Reversed inequality

Recently, Mike Lawler posted a challenge base on the first question from the recent European Girls Math Olympiad (side note: what is "European" about this contest now that a US team participates?)

I enjoyed thinking about this problem and wanted to come up with something related to do with our kids. One way to see this problem is that it appears to reverse the direction of the inequality between the arithmetic mean and geometric mean. My version for younger students involves exploring this inequality first.

Discovering the Arithmetic Mean-Geometric Mean Inequality

I had J1 and J2 select two dice from our pound-o-dice. Initially, J1 chose a d20 and d6 while J2 chose a d12 and d6. I asked them to make a table with 6 columns and 7 rows (we ended up adding more, so probably better to ask for 8 columns and 10 rows). I had them label the columns:

1. A
2. B
3. AxA +  BxB
4. 2xAxB

They rolled their two dice and put the values into columns A and B, the calculated the other two columns as labeled.

J1's first results were 16 and 5. After filling in the rest of that row, he paused and then switched to 2d4. This wasn't a problem for our investigation, but he did miss some calculating practice with more difficult seed numbers.

After filling in 6 rows of data, I asked them what they noticed.

Here were some of the observations:

• when A and B are the same, our calculated values are the same
• when A and B are different, our calculated values are different
• When A and B differ by 1, the calculated values differ by 1 (also vice versa)
• AxA +  BxB > 2xAxB
• We are only using positive integers

J2 really got into the spirit and asked for some more suggested columns. I told him to add:

AxA - BxB and (A-B) x (A-B). J1 also added the square of the difference.

With that, they noticed two more things:

• AxA + BxB was larger than (A-B)x (A-B)
• AxA + Bx B = 2x Ax B + (A-B)x(A-B)

You can see that they also added some negative numbers on J1's sheet and challenged one of these observations.

A picture proof

To finalize our exploration of the inequality, we used tiles to build intuition for a picture proof of this identity:

AxA + Bx B = 2x Ax B + (A-B)x(A-B)

This is one of the pictures we liked the most. In this case, A = 5 and B = 2. Along the left side and the bottom are rectangles AxB (the green + yellow and the blue + yellow regions). These two rectangles overlap in a BxB square (the yellow region) which they also highlighted with the 4-color square. The remaining red square is (A-B)x(A-B).

Side note: In our discussion, I was delighted that J2 would correct me whenever I slipped and said "greater than" instead of "greater than or equal to" when discussing the inequality.

Two quick algebra extensions

For kids who have already done a little algebra, proving the identity using the distributive law should be fairly straightforward. The other little extension is to show that our expression A²+B²≥ 2AB is equivalent to the arithmetic mean-geometric mean inequality.

Dice mining (grades 2 and 3 math)

Notes for Grade 1 will come later

This week, we continued our run of dice games. This is a simple game we found in Marilyn Burn's About Teaching Mathematics. She calls it Two Dice Sum Game, but I like Dice Miner better (to pair with Dice Farmer, of course). Here's how to play:

1. Students make a number line with slots for each integer 2 to 12
2. Everyone gets 11 blocks or other counters (we used unifix cubes which were nice for stacking).
3. Players put their counters on the numbers, distributing them in whatever way they want. In particular, they can put more than one counter on a number or none on a number.
4. Everyone takes turns rolling two dice (normal 6-sided). Players can take one counter off their board for the sum of the two dice. For example, if I have 3 counters on 7 and roll 6+1, then I can take one counter off and now have 2 counters remaining on 7. If I had no counters on 7, then I would not take any action.
5. The first player to have all counters removed is the winner.

Any activity with construction cubes is bound to create opportunities for other mini-conversations:

• using the cubes as a ruler to make the number lines
• comparing how many cubes one person has to another by measuring lengths (before we made sure everyone had exactly 11)
• using the cubes to form a ruler to measure other things (our whiteboard, a table leg, a friend)
• making a pattern with light and dark colors from the cubes

As to the main game, there is some basic arithmetic practice through all the addition, but the real interest lies in trying to figure out how best to arrange your counters at the start. All the kids had their own hypotheses, but it was interesting to see these highlights:

• One student realized that 7 was the single most likely result and put all his cubes on 7 (this did not win that round)
• Most students started with a uniform distribution, one cube on all numbers
• There was a surprising amount of enthusiasm for 2 and 12, with many students initially playing multiple cubes on those extremes
• One student tried to be tricky and put cubes half on 6 and 7, presumably planning to take them off if either number came up.

Toward the end of the class, there were two comments that were really interesting. First, one student suggested rolling the dice many times, recording the results, and seeing how often all the numbers came up. This would then inform his strategy for how to distribute the dice.

We asked the students how many times they would have to roll. Would 1 or 2 rolls be sufficient? No, everyone was sure that wasn't enough. What about 25 times? One student pointed out that there are 11 slots, so 25 times is only an average of 2 per slot, so that didn't feel like enough to get a good sense of the "true" answer. Most students were eager to see for themselves, so this became their homework.

Homework

1. test the distribution by rolling two dice 100 times (or more) and tallying how often each number comes up as the sum of the two dice.
2. Play Dice Miner 5 times with your friends/parents at home. Record your initial starting position for the cubes, how many rolls to take them all off and who won each round.

Seasonal math and dice, dice, dice

who: J1, J2, J3
what did we use: mangosteen(!), dice, a checkers set.

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Queen of mathematical fruits

My condolences to everyone outside of Thailand, you are current missing some delicious pre-school math: Mangosteen!

The mathematical secrete is on the bottom of the fruit. The flower shape on the bottom has petals that correspond to sections of the fruit inside. This one will have 7 sections

Just to trick you, this is a view from a different one, with 6 fruit sections.

J3 has been having a lot of fun counting the "flower" petals, opening the fruit, then counting the segments to check.

J2 has been using these to practice subitizing (recognizing number groups without counting them). J1 has just been gobbling down the mangosteen.

Dice Games

A dotty dice puzzle
In school for the 2nd and 3rd grades, we introduced a tic-tac-toe variation using dice where the winner is the first player to make a straight line that is a multiple of 5 (more notes here). For homework, the kids played this game with other multiples, in our case 6. During one game with J1, there was an interesting scenario that I saved to discuss with J2. The board has the numbers as marked and the player has just rolled a 4. Where should you play (you can add 4 to any cell of the grid, remember that you don't want your opponent to be able to make a multiple of 6 on their turn):

Race to 20
For J3, we modified one of the games that we had played in the 1st grade class last week. In this game, two players race to be the first to 20 on a 100 board. In this version, both player has a token/marker of their own. For each move, the players roll 2 dice and choose which value to move, then count up their position on the board. First player to 20 or more wins.

The strategy here is very simple, but it was a good game for J3 to start learning about dice and the 100 board. Once she got the point of the strategy, though, she had her own idea:"I'm going to add both dice together and move that amount!"

Snakes and Ladders experiment
J1 and I tried an experiment with snakes and ladders: a race between two different teams.
Team 1: a single marker that has to move according to the roll of one hexahedral die.
Team 2: two markers, the player can choose which one to move on each turn.
The winner is the player who gets a marker to 100.

Which team has an advantage? How big is that advantage? What if you play so that the winning marker has to land exactly on 100 (otherwise bounces back)?

Knights Move

SolveMyMaths recently tweeted about Knights Move (aka Razzle Dazzle.) Since we are always looking out for more/new games to play, it sounded interesting. Also, we could repurpose existing material to play. The game is a cross between chess and basketball. Below, see J1's set-up:

Yes, we are using icosahedral dice as the balls. Also, the chess pieces on the side are spectators, they aren't part of the action (since this is played on a 7x8 board).

One of the most interesting parts was our ability to experiment with rules. Since we didn't have a definitive version, we felt free to try slight modifications. In particular, we experimented with:

• Passing: allowing multiple (chain) passes or just a single pass on a turn.
• Defense: moving into the 8 squares around the ball holder forces them to pass on their turn
• back-and-forth passes: player who passed the ball has to move again before they can receive the ball (we turned the checkers upside down to indicate they couldn't receive a pass). 

Cake is wonderful

who: J1 and J2
where: kitchen
when: afternoon during J3's nap
what material did we use: dish soap, water, a plastic drink bottle, zometool set, and two small blocks of solid CO₂

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Last weekend, a generous uncle brought two pieces of chocolate cake for the older J's (littlest J can't eat dairy, so has to make due with dark chocolate squares). The cake was nice, but the real delight was in two small pieces of dry ice that we part of the packaging to keep the cake cool. While J2 had his violin lesson, J1 and I looked for good activities to take advantage of this bounty.

We hit on the Crazy Russian Hacker's 8 dry ice activities (video linked below). Of those, we selected the smoke bubbles and the smoke rings since they seemed cool and possible with our resources. These turned out to be really easy and fun activities.

Here are some videos of the smoke rings:

By coincidence, our friend Pongskorn Saipetch was also playing with dry ice recently, you can see his video here. From the size of his piece of dry ice, I have to guess he had a lot of cake!


After the dry ice had melted, we made bubble wands out of zometool and continued to play with the soapy water:

Our video inspiration. While our photos and video didn't turn out this well, the actual experience was at least as much fun as what you see here:

The high chair for learning inequalities (also, a broken calculator)

who: J2 and J3
when: at lunch
where: local Japanese restaurant

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Who is taller

While eating lunch today, we found a good excuse to talk about (mathematical) inequality. Next to our table were two spare chairs, a kid high chair and a standard adult chair. The natural questions:

• if J2 sits in the high chair and J3 in the adult chair, who will be higher? 
• Are you sure and why do you think so? 
• What if you switch with J2 in the high chair and J3 in the adult chair? 
• How confident are you of the answer now?

In the course of the conversation, they talked about who is taller standing (J2) and which chair has a higher seat. It made intuitive sense to them that the taller person in the higher seat would end up higher. Still, it was good to test:

For the question about switching seats, they weren't sure, but thought J2 would still be taller (he was). Finally, I asked J2 if this would always be the case: if he sat on a lower seat, would he still be taller? After a minute's reflection, he said it could be either of them. Could they happen to end up the same height?  Also, yes!

With these simple props, it ended up being a surprisingly good conversation.

A broken calculator

After reading Mike Lawler's post about of Dan Finkel's Broken Calculator puzzle, I had to share it with J2. He was asleep at the time, so I made my own in pencilcode (a souped up version here). This morning, after breakfast, I showed it to J2, gave him the back story. We briefly talked about square roots to remind him, and then he was hooked.

You can see his current progress here, working toward finding a way to get every integer from 0 to 109:

Mike's post and videos are very good, so I only want to make a couple points to complement his discussion:

1. Playing with the calculator first made the problem much more accessible. For J2, it helped him see that the +5 and +7 buttons could only make the value larger. It also helped him recognize that he needed square numbers for his square root and to strategize about how to make them. Finally, it led him to discover the trick for making 1.
2. Making other numbers than 2 became a very natural extension that he asked on his own. At first, he started recording (or having me record) the numbers he had made on a paper, then I added the table to our program to keep track automatically.
3. He had fun the rest of the day asking other people, mostly his mother, if they could figure out how to make 2.
4. It was also very easy to extend this by asking about other combinations than +5 and +7. We played with a +6 and +7 version that is, conceptually the same, but practically much more difficult since you lose the ones-digit preservation.

For anyone who wants to sneak in some calculation practice, this served that purpose, too. Why, you might ask? Even though he could always see an answer by pressing the button, there was a cost if he pressed the wrong one because then he would have to go through his sequence again. As a result, he would pre-calculate each operation to make sure it was taking him along the right path.

Finally, this same framework could be used easily with other operations. In particular, for kids who aren't yet ready for square roots, the reduction button could be division (e.g., divide by 4) or even subtraction (e.g., subtract 19).

Pseudosphere Hat and our Robot begins (some arts and crafts)

Who: J1
When: around lunchtime
where: on the floor

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Today, we tried making a couple of things. First up, was a pseudosphere.  The inspiration for this is a really nice post from Daniel Walsh: Sudo Make me a Pseudosphere. By all means go to the original post as the pictures, animations, and video he posted are far better than what we managed, but it was still a fun conversation about shapes, angles, slope, and fractions.

The process is easy:

1. cut out a bunch of equally sized circular discs
2. cut the discs into different sized (different angle) sectors
3. make all the pieces into cones
4. stack them from shallowest to steepest

Daniel mentioned something about calculating the optimal sizes, but I didn't really know what he meant. We went for a child's dinner plate for our circular template and cut sectors in multiples of 45 degrees.

Only the finest used newsprint for us!

One good question came up along the way: if I cut out a larger angle, will the cone we make be steeper or flatter?

Pseudosphere taking shape

There was another point where I'd cut out a 135 degree sector and J1 said: that's 1/3.  When he measured very roughly, it did seem to be a third, so I asked him to try it more precisely. He had a sudden realization when he saw the 90 degree remainder.

The cone of our dreams!

We went one step farther and permanently attached all the cones together, then re-purposed the whole thing to create 2015's fashion must-have item: the pseudo-rocket pseudosphere hat:

Starting our Robot

Our other activity is really the beginning of a longer project.. J1 has been talking about making a robot and we are starting to work on some of the main functions. Of course, he is really excited about camera eyes and a laser cannon, which we'll get to eventually (will we?) For now, I have some ideas about how to get the robot to walk.

My plan is to connect a basic rotating electric motor we have, so that leaves us solving an old problem: how to convert rotational motion into straight-line motion. Of course, wasteful methods are easy, but we want our robot to have the maximally powerful stride. For now, we are investigating multiple linkages, in the footsteps of Chebyshev.

Below is a first test: three linked arms:

1. Arm 1 has a fixed end and is intended to rotate in a circle
2. Arm 2 has one end fixed to the rotating end of arm 1. The other end of arm 2 is the motion we want to examine
3. Arm 3 has one end fixed and the other end attached to the mid-point of arm 2. This constrains that midpoint to travel along a circular arc

You can see our ultra-high tech implementation below, using card paper as the arms, a large cardboard box as the base, nails (into the base) to create fixed points, nails point up to create hinge joints, and extra bits of cardboard to cover the point ends of our hinges and past muster with  the health-and-safety inspectors:

The action of the multiple hinges is pretty wild. J1, J2, and I enjoyed cranking arm 1 and watching arm 2 fiddle around. Carefully holding a pencil in place, we managed to draw he path of arm 2's free end. It is the rounded wedge that looks very close to a circular quadrant.

If you want to see some great animations of multi-hinge contraptions, check out the animations at Mathematical Etudes. I'd be delighted if we could get close to this one.

Avocado update

I'm pleased to announce that another family member, D, has started sprouting her own avocado pit and, apparently, has made this into a race.  When told the news, J1 and J2 immediately started guessing what type of sabotage techniques would be employed by D. I think this says more about them than her.

Total mass: 67 grams
Length from root tip to sprout top: 21 cm
Length from pit to sprout top: 12.3 cm

A christmas eve mystery

Who: J3
Where: at school
When: over 2 weeks

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This is actually something being done at school, but it matches our seed growing at home very nicely.
They did a couple of experiments to test the effect of different conditions on plant growth:

• with and without water
• in different potting media (soil, sand, rocks)
• with and without sunlight.

Here is the picture of the plants without (left) and with sunlight:

Sorry about the blur, this was my only shot through a swarm of excited toddlers

Thus, the mystery: why did the plants grown in the dark grow so much taller? Add your hypothesis in the comments!


This is a special day for our family: Grandpa G's birthday.  So, in the spirit of celebration and birthday wishes, we send some powers of 2 (and square relationships):

Sometimes 6s got to get a bit crazy, right?