## Sunday, May 29, 2016

### Memorial Day Miscellania

#### Pedagogy Riff

I see a lot of "answer seeking" as well in my kids from time to time which is rich since Math club is just an optional after school activity and clearly has no grades.  The behavior I most want to alter is a student asking "Is X the answer?" and then if you tell them that's its not correct, they immediately and blindly guess a nearby number rather than analyzing their work for errors.  My first instinct is that this is not an issue with being advanced. There's this false narrative that only being quick to find the result is advanced but there's a reverse one that advanced students are only good at calculation and somehow lack conceptual understanding or appreciation for Math. Focusing on finding the correct answer is a natural instinct and built into the type of activities Mathematics is founded around.  In fact, I believe that concrete structure is what draws many initially to Mathematics: Here's a powerful tool for discovering truths.  I still personally derive much pleasure from carefully solving something that's difficult. However, as noted in the original conversation this can easily go awry.  On reflection, I tend to often pick puzzle type problems which lend themselves to bad habits if you're not careful.  Its the end of the year so I won't probably pick up on this fully until next time but;

• I want to emphasize this more in the initial charter discussion.
• I need to remember to also talk about answer seeking on a day to day basis My standard response is always "I'm interested in your thinking not the answer per. se."
• Should I thread more open-ended exercise in to try to break down this habit?
• I think part of the antidote here is to appreciate the math itself in a problem. What I find most satisfying for this is to riff on an already completed problem.  Now I need some good examples to use in a group setting.

On a different note, I've also been thinking about this post: https://samjshah.com/2016/05/25/merblions/which had an interesting approach to experimenting with inscribed arcs using geogebra. What I like here is the sense of  free play this encourages with the geometrical constructions while heading towards an important property.

I'd love to do something similar with other geometry topics like similar triangles or scaling but I don't have access to large numbers of computers. My thought experiment for now: could we do this as a group with a single laptop or go quickly enough using graph paper to approximate the same sense of experimentation?

#### Planning

There are only 3 session left to plan. The last will definitely be a party/game day. So on my remaining idea list so far:

I've seen a fun worksheet from nrich with shapes times shapes: https://nrich.maths.org/5714 that  I'm fairly sure I'll be using as a warmup.

#### Geometry Binge

Maybe it was because of the holiday, or seeing the twitter feed for gogeometry but I ended up doing a few proofs in a row.

This one is a good example of the power of  common inscribed arcs from above. After finding all the congruent angles and similar triangles. I went with a straightforward approach of overlaying the line segments that actually worked on the first try.

This one is actually a very similar variant to the prev. one and has a similar solution as  well. First we  divide the line into 2 pieces one of which is equal to the first segment and then prove the remaining piece equals the second one. This  version has a particularly pleasing tiling in the proof. Where it divides into an equilateral triangle and parallelogram.

Like most ratio proofs this one starts from similar triangle ratios. After identifying all 3 in the picture it was then a process of working backwards to get the desired ratio to look like one of 2 sides of one of the triangles.

This was a quick extension on the previous problem. I saw a version using only area addition (no similar triangles)  from @five_triangles later on. Basically you extend the picture to be a full rectangle and since the the 2 triangle halves are the same area and the sub triangle also are, the remaining two little rectangles (s^2 - bs and ab - bs) must be equal.

## Thursday, May 26, 2016

### 5/26 Magic Squares

My new numberphile t-shirt from the latest video arrived in the mail last week.

I decided this was a sign to do another magic square themed Math club properly dressed in the new shirt:

#### Preliminaries

P.O.T.W participation went up quite a bit this week. I'm pleased that everyone applied the Pythagorean theorem really well. In a related note, the kids were very excited to  report a Pythagorean error in one of their worksheets from class. This involved  a problem finding the volume of  a triangular prism where the values for the triangle face's side lengths were not consistent.  I'm glad to see this really stuck with them and that they tried it out in other contexts but it did necessitate a short "sometimes mistakes happen even in printed textbooks" speech.

I went with another battleship puzzle this week. I didn't check it as closely and the kids finished a bit quicker than I wanted. I'll either up the difficulty next time or more likely move onto to something  new.

#### Main Activities

For comparison last year went like this: http://mymathclub.blogspot.com/2015/04/428-magic-squares.html

While the warm ups were ongoing I struggled in vain with a video projector. In the end I had everyone crowd around my laptop to show the above video and the followup one since they're short.

Interestingly, some of the kids really latched onto the idea of finding a 3x3 magic square of perfect square and continued to talk about it all through the hour. Also I hadn't planned to do the following (assuming it wouldn't be interesting) but several kids also wanted to just find the the base 3x3 and 4x4 squares by themselves. So I  ended up telling them the sums (15, and 34) and that made them happy for a good period of time.

I then handed out the worksheets I had actually prepared:

Interesting this year, I learned something about internet material. There was a magic hexagon problem from the last year I wanted to reuse here but I had only stored the link. By this time it was no longer there and I ended not quite finding as good of a version as I remember. Its better to store a copy of your own of any material that looks worth reusing in the future.  This all sounds a bit slapdash in  recounting it but generally worked very well in the room. If I do this again, I think I'll  substitute in a magic square transformation type problem.

#### Problem of the Week

I'm going back with another MathCounts set:  https://www.mathcounts.org/resources/problem-of-the-week/taking-train

## Wednesday, May 18, 2016

### 5/17 Battleship

The kids surprised me twice today during Math Club. We started by going over the Math Counts problem of the week I tried out: https://www.mathcounts.org/sites/default/files/images/potw/pdf/PoTW%20021516%20Solutions.pdf

I had assumed that most of the kids wouldn't finish the four parter and I'd be lucky to get some discussions about even the first section. Instead I had kids volunteering to answer every part on the whiteboard and they even caught the leap year logic. I'll definitely have to try some more sets from here.

Secondly: I decided to talk a bit about the Math Counts final problem: See: http://mymathclub.blogspot.com/2016/05/httpq13fox.html  This was fun since everyone is in fourth and fifth grade, no one even had heard of Math Counts. So it was a good chance to broaden their horizons and we had the local connection with the winner being from Seattle. I'm hoping that a good number of  the kids will participate when they are old enough. I had expected to do a longer white board session with some breakout brainstorming. But about 30 seconds in one of the kids came up with key insight and announced the answer. There was nothing more to do but have him explain the logic to the room and write up some of the numbers on the whiteboard.  This was another high point for me for the year.

I found a new warm up logic puzzle which we tried out next: https://www.brainbashers.com/showbattleships.asp?date=0506&puzz=A&size=15   These puzzles are built around battleship and involve figuring out the ship locations given counts from the rows and columns.

Based on partial feedback these were a big hit and I'm going to bring another more difficult set next week.

Finally, I gave the kids a choice on the main activity again. I printed up another set of AMC8 questions to try out and also brought more graph paper for those who wanted to finish last week's project and make their own coordinate plane connect the dot puzzles.

Some sample products.

This session there was enough time for some to finish and I had a few that said they'd work on them and bring them back next time. On reflection the one improvement here is I need to provide more feedback for the sample AMC8 tests. I think I'm going to repackage them in groups of 5 with an answer key. What I'd really like to do is to treat them like a MOEMS test and have the group go over them together but that requires me to not run two simultaneous activities.

For the problem of the week I went with an easier  @five_triangles geometry puzzle:

Find the length of AB.

## Monday, May 16, 2016

### A quick note on the MathCounts final question

http://q13fox.com/2016/05/09/seattle-13-year-old-wins-national-math-bee/

"Competition officials said in a news release the 13-year-old won the final round by answering the question, “What is the remainder when 999,999,999 is divided by 32?”
Wan gave the correct answer of 31 in just under seven seconds."

I heard about the MathCounts results a week ago and my first reaction to this  question was "Huh doesn't seem like a very deep question, its so computation focused."  My second thought was I wonder how long just doing the long division would really take and then I immediately followed that up with  I could type this into python (or the google address bar nowadays) and find the answer very quickly.

However, I thought about it again while trying to fall asleep last night and there is actually a lovely (and accessible) piece of number theory contained within it.  This time, I started by musing  "How did he do it so quickly."  The easiest way I thought of was to build up a divisibility rule for 32.  Since 32 is 2^5 multiplying it by 5^5 produces a nice power of 10. But I was in bed so I actually went 32, 160, 800, 4000, 20000, 100000.  This means, you can ignore all the digits past the 5th one i.e. this is the same as 99999 mod 32.  I then thought let's subtract the largest multiple of 20000 to get 19999 left over. Then lets take the largest multiple of 4000 to get 3999 leftover. At which point I noticed the pattern (mind you I didn't have any paper).

99999  remainder for 100000
19999  remainder for 20000
3999 remainder for 4000
799 remainder for 800
159 remainder for 160

What's occuring here is that 999,999,999 is 1 less than 1,000,000,000 which is divisible by  32 based on my first observation since its a multiple of 100,000.  So clearly 999,999,999 mod 32 is -1 or adding 32 to it the remainder must be 31. This basic fact was showing up in all the partial sums as well.  In fact, with any pure power of 2 a sufficiently large number made only of 9's will have this property! 999,999,999 mod 64 is 63, 999,999,999 mod 16 is 15 etc.   So I've completely changed my mind, I think this will make a fun white board group exercise where the kids can look for patterns to solve the problem an easier way. (Maybe I'll even show a video snippet of the Math Counts finale)  Oh and congratulations to Wan who still totally creamed me.

## Wednesday, May 11, 2016

### 2 Fun 3-4-5 Pythagorean Triples

I'm going to warehouse these problems from @five_triangles here. I really like how they both show constructions for a 3-4-5  Pythagorean Triple. My plan is to keep looking for other variants on this theme. This is almost enough material to make a followup day after the Pythagorean Theorem although it still needs a few slightly easier problems to go with it.

#### Discussion

Examine triangle DCO.
• DC is R + r
• DO is 2R - r
• OC is R
Apply the Pythagorean Theorem:  $$(R+r)^2 = (2R - r)^2 + R^2$$
After simplifying this reduces to $$3Rr = 2R^2$$ and then assuming  R is positive $$r = \frac{2}{3} R$$ Or put another way: r is 1/3 of the side and R is 1/2 of the side. Plug those values back into the original triangle and voila you find the 3-4-5.

#### Discussion

This construction is a bit trickier.
1. Note the congruent segments in the two big triangles
2. Angle chase to find the 2 sets of similar triangles and the right angles.
3. From the first set AFC and AFE and the given hypotenuse lengths You can immediately see the areas are in a ratio of 4:1 as is BF:EF since they share an altitude AF.
4. Let H be the intersection of GC and the line from B. This is a right angle from the initial angle chase.  This second set EFG and EBH are also similar.   For EBH you know have a right triangle with a hypotenuse of 5x and one side of 4x from step 3 where x is the length of EF. The Pythagorean theorem immediately says EH must be 3x.
5. Its easy to do the 3:1 scaling if you want to find the area of EFG from there.

## Tuesday, May 10, 2016

### 5/10 Connect the Dots

Around the middle of last week I received a surprise phone call from the school office: "Your package arrived and we're holding it for you."  As expected, it was a small cardboard box containing the MOEMS bling for this year. I decided to hand out most of it as the students arrived to make the process a bit more private. Mostly, because not all of the kids in Math Club participated in all of the 5 tests, not all of them received anything. However, I did think it was important to recognize the top scoring student for the year whom was  given a trophy at the start of the session.

This was a natural point to also recognize the team that went out the WSMC Olympiad last Saturday. I had all the kids who went give a short report on the contest and what they thought of it. As usual I'm hoping that this will encourage others to participate more next year.

My main planning goal for this session was to do something less formal after the last two weeks. I found this awesome chair exercise from Lisa Winer: http://eatplaymath.blogspot.com/2016/05/puzzle-day-human-jumping-puzzle-riddle.html
that I thought would fit perfectly.

First I had the kids collect two sets of 7 chairs from the hall and lay them out in rows.
Then I split the kids and set them up 3 on each end of the rows with the extra kids helping to make decisions and counting moves.

The basic rules are simple and are very similar to checkers.

1. The kids on each end could only move in one direction.
2. You can move forward to an empty seat.
3. You can jump over 1 person in front of you.

The goal is to get everyone to the opposite side of the row of chairs. This was a lot of fun. I ended up talking a lot about the concept of clumps and how every time they occurred was a dead end. After 5 -10 minutes both groups eventually found the solution.

This didn't happen at the same time so I took the half that was a bit quicker and gave them the second puzzle of the day on the whiteboard inspired from Nat Banting. Given the 8x8 checkboard below: how many square are there that do not include the 9 red ones?

We ended up with groups of kids trying out counting strategies around the whiteboard. I floated in and out to give hints about strategies i.e. how many 1x1, 2x2, etc. squares are there in the checkboard before you add any complications.

Finally the main activity was inspired by some math homework my 2nd grader took home. The sheet was a connect the dots picture made up of coordinate points. I found helping him out to be fun as an adult and I thought the fourth and fifth graders would probably also like them. So I found a sample site: https://www.worksheetworks.com/math/geometry/graphing/coordinate-pictures.html that would generate some.  To make this more interesting I brought graph paper and told the kids after doing the sample one, I wanted them to generate there own picture and puzzle. Sadly, we didn't have enough time for anyone to get quite that far so I'm going to see if any of the kids will finish one over the week.  If I repeat I'd budget 35+ minutes to allow enough time for the creative portion. (Basically I should have skipped one of the warmups)

For the   Problem of the week I'm experimenting with the sets from Math Counts: https://www.mathcounts.org/sites/default/files/images/potw/pdf/PoTW%20021516%20Solutions.pdf

Small Procedural Tweak:
I'm having everyone store their backpacks on the side at the beginning of Club. That's been a win,

## Monday, May 9, 2016

I brought a small team over the weekend to the WSMC Math Olympiad down in Maple Valley.
The contest is sponsored by the a math educators organization: Wa. State Math Council which is an affiliate of NCTM. This was also my first time attending and I haven't fully debriefed the kids yet to see their impressions. I'll add those on after this Tuesday's Math Club meeting.

#### Organization

• Unlike all the other offsite events, this one fills up very quickly.  Next time if I go again I will register immediately when it opens rather than my normal procedure of finding team(s) and then registering. Hopefully that would procure us a spot in the much closer Seattle U. site.
• No food is provided at lunch which is a bit on the late side.
• Calculators are allowed but looked to be completely unnecessary and I think I would not encourage the kids to bring them.

#### Pros

• Standards based scoring. If you get over a certain number correct then you get a ribbon. Over another threshold you get a medal. There is no limit to the number of winners and no team rankings.  This makes for a more low key experience where more kids get awards.
• The test is fairly generous with the amount of time given per problem which also is great for kids who are not super fast (unlike all the other local contests).
• There's one long form problem for 1 hour and a series of 20 minutes short form sections. The long form problem in theory is a chance to do something deeper.

#### Cons

• I proctored a sixth grade room. Most of the kids finished the short form sections very quickly. There needs to be more problems, or harder problems or less time.  This is an issue just from a test proctoring perspective.  You really don't want middle schoolers sitting around idle. If I go again, I might bring some puzzles for them to occupy the dead time just like I do for the MOEMS tests.
• There's a lot of potential in the long form problem. But the sixth grade version I administered was structured around calculating the profit in cutting down a tree for lumber. In practice this meant they gave all the formulas and the kids really just had to plug the numbers into their calculators and do a write up. There should be a lot more room for problem solving in an hour long exercise.
• There's no individual section. I like events where kids who like to work together and those who like to work by themselves both have a chance to shine.

Overall, this format improves on several of my concerns with the other off site events. And I think the kids like winning medals which is much more likely here. But the problem format is not challenging enough. I could partly compensate by entering them in the sixth grade rather than the 5th grade level.
In my ideal world, this contest would have more (at least 2x) problems in the short form section and something less mechanical for the long form part. I'd also like progressively more difficult problem sets where there is enough doable problems to not be discouraging but enough interesting problems to provide some challenge. I  do really like the generous time limits.

## Thursday, May 5, 2016

### 5/3 Square Roots

This week's theme was inspired by some of  questions the kids had while working on the Pythagorean Theorem about finding square roots. I have a small collection of subtopics around square roots and I thought we could go over some of them in Math Club as a group.

To start things off one of the girls showed her (really well done) solution to the Problem of the week:
See (https://drive.google.com/open?id=1R2gM2gyvIyrSdRK-y_wlYXrAZKTGY3KYcD63UZeZfVM)  which used an external equilateral triangle rather than the interior one I used.  Participation has been winding down a bit more than I like so I'm going to think if I can come up with an incentive to encourage a few more kids to work on the problems.

From there  I used a warm up riddle from Joshua Greene's BlogNumberPlay
"You’re creating a new coin system for your country. You must use only four coin values and you must be able to create the values 1 through 10 using one coin at a minimum and two coins maximum"
Everyone split it up into several groups of 3-4 kids and after 5 minutes of brain storming the kids found 3 different solutions. Having all the groups writing and verifying their answer simultaneously on the whiteboard made for some extra excitement.

Next I bridged into the square root discussions.

Firs we talked about bounding a square  root by using landmark numbers and then binary searching.
So for example the square root of 60 should be between 20^2 and 30^2 (which are easier to calculate) and then you an try 25^2 which is slightly larger, then perhaps 23^2 to finally arrive at the answer of 24^2 < square root (600) < 25^2 .

We then talked a little about irrational numbers.  As I found last year most kids at this age when asked to define a irrational number mention repeating decimals and often can't define the converse. So after having everyone come up with their ideas. I gave a formal definition i.e. the ratio of two integers.

With this preamble out of the way,  I mentioned most square roots are irrational. I had prepared two different proofs to show for why the square root of 2 is irrational.

1) Even/Odd proof by contradiction: http://www.math.utah.edu/~pa/math/q1.html

2) Geometric proof by contradication: http://www.cut-the-knot.org/proofs/sq_root.shtml#proof8ppp

I knew that these were both going to be a stretch, So I started by talking about how a proof by contradiction works and what we were assuming: that a rational number existed equal to the root and therefore at least one of the numerator and denominator were odd and how we were going to prove both were even.  We also worked through the basic rules for multiplying first:

1. even times even = even
2. odd times odd = odd
At last, I walked through the proof twice. Even so I think this was the most complex topic for the year. I decided to skip the second proof based on the room's reaction. The first was more than enough to absorb. I'd still like to present this next year but I'm thinking a worksheet format might work better where each step was guided.

Finally for the last piece of the lecture I asked if anyone knew how square roots were found before we had calculators?  As expected no one knew the answer. I explained that I was going to show one of the old algorithms for fun but I didn't expect anyone to memorize it.

I then reworked out the square root of 600 to a few decimal places by hand on the board.  Again if I repeat next year I'll have the kids try it once with some simple values like the square root of 2 or 3.

My original plan was to have the kids try out some  Sample AMC8 questons for the back half of the session:

https://www.artofproblemsolving.com/wiki/index.php?title=2014_AMC_8_Problems

But I thought everyone could use a break so I ended up handing out another sudoku puzzle. This occupied everyone to the end of the hour.

Overall:
1. I need to tinker with the format a bit. I think a work sheet for some of the square root material would be great for next year. I want to keep the proof but perhaps lead up to it with a simple proof by contradiction if I can find one.

2. I'm  going to come back to the sample AMC8 material in the next next few weeks. I'm toying with turning this into some kind of team contest.

3. I really want a lighter game/puzzle for next week to switch things up.

Problem of the Week: April's problem from http://www.moems.org/zinger.htm