Tuesday, April 25, 2017

4/25 Platonic Solids

This week's inspiration started with a very late school bus. My son's bus driver has been on vacation and the substitute drivers have been really, really tardy. So much so that he missed most of Math the other morning when the class was going over the volume of a pyramid. I checked with his teacher and seeing he had missed the explanation for the formula decided to try out some activities at home to make up the gap.



To start, while I love much of Geometry the introduction of sundry area, and volume formulas in the middle school sequence seem pretty pointless to me. They don't connect with much before or afterwards and are often taught without sufficient explanations. Frankly you can go really far even in pure Mathematics without ever missing the pyramid formula. (Brainstorm topic: where would this fit more naturally? The calculus connection is fairly compelling ...)  The missed experiments in class compared prisms and pyramids and the volume of rice they held. As an experimental process this is not bad but as a mathematical foundation it doesn't totally satisfy me. Its neither universal "How do you know that if the pyramid dimensions shift the relationship stays constant?" nor does it speak to "Why is this happening?"  The question I want to provoke is "Why 1/3 and not 1/4?"

So at home, we started looking at the formulas and  I asked "What does the similarity between the volume of a cube/prism and pyramid suggest to you?" I was lucky that was enough of a prompt for him to suggest "Is there a way to cut a cube into 3 pyramids?" From there we printed out some templates and built some 3-D models to show the trisection.    This was enough fun that I thought I'd build a day out of it for the whole Math Club. To round things out I thought I'd bridge from there to an exploration of platonic solids.   At this point I worried a bit about timing and decided to have some Sudoku puzzles in reserve. But I stayed firm and left them for the end if needed which as it turned out was not the case.

The afternoon began for real with  me handing out spiced gum drops for reaching our problem of the week target. I also left out a sample tetrahedron I had built to see if I could garner any questions. (Nope)  Once we were upstairs I decided to have a short debate about last week's problem. Infinity Link.  I asked everyone to pick a corner of the room. One side for those in favor of Courtier A's offer, the other for Courtier B. A group of students actually remained off to the side and I asked what they supported. Their answer was they thought both offers were equal and since that seemed interesting I setup a 3rd corner for them.  We then went around the room with everyone offering positions on why their side was correct and rebutting the other side's idea. This went on for may 5-6 minutes which was fun . The disadvantage was this format really makes universal participation hard to achieve so I wouldn't rely on it a lot. (To be fair: repeated usage could make it more natural for more kids to speak.)   Secondly, a group of kids really wanted me to rule on the "correct" answer which  I demurred on.  Next time, I should also remember to close this with a final vote.

From there I did a version of my initial process with my son and  we bridged to building the pyramid templates. I used some cutout templates from here:
  http://www.korthalsaltes.com/model.php?name_en=three%20pyramids%20that%20form%20a%20cube and spread them among the various tables. We eventually cut out 3, folded and assembled them with scotch tape, and confirmed they were identical and formed a cube.



I was hoping that someone would complain that the pyramids weren't exactly the same as the regular ones we started with. That didn't happen so I prompted "Is there anything that doesn't seem quite right in this explanation?" That eventually brought out the idea and I gave a brief hand waving explanation of slicing the pyramids and rearranging them to have the same volume but centered rather than offset to the corner.

Next: I handed out a combination of further platonic solids from the site above (cube, tetrahedron, octahedron, dodecahedron,  icosahedron)  and from  http://www.senteacher.org/worksheet/12/Nets-Polyhedra.html.  Each table had a different one to assemble. At the same time I brought some pipe cleaners and straws to make companion wire models.

I had everyone work on the models and to tie things together chart the edges, faces and vertices per shape on a communal white board. My hope was to have the kids observe the the Euler Characteristics.patterns and I seeded things a bit by arranging the chart  V / E / F.


Some of the kid's handiwork.


This worked fairly well. Engagement was good among the modellers (which I had to rotate due to limited tape and scissors) but I had to work a bit to keep the other kids counting edges and faces and thinking about patterns. 

At the end I gathered everyone back at the board to discuss the data. There were a few fun observations.
  • All the numbers were even. (Followup for another time: could any characteristic ever be odd?)
  • You could more quickly and accurately calculate edges and vertices than counting by multiplying the number of faces time edges/vertices per shape and dividing by the number of faces that met at an edge/vertex.
I had to have everyone think about a numeric relationship for a few minutes but lucked out and one boy discovered   V - E + F = 2.  At that point my time was almost up so I left with some closing questions:
  • Is there an equivalent relation in 2-D?
  • Why do you think this is happening? 
  • Are there any other platonic solids you can discover?

In the end, this was a lot of fun. I actually had templates for stellated polyhedrons and Archimedean solids we never got to in my back pocket. We could easily do a followup day on the topic although my general style is to zig-zag around subjects.

POTW:

This one from an old Purple Comet is interesting because its algebraic but linear not quadratic like I first suspected and you don't need to every find the exact dimensions to discover the perimeter.

https://drive.google.com/open?id=1TUxHXM0j2PvwDMY1k7VGR2WB9YEqo1M7w2G78W89D3U


Related Session: Euler Characteristic

Wednesday, April 19, 2017

4/18 the series "Infinite Series"

Spring break really flew by and yesterday to my surprise Math Club was already resuming. Things started with small snafu, the door to our room was locked. While we were waiting in the hall for the custodian I went over some administrative items. I'm still looking for a few kids to round out the group going to the upcoming WSMC Olympiad, I wanted to acknowledge the high participation in the problem of the week and that I'd bring candy in next week. Finally, I also started laying the groundwork for the talk next month and asked the kids to start thinking about questions to ask our guest mathematician. 

If only there was a whiteboard in the hall I would have gone over the previous problems of the week but sadly we waited a few extra minutes instead.




For this week I wanted to try out the Infinite Series youtube webcasts with the kids. I thought the above video on proofs was a good first choice since one of my priorities is to emphasize understanding why things work and how it will become increasingly important (and computation less) for the kids as they move forward. In fact, I'm trying as much as possible to add in comments about the math progression whenever appropriate. This is one of those areas I feel is not well understood in 5th grade.  Most of the kids know they're working towards algebra, geometry and probably Calculus. They don't necessarily know what Calculus is about even in the most broadest sense and they don't often think what happens after they finish that sequence. I also think they take it for granted that Math topics are all a roughly linear sequence which is not truly the case beyond school math.

What's also nice about the video is it structured around several problems and even has breaks where you're supposed to try them out first.

I took full advantage of that format and stopped 3 times:

1. The chessboard / domino coverage question was the easiest and one of the boys came up with the standard reasoning in a few minutes.
2. Probability of sticks forming a triangle. I wasn't sure if the kids had been exposed to the triangle inequality so I played that part before pausing. Interestingly everyone said "Oh yeah" even if they didn't recognize it by name.   No one came up with he answer but there was a lot of good discussion before I resumed.
3. Sum of odds formula:  Again no-one fully came up with an answer but I was satisfied with the thinking along the way.

In general this was a bit of a balancing act on how long to let the kids grapple with each problem, knowing they would probably not crack them. I wanted enough time so that the explanations really resonated afterwards but still allowed me to finish the video. In the end I had about 10 minutes of the session left. I thought the quality of discussion was particularly good even though everyone reasoned at their group of tables. Perhaps this was a residue of our work on the whiteboards the last few weeks.

Finally, to round things out I brought two sample Sudoku puzzles and an older purple comet problem set: http://purplecomet.org/welcome/practice.   I thought most kids would prefer the Sudoku but I was pleasantly surprised that many asked for both so they could try them out.  This represents a shift in my organizational thinking. I'm tactical about this but especially with new activities I'm not sure the length of, I'm jumping right in and saving my old warm-up ideas for the end instead.  I see more benefit from having a light weight activity for those whose focus is used up than a transitional one at the beginning and it means I'm shorting my main focus much less often. If the activity takes the whole time and everyone is engaged I'll just save the extra puzzle for another week.

P.O.T.W:
I went with an infinite series conceptual riddle. My hope is to have a group debate next week.

You’re a venal king who’s considering bribes from two different courtiers.


  • Courtier A gives you an infinite number of envelopes. The first envelope contains 1 dollar, the second contains 2 dollars, the third contains 3, and so on: The nth envelope contains n dollars.
  • Courtier B also gives you an infinite number of envelopes. The first envelope contains 2 dollars, the second contains 4 dollars, the third contains 6, and so on: The nth envelope contains 2n dollars.

Now, who’s been more generous?

Courtier B argues that he’s given you twice as much as A — after all, for any n, B’s nth envelope contains twice as much money as A’s.

But Courtier A argues that he’s given you twice as much as B — A’s offerings include a gift of every integer size, but the odd dollar amounts are missing from B’s.

So who has given you more money?


Friday, April 7, 2017

Spring Break Geometry

[In exciting real news, I almost have a guest speaker from the UW Math department lined up for May. My hope is that this will be helpful in showing the kids that Math is a living field where research is still going on. My goal is to collect some questions ahead of time to prime the pump.]


In the meantime while we're on break, here is one of the latest problems  I've looked at from @go_geometry. This is a good example of the power of cyclic quadrilaterals and approaches to more difficult ratio problems. (original problem)



My first thought was that all segments in the ratio were on the same line. That's a problem because we only have a few tools to use that create ratios and they all need polygons.

1. Combinations of well known triangles.
2. Similar triangles.
3. Cyclic quadrilateral diagonals (which really are just similar triangles).
4. The angle bisector theorem (although I didn't initially think much of this one.)

  • My second thought was that BC is congruent to every other side of the square so that could at least give sides to one triangle CD  and CG for instance but FG still looked hard.
  • Triangle EFG is similar to ADE  which does generate some ratios involving FG and AD but I wasn't sure I could do much with them. The algebra looked fairly complex when playing with such ratios.
  • It looked clear from everything so far that it would be a combination of ratios to produce the result.
  • I then noticed ABEC was a cyclic quadrilateral since angle ABC = angle AEC = 90 degrees. That's useful for angle chasing and produces a set of similar triangle including ABF and CEF.
From those triangles one gets:

\(\frac{BF}{AB} = \frac{EF}{EC}\)  Since AB = BC that converts to \(\frac{BF}{BC} = \frac{EF}{EC}\)

That's about half way to the desired ratio \(\frac{BF}{FG} = \frac{BC}{CG}\) so I rearranged the goal  to the same form on the left side:

\(\frac{BF}{BC} = \frac{FG}{CG}\)   which meant  I still had to show  \(\frac{EF}{EC} = \frac{FG}{CG}\) 

  • My next observation was that angle DEB sure looked like a right angle also.  I then stopped to measure and check in geogebra. That appeared correct so I looked around some more for reasons why this was the case. I started angle chasing and found BECD was also a cyclic quadrilateral since angle DBC = DEC = 45 degrees. This could be used to show that the original intuition DEB was in fact a right angle.

At this point I stopped and had a "duh" moment. If you add in the diagonals of the square and the circle that circumscribes it ABECD are all on it.  The diagonals of the square are the diameters of the circle and meet at its origin and its obvious why DEB had to be a right angle since its a triangle made of the diameter and a point on the circle.



This gives a lot of underlying structure for angle chasing. I could find all the angles at the top in my triangle of interest CEF including that CEG = FEG = 45 degrees.  (FEG inscribes the same arc as ABD which is a 45 degree angle in the square, then its simple angle subtraction)

I then stared at \(\frac{EF}{EC} = \frac{FG}{CG}\)  and realized the form looked familiar. This is a slightly rearranged version of the angle bisector theorem and EG does bisect angle FEC!  So
\(\frac{EF}{FG} = \frac{EC}{CG}\) and when everything's combined you're done. Looking back this flowed fairly quickly from intuitions and observed patterns. The whole process was actually a bit chunky and done during various points in the morning when I had a moment.

Tuesday, April 4, 2017

4/4 Spring Quarter Begins

This quarter began with a seamless transition the week after the old one ended  However, I had a little bit of turnover with 2 kids leaving and 2 new boys and 1 girl joining.  I always want a math club session to be compelling but knowing its the first time for some of the audience adds a bit of pressure to get the balance right. So this week, I spent a lot of my planning time work deciding on what to do as an icebreaker and where to focus our main activity. I actually made several adjustments along the way until I settled on what occurred and still hope that I tuned the difficulty level correctly.

Intros

To start off, I had all the kids gather on the rug in the front of row and introduce themselves. As usual I had everyone state their name, homeroom teacher and either their favorite activity from last quarter if they were returning or why they decided to join if they were new.  Interestingly, there was a strong consensus that Pi Day was the favorite. I'm hoping that it wasn't just the literal pie I served that influenced everyone.

Human Knot

I really wanted to do something physical at the start and I had used up most of my ideas already in previous quarters. After looking around I didn't find anything new that was really satisfactory. There's a lot of ideas that revolve around Simon Says or Duck Duck Goose that just don't feel very authentic to me. So I went with a short team building exercise I used in cub scouts. http://www.group-games.com/ice-breakers/human-knot-icebreaker.html  Basically, you have the kids stand in circle grasp hands and then cooperate to untangle the resulting knot., If you're being generous you could say this relates to topology or knot theory but really its about having the kids interact together and practice cooperating. I found that my initial knot was  too difficult  so I split the group in half (6-7 kids per knot) which worked better.  [I'd actually like to come back to knots from a mathematical perspective at some future point in time.]

Charter

Afterwards I went over the the serious part of the day, the basic rules for the club. This time I boiled it down to the 3 core values:

  1. Respect  - As guests in the classroom, towards each other etc.
  2. Listening  - To me and to each other when they are sharing, I like to stress this is both hard and really important.
  3. Perseverance -The only section where I solicited opinions this time. I went around and had the kids talk about how they handled getting stuck. As I remember I went off on a short tangent about how long it took to solve Fermat's Last Theorem for my real life example.
Math Carnival

For the main activity, I decided to explore using the whiteboard more this week. I went back and forth on leveling and finally settled on the following 3 problems which I wrote on three different sections of the board. After explaining each problem, I  handed out markers and told the kids to pick which problems they wanted to work on.

Cue Ball
http://mathforlove.com/lesson/billiard-ball-problem/   This one flowed really well so I spent most of my time asking questions like "I see you have a pattern for even numbers, what about the odds" or "What happens when you grow or shrink this row by 1?" I also worked a little on emphasizing charting results to look for patterns. Kids in the group tended to stay put the entire time in contrast to the other 2 problems which were a bit quicker to crack.


Letter Magnets. A store sells letter magnets. The same letters cost the same and different letters might not cost the same. The word ONE costs 1 dollar, the word TWO costs 2 dollars, and the word ELEVEN costs 11 dollars. What is the cost of TWELVE?

Interestingly, most kids found the solution to this through a combination of guess and check rather than equations. This was actually easier to do than I realized. So where algebraic approaches sprang up I tried to encourage the kids to go down that avenue.

The geometry here was a bit harder than I expected for everyone. I ended up scaffolding a bit and ran into some issues with knowledge about calculating the area of obtuse triangles. I was pleased that one group came up with the idea to split the shaded shape in half on its own. On the downside this one in particular was a bit susceptible to encouraging answer seeking. Next time, I need to remember to tell the kids to check their answer with another group when they think they have a solution.


Once again I was pretty happy with overall group engagement and thinking during this process. The whiteboards proved superior to paper in keeping the group fully involved. Also I noticed that they make it a bit easier for me to drop in as I walk between groups and absorb where they're at. The larger format is easier to access.


POTW

I couldn't decide between the following 2 problems so I gave them both out. We have a week of Spring break before the next meeting so that seemed reasonable.

From AOPS:
Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. Te trip from one city to the other takes 5 hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-Bound bus pass on the highway (not in the station).


From Blaine:
Suppose that N is an integer such that when it is divided by 3, it leaves a remainder of 2, and when it is divided by 7, it leaves a remainder of 5. How many such possible values of N are there such that 0 < N ≤ 2017?


Friday, March 31, 2017

Not so Innocuous Quartic


\(x^2 - 16\sqrt{x}  = 12\)

What is \(x - 2\sqrt{x}\)?

The above problem showed up on my feed and my first thought was that doesn't look too hard it's either a factoring problem or you need to complete the square.  That's the same reaction my son had too when I showed it to him.

But a little substitution (\(z = x^2)\) shows that its actual a quartic equation in disguise:

$$z^4 - 16z - 12 = 0$$

The wording strongly suggests that \(z^2 - 2z\) or some variant is a factor which is a useful shortcut but that led me down the following path on how to generally factor a quartic.  The good news here is that the equation is already in depressed format with no cubic terms.

Some links for the procedure:

http://www.maa.org/sites/default/files/Brookfield2007-103574.pdf

A little easier to read:

http://www.sosmath.com/algebra/factor/fac12/fac12.html


How it works:
1. First we need to find the resolvent cubic polynomial for \(z^4 - 16z - 12 = 0\).
That works out to \(R(y) = y^3 + 48y - 256\).

2. Using the rational roots test we only have to look at \(\pm2^0\) ... \(2^8\) for possible roots but since we only can use roots that are square we only have to test \(\pm2^0, \pm2^2, \pm2^4, \pm2^6\) and \(\pm2^8\).   Plugging them in we find \(2^2=4\) is indeed a root. So there is a rational coefficient factorization for our original quartic.

3.  Now we can use the square root of the resolvent root i.e. 2 and its inverse  to get the following factorization (they are the coefficients of the z term): $$(z^2  - 2z - 2)(z^2 + 2z + 6) = z^4 - 16z - 12 = 0$$

4. At this point we could factor the 2 quadratics and plug the solutions  back in to find  \(x - 2\sqrt{x}\)  which in terms of z is \(z^2 - 2z\).   But we can shortcut slightly for one of the solutions since the  if the first factor is the root then \(z^2 - 2z - 2 = 0\) which implies \(z^2 - 2z = 2\)

5. Interestingly for \(z^2 + 2z  + 6 = 0\) we have the two roots \(1 \pm i\sqrt{5}\)  Plugging either
one into \(z^2 - 2z\) and you get -6 anyway!



Tuesday, March 28, 2017

3/28 #VNPS

Today was a fascinating learning experiment for me. I recently watched the following lecture:

https://www.bigmarker.com/GlobalMathDept/Building-Thinking-Classrooms by Peter Liljedahl.

Several of the ideas seemed relevant but I was particularly interested in his talk about the value of whiteboards  or VNPS (Vertical Non-Permanent Surfaces in his parlance) for working problems. I've talked previously about how I've been learning to more effectively use the double whiteboards in the room this year. Like previous years, I always have the kids demonstrate the solutions to problems on them like the Problem of the Week and after Olympiads I've taken to writing the problems across all the boards and doing a review  by moving among them rather than erasing and I'm more mindful of switching orientation and moving between the front and back ones for various transitions. But for the most part most group work I give out is done at the desk pods in groups with paper and pencil. Liljedahl's research suggests you can get much more effective engagement having kids work standing up on the boards. This is something I hadn't considered although I have always noticed the kids are irresistibly drawn to try and write with the markers.

So I decided to dive right in and try out an experiment. I looked through some of the suggested problems on his website: http://www.peterliljedahl.com/teachers/good-problem and noticed the four 4's one.  I use the game of 24 cards from time to time and actually had tried this exact exercise 2 years ago: http://mymathclub.blogspot.com/2015/06/62-pentagrams-and-some-inspirational.html. The problem involves using four fours and any operations you'd like to derive the numbers 1 .. 30. For example:  (4 / 4) + (4 - 4) = 1 and  ( 4 / 4 ) +  ( 4 / 4 ) = 2.  Last time, I wasn't entirely happy with how things went. That gave me a baseline to compare today with.  So after a quick review of the problem of the week I decided to dive in.  First I gave out a blue marker to everyone and told them to form into group on the board and then I talked through the challenge.

Results





In the end, I thought this was a total success. All the kids worked excitedly at the boards this time versus two years ago. There was a fair amount of cross communication between the sides of the room as answers were discovered, A few times. I thought a kid was sitting down in a char to disengage, but in each case they were only thinking and then got up and went back to the board to write down a new idea. Afterwards even though I had brought boards games for an end of the quarter celebration some of them  even continued to work on the problem looking for solutions to 31, 32 etc.   I'm definitely going to keep playing with this format. Perhaps this is also part of the answer for middle school next year.

I actually had my end of quarter / game day activities planned as well for the day. Since the kids had seen all the materials (pente, prime climb, terzetto, rush hour,tiny polka dots) and were excited to play with them the previous experiment was even more impressive. There was very little attempts to break out during the 20 minutes or so. In addition to the above mentioned games I also had https://en.wikipedia.org/wiki/Sprouts_(game) in hand to try out on the board.  This game was new to the group I thought this would dove-tail well with the previous activity.



We were a bit short on time due to being temporarily locked out of the room in the beginning so rather than having the entire group play, I strategically pulled pairs of kids out showed them the rules and had them try it out. In the end I probably drew about half of the Math Club in. We will be looking at Sprouts more in the future to look for patterns and strategy.

Wednesday, March 22, 2017

3/21 Graph Pebbling

This week I went back to a pure math circle format with my favorite activity from the recent Julia Robinson Festival: Graph Pebbling. Based on my experiences at the festival I thought it would occupy 30-40 minutes so I decided to do a warm up puzzle as well. Initially I had considering doing a battleship puzzle (see: https://www.brainbashers.com/battleships.asp) but I found a tweet from Sarah Carter that looked interesting about slant puzzles: https://mathequalslove.blogspot.com/2017/03/slants-puzzles-from-brainbasherscom.html.  These have a fairly simple set of rules: put a line through every cross and make sure to have the requested number of lines connecting to each square with a number. Unmarked square are free and can have any number of connections.





Simple is often good though. All the kids really liked them:



We then transitioned to graph pebbling: The full rules are here: https://drive.google.com/open?id=0B6oYedIeLTUKc1hWSWtHMi1vbHM   A series of graphs are included as well as 5 variations. For Math Club I used lima beans again as "knights"


My only issue was I have one table of boys that are harder to keep on task. I tried separating them a bit this time which didn't quite work but I may do it again next week but from the start. They're not disruptive per. se but they are distracting each other and only stay on task when I come over and work with them.

P.O.T.W

A fun factoring / number theory problem for this week:

https://drive.google.com/open?id=1v9i-1YP9OeAjI7i0lZd5BcMUXi3B8OSavtzS-riEHEY