## Monday, December 21, 2015

### marbleslides and desmos for 6 year old

J2 and I were playing around with Marbleslides last night. He had a lot of fun and clearly validated that the Desmos team has achieved their objective to "design for delight" (see their blog announcement of Marbleslides here).

 Our version of bedtime math

While J3 has seen axes and graphing before, this was his first exposure to equations for lines. Really cool to see how Desmos lowers the threshold for exploring enough to make this easy and (again) fun for a young child. Now, I'm not under any illusion that he fully understood everything he was doing, but I believe these types of experiences build a comfort and background that is helpful whenever he formally encounters these ideas again. Feel free to disagree with me and let me know in the comments!

# Use of Sliders

For some of the challenges, I gave him a generic equation and added sliders for the coefficients. Here's an example:
 His choice of variable names
I wasn't really sure whether it was better to have him change the coefficients and parameters in the equation or via a slider. The great thing about the slider is the animated, continuous (looking) change in the graph as the parameter changes. The cost is that the extra variable introduces another level of abstraction ("this slider changes A which is a parameter in this equation which describes this line.") I'm not convinced either choice dominates, so we did some of both (with and without sliders).

J3 asked me to post his solutions to the first three challenges. Can you solve them with fewer lines?
 Four lines for the first challenge

 Can you even see how this solution works? Some kind of quantum effect, I guess? Solution has 2 lines

 Whoa, a new kind of line! Solution has 2 lines

## Sunday, December 20, 2015

### Just in time for christmas: some valentines math

Note: The point of this post is that you should go take a look at the CSMP storybooks and read them with your (elementary age) kids. They are hard to find, but I've done that work for you: CSMP at the Wayback Machine

You might call us hopelessly out of sync with the rest of the world: we haven't seen the new Star Wars and we are reading math stories bout Valentine's Day instead of Christmas. Oh well, at least we manage to link in at least one New Year's tradition.

# The Story

In the world of numbers, everyone sends just one Valentine card. However, they arrange it so that everyone receives ten. The fact that they can do this and their technique are explained in the old (old, old) CSMP storybook: Valentine Mystery.

 Just look at those dandelions!

I grew up on the CSMP workbooks and storybooks and they still have a large place in my heart. Now, they are hard to track down; this link is via the internet archive: Wayback Machine.

# Our investigations

J3 read the story to me and then we talked together about what was going on. He really liked that the diagrams created multiplicative fractals. We had seen these before, in May when we did activities from Natural Math's Multiplication Explorers, but he didn't remember until I showed some old pictures.

It seemed that he understood the story well enough. What we wondered: what will the numbers do for New Years? I suggested that they would play a game where each number would give one red envelope (filled with cash, of course) but that every number would receive two. Was that possible?

Well, he wasn't so enthusiastic because: "that means every number will just get one envelope." Meaning that, net, they will send out one envelope and get one back. Not so exciting.

His suggestion: In the valentine's mystery, basically the numbers drop their one's digit to figure out who will receive their valentine. Maybe now, they should drop two digits, their ones and tens. Thinking for a bit, he realized this was great: now all the numbers will get 100 ang bao.

Why stop there? Can we get everyone to receive 1000 red envelopes? What about one million? Or more?

Conclusion:  the whole numbers are the ultimate Ponzi scheme!

# Detour via 5

After realizing that we could get the numbers to receive an arbitrarily large number, we went back to the formulation in the book. Namely:
25 = 2 x 10 + 5, so 25 sends a note to 2
306 = 30 x 10 + 6, so 306 sends a note to 30
So, J3 wondered, what if we used 5 instead of 10? Okay, we did a little exploration where he would call out numbers, I would expand the division relationship, then we'd conclude who would send a present to whom:
368 = 73 x 5 + 3, so 368 sends a present to 73
306 = 61 x 5 + 1, so 306 sends a present to 61
5 = 1 x 5 + 0, so 5 sends a present to 1
0 = 0 x 5 + 0, so 0 sends a present to itself
So, how many presents does each number receive?

# A little binary

Not giving up on the idea of receiving 2, I asked J3 if he knew about binary. "No, but I've heard of it." These are our notes (his handwriting is now almost the same as mine!):

For our table of decimal and binary versions, he would try to figure out the next number. For the first couple of guesses, I would decompose the number and he would figure out the decimal version. After a couple, he did the decomposition himself. Finally, he caught on and understood some of the patterns of how to count up in binary.

We still haven't gotten to the punchline for our New Year's story, but it seems close.

# Some snaps

 Famly fun working on Find the Factors puzzles together

## Friday, December 18, 2015

### Modern math competition

Saw this brief exchange and thought it would be a good time to write down some ideas for a new type of math event:
It will be clear below that these are all other people's ideas, I've really just added ideas for how you might score this.

# Fold and Cut

Everyone gets a pair of scissors and three pages with a straight-line figure (a shape that has a boundary made up of polygons) in the middle of the page. For each shape, players try to find the smallest number of straight cuts they need to cut out each shape.
 One example of a shape to cut out
 Another example: cut the house and the tree at once (hard!)

Scoring for each shape:
10 points for least cuts
5 points for second
1 point for all who cut out the shape successfully, regardless of how many cuts it took

Reference: Fold and Cut Theorem.

# Estimation

Set up an estimation challenge (number of marbles in a jar, weight of a collection of books, height of a blown-up picture of your favorite mathematicians, etc). All players write down:

1. Their estimate of the amount
2. An explanation of their reasoning
3. A value that they think is too low
4. A value that they think is too high
Scoring:
Each estimators performance is based on the absolute size of their error (|estimate - actual|) multiplied by the size of their low-to-high range, divided by 2 if the actual value is within their range. The lower the resulting value, the better.

20 points for best performance
10 points for second
5 points for third
1 point for everyone who estimates

Reference: Estimation180 has tons of great prompts for estimation.

# Which one Doesn't Belong

Really simple: show players four things, then they figure out reasons why each of the figures could be the one that doesn't belong.

 This is just a small sample of the ideas for Which One Doesn't Belong

Scoring:
1 point for every different reason + 5 bonus points if there is a reason given for each of the 4 shapes

Reference: Which One Doesn't Belong has a huge collection of great ideas for shapes, numbers, graphs.

# Notice and Wonder

Pick a 3 to 5 math pictures, gifs, or videos to show on a large screen. Players write down the things that they notice ("I notice that ...") and wonder ("I wonder [what, why, when, how many, ...]..?").

 From SolveMyMaths

Scoring (similar to WODB):
1 point for every notice or wonder + 3 bonus points for having both a notice and a wonder.

Reference: SolveMyMaths and MathHombre both have really nice collections. Mathematical Etudes has some excellent videos that don't seem to be as widely known among Anglophones. I'm sure there are others

## Thursday, December 17, 2015

### Big numbers and BIG NUMBERS (emulating classic TMWYK)

During a recent chat with J3, 3.5 years old, I decided to test some ideas about her number sense. She was playing with 3 and had just counted them.

me: How many fingers do you have?
J3: 10, here's five (shows one hand) and here's five (shows the other hand)
me: Let's count them?
J3: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
me: I'll try mine. 1, 2, 3, 4, 5, 7, 8, 9, 10, 11
J3: (laughs) That's silly. You skipped ... you don't have 11. Let me show you. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
me: Oh, thanks for helping! I have something else to show you. (I write down the numbers below). Do you know which is bigger?

 Why is it so dark? Oh, yeah: winter

J3: this one (points to 306).
me: what number is that?
J3: 3-zero-(pauses) 9.
J3: 3-(pauses) 8

I  realized that I had drawn the 306 a bit larger than the 36. Maybe that was why she pointed to 306? I drew two more.

me: Hmm, what about these two numbers?
J3: (points to the 2). That is a really big number (stretches her arms wide)

we then drew faces for each other.

# A J2 project and challenge

J2 was arranging the pieces from a checkers set. He wanted me to pose this challenge to all of you:
what are your favorite ways to arrange 24 checkers, split equally into two colors? here is one of his designs, to get you started:

 We are working on imitating JJ Abrams lens flares, but only have gotten lens glare

### Some easy logic grid puzzles

In our math games classes recently, we've been introducing more puzzles. One class of puzzle I loved as a kid is the logic grid. There are some nice websites that specialize in these puzzles, namely:

I wanted some easy puzzles to use as introduction and was surprised not to be able to find any two factor puzzles, so I went ahead and started creating some. Here are three, for your amusement. If you know a source of others, please let me know!

Some reasons why two dimensions aren't popular
I see two weaknesses of the small puzzles.

First, there is a strong temptation to make puzzles about people. That means names are one common dimension. If we restrict to two dimensions, then the puzzles are all of the form "match the people on this list with one of their attributes from this other list." That quickly feels repetitive.

I'm open to the idea that this is just my own weakness as a puzzle writer, maybe exacerbated by the fact that I've just started trying to write these. If so, then I should be able to create more interesting contexts over time. Puzzle 3 is in that spirit.

Second, structurally, 2 dimensional puzzles are missing some important and difficult inference rules. Let's use the terms (names and instruments) from my Puzzle 1 and add another dimension where the kids like different sports (football, tennis, volleyball, and biking). Taken from this paper, the key inference rules are:
1. Related by elimination. For example, if we know that Aaron doesn't play violin, trumpet, or drums, then he must play piano.
2. Not related by exclusion. For example, if the violin is played by Carly, the it isn't played by Aaron, Benjamin, or Diana
3. Related by transitivity: No example from Puzzle 1. Using the extra dimension, if Benjamin plays violin and the violin player likes biking, then we can conclude Benjamin likes biking.
4. Not related by transitivity: Again, our basic puzzles can't illustrate this. Illustrating with the extra dimensions, say Carly doesn't play piano and the piano player likes tennis, then Carly doesn't like tennis.

Empirically, the transitivity rules make puzzles much harder, partly because the grids make the other two rules visually clear.

Anyway, I'll test these out with the kids and report back.

## Tuesday, December 8, 2015

### Skittles and middles (a 3 act investigation)

Some really quick notes about a notice & wonder/3-act activity we did, piggy-backing on Graham Fletcher and Mike Wiernicki's nice presentation to a regional NCTM conference.

In particular, I was struck the trouble that the teachers were having calculating the midpoint of a number line and wanted to see what my older two would have to say. I thought this could prompt some interesting "notice and wonder" from the two older J's. Overall, there was a lot to discuss from this presentation with many great mini-conversations.

Quick background
If you don't watch the video, they go through an estimation problem involving skittles poured from snack-sized packages into a large glass jar:

 To be clear, these are the same jar!

# Notice and wonder

My guys' notice and wonders:
• that's a lot of candy
• how many bags is that?
• why did they pour it in a jar? It is easier to take as a snack when it is in the little packs!
• how many skittles is that?
• who is talking (giving this presentation)?
• are those chocolate?
• it was a man pouring the candy.
• How do we know it was a man?
• how big is the jar?
• are there the same number of candies in each pack?
• why does he pat the candy at the top?
• could we have some candy? (quickly correct to: could we have some candy, please?)

# Their high/low/estimates

1. J2 made guesses first: 30 (low), 600 (best guess), 1000 (high)
2. J1 said 50 (low), 800 (best guess), 1000 (high), but then smiled and changed to 0 (low), 800 (best guess), and 5000 (high).
I forgot to ask them to draw a number line and put their best guesses on it. However, we did talk a lot at this part of the presentation:

They quickly agreed on 525 as the middle of this number line. Then, we talked about where the teacher's answer 475 might have come from. This led to a bit of confusion and frustration, basically centering on the distinction between the number at the middle and the distance to the middle.

# Student work and a quick multiplication

They did the multiplication calculation a couple of ways. First, they used a calculator. They would have been satisfied with this, but wanted to understand the student work, so were inspired to try some other approaches. It was really interesting for me to see how they responded to something when they were told it came from other kids:

Starting with the bottom right, we drew split rectangles/area model with 50 + 8 along the side and 10 + 4 on the top, then talked about how the student had just calculated the areas of the top left and bottom right rectangles. Some extra comments/mini conversations:
• this is like the students who thought 11 x 11 = 101
• Oh, I see someone else did their rectangles the other way: 50 + 8 on top and 10 +4 on the side
• It doesn't matter which way we do it
• like addition: 5 + 4 = 4+5
• Does that always work? No, for subtraction 4-5 isn't 5-4. Also division. Right, 100 candies shared by 2 boys is really different from 2 candies shared by 100 boys (both then fall on the floor laughing)
• Hey, look at that student who added fourteen 58s in a tree!
This grand reveal (which actually came before the student work) also gave them a nice counting challenge:

# Contextualizing

At that point, we'd caught up to the part of the video I had watched and didn't know what came next. They got surprisingly excited about this slide:
They wanted to understand what "contextualized" and "decontextualized" mean. I gave them the following comparison:

• 5+4 = ? is a calculation without a context
• I'm measuring the length of this room. My tape measure is 4 meters long. I've measured twice, 4 meters and then another 5 cm. What is the total length?
The context helps them understand what we are doing, allows them to use their intuition to estimate the likely answer, and determine the appropriate relationship between the numbers.

### Math of Duolingo (100k Lingot Challenge)

Well, at least about the lingots . . .

During our recent long weekend, the two older J's and I were talking about lingot acquisition on Duolingo.  This is a summary of our conversations, plus a bit of background for Duolingo non-users.

What is the point of this post: opportunities for mathematical modeling are all around us!

# Our streak obsession

Some of us have become heavy, frequent, users of Duolingo. While others debate whether it is the best tool for language learning, we've at least been finding it a good place to practice languages we already know and learn the basics of some new ones. And accumulating XP, experience levels, translation tiers, and lingots makes it into a fun game that tweaks our greedy instincts for more, more, more.

Especially those lingots:
 Oooh, shiny, shiny!
You know our penchant for finding math in all things, so it won't surprise you to know that we've been thinking about the math of lingots. Especially when our curiosity took us to the Lingot Hall of Fame (Thousadaire's row) and we saw this:

There are people with over 100,000 lingots.  Hmm, how did they get so many? How long will it take us to get that many?

# Some basic rules

Basically, we know 6 ways to earn lingots:
• Pass a skill: every skill passed earns 2 lingots (or 3 if you test out with a perfect test)
• Go up a language level: rise to level N and you earn N lingots.
• Extend your streak: consecutive days of play build a streak, every ten streak days, you earn S/10 lingots (where S is the length of your streak)
• Wagers: you can bet on maintaining a streak and (net) earn 5 lingots every 7 days.
• Upload a widely liked document to immersion, more on this below
• Direct grants
Data on skills
Each language has a tree of skills that you complete to unlock the rest of the tree and earn lingots. Oh, yeah, to learn vocabulary and grammar skills, too, I guess:

 A section of our German skills tree
There are 120 skills on the German tree. Assuming this is typical, that gives us a max of 360 lingots/language from completed skill trees. There are currently 50 live courses listed on Duolingo, so that means a user could earn 18,000 lingots by completing all the skill trees.

Not bad, but a huge amount of work. To give you a sense, over the past 150 days, I've managed to complete the German and Spanish skill trees, but those were both languages in which I was already quite proficient. Suffice it to say, my progress through the Russian tree has been much slower.

For our time estimate, we assumed 100 days to complete a tree with the possibility of working on 3 languages at a time, but we guessed this is a bit on the optimistic side. This gives us a lingot rate from skills of 10.8 lingots per day (3*360/100).

Also, our top L-tycoon, KcaJP is only studying 8 languages, so even with full trees (which seems unlikely, see below), that would be 2880 lingots from skills.

The math on lingots from levels is pretty easy. If we have levels {ni} for m different courses, then we received this many lingots for our level progression:
n1(n1 + 1)/2 +...+ nm(nm + 1)/2 - m

The subtle -m comes from the fact that we start each course at level 1 and don't get a free lingot from that.

I've seen one claim that the max level per course is 25, so that would be 324 lingots/course, or 16,200 total lingots. Making similar assumptions to that for the skill trees, we estimated 150 days to get to level 25, 3 languages running in parallel, for 6.48 lingots per day from this source.

Based on the KcaJP's levels in the 8 courses, I think this source only provided 514 lingots. Also, because many of these 8 courses have very low levels, it is unlikely KcaJP has completed many of the skill trees.

This is very similar to the language levels, but is open-ended. In this case, lingots accumulated are S(S+10)/200.

Betting on keeping mini-streaks (each of 7 days) gives us 5 lingots every 7 days, so we will accumulate 5*S/7 from this source.

Data on likes
The last two ways to get lingots are to be loved. If you are loved for yourself, people can give you lingots directly. The most extreme case we've seen was a user who was gifted about 4000 lingots. However, even gifts of 100 lingots are extremely rare. Because of this, we aren't counting on donations and think it lingot gifts can be ignored for understanding lingot tycoonhood.

One other way is to be loved, indirectly, is to upload a document to Immersion that then gets upvoted. Typically, this earns the uploader 0-5 lingots. There is one extreme example (a Harry Potter page) that got about 1500 upvotes. Again, we think this source can be safely ignored for our analysis.

# How long for our 100k badge?

So, in summary, we have the following as the key factors for earning lingots:
• Pass a skill: estimate 10.8 lingots per day, source of 2880 lingots for KcaJP
• Go up a language level: estimate 6.48 lingots per day, source of 524 lingots for KcaJP
• Extend your streak and wagers: 5*S/7 + S*(S+10)/200
• Total forecast lingots on day S: S2/200 + 18 S
We rounded the coefficient of S since our estimates for passing skills and level increases weren't precise to that level anyway.

Solving this quadratic for 100,000, gives is about 3020 days for us to reach 100,000 lingots, or 8.3 years. Assuming we keep our streak every day for that whole time, our assumptions mean we would have:

1. Earned about 32k lingots by completed 90 skill trees (40 more than duolingo currently has, but there are another 29 at different stages of development and we're gonna take 8 years, so, maybe?)
2. Earned about 20k lingots having gotten to level 25 on 60 courses (not sure how this links with 90 completed trees, but, oh well)
3. Earned about 48k lingots from sheer persistence

# An obvious conclusion

Having explored our estimates of how long it would take us to get 100,000 lingots, we want to turn back to our favorite Ltycoon, KcaJP. While we can't guess about how many lingots KcaJP has been gifted or upvoted, the visible sources of income are:

1. less than 2880 from skills
2. 514 from levels
3. less than 12000 from streak and wagers (this is the max available since Duolingo launched on 30 November 2011)

In total, that would still be an impressive 15k lingots, but we've still got 100k unexplained!

So, how did this lingot tycoon reach this pinnacle of virtual wealth? Frankly, we don't know, but can conclude that there must be/have been other ways to earn lingots. We have found parchment fragments suggesting a distant time when it was easy to earn loads of lingots from translating on immersion. Is that the key?

## Tuesday, December 1, 2015

### SolveMe Mobiles and BlockBlobs

Two activities that worked really well in class yesterday:

# Mobile Balance Puzzles: from EDC

Several months ago, the great folks at EDC released SolveMe Mobiles, a collection of balance puzzles. The idea is pretty simple; you'll probably get the idea from just a couple of screenshot examples.

In many puzzles, some shape values are given and we need to find the ones that are missing so that the arms of the mobile stay in balance:

Sometimes, we are told the total weight of the shapes in the mobile. We then have to find the weights of the shapes so that the arms stay in balance:

Since we were introducing this idea for the first time, we did a simple version together, then gave the kids size more as a warm-up.

the most advanced puzzle we gave them had a side with three descending strings, like this:

We had a nice discussion about what this means: in real life, do all three of the hanging strings have to have the same value or not? What do we think for these puzzles, does it work like real life or not?

Side note: to online or offline?
For our class, we just had print-outs of the puzzles. This worked really well because it was easy for the kids to make notes and helpful drawings on the side of the pictures. The online system has a tool to write notes and draw on the screen, but it is slightly awkward to use on a desktop. Probably it works well on a tablet or smartphone.

Also, we brought out the unifix cubes as manipulatives for several of the kids to work through the puzzles. Of course, nothing would stop someone from doing the same when they are solving puzzles on a computer, but, in practice, I find that the idea just doesn't occur to them in the same way.

With the online version, you have two options that can give additional help. One is an animated test of whether the mobile is in balance. The other options shows numbers at the top of each string that represent the total weight under that position. Each of these makes the puzzle a bit easier by offering a guess-and-check strategy and a nice form of feedback.

I always love getting kids to make their own versions of the puzzles. In the class, several of them finished a bit faster and I asked them to make puzzles for me. This feature is also built into the on-line version, which is a nice touch.

# Block blobs

Our second activity of the day was Block Blobs. This comes from Beast Academy, though we made some small adjustments to their game.

The basic playing board is graph paper with a central dot. To make each game a manageable length, we split our paper into four sections for four games. Each section is about 20x30:

This was one slight deviation from the BA version: they use a 12x12 board. Each player has their own colored pencil to use as they mark out territory.

On a player's turn, they roll two dice, then they try to draw a rectangle on the board that uses those  two numbers as side lengths. For their first rectangle, they have to put one of the vertices on the center dot. For all subsequent turns, their rectangle must share at least 1 unit common boundary with an existing rectangle, may not overlap any existing rectangle and may not go off the board. The growing collection of linked rectangles forms their Block Blob.

 Hmm, can you spot the illegal rectangle?

 Two against teacher

Play ends when one player is not able to add the required rectangle to their blob. Then, each player figures out how much area they have covered and the one who covered the most wins. To make this final calculation a bit easier, we had them write the area inside each new rectangle as they added it.

Differences with the BA version
As mentioned above, BA suggests playing on a 12x12 board. We originally thought we would us 2d10 instead, so wanted boards that were at least two times larger in each direction.

Second, we required each player to start their block blob with a vertex on the center dot. The BA version requires this of the first player, but allows the second player to start their blob anywhere on the board. Since we were using larger boards, we wanted to facilitate the issue of competing for territory by having the blobs share a vertex. Another option would be to have each player start their blob in opposite corners of the board. This idea is similar to the game Blokus.

Third, we ended play when one player could not add a required rectangle (they pass their turn). The BA version stops play only when there are 4 passes in a row (two for each player). Though seemingly subtle, this was a major change. In our version, the player who gets squeezed out, either through weak play or bad luck, doesn't get penalized as harshly as in the BA version. This turned out to be very helpful because it meant that ending scores were much closer than expected, so (a) it was actually necessary to total the areas to see who had won and (b) the loser got a pleasant surprise when they saw that they were still very close.

For what it is worth, all credit for this important rule change goes to PK!

Of course, each of the versions can make for a fun game with slightly different strategic considerations.