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:

A little easier to read:

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: 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: 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: 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.


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 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: but I found a tweet from Sarah Carter that looked interesting about slant puzzles:  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:   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.


A fun factoring / number theory problem for this week:

Wednesday, March 15, 2017

3/14 Pi Day

Every 7 years or so accounting for leap years, Pi day actually occurs on a Tuesday. Yesterday was the first time that occurred while I've  been running the Math Club. Because most of the kids were here last year I did not go over my usual conceptual question "Why is the circumference of a circle in a constant ratio with its radius, and why such a funny value?"


I fall into the camp that its fun to celebrate as long as something mathematically meaningful occurs during the party. I also try to de-emphasize anything to do with memorizing digits. So due to all the apple pies being taken this year I picked up a strawberry rhubarb pie at the local super market which I served as everyone arrived in the cafeteria. This kept the mess to a containable minimum and as expected the kids were all very excited by the treat.

Like last year I decided to also do a pi day themed video after the following one showed up in one of my feeds:

After we were done I had another NASA packet to try out:
I tried this type material once before (space map session).  Since some of the kids liked it before, I thought 20-25 minutes would be about the right amount of time to try a similar activity again. I'm not completely keen on the formula plugging involved but in watching the kids, its actually useful every once in a while to use real, messy physical values and reason a bit how to apply basic geometry.

Overall everything went smoothly including setting up the video (cabling + wifi). The setup time did mean the kids fooled around for the 2 minutes before I could start but that just took a little extra talk to get the room's attention and settle in.

A not too hard but perhaps counter-intuitive circle property from

Wednesday, March 8, 2017

3/7 Olympiad #5

Today started with a small mix-up. A boy I recruited at the Julia Robinson Festival to join Math Club showed up. But the next quarter doesn't start for 3 weeks. I offered to let him join us anyway but I think he was too embarrassed. Hopefully, he'll still come on the real first day. The whole incident is a reminder that even though I assume I know most of the "mathy" kids in the grade, hidden depths are out there.

After that, the rest of the day went  more smoothly and had several small rewarding moments. We started by running down the  Problem of the week as a group.  I only had one student demonstrate how to divide the boards (its a stair like cut) and unfortunately this didn't generate as much problem solving discussion as I prefer.  From there, we completed the last MOEMs Olympiad for the year. Looking this one over, I thought it was among the trickiest of the series. We'll see how the scores go but several of the problems had fairly complex instructions to deduce the answers and I think the general trend will be a bit lower than the last one. I did have a good group problem solving session afterwards and had the kids show solutions for all the problems.  One small tweak I've implemented is to write the problems on all the whiteboards while the kids are working so we're set to go for the group discussion.  Kids were well focused through the entire time with the only extra chatting being about how to solve the problems differently. The one future topic I noticed among the problems was to work a bit on explaining how choosing unordered sets work i..e  \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)  This pairs well with a dive into Pascal's triangle. I'm going to take a look at Arthur Benjamin's book to see if he has an approach that is adaptable for a group.

For the light activity I had all the kids who finished early working on an Euler path exercise from "This is not a Maths Book"

The kids found this very interesting and it again could be a topic for a whole session.

Other Ideas from around the web I'm thinking about for future meetings:

Problem of the Week

Wednesday, March 1, 2017

2/28 Infinite Series

For this session of Math Club I wanted to revisit one of the ideas from the "free the clones" games: (See:

What is the sum of the infinite series  1 + 1/2 + 1/4 + 1/8 ...

On reflection, I decided this would make a nice connection with converting repeating decimals back to fractions. I had actually tried this 2 years ago and it went okay. Most kids can convert fractions to decimals but can only handle non repeating decimals in the other direction. But in the intervening time I had lost the worksheet I used back then.  This time, I wanted to risk it and just work on the whiteboard, have the kids go off and experiment and come back and discuss what they found.

Planned Questions

1. What is .999999... equal to and why?

2. How can we represent .99999.... as a series of fractions.

3. Warm up with some easier ones.

S = 1 + 1/2  +1/4 ....
S = 2/1

S = 1 + 1/3 + 1/9 ....
S = 3/2

S = 1 + 1/4 + 1/16
S = 4/3

4. Find the pattern and then come up with the general case:

S = 1 + 1/n + 1/n^2 .....
S = n/n-1

5. Ok let's go back to decimals

S = 1/10 + 1/100 + 1/1000 just like above. Can you use the same technique?

How about if the digits differ

S = 12/100 + 12/10000 + 12/100000

Final Conundrum

1 = 2 / 3 - 1 vs  2 = 2 / 3 -2 as continued fractions.


I actually started by having everyone talk about the Julia Robinson festival. A couple kids mentioned the final flatland talk and this was of sufficient interest that I ended up spontaneously repeating a huge section of it for those who weren't there.  My retelling was accurate except I didn't have any klein bottle pictures on hand other than one on my phone. This ended up taking at least 10 minutes and I would repeat and make a day of it based on how it well it was received.

Basically you have a town in a 2 dimensional world  and the inhabitants assume they live in an infinite plane but have never explored it. Then finally one tries it out and discovers if he goes north and leaves a trail he arrives back in the town from the south side etc. Given the behavior when the inhabitants go N, E and then NE you conjecture the existence of a sphere, torus and then klein bottle. 

As I result I ended up skipping my planned kenken warm up. We made it through about question 5 from above but by this time I had exhausted the focus of the group, it was getting harder to keep everyone on task. So I made the executive decision to pull out the kenken puzzles after all and "cool off". Fortunately, that pulled everything together again.   My take away from this is:

  • Kids were aware that .9999 =  1 but the explanation was a bit fuzzy (no numbers between 9 and one) but I didn't have enough time to circle back at the end and show why this must be the case.
  • This was still too much material, I need to break it up with something "lighter" if I try again. I think I want either a visual interlude (color in one of these infinite series?) or to gameify the middle somehow.


I went with this puzzle from The Guardian:

Julia Robinson Festival

(The flatland talk at the end of the afternoon.)

For the second year in a row, I volunteered at the Julia Robinson Math Festival over the weekend. This is among my favorite mathematical activities to do for the whole year. This time around  I went to the training session before hand. That was useful, since I had a chance to look at the problems I would be facilitating prior to actually jumping in.

My first one was a bit daunting from the perspective of maintaining interest. The first part was to figure out the brain teaser: What comes next in this sequence?

      1 1
      2 1
   1 1 12
   3 1 1 2
2 1 1 2 1 3
3 1 1 2 1 3

This took me almost 25 minutes to see by myself and I worked through a bunch of different ideas. My goal was document all my wrong approaches so I could anticipate what students my do. I also knew it involved some lateral thinking. As I remember my main thought was "Gosh I hope this isn't something silly like number of curves and lines in the numbers."

At any rate, I was pleasantly surprised during the actual Festival.  Based on the prep work I managed to keep multiple students occupied for 30+ minutes in the productively stuck state. The main thing I did was to have folks work together, keep close tabs on everyone and ask about what they were trying. I also tried to emphasize regrouping the pyramid as a triangle and looking for patterns.

My second table was a really cool graph theory game.  I'm going to use this in Math Club and I will talk about it more then.