Math Teachers At Play Carnival #110 Summer Vacation Edition

Hello again math folks! I've been in the middle of a major transition, moving between Asia and North America, so haven't really had time to post recently. Putting together this month's carnival was a nice opportunity to see what everyone else has been writing about and get some new ideas!

As you scan through the links I've highlighted, please don't get too fixated on the grade level splits. These are really approximate and I expect you will find worthwhile activities for all ages in every section.

In Memoriam: Maryam Mirzakhani

On the 14th of July, Maryam Mirzakhani passed away. She was the first woman to win the Fields Medal. It would be wonderful if you could do some exploration in her honor this month. One of her areas of research was on pool tables. Here are some places to get an idea of the way mathematicians have been inspired by this game:

• Elliptical pool (from Numberphile)
• The Illumination Problem (also Numberphile)
• A snooker (problem 3 from the 2013 International Tournament of Young Mathematicians)
• Wolfram MathWorld Discussion of Alhazen's Billiard Problem that indicates the link with optics

If you find other kid-friendly projects related to Mirzakhani's work, please tell me in the comments!

Some 110 facts

This was the best number carnival to be able to host, because:

• 110 = 10 * 11. That means it is pronic, the product of two consecutive integers.
• 110 looks suspiciously like a binary number. Binary 110 = decimal 6. Decimal 110 = Binary 1101110, which I like to read as 110 1 110
• Because it has an odd number of 1s in its binary expansion, 110 is odious
• 110 is a Harshad number because it is divisible by the sum of its digits
• The element with atomic number 110 is Darmstadtium (Ds).
• 110 is the number of millions of dollars spent in March for a Basquiat painting, the highest amount paid at auction for a work by an American artist.

A number talks picture that caught my eye

I'm not sure there is anything especially 110 about this picture, but there are a lot of mathematical questions to ask and things to observe here:

In a related vein, if you and your kids need some mesmerizing math gifs, take a look at Symmetry.

Elementary skills

Denise Gaskins has written a lot to help parents engage playfully and mathematically with their kids. In this blog post, she has collected highlights that are great with young students and worth remembering for older ones, too: How to Talk Math with Your Kids.

I love board games and think there is still tremendous value in the physical games that electronic versions miss. Here's an example from Sasha Fradkin, where cleaning up after playing gives us a chance to think about whether skip counting is just a chant or if the words mean something: Skip counting or word skipping

While she's at it, Sasha Fradkin also has a nice puzzle activity with Numicons. I would think of this as a progression step toward tangram and other dissection puzzles.

Which one doesn't belong is a math meme you should know already. If you don't, ask in the comments and I'll point you in the right direction. Christopher Danielson has recently introduced Which Poster Doesn't Belong? While you are visiting his blog, enjoy his story about The Three Year Old Who is Not a Monster.

Exploding Shapes is a catalyst for notice and wonder from The Math Forum. I really like this because here are many different directions to go and no single "right" answer. Also, let's give a cheer because it looks like this recent set of posts shows the math forum folks have returned to posting nice conversation starters.

Swine on a Line by Jim Propp is a nice game/puzzle that seems a great companion to James Tanton's Exploding Dots. Hmm, maybe July 4th inspired me to look for lots of explosions...?

Middle school(ish)

Rupesh Gesota starts with a nice puzzle and shows us how it was analyzed by several different students: One Puzzle, Many Students, Many Approaches. I particularly like how the introduction to the puzzle encourages us to think of different methods.

Mike Lawler has done a huge number of really great explorations with his kids. Here are some recent projects with books from the Park City Mathematics Institute: Playing Around. If you haven't been following Mike and his kids, I really encourage you to go through his past posts.This blog is fantastic for great projects and connections with other resources.

Curious Cheetah shows us several ways to calculate square roots. I would say, like long division, the value isn't in memorizing the algorithms, but understanding how they work and using them to play with numbers.

Manan Shah has a couple of nice summer explorations. The first is an excursion into the digits of prime numbers: Prime Numbers. The second is a coin flipping and gambling game to ponder during these warm vacation months: Summer Excursion Coin Flipping.

There are other, problem-based, posts on Benjamin Leis's blog, but this one made me jealous of his recent purchase of the A Decade of the Berkeley Math Circle.

High school/more advanced

Thinking Inside the Box, Simon Gregg takes a new look at a familiar shape, the cube. His comment about the exploration really nicely captures something that is beautiful about mathematical exploration: "I came back to a familiar place from an unfamiliar starting place."

A cute absolute value game now appears as a nicely animated game: Absolute Value. I think this is a nice simplification and implementation of the original game.

There's an improv game where the players have to switch between movie genres. Film noire or "hard boiled detective" comes up every time. This TedEd video could introduce fractals and this film genre at the same time.

Michael Pershan puzzles over two measures of steepness in his trigonometry class: When Measures of Steepness Disagree. I really like the questions he raises about how to use two different scales that measure the same concept, but are not linearly related.

Also, Michael links to the New Zealand Avalanche Advisory, with a nice graphic showing a case where the greatest danger of avalanche is in the middle of a slope range:

Dave Richeson breaks down an impressive rainbow photo:

Fair sharing is a really interesting theme to motivate a lot of great math. Tanya Khovanova looks at a couple of fair sharing problems and strategies in Fair share sequences.

Also, check out the sister carnival to this one: The Carnival of Mathematics over at The Aperiodical.

Techniques for teaching

This post is an old classic, but I've been reminded of it because it is used in a workshop that I frequently attend: using student reflections.

Have you visited NRICH recently? No?!?! Go over now (here's the link) and find a really cool activity to do with your kids. Seriously!

Some tips on giving feedback: Effective feedback for deeper learning.

Using Desmos to check your work: Desmos is the new back of the book.

A thought piece on the modern role of teachers: Teachers Sow Thirst for Learning. If you can read Indonesian (which I can't) you may find some other interesting pieces here on math education.

A special announcement

James Tanton is leading a project for a world-wide week of math this fall. Please take a look at the project page Global Math Week

Leftorvers with 100 game

In Grades 3 and 4, we played a nice game that (I think) we got from Marilyn Burns. Looking for a reference after the fact, I see it explained in her book Lessons for Extending Division.

Basic play

• Start with a target number (we used 100) and collection of available divisors (we used integers 1 to 20)
• Players take turns choosing a divisor from the remaining available options. They divide the current target by that divisor and keep the remainder as their score for the turn. They also subtract the remainder from the target to create a new target for the next player.
• Each divisor gets crossed out when it is used, so it can only be used once.
• The game ends when the target is reduced to 0 or when all available divisors are exhausted.
• We played as a two player game.

Here's an example of a game play:

Player 1 chooses 17. 100 = 17 * 5 + 15, so player 1 scores 15 points, the target is reduced to 85, and 17 is no longer available as a divisor.

Player 2 chooses 20. 85 = 20 * 4 + 5, so player 2 scores 5 points, the target is reduced to 80, and 20 is no longer available as a divisor.

Player 1 chooses 14. 80 = 14 * 5 + 10, so player 1 scores 10 points, the target is reduced to 70, and 14 is no longer available as a divisor.

Player 2 chooses 18. 70 = 18 * 3 + 16, so player 2 scores 16 points, the target is reduced to 54, and 18 is no longer available as a divisor.

Player 1 chooses 19. 54 = 19 * 2 + 16, so player 1 scores 16 points, the target is reduced to 38, and 19 is no longer available as a divisor.

Player 2 chooses 13. 28 = 13 * 2 + 12, so player 2 scores 12 points, the target is reduced to 26, and 13 is no longer available as a divisor.

Player 1 chooses 15. 26 = 15 * 1 + 11, so player 1 scores 11 points, the target is reduced to 15, and 15 is no longer available as a divisor.

Player 2 chooses 16. 15 = 16 * 0 + 15, so player 2 scores 15 points, the target is reduced to 0 and the game ends.

Player one wins 52 to 48.

Our experience

We found this to be a fun, interesting, and engaging game. The practice with dividing and remainders was pretty obvious. In addition, it opened up some opportunities for strategic thinking, particularly at the end-stage of the game. I think there are also several good extension explorations.

Extensions

First, I created a simple pencilcode program for two players to play this game against each other. Here's a playable version (and here's the code).

Second, you'll notice that the first player in our sample game followed a "greedy strategy."  At each stage, that player chose the divisor that would give the most points on that turn. If you look closely, that isn't the best strategy at the end of the game.

So, a natural exploration is to find the best strategy for different starting targets. One specific point of about which we're curious: is it ever desirable to skip your turn (choosing 1 as the divisor is effectively a turn skip)? 

Some other areas for investigation:

• must the game always end on 0 or can we run out of divisors?
• given a target and collection of starting divisors, what is the shortest (number of turns) game possible? What is the longest game (number of turns) that does end at 0?

Times square variations (math games classes)

In grades 2 and 3, we have been playing with variations of NCTM's game Times Square, one of their offerings on Calculation Nation. This is one of my favorite multiplication games because, like the puzzle Bojagi, it is fun and multiplication is integral to the game, it isn't just a set of flashcards in disguise.

Here's a basic Times Square board:

The AI doesn't understand edge vs center!

Players take turns moving one of the square windows at the bottom to select two numbers, then get to take possession of the square that is the product of the values the windows are on. In our starting game, the AI moved the first window to 6, I moved the second to 5 and captured 30 (5x6). The AI then moved from 6 to 1 and captured 5 (1x5). On their turn, the player can move either window, but has to capture an open area (you can't duplicate a product you've already captured or take over an area your opponent has previous captured). The first person to get 4 in a row wins.

A pen and paper version
We didn't have (or want) computers for all the kids to play online. Instead, we created a simple paper and pencil version. We made many copies of the board on a piece of paper, with the numbers 1 to 9 at the bottom. We then used small rubber bands (loom band left-overs!) to select the factors and players used colored pencils to claim their territory.

It was an easy, colorful, and fun implementation of the game:

Notice the sad faces where mom/dad won that round?

Noticing the structure
As usual with this kind of activity, there are many possible extensions, with two obvious groups being strategy (how do you win the game) and structure (what do you notice and could change about how the game is set up).

So far, we have been looking at structure. Here are some of the things we discussed relating to the basic board:

1. What shape is the playing board? How many numbers are on it?
2. What is the largest number? Why aren't there that number of spaces?
3. What is the smallest number (positive integer) that isn't on the board? Why?
4. What numbers are missing from the board?
5. If we say an integer between 1 and 81, can you tell, without looking, whether it is on the board?

Make it simpler

The next iteration was an exercise in simplifying. Do we need to use all factors 1 to 9? What if we made an easier game with factors 1 to 4? Here is the version we came up with:

Surprise, surprise, we can still make a nicely shaped grid! Now, we aim for 3 in a row, like standard tic-tac-toe, but with a constrain that means we can't always move where we would like. With this simplified version, maybe we can go back to our strategy questions and gain some wisdom that will help for the 1 to 9 version?

Make it more complicated

What if we wanted a harder (calculation) challenge than 1 to 9? Are there other collections of factors that would give us nicely shaped grids? We had them work out creating a grid based on factors 1 to 13.

It was really interesting to see the different strategies that the students took to determining what would go on their boards. Some people tried creating full multiplication tables and then removing duplicates. Other people counted up from one and tested each number as they went along. Some people identified patterns, essentially working with the diagonal and upper half triangle of a multiplication table.

Here's a student, hard at work calculating the 1 to 13 board:

In this case, there are 72 distinct products, so the students also had a choice of making near-square boards that are 8x9 or 9x8. We didn't guide them to these shapes, but it was interesting that no one made a 6x12, 4x18, 3x24, 2x36, or 1x72 shaped board, 

For the 8x9 and 9x8 boards, we had them take some time to play on each version. Does play feel different on the two different boards? Is there a different strategy for the two boards? Perhaps you will also experiment with this.

Further exploration

A sequence

How is the sequence 1, 3, 6, 9, 14, 18, 25, 30, 36, 42, 53, 59, 72, 80, 89, 97, 114, 123, 142, 152... related to this game? These are the numbers of distinct products of the integers 1 to n as n grows. What can we say about this sequence? For example, how quickly does it grow with n? Is there a closed form for the nth term?

Can we see anything interesting if, instead of using integers 1 to n, we use a different collection of n integers? For an easy one, try using the first n primes. Maybe using n integers that are in an arithmetic sequence would be interesting?

This simple pencilcode program could get you started on gathering some data: TimesSquareBoards.

For some light, related reading: Number of Integers with a divisor in a given interval (Ford 2008) which was linked on this Math Stackexchange question.

Strategy and Structure

Ok, so we can take factors 1 to n, then create a board that arranges the distinct products into a rectangle. Because we see primes in the sequence above, we know that some of these rectangles are just 1 x p (or p x 1) shaped.  Even so, what can we say about winning strategies:

• When does the first player have a winning strategy?
• When does the second player have a winning strategy?
• When does optimal play by both players lead to a tie (like classic tic-tac-toe)?
• Are there n for which differently shaped boards have different winning strategies? Is there an n which has 3 differently shaped boards that cover each of the different strategy outcomes (one that is a first player winner, another that is a second player winner, a third that ends in ties?)

In particular, I think it would make for a delightful bar bet if, say, the first player had a winning strategy for the 8x9 board, while the second player has a winning strategy for the 9 x 8 board!

A picture, just for the heck of it

Having nothing to do with any of this, what estimation and math questions do you have about this picture:

Yes, these are gold, but just covered with gold leaf, not solid!

We are totally flipping out!

Who: J1, J2, J3
when: dinner time
What did we use: 4 coins

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In Sue VanHattum's linkfest post, there was an intriguing puzzle: given four coins in a square, can you call out commands to flip coins so that you are guaranteed to eventually get them all up or down? More precise instructions/rules are in the link.

We saw very quickly that this is an excellent puzzle:

• complex terminology? Not at all.
• difficult to find equipment? Nope.
• low threshold? Yes, Even J3 was able to investigate along with the older ones
• high ceiling? Yes. Solving the initial puzzle is hard enough, but there are plenty of possible extensions. 
• Fun and deceptively tricky challenge? YES!

Everyone got to have fun taking a turn as the coin master (following the flipping instructions and rotating the square) and most of us took a turn as the (attempted) solver. Here are our tools:

For most of the conversation, I just listened. The kids naturally focused onidentifying states: what are the possible conditions for the coins, which of these are, for purposes of the puzzle, identical and which are truly different? One other part of the discussion was about moves, again, which are identical and which are different. Of course, the didn't quite use this terminology.

Some extensions

Simplify?
I suggested they try the puzzle with a smaller number of coins. 1 coin, no problem, it always starts solved. 2 coins were pretty easy again. What about 3 coins? Here, there was a little thinking about what the equivalent version would be for 3 coins. In particular, do the coins have this configuration:

and the coin master can only swap the outer two coins, or are they in an equilateral triangle like this,

and rotations are the only transformation?

Well this is math, so there are no right or wrong answers, you just have to try out your different ideas and see what is the most beautiful.

Another simplification we thought about was to eliminate the rotations.

Probabilistic Attack vs Strategic Game
The problem asks for certainty, but let's assume the rotations are done at random. What if  you are happy with a high probability of getting a solved configuration? Is there a different strategy that has a faster expected ending time? Can you say anything about confidence levels (probability of ending at n or fewer moves)?

In contrast, what if the coin master is playing against you, trying to increase the number of moves you take by selecting tricky rotations. Does that alter how you think about the game? How you play, in practice?

See, I told you the ceiling could be pretty high.

Make it bigger
Okay, you've solved the 4 coin puzzle, what about 5 coins? Are there interesting versions for larger numbers of coins?

A humble suggestion

Whatever you do, I encourage you to try the 3 coin triangular version with rotations.

Advancing the calendar trick

I've written twice (first and second) about classroom experiences with a simple calendar trick that I originally got from Calendar Puzzles via Denise Gaskin's monthly newsletter. As happens so many times with these things, ideas from the kids make these activities into deeper and more interesting than I could have imagined on my own.

Tricked
On Tuesday, some of the first graders gave me their sums: 168 and 198. I immediately knew something was up. In the original calendar game, the square with the largest possible sum is the 23-24-30-31 square:

23	24
30	31

This has a sum of 108. I asked the students if they were sure of 168 and 198. They giggled, then the teacher smiled and told me she had checked it. What was going on?

My homework
I didn't have any immediate ideas, so I promised the kids that I would work on their puzzles. I told the kids that it was great to get my homework from them this time!

See through paper
One clue was that we were using a special calendar today and the paper was slightly see-through. This gave me an idea that the kids had turned the paper over and were seeing the numbers through the page with digits reversed. At first, I thought they were transforming 2s to 5s and vice versa, but was able to find 168 just by reversing digits.

How many carries?
For the original game, the crucial insight is simply that there are 7 days in a week and the calendar is organized into weeks. That means there is a simple relationship between each of the numbers in our 2x2 squares. Add a bit of simple algebra and you have an easy formula relating the upper left square of your 2x2 matrix to the sum (or, if you want to be fancy, a different formula relating whichever square you want to the sum).

For the reversed game, though, it isn't quite so easy. The relationship between the numbers can take one of several forms and is rather messy. I did manage to get 168. Can you?

But, I still couldn't get 198.

Two little helpers to the rescue
Last night, I "cheated" and asked for help. As J1 and J2 got ready to sleep, I asked what they thought their friend might have done to get 198. Their ideas from brainstorming:

• maybe the friend made an addition error
• maybe the friend also transformed 2s to 5s when reversing the paper
• maybe he summed a 3x3 square instead (which quickly gave rise to 4x4 and 5x5)

3x3 square? Interesting! Work through the algebra again and you can quickly see that there is (always!) a 3x3 square whose contents sum to 198.

Some further exploration, for you
More fun follow-on questions:

1. If the kids are allowed a choice of 2x2 or 3x3 section, but they still only tell you the sum and not the size of their square, can you still figure out which days they chose? Are there any conditions you might put on which month is chosen that allow you certainty in finding the square?
2. What if you allow 4x4, too?
3. Why stop at 4x4? What size squares are possible on a 1 month calendar?
4. If you make a year calendar instead, what sums are possible? If you are given the size and sum of a square, how close can you get to finding the source? In other words, how many squares have the same sums?

If you have other ideas, please let me know in the comments!

Pandemic: different types of randomness

who: J1
where: my home office
when: after lunch

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We've been playing pandemic a lot recently.

For this post, you don't really need to know the game, just that there are a deck of playing cards from which each player draws on their turn. Usually, these are resources that are needed to win the game, but occasionally there are epidemic cards that really stink. Oh, also know that everyone is playing together, cooperatively, against the game itself.

J1 noticed something very interesting by varying the set-up:

• Normal set-up: split the playing cards into equal piles and put one epidemic card into each pile (number of piles is the number of epidemic cards you are using). Shuffle those piles and then stack them on top of each other.
• Modified set-up: put the epidemic cards you are using in the deck and then shuffle the whole deck.

Focusing just on the epidemic cards, these result in very different distributions. Here are some challenging questions (assume there are 5 epidemic cards and 50 non-epidemic playing cards):

1. For both distributions, what is the probability that the n-th card in the deck is an epidemic card?
2. For both distributions, what is the probability that two cards in a row are epidemic cards?
3. For both distributions, what is the probability that three cards in a row are epidemic cards?
4. For both distributions, what is the probability that all epidemic cards are in a row?
5. What have we learned about how these distributions compare?

Ok, these are hard questions, particularly if you work through them in order 1 - 5. Without giving away the answers, I think it is striking that:

1. the answer is the same for both distributions, but what follows is very different!
2. left for reader
3. much higher for modified set-up
4. actually not too hard to calculate for each distribution; again, much higher for modified set-up
5. very different for potential impact on game play

Conditional probabilities
I'm open to other suggestions, but currently think that the key way to see the difference in these two ways of shuffling is to focus on conditional probabilities. For method 1, say p1(n|i) is the probability that the nth card will be an epidemic card, given that there have been i epidemic cards already drawn up to that point and similar for p2(n|i). For a fixed i, even the domains of definition of p1 and p2 aren't the same!

What happened when we played?
There's one key fact you need to know to understand our actual game: J1 hasn't learned how to really shuffle yet. That's right, all the epidemic cards were in one cluster!

J1 asked me to include some other information about our game play:

• J1 really likes to be the medic. At the start of the game, it is the most helpful piece
• Dispatcher plus medic is a very powerful combination. The dispatcher becomes even more powerful than the medic once cures have been found.
• We don't really follow the hand-limit rule.  This feels like an unnatural restriction that doesn't fit with the story of the game.

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

Calculating and computers

Another post that isn't about the kids, oh well.

I got a bit carried away in a comment over on Dan Burfiend's blog: Quadrant Dan. As an opener for his geometry class, he asks about some large numbers. I suggested a couple of follow-up questions, for those who wanted to pursue the opener further:

Exponent Investigation
Are there any numbers (feel free to restrict to integers) where a < b but a^b < b^a? What are they?

I will leave you to play with this one.

Approximating big powers
A rough approximation that can be really helpful is 2¹⁰ is close to 1000, aka 10³. For 42³⁴, you could approximate:
42 is approximately 40 = 4 * 10
So 42³⁴ could be close to 2⁶⁸ * 10 ³⁴
Using our approximation of 1000 for 1024, replace 2⁶⁸ by 2⁸ * 1000⁶, so we get
256 * 10⁵² or 2.56 * 10⁵⁴

Of course, that's still only about 1/6 the precise value calculated by worlfram alpha, but seems pretty good for such simple calculation.

Approximating compound interest
Let's say you want to do better than the previous approximation (we do, we do!) Can we make a useful adjustment to correct for replacing 42 by 40? Well, 40 = 40 * 1.05, so 42³⁴ = 40^34 * 1.05³⁴.

That second term looks like a calculation for compound interest, right? One rule of thumb (the rule of 70) is that a compounding process will double in approximately (70/rate) periods. In other words, the time it takes your money to double at interest rate r% is about 70/r years. At 5%, about how many doubling periods do we get when we compound 34 times? About 34/70 * 5 which is about 2.5. So, we can approximate 1.05³⁴ by 2^2.5

Depending on your love for √2 , you ignore that bit and end up with a final estimate of 10⁵⁵ (approximately 2.56*10⁵⁴ * 4). However, for those playing along who want to say √2 is close to 1.5, then we get a final approximation of 1.5*10⁵⁵.

Exercises for the reader
Try to approximate 34⁴². Is your approximate result larger or smaller than the approximation we got above? How confident are you that this allows you to determine which is larger, 34⁴² or 42³⁴?

What did we learn?

Well, in cleaning up this post, I learned how to do exponents and square roots in html, so that's cool.  More seriously I feel this example shows something important about the roles of manual calculations and computer based math.

First, this wasn't blind calculation following an algorithm. At each step, we were thinking about relationships, albeit approximate ones, and ways to short-cut the direct calculation.

On the other hand, the sequences of approximations could easily have taken Dan's whole class. Would it have been fun for the students? The use of the calculation engine brings this into scope as a 5 minute class opener for a class that will eventually be about something else entirely (I guess).

Even if you wanted to talk about the approximations in class, I think seeing the answer from Wolfram Alpha actually makes the hand calculations a lot more fun. The kids would be thinking something analogous to this: "sure he can fly over that building in an airplane, but can he really jump over it?!"

Multiply my love

Who: J1 and J2

When: while not hacking minecraft on raspberry pi

where: in my home office

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Recently, we found a couple of nice games and puzzles built around multiplication that I wanted to share with y'all.  The important point is that these aren't drill-in-disguise where the primary objective is reciting multiplication facts, they are games with their own goal that is facilitated by multiplication.

Bojagi 

In this puzzle, you have to cover a grid with rectangles. The trick is that the grid has numbers sprinkled throughout and every rectangle you draw has to contain exactly one number that is equal to the area of the rectangle.

Solving (left) and make-your-own (right)

Here's the game and a collection of puzzles: http://bojagi-gotmath.rhcloud.com/

For credit, the game was built by David Radcliffe (@daveinstpaul) and I originally read about this game on Moebius Noodles.

Why do I love it?
(1) Fun and challenging puzzle worth doing on its own
(2) Great reinforcement of the area model for multiplication
(3) Tremendous scope for further investigations
(4) YOU CAN MAKE YOUR OWN PUZZLES!

The last two points are related. Creating a puzzle helps stimulate thoughts about the structure of the puzzle.

Examples of things you might want to explore
a. Is there always a single solution or could there be several?
b. How many ways are there to partition a rectangle into sub-rectangles?
c. If there can sometimes be multiple solutions, can we recognize this or recognize that the solution will be unique in advance (before we solve the puzzle)?
d. If you put numbers into the square grid, will they always form a puzzle that can be solved? If not, what conditions are necessary? What conditions are sufficient?
e. Is there an algorithm that will always find the solution (when one exists)?
f. Are any game versions fun (and what mathematical structure do they have?

Times Square
This game comes from Calculation Nation. They have several good games, so it is worth taking a look at their collection. Other than this one, I particularly enjoy Nextu.

The objective here is to get 4 in a row before your opponent does. On each turn, you move one of the sliders below the playing square and capture the square that is the product of those two.

Why do I love it?
(1) The game is fun and challenging (this is the sine qua non of games, no?)
(2) Players have to use multiplication and division when planning their next move
(3) The interaction with  your opponent is not straightforward
(4) J1, J2 and I had a delightful conversation about which numbers are in the playing board, which are missing, and why.

In this case, I don't see as much scope for further investigation, but  you might have more ideas than I do. At least I will leave you with one:

Why should we have expected that the playing board would be a square, thus justifying the name?


Some other things related to multiplication
Prime Climb
A beautiful multiplication table emphasizing prime factorization, from Math4Love:

This is linked with their game Prime Climb (which I don't have, so insert unhappy smileys here!)

Factor Game

As the name suggests, a game related to factorization.  Rules here.  I will blog about this when we play the game at home or with one of the school classes.  Also, I noticed a pencilcode user that started building a program related to implement this game (Introbot's FactorGame).

An alternative algorithm

A video (here) and picture (below) are getting pushed around the web. My take:

thoughtlessly teaching/learning/applying any algorithm isn't very useful, but playfully investigating and thoughtfully considering why it works is always worthwhile.

A little question about squares (SQ1TV warm-up)

Remember the perfect squares song from Square One TV? No, then try this link. 

Did you catch the part about 14?  Not a square number, but the save is to add an extra digit to make it 144. Let's call that a square-save. Here's the section of the song, if you want to enjoy the 14 square-save in song.

With that as the inspiration, can you always square-save any positive integer? In other words, can you always add some extra digits to make a perfect square?

Extra credit

What do you make of the following sequence:

1, 5, 6, 2, 23, 8, 27, 9, 3, 10, 34, 11, 37, 12, 39, 4, 42, 43, 14, 45 . . .

What number comes next?

Do you notice any patterns?

Extra Extra Credit

One square-save of 10 is to add a 0, making 100. One hundred, of course, is 10 * 10. Are there any other numbers that can do this, e.g., their own square is a square-save?

Taking Mr. Men too seriously

Who: J2 and guest appearance from G1 (grandpa)
Where: in bed
When; at bedtime

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There are a bunch of ways to take the Mr. Men too seriously, and I'm not even talking about this.

Reading out loud
We recently got the full set of Mr Men and Little Miss books.  J2 has especially enjoyed reading them and has his own routines for extracting the books that will be read each night, then collecting the books at the end and flipping the first book for the next night upside in the box.

He really seems to enjoy these books, whether we are reading to him, he is reading to us, or he is reading to his sister.

As he was reading to us tonight (Mr Impossible!) I was wondering about how to be a good listener when the Js are reading. A quick scan of literacy sites suggests that it is both easier to get this right and easier to mess it up than I had thought.

Mainly, I think you need to have the right attitude and, like so much of parenting, the answer here is to be playful and focus on enjoyment.  Choose books, talk about them, help with the reading, let the kids struggle, but all to a degree that it is fun for you and them.

More specifically, the 5 finger test: as the child is reading, have them hold up one finger whenever they encounter a word they don't know/can't read. If you have a full hand up, then the book is too hard for them.

Two nice references, I found are Trevor Cairney's Blog and a New South Wales schools brochure, if you want to pursue this further.

Estimation
When I started writing this post, it was only a reading note, but you know that I'm bound to see a math activity, exploration, or discussion in anything. There were a bunch of counting opportunities in Mr Strong, then we hit this picture:

So, what is the mass of the water in the barn carried by Mr Strong? We had to investigate.
To be honest, I was more interested than the little one who was absorbed in the story, so I'll leave you to come to your own conclusions about how much water there was.

Further exploration: how much pressure does Mr Strong exert on the ground when he walks?
Further further exploration: what happens to soil under that much pressure?

Attitudes

Ok, so further, further explorations about the Mr Men books:

- Is Mr Men vs Little Miss sexist?
- Does the whole series reinforce a fixed mindset?

the measure is 27

Who: J3 (also something for 13+)
Where: at home on the reception floor
When: just after breakfast

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A quick picture to show some standard activities in our home.

When in doubt (i.e., too tired to think creatively), I reach for the trio blocks and polydrons and start putting them together. Inevitably, the children will join and take over the activity. We had our tape measure lying around, so J3 started measuring our creations and Ms Rabbit.  After putting the tape measure up to something and looking carefully at the numbers, she would proclaim: "27." She did this several times, each time announcing the same length: 27.

I guess this is similar to her lack of 1-1 correspondence when she's counting: just a developmental step she hasn't yet taken.

A further exploration
Did you notice the star-shaped polydron construction?  It is a cube with the faces replaced with square pyramids. Though it is pretty obvious, I was delighted when we realized that square faces in our constructions could be replaced with 4 triangles arranged as a square pyramid and equilateral triangles could be replaced by 3 sides of a tetrahedron.  Here's an NRICH exploration I found when trying to determine the name of our construction (the cube with pyramids instead of faces).