementing richer problems.

Kendall posed "Math Magic": pick 2 positive integers in a vertical fashion.

456

987

1443

2430

3874

Then add the sum to the 2nd addend to get a 4th number. Now add the 3rd and 4th to get a 5th, keep going until you have 10 numbers. How can you quickly get the sum of all 10? Trick: 11 times 7th number = sum of all 10. Kendall’s problem is a challenge for his 13-year-old son Benjamin. Kendall also asks for help to decide favorites from Jackson to take to ND where he’ll present about TCs, and for suggests about other patterns that are like this, or ways to extend this. Maybe there is a similar pattern with 3 numbers to start.

Lynne & Kendall have a brief exchange about how people, even people who don’t think of selves as good in math, are drawn to this sort of problem.

Steve has us view a YouTube clip of Dr Who where we need to enter a Happy Prime:. Had his students exploring digit squares and looking for happy primes and happy numbers. Square the digits of a number, sum and then repeat. If this process reduces to 1, the original number is a happy number. Steve likes this for the training it gives for the 2-step process of squaring and adding. Examples of happy numbers are 13 and 19. Note that permutations of the digits will give another happy number, e.g. 91 or 31.

13 -> 1^2+3^2=1+9=10 -> 1^2+0^2=1

19-> 1^2+9^2=1+81=82 ->8^2+2^2=68 -> 6^2+8^2=36+64=100 ->1

Ariane asks for more information about Kendall’s pattern, and Lynne suggests understanding the pattern using letters instead of specific numbers. Combine with multiplication, and it all works equally well.

Steve relates that there are things we know about finding happy numbers. For example, every number along the path of converging to a happy number is in turn a happy number. For example, in expanding 19 we generated 82, 68 and 100.

Lynne suggests we return to our rubric for judging problems. Steve suggests that last two problems are not relevant to general population, but that it would be good for grades 4-5. Seem to rank well on all other points. Steve suggests that perhaps #8 is not perfect, and maybe we need a point about engagement.

Ariane suggests that there is a challenge in defending problems that don’t directly relate to her standards. Struggle will be to align and then get back to the book = boring things saying that the other part can be fun too. Steve agrees about the importance of some routines.

Bead Bracelets with fifth graders.

Taking an idea from Ariane, I modified the activity to multiplying beads rather than adding beads. I did this simply to relate to what we are doing in class. So most of the class was working with multiplication. However, I did have a group that did the addition as written up. The questions were posted and I added a few questions:

What patterns do you notice?

If you used only odd (or only even numbers) could you make a prediction about the ones digit in the next answer?

If you used odd and even numbers could you make a prediction about the ones digit in the next answer?

All of the students enjoyed the activity. The higher students, who tend to be bored during regular math instruction and need to be challenged, enjoyed looking for patterns and relationships. The students who tend to struggle enjoyed that they had an entry point.

The answers to the questions were not as thoughtful and deep as one would hope. That was to be expected due to the age of the students and the lack of exposure of problem solving and justifying or proving their reasoning.

I also did a quick poll as to whether the students would like to do this activity again, never again, or if I made them. The class was pretty much divided.

Bead Activity

My students used the bead activity as an introduction to patterns/algorithms;i.e. order of operations. The students so enjoyed the activity. I had them write down their two numbers on a sheet of paper and do the algorithm to find when the two numbers repeat. After completing the pencil/paper task the students actually made their necklace/bracelet from beads and pipe cleaners. The task became quite competitive and students worked diligently!

It gave me chill bumps to see the interaction and chat regarding algorithms.

This also led into the "Math in Action" photo for the new concept. The students were to take a picture of an algorithm and use pencil/paper to describe the algorithm. . . rec'd some real unique ideas!

Math Bracelets

Wow! My students were hooked on the math bracelet activity! They wanted to try multiplying the numbers after they were successful with adding the numbers. Two students now have a contest going to see who can make the longest bracelet. The numbers were great to set the pattern and practice some basic computation. My enrichment class expanded the activity by creating a color code for the beads. One student suggested using to beads when we get to double digit answers. I was able to get some digital photos of the colored bracelets. Hopefully, I will be able to share them next time we meet.
One of my goals was to share this activity with our special education teacher. I was also able to share it with my team. They thought it was great.

Divide and Conquer

Divide and Conquer is a game played by subtracting proper divisors from the initial number. One is always a proper divisor and no number can be divided by itself. The game is won when a player presents his/her opponent with one.
As we worked with 100, which has several proper divisors, we tried strategies that included making a list and drawing a diagram on graph paper. In the list we found a proper divisor of 100, and then subtracted the divisor from 100. We repeated the process with each new difference until one player could present his opponent with one. Similarly, we drew a 10 x 10 grid on graph paper and took turns crossing off proper divisors until one square remained.
We discussed that many middle school students will learn how to play this game with the initial number of 30, They will memorized the specific steps for 30 and attempt to generalize those exact steps to other numbers. The conclusion was that in order for students to truly demonstrate their understanding of factors in this game, they need to play it with other numbers.
I shared this game with my summer school students. They were thrilled to be able to beat the teacher.

Finding the last digits of 7 to the 9999th power

Now here is a question. Suppose you took 7 to the 9999^th power. That would be a number with 8450 digits. In 12 point type, that is a number about 70 feet long. What are the last three digits of that number? Why would we ask such a question? No reason, that’s just what we do.

One problem solving strategy is to create an organized list and look for patterns. Using Excel, here are the first 10 powers of 7 (formulas are listed in column C).

	A	B
1	7	7
2	49	=A1*7
3	343	=A2*7
4	2401	=A3*7
5	16807	=A4*7
6	117649	=A5*7
7	823543	=A6*7
8	5764801	=A7*7
9	40353607	=A8*7
10	282475249	=A9*7

No obvious patterns seem to jump out and the numbers are already getting pretty big. Is there a better way? Asking for the last three digits is the same as asking for the remainder when you divide by 1000. Excel has a modulo function we can use to find the last three digits of the powers of 7. We only need to multiply these values by 7 to find the last the last three digits of the next power of 7 and the mod function returns only the last three digits.

As a shot in the dark, what is the value of the first 25 values for the powers of 7 mod 1000? It would be very cool is some relatively small power of 7 ended in 001 (or written in mod notation: 7^x mod 1000 ≡ 1) since powers of that number would also end in 001.

A	B	C
1	7
2	49	=MOD(B1*7,1000)
3	343	=MOD(B2*7,1000)
4	401	=MOD(B3*7,1000)
5	807	=MOD(B4*7,1000)
6	649	=MOD(B5*7,1000)
7	543	=MOD(B6*7,1000)
8	801	=MOD(B7*7,1000)
9	607	=MOD(B8*7,1000)
10	249	=MOD(B9*7,1000)
11	743	=MOD(B10*7,1000)
12	201	=MOD(B11*7,1000)
13	407	=MOD(B12*7,1000)
14	849	=MOD(B13*7,1000)
15	943	=MOD(B14*7,1000)
16	601	=MOD(B15*7,1000)
17	207	=MOD(B16*7,1000)
18	449	=MOD(B17*7,1000)
19	143	=MOD(B18*7,1000)
20	1	=MOD(B19*7,1000)
21	7	=MOD(B20*7,1000)
22	49	=MOD(B21*7,1000)
23	343	=MOD(B22*7,1000)
24	401	=MOD(B23*7,1000)
25	807	=MOD(B24*7,1000)

Holy Math Smack! 7²⁰ ends in 001 or 7²⁰ ≡ 1 mod 1000, then the pattern begins to repeat. That means that 7²⁰ ≡ (7²⁰ )ⁿ ≡ 1 mod 1000. So, think of 7⁹⁹⁹⁹ as 7²⁰ x7²⁰ x7²⁰…7¹⁹ . so 7⁹⁹⁹⁹ ends on the 19^th step of a repeating 20 step cycle. That value, according to the Excel sheet, is 143. QED. (that is Latin abbreviation you put at the end of a proof. I think it means “Thank God its over” but you should google it to be sure).

So even though 7⁹⁹⁹⁹ is impossibly large on the scale of 8^th grade math, we can make some statements about it, such as the last three digits have to be 143. Now, can you find the first three digits?

Clock Arithmetic

The problem above makes use of a repeating pattern in the last three digits of the powers of 7.

Repeating patterns such at this come up often enough to be interesting. Here is a simple case that would be accessible to 4^th or 4^th graders.

My friend, Mary O’Dell works in a candy shop. Part of her job is to bag the candy when a fresh batch comes out of the kitchen. As one of the benefits of working at the shop, she gets to keep any candies left over after bagging – that is, if there are some candies left but not enough to fill a whole bag, she gets to keep them. Today she is bagging candies 9 to a bag and the cooks made 665 in this batch. How many candies does Mary get to keep? What are some numbers that would leave Mary several candies? What are some numbers that would leave her with no leftovers at all?

Clock arithmetic is a common example. If it is 8:00 am now, what time will it be 133 hours later?

A mathematician would ask, “what are the patterns in these kinds of problems?” The first thing we’d need is a notation – a way of writing these ideas down. In this case us a mod notation. If we want to look at Mary O’Dell bagging 9 candies per bag then we can start with the number of candies which was 665 in the example above, then designate the number is a bag (or the number in any group) by using the notation “mod 9”. The notation would be 665 mod 9. We use a congruence notation rather than an equal sign because we are trying to say these are numbers with the same remainder when divided by 9 and these numbers are not equal, their remainder is when they are divided by 9. We use the notation ≡ and we read it as “is congruent to” so we would write 665 mod 9 ≡ 8 and we read it as “666 mod 9 is congruent to 8”