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:

Hey dad, there's a diagram!

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.

Advancing levels
The math on lingots from levels is pretty easy. If we have levels {n_i} for m different courses, then we received this many lingots for our level progression:

 n₁(n₁ + 1)/2 +...+ n_m(n_m + 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.

Extend your streak and wagers
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?

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?

Baking math

Who: J3 (2 year old)
When: late morning, after J1 and J2 had gone to school
What did we use: water, yeast, flour, sugar, salt, oil, measuring cup, measuring spoon, scale, bread machine
Where: kitchen

See how little of the finished product was left before I remembered to take a picture:

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I'd posted before (here!) about some of the (mathematical) reasons to bake with kids.  Here is the math we did today while making bread:

1. counting
2. measuring volume and mass
3. estimating
4. next steps: ideas I forgot, but you can include them!

Counting
Maybe I go overboard, but my habit is to count everything around the smaller children.  Based on the recipe alone, there wasn't much scope for counting (2 eggs, 2 tsp salt, 2 tsp yeast) but we also counted the number of measuring spoons in our set and how many scoops of flour we needed to get our target mass. Along the way, we probably counted fingers and maybe even toes, too.

Measuring volume and mass
This is the obvious place to develop number sense while cooking.  We end up doing a lot of extra measuring during the process, playing with the different tools, and talking about comparisons between them.  For example, flour (21 ounces, about 600 grams) was the only thing we needed to weigh. What we actually weighed: flour for the bread, flour not used for the bread, the mixing bowl, the measuring cups (full and empty), a water bottle and box of crackers near us on the counter, the measuring spoons, and the sugar jar.
That allowed us to explore a range of weights from about 20 grams up to 1.5 kg.

We talked about metric and imperial units and looked at all the quantities in each. In particular, I use the imperial units as a good place to talk about fractions.  For today, I just read out the fractions close to what J3 measured, and included those in our chatter about comparisons.  My goal is to head off any (distant) future anxiety that comes from seeing fractions as a different "type" of number.

Estimating
I try to encourage them to estimate measures before they see the result.  This is easiest for mass, I just say "I think that will be 100 grams (or whatever I guess)" before they put the object on the scale. For the older kids, they will almost always respond "I think it will be less/more/[x grams]." J3 will usually just copy me, but she makes a delightful sound when we read the measurement together and comment on how close our estimate was.

Next steps
In truth, I didn't really go through every step with J3.  We measured the ingredients into a bread machine pan and then let the machine do the mixing and kneading.  Of course that saves a lot of time and effort, but we didn't see the exciting phase transitions where the dough goes from  separate liquid+dry flour to lumpy/sticky to smooth/elastic.  For older kids, it is interesting to ask them why they think the dough changes. Also, I did the shaping myself and that could be an opening to talk about braids/twists/knots.

As I wrote up these notes, however, I realized that I missed an opportunity with J3 to talk about temperature and time.  I think time was the big one: time for mixing/kneading the dough, time for a rise (actually there are 3 rises in this recipe) and time to bake.  With that in mind, I'll leave you with a picture of our kid-friendly timer: