Tuesday, October 31, 2017

10/31 Put a bird on it.

My surprisingly well read retweet from today.

I joke (or recycle Ed's joke) above but that's essentially what I did today in Math Club.  It's about 2 weeks until AMC8  and I really felt the need to have the kids do one sample test before hand so they are familiar with the format. At the same time, I also know from experience handing kids a multi-page set of problems doesn't work that well and the group tends to get unfocused.  So I went through several options beforehand:

  • Select 3-5 representative questions. 
  • Do a few problems over several weeks.
  • Create some kind of competitive relay.
  • Slog through a larger selection and just work really hard at keeping everyone focused.
There are drawbacks to all of these ideas. What I finally went with was a 30 minute practice. I printed out the entire 2011 test from https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_8 
but stripped away the multiple choice answers.  I also slapped a Halloween pumpkin on the sheet. 

I then had all the kids put their names on the whiteboard and had them work through every third problem individually.  After each problem was done I told everyone to put the problem number on the board under their name and find someone else who had finished the same problem to compare answers with.

This structure worked out surprisingly well. A lot of the kids got into running up to the board to write down what they finished but it didn't feel super competitive.  At the same time, this farmed out the answer checking so I didn't need to act like a living answer key while there was still feedback for everyone. I was also able to keep the process going with a bit of nudging over multiple problems and circulate to help out with questions. Further by skip counting questions the kids were able to try  the range of problems from the easier ones in the beginning to the more difficult ones at the end.  I didn't really need it but by not condensing the questions there were enough to not worry about running out.

To keep things interesting, I interrupted several times during this process to tell some corny Halloween Jokes:

"Why do mathematicians mix up Halloween and Christmas. Because Oct. 31 = Dec. 25" and to take a field trip to my brand new Math Club Poster Board:

After the 30 minute mark I did a cool down / party. First I handed out candy to celebrate Halloween. This was strategically delayed until after the serious practice was done and then I found a lovely set of Halloween themed logic puzzles from here http://geekfamilies.co/halloween-math-and-logic-puzzles-for-kids/   Today confirms these are still irresistible even to older students.

A final holiday themed geometry problem, I found from @five_triangles:

Tuesday, October 24, 2017

10/25 Strange Bases

This was a funny week. All the eighth graders were out on the class trip so Math Club skewed younger. That played a part in my planning. I aimed a bit less complex this time and hoped to draw all the sixth graders in.

My main inspiration was an article by @RobJLow on  balanced ternary number systems. This got me thinking about the Global Math Project and the exploding dots work James Tanton has been doing. I've seen a lot of folks participating and it looked like it might be a fit.

Then last Friday, our official MathCounts team packet arrived with the lovely poster from my last post.    My own son found the poster intriguing and I was fairly sure everyone else would too. So I started with a group whiteboard attempt at the problem.

With closer to 9 kids,  I could have everyone work in to two groups up at the board. I brought in some magnets and tacked the poster in the middle.  We worked on the problem for about 10 minutes and  I circulated making sure to keep stragglers at the board working.   What was nice was the first group to solve was fairly collaborative so I could have the kids take turns explaining their thinking to the other half of the group. (This was a theme for today: making lots of opportunities for kids to demo to each other.)

This week I remembered to tackle the P.O.T.W early on. After 3 or 4 kids presented it was clear we didn't have a solution yet to the problem:

So I made the executive decision to spend some more time engaging with it on the whiteboard. Once again, I had everyone come up. This time I seeded the groups with a few ideas after a few minutes like: try adding the origin of the circle and radii from there to all the points on the circumference.

One of the topics I worked on a bit with several students was applying the Pythagorean Theorem to find hypotenuse side lengths. At the end, I lucked out and had two different solutions from two students that again I had each explain to the larger group.

As a consequence, we were now half way into the time and just about to start working on my main focus: alternate bases. It  was  clear, we'd only have time for one of the two examples. So I  told the group "We can work on base 3/2 or a balanced ternary base system, which would you like to explore?"  The room voted for base 1.5, probably because it sounded less formal.

In looking through the source material: I found Tanton's videos not quite to my taste and I wasn't sure if I'd have a projector. So I decided to focus on topic 9: source link but work the material directly. I started by asking who had played with alternate base system before. Most of the room had some experience. I also asked if anyone had seen exploding dots (nope).  So I spent a few minutes explaining how the model worked on base 2 and then jumped right into base 3/2.

First we started by counting up and recording the numbers. Here I enlisted one of the students as a scribe which freed me up to talk more. We then worked on what was the meaning of the place values, why were some values missing, and was there any pattern to the numbers. Just like the source material, I had the kids verify that the number worked and converted correctly back into the familiar decimal system. This all went fairly well. The back half of the exploration, we built multiplication and addition tables and figured out how to manipulate numbers in this system  With a minute to go we started to look for divisibility rules: like the one for multiples of 3.

All told this material worked really well. Building the tables up was particularly satisfying. If I had another half an hour, I would have loved to have broken into the second system but I think not rushing and digesting the topics we already had was necessary today.

I took a page out of the MathCounts manual so the kids would have an initial exposure to some of these style problems

Monday, October 23, 2017

This year's Math Count Poster

The poster arrived in the mail with the rest of the contest materials. I think its charming and hopefully I can get permission to post somewhere in the school.

Friday, October 20, 2017

Heron's Formula

The MathCounts guide for the year arrived today and I was looking over the problems. The following one caught my eye.

Given a triangle of side lengths 13,14, and 15. What is the radius of the largest circle inside it?

This is a slightly convoluted way of saying what is the radius of  the incircle. I then thought about it for a moment and came up with the following approach.

  • Find the area with Heron's Formula  = \(\sqrt{s(s-a)(s-b)(s-c)}\)  $$\sqrt{21\cdot6\cdot7\cdot8} = 84$$
  • The area is also \(\frac{1}{2} r \cdot 2s\)  So in this case that means r = 4.

My second thought was I don't think any of the kids in Math Club know Heron's formula. How would they approach this?  And then I realized they could drop an altitude and compute it via the Pythagorean theorem.  That would get them to being able to find the area from the standard formula \(\frac{1}{2}\)base x height.

Note: if you were lucky you might pick the base to be 14 and then find its made up of 2 Pythagorean triples the 5-12-13 and 9-12-15 but I'm going to ignore that possibility in favor of some more investigation.

From here   \(14^2 - x^2 = 15^2  - (13-x)^2\)   All the x^2 terms cancel out and you're left with a simple linear equation that reduces to \(x = \frac{70}{13}\) .

From there you reapply the Pythagorean theorem to find:
$$h=\sqrt{14^2 -\frac{70^2}{13^2}} = 7 \sqrt{\frac{4\cdot 13^2 - 100}{13^2}}$$
$$  =  7 \frac{\sqrt{576}}{13} = \frac{168}{13} $$

Now you can compute the area =  \(\frac{1}{2} \cdot 13 \cdot  \frac{168}{13} = 84 \) again.

And then I had an epiphany (which I'm sure has occurred in many textbooks.)  This would be a great intro to actually derive Heron's formula.  After finishing the concrete problem redo it with side lengths of a, b, and c.    

Repeating our steps. 

  \(b^2 - x^2 = c^2  - (a-x)^2\)     The x^2 terms cancel out again leaving:

$$ x = \frac{a^2 + b^2 - c^2}{2a} $$

You again apply the Pythagorean theorem to find the altitude:

$$ h = \frac{\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2}}{2a}$$

This looks complex at first but if stare at it long enough there are a lot of differences of square here that we can take advantage of.  This would be a good time to review 

$$a^2 - b^2 = (a-b)(a+b)$$

So first  \(4a^2b^2 - (a^2 + b^2 - c^2)^2 = (2ab + (a^2 + b^2  - c^2))(2ab  - (a^2 + b^2 -c^2))\)
You then can combine the a and b terms to get:

$$ ((a+b)^2  - c^2)(c^2 - (a-b)^2)$$

That's nifty because we can reapply the difference of squares formula again:

$$((a+b)+c)((a+b)-c)(c + (a -b))(c - (a - b))$$

Putting this back into the height formula and with a little rearranging we get

$$ Area  = \frac{1}{2} a \cdot \frac{ \sqrt{(a+b+c)(a+b -c)(a+c-b)(b+c-a)}}{2a} $$

We're now ready to substitute the semiperimeter s  = \(\frac{(a+b+c)}{2}\) in which results in:

$$ Area  = \frac{1}{2} a \cdot \frac{ \sqrt{(2s)(2s -2c)(2s-2b)(2s - 2a)}}{2a} $$
$$           =  \frac{1}{2} a \cdot \frac{ \sqrt{ 16 \cdot  s(s -c)(s-b)(s - a)}}{2a} $$
$$           =  \sqrt{s(s -c)(s-b)(s - a)} $$

I'm guessing this would take almost the full hour especially if I let the kids experiment on their own first.

As a followup: something nifty happens when you investigate the Pythagorean triple triangles. The numbers in the formula are always the side lengths of the 2 triangles and the square/incircle radius that make them up.

This is not a coincidence and its easy to prove. Another good extension for a right triangle's in circle radius:

$$r = \frac{ab}{a+b+c} = s - c $$  

Wednesday, October 18, 2017

10/17 Pythagorean Triples

http://mymathclub.blogspot.com/2016/10/12-triangles-and-their-link-to.html The inspiration for this week was a puzzle from the recent Pythagorize Seattle event thrown by MoMath that my friend Dan recommended.

Link: https://drive.google.com/open?id=0B6oYedIeLTUKOFpYR1VZek80Y1RYeEJqLUN4ZmtVWHVqRTVN

Each of the triangles above is a Pythagorean triple with one edge given. To solve the puzzle, you need to find the length of the side marked with a question mark.  Then translate each number into its position in the alphabet. For example if you find a side of length 5 that becomes the letter "E". Altogether this forms a 8 letter word scramble.

With 12 kids, I printed and cut out 3 sets of these triangles so we could have 3 groups working at the same time. To start off I quickly reviewed the basic Pythagorean Theorem on the whiteboard. Because I knew we would talk about it more in the middle I deferred any review of why it works.

Then I let each group work for about 10 minutes. What I found was that a lot of kids took out calculators and started plugging numbers. Walking around, if kids looked stuck I suggested estimating the missing sides edges and just trying out numbers near the estimate.

At this point I paused everyone to watch the following video:

Some of the parts here are a bit advanced so I stopped in the middle a few times to go over ideas like the Complex plane. And at the end I emphasized the generator functions:

  • 2uw
  • u^2 + w^2
  • u^2 - w^2
I then sent everyone back to finish working on the puzzle. From here the groups fairly quickly finished although not at the same time. I gave out a followup Pythagorean Triple geometry problem for the last few minutes:

See the first proof here:

No one had fully solved this yet before we left for the day so I'm tempted to come back here. I'd also like to have the kids find some of the patterns in the triples i..e one of them is always a multiple of 3, 4, and 5. This might make a good bridge with a day on modular arithmetic as well.

Continuing with the Pythagorean theme:

When handing this out I told everyone to be on the lookout for another hidden triple. I think a few kids found it already before they left.

I'm getting to know the kids a little better. This week I was hoping the initial puzzle would have enough entries for everyone. That was mostly the case but I ended up working with one student a bit to get him started.  Now that I've heard what class everyone is in, I realize I have a larger gap than was indicated in the entry forms from Math7 to Algebra II.  If I'm going to throw algebra into the mix I'm going to need either to run different activities at the same time or plan how everyone will be able to approach and work on the problem at the same time.

Tuesday, October 10, 2017

10/10 Middle School

In a careless move I deleted my original post on this one. Here's a skeleton version until I have time to rewrite it.

  • Bylaws and ASB officer election
  • Demographics
  • 4 color map problem game.
Divide kids into 2 teams. I had group of 2-3 per team. Give each one a different colored marker. Then
take turns drawing dots on the whiteboard and then connecting the dots to form a "map". It works best if there aren't any overly small shapes.  After that the real game starts. Take turns now coloring in the map segments. The only rule is you can't touch a segment of your own color. The first team that can force the other side to not have a move wins.
  • 3 squares: pick 3 numbers such that any 2 add to a square.
  • Leveling/Behavior
  • Outlook so far