Friday, May 29, 2015

The problem I'm thinking about right now.

I worked with my son on this problem last night from AMC8 and I think its really interesting and plays off the previous pentagon problem I had given to the kids in the club.


In the pentagon above $\angle A = 20 ^{\circ}$  The triangle containing A is isosceles. What is $\angle B$ + $\angle D$?

General Steps.
1. Determine the 2 other angles must be 80$^{\circ}$ since its an isosceles triangle.
2. Find the supplementary angles on the interior.


3. Continue tracing the angles for the two triangle B and E since you have one corner already. I assigned x to $\angle B$ and that means its final angle must be 180 - (x + 80) = 100 - x. While I'm at it I added in its supplement on the interior 80 + x.


4. Repeat the process for $\angle E$ letting its value be y.  Which also derives an angle of 100 - y and a supplement of 80 + y.

5. Applying the fact that sum of the angle of the interior pentagon are 540 and we now know 4 of the angles, the fifth must be 540 - (100 + 100  + (80 + x) + (80 + y)) =  180 - x - y.  Note: its not a coincidence that the sum of the angle pairs both contain 180 degrees.

6. Now attack the final triangle spoke anchored by D.  Two of its angles are supplements of ones we've already found: 100 - y and x + y.  So $\angle D$   = 180 - ((100 - y) + (x + y)) = 80 - x.

7. Lo and behold when you add $\angle B$ + $\angle D$ the x cancels out: x + 80 - x = 80.




In fact you can generalize this process to see the relationship between any interior angle of a triangle and 2 exterior spokes of the pentagon.

I think this is one of those interesting inherent relationships and would make a nice progression but I'm not sure if I can cram everything into a worksheet session. To make it work I'd need to:

  1. Review basic angle tracing for two intersecting lines and triangles.
  2. Review finding the sum of the angles of a polygon at least for a pentagon by breaking it into triangles.
  3. Walk through some more abstract angle tracing examples with variables. The symbol manipulation is a little abstract here.
  4. Give the problem and tell the kids to start tracing.
  5. Be ready to hint about assigning a variable to the two vertices or maybe just supply that from the start.

If I'm not satisfied this will work out I may just go back to my original ideas about doing a intro graph theory problem like the bridges of konigsberg.

Tuesday, May 26, 2015

5/26 Gamification

Today I tried something completely different with the math club. We participated in the Washington State Algebra Story Problem Challenge. http://wa.algebrachallenge.org/  To start with, I wasn't sure what to expect from the process. I believe there is some association between the game science group at UW and the developers of dragon box based on some of their past projects and the fact the game shared the same background music. So if a game "Book of Riddles" shows up some day, we helped test it while it was being developed. Overall I was hoping that this would be similar in interest value to dragon box or dragon box elements.

For the  beta testing I had my son try out 5 minutes of the various grade levels to try to decide which one to give everyone in the club (I couldn't set more than one for the group). Based on that experience I went with setting everyone to 7th grade.  It also appeared interesting from that initial sample so I had high hopes.

However, in practice I ended up having a mixed experience with the game. First it tended to hang or crash repeatedly over the session. I ended up having to continually log kids back in again. This required remembering a non-mnemonic bit.ly url and team code. Secondly, the actual controls were fairly non-intuitive in the middle section of the game. (Of course this was the part I hadn't sampled before hand.) Four or five kids  became frustrated trying to manipulate the controls to express an equation they clearly understood. I also found finding the correct input form to be fairly exasperating and it was  often easier to have the help show the final solution in several cases. So it only maintained the interest of about 10 of the kids over the entire period with the balance going over to cool math games instead. I let that pass this time but for future reference its hard to keep everyone on task when they can type URL's in themselves to the browser.

Overall, I would have been better off borrowing 15 iPads and just having them play dragon box  So I think I'm going to probably skip this activity next year unless on investigation it has radically improved.

Wednesday, May 20, 2015

Cool Geometry Problem

I ran into another problem that I think demonstrates an unexpected and interesting fact

(from http://matharguments180.blogspot.com/2015/05/468-fields-of-green.html)


The following 3 squares are connected to form four triangles. What is the area of the whole hexagon shape?

The full solution requires the use of heron's formula as far as I can see which puts it out of reach for the kids. But the more fascinating part to me is the fact that the four triangles above all have the same area!

Not only that, it's not too hard to prove:




Let's look at the inner and lower triangle. They already share a base (let's pick the square with sides of length $S_2$).  So to show they have the same area we just need to see if they have the same heights. So I drew in the 2 height perpendicular lines $h_1$ on the interior and $h_2$ which is a continuation of the square for the exterior.  Let x be angle inside $\angle EDA$ then $\angle BAC$ must be 90  - x since they are on the same line and have a 90 degree corner of a square in the middle.  Since these are both right triangles we can find the 3rd angles which are just 90 and 90 - x again. So clearly the two triangles $\triangle ABC$ and $\triangle ADE$ are similar. But they also both have hypotenuses along the square with sides of  length $s_1$. So they are completely congruent and $h_1$ = $h_2$.  You can repeat this construction on every side to prove all the triangles have the same area.

Addendum: Another way of seeing the same thing is that you can rotate the inner triangle so its $s_2$  length edge lines up to the same edge on the outer one. These will form a larger triangle with the shared edge exactly a median so the 2 halves, the original triangles are the same size.

I'm saving this one for a future session. I think I'll start by having the kids cut out different size squares and measure the triangles to organically discover they appear to be of the same area regardless of the square sizes and then move onto to figuring out why this must be true. This will work especially well paired with Pythagorean Theorem work given its similarities.

Monday, May 18, 2015

Virtual Math Club

Tomorrow our district is shutting down while the teachers go on a one day walk out. As a result there will be no math club session.  But that's no problem - we have an app for that. Well actually its just email but I'm going virtual for the week and sending out two problems for the kids. Hopefully I'll hear back that some of them gave them a try.

The first is a continuation of the decoding work we did last week. This one is a long division problem with missing digits. I think this should flow fairly easily from the strategies everyone was employing.

http://fivetriangles.blogspot.com/2015/05/247-longer-division.html


The second is from a new site I found and explores some more of the pentagon/pentagram geometry. The full question is a bit hard for the knowledge level we have but the more basic question: find the sum of the acute angles in a regular pentagram is really good practice.





Saturday, May 16, 2015

Interesting Tool



I've been looking at  a series of videos on https://mikesmathpage.wordpress.com/tag/zometool/  recently where Mike uses zoomtools with his sons. In  several there are a really fun stab at the golden ratio via pentagons and pentagrams. This is a topic I've mulled over doing before but the need to solve a quadratic to derive the golden ratio i.e. x^2 - x - 1 = 0 has stopped me for now.  I also like the ones where they work at deriving the volume of pyramid is 1/3 of a cube.

I'm starting to think this might be fun for us as well next year. So I've tooling around the company's site looking at the suggested lessons as well: http://www.zometool.com/content/ZGB-TOC.pdf 
I have a mail out asking about how many kits I'd need to support say 15 kids at once. But assuming I don't need to buy their 1500 workshop kit I'm very tempted. ($40 per kid doesn't go very far) Geometry is continually short shrifted in the regular curriculum and this has the bonus of being tactile. I'm imagining we could thread some of the suggested lessons throughout the year which would give me a whole new class of activities.

Prior to seeing these demos I'd been thinking about more formally organizing geometry next year anyway. At this point I have seen/worked over a large number of proofs and I think I could start to build up some more logical sequences of activities that would build on each other if I look over the whole set. A series of progressively more complex angle related proofs for example.  In the meantime I have my own retrospective on geometry problems (I've done many more than the kids in an attempt to try out and winnow down the best candidates) which I'm drafting and hopefully will post in parts soon. With only 3 sessions (!) lefts of math club for the year I'll have more time this summer for this type of planning if I'm ambitious.

Tuesday, May 12, 2015

5/12 Dog + God

I have a lot of interesting observations from math club today. First up, for the warm up I went with some decoding problems:

    D O G                  
+  G O D          
-----------
  D E E P

This one we mostly solved together on the whiteboard in order to give everyone a sense for how to do this class of puzzles. I asked everyone to raise their hands if they had an idea what to do next. First came the suggestion that D must be 1 since that's the only value you can get from a carry. Next someone noticed that means G can only be 8 or 9 or there won't be a carry at all. A few moments later there was the idea that either E was 1 or 0 with those values but we had used up the 1 so only zero was left.  Things went fairly quickly from there.

One lesson for me: My handwriting is messy and I should be careful to always read the problem aloud slowly to avoid kids thinking my G's are 6's.

Then I went for the slightly harder one courtesy of fivetriangles:

   A B C D E F               A B C
x                6   and    + D E F
-----------------            ---------
  D E F A B C                9  9 9

The group exercise was very helpful here. Everyone solved the problem within 15 minutes with only a little prompting for some of the later kids on my part. Better yet: I had hit my internal time budget for this section.

I then proceeded to give out the triangle number worksheet I wrote a week ago:
http://mymathclub.blogspot.com/2015/05/triangle-number-worksheet.html

Second lesson for the day: I should have reviewed this one more closely or written some of the answers down originally. As a result I was a bit rusty and had to do more off the cuff recalculations like what is 1 + 2 ... + 685?  than I would have if I had done it last week as intended.

Most kids derived the basic triangle sum number on their own. About a third needed a hint to try regrouping the sums to make them easier.

Another consistent observation I saw was a weakness in notation. I noticed a lot of  samples that looked like : N + 1 x N / 2  missing the parentheses. This didn't seem worth correcting today since my main goal was to walk them through the derivation of the formula.

Also because I started the kids at different times I never regrouped everyone to do the proof without words I had originally intended. That's still pretty neat and worthy of a mention next week.From there most kids made it to 2 or 3 of the problems. If I repeat this next year it could probably be done as the entire activity for the hour.

Tuesday, May 5, 2015

5/5 Missized Logic Puzzles



I probably should have known better especially after my experiences with previous puzzles. For this week's warmup I planned for the math club to work on the following logic puzzle: http://www.puzzlersparadise.com/article1008.html.  I haven't tried these out with the club at all since a take home version I gave before we could really get started.  The plan was to do this for about 20 minutes and then to move onto a triangle number worksheet I've been planning for a few months. [See prev. blog post]  Because I was so wrapped up in the worksheet planning, I didn't test out the puzzle myself and also didn't think to do multiple levels of difficulty.  What happened in reality was that we spent the whole hour working through it. On the bright side, as soon as I handed it out, I heard comments like "I love these type problems". Except for a few holdouts some of whom gave into the charm by the end this activity was really successful.  A few times in the middle of the process I asked for a room vote on whether to table the puzzle and move onto the main task and the kids insisted we keep working.  So I'm saving the triangle numbers for next week and making a note for next time to find a simpler puzzle if its meant to be only a partial activity. Also for next week I'm going to have to do a longer discussion on problem solving strategies around what the kids discovered while working on it  than I originally planned. From my work I think the key is to stress boy/girl splits and finding their ages after which everything else falls out fairly quickly. We'll see how the kids actually thought about the problem though. The ultimate irony here is that I was worried about running out of material and actually prepped a few extra activities right before I left for the session just in case.

Triangle Number Worksheet

Triangle Numbers

What is the sum of 1 + 2 + 3?  

What is the sum of 1 + 2 + 3 ....   + 10?

What is the sum of  1+ 2 + ....  + 100?

What is the sum of 1 + 2 +  .... + 568?

We call such sums triangle numbers because they can always be drawn as follows:




Can you find a general formula for 1 + 2 ...  + N based on what you did above?  After that 
see if you can come up with a way to prove this is true. Hint: there are multiple ways to do so. See if you can a find a geometric explanation that takes advantage of the fact that triangle numbers are actually triangles.




Raise your hand once you've thought about or solved the above question.

Problems

1. Find an expression for the first n even integers: 2 + 4 + 6 + 8 + . . . + 2n

2. If there is a room full of 50 people and everyone shakes hands with every other person how many handshakes are there?  Can you generalize to if there is a room full of N people?

3. Amy has a box containing ordinary domino pieces (up to six dots) but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether.
Which of her domino pieces are missing?


4. One line can divide a circle into at most 2 regions, Two lines can divide a circle at most into 4 regions. Three lines can divide a circle into at most seven regions. What is the maximum number of regions that a circle can be divided into by 50 lines?

5. Try multiplying a triangle number times 8 and adding 1. What kind of number do you seem to find?  Can you figure out why this is happening?


Bonus

Try making a chart of the square of the triangle numbers and see if you can find a formula for it.
For example: (1 + 2 + 3) squared is 36.  Hint: take a look at the list of squares and cubes.