5 minute sharing

Who: J1
When: after dinner and before brushing teeth
Where: bedroom

I've talked before about Peter Liljedahl's Numeracy Tasks as explorations in fair sharing. Tonight, J1 and I briefly discussed the cookie question and the cupcake conundrum.

Sharing Cookies

Six cookies, 3 friends (J1, Ji-Ping and Tanya), sharing is easy peasy, right? Well, one mom insists that her child (Ji-Ping) can eat only one cookie, so what do you do?

J1 responding quick: Ji-Ping gets one, Tanya and I each get 2 and a half
J0: Is that fair?
J1: Its fair because he gets to eat as much as he is allowed and then Tanya and I get the same amount.
J0: How will Ji feel if you get so much more?
J1: Well, his mother probably only wants him to eat one because he is going to get more treats at home, like birthday cake, so that's fair. (By chance, today is Ji's sister's birthday!)
J0: Are there any other ways to split?
J1: We could all have one today and save the others for tomorrow. We could ask Ji-Ping's mom to let him have more.
J0: Maybe you could share with other friends?
J1: yeah, and then in the future, they might share with us. At first, I thought you were going to say that the snack was yo-yo bear <laughs>
J0: What if it was yo-yo-bear, how would that change the situation? Each pack has two strands, but what if Ji-Ping was only allowed to eat one?
J1: <thinking> I guess it wouldn't change it.
J0: So, what about your first idea with the cookies?
J1: Oh, Ji-Ping would get one loop and the treasure card, Tanya and I would each get a treasure card and 2 and a half loops.
J0: how does that feel compared to the cookie split?
J1: it seems pretty fair

Sharing cupcakes

Again, we've got three friends and six treats, but this time 4 cupcakes have delicious chocolate frosting and two do not. This time, J1 had a clear sense that the right answer was to cut in equal portions.

J1: well, we each get 2/3rds of a cupcake with no frosting.  And we get two with frosting. Wait, how many had frosting?
J0: 4
J1: I thought it was 6 <laughing>. Hmm, then we get .... 1.5.....no......1 and 1/3rd with frosting.
J0: any other ideas about how to split them/
J1: Well, this is fair, we all get the same treats and we don't have any left over so that's got to be best.

A bedtime math confession

Recently, we've been talking about the "fun nightly math" activities on Bedtime Math. Both J1 and J2 enjoy the scenarios and they have fun calculating to answer the questions. Frankly, I'm not in love with the questions as they usually seem a bit artificial to the story and are often just a single arithmetic calculation. However, it is a very handy resource to easily add a couple minutes of number thinking to the end of a day.

Tonight, the 1000-year Rose led to some good diagrams and a fun discussion about very long times (hundreds of years). J1 made a number line to answer the "sky's the limit" question and a labelled array to answer the big kids bonus question.

Hardly a thrilling photo, but some evidence I'm not making all of this up

So, I hereby officially give you permission, nay encouragement, to open bedtimemath.org the next evening when you don't have time or energy to have a more extensive TMWYK conversation.

Simple splits: sharing money

Who: J1 and J2
When: just after dinner
Where: dining room floor

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Taking the skytrain (BTS) yesterday, I ended up with interesting change: 48 Baht comprised of four 10s, one 5, and three 1s. Why is this interesting change, you ask? While 48 has a lot of factors, this set of coins makes it impossible to evenly divide into any smaller amount!

Just chop it right here!

This gave me an idea for a sharing discussion with the kids. Here's the intro to our conversation:

• J0: Hey, I just realized I have 48 baht
• J1: Can I have it?
• J0: Not yet, I want to ask you a couple of questions. If you were going to split it equally with J2, how much would you each have?
• J1: 24. Can I have it now?
• J0: (I write down 48 split for J1 and J2 means 24 for each). What if you were going to split it with your sister, too?
• J1: still 24
• J0: Oh, I mean you split with J2 and J3. All three of you get the same amount
• J1: (losing interest, the coins don't seem closer to his grasp) Uhh, I don't know. Let's do something else.
• (proceeds to wander around the room for a bit, does some other activity for a while. I have a guess he is thinking about the question and avoid pressing)
• J1: 16, we each get 16 baht!
• J0: Good! How did you figure it out?
• J1: I divided 48 by 3. Can I have the money now?
• J0: Yes. Here are the coins. Can you show me how you would split it like we said? Show me how to share it with J2.

He played for a while until he realized that it couldn't be split evenly. We talked about why (easiest path to seeing this was to realize a split means making 24 baht and that isn't possible with those coins). Then we talked about whether it would be possible to split them fairly in some other way. Here were ideas, mostly his, but this was a collaborative conversation:

1. split them as close as possible and randomly decide who gets which pile
2. split them as close as possible and then J1 gets more because he is older
3. J1 splits them as close as possible, gets more and gives the rest to J2. This is considered fair because every child ends up with more money than they had at the start, so they should be happy. This is a version of the ultimatum game and I was really surprised that J1 came up with this reasoning on his own.
4. Split evenly what we can and give the rest back to daddy.
5. Ask for change for the 5 baht coin and then split evenly.
6. Just cut some of the coins in half (physically cut them)
7. Buy something with the money that we both want and can split evenly (ice cream, yogurt drinks, etc)
8. Split 25/23 and J1 gives J2 something of value to balance

Do you have any other ideas for how to tackle this sharing problem?

Determining value

Point 7 led to a mini-conversation about how much value the extra item should have: 2 baht or 1 baht. J1 gave a bunch of examples (used toys, some services) and asked me if they had the right value, and I explained it would really depend on whether they both agreed because there wasn't a separate way to determine the value.

The conversation propagates

J1 had so much fun with this conversation that he then got J2: "hey, i want to show you something. How can you split 48 baht into two?" He didn't completely recreate the discussion, but the two got a lot of the same ideas out together and it was great fun to watch.

Important Lessons

• When they don't seem to be focusing on a question or challenge, (sometimes) they still are thinking about it.  Let them have space and don't force it.
• Even mundane items and observations can be gateways to deep ideas

Fair sharing Exploration (warm-up)

who: J1 and J2
when: over a course of weeks (this is a plan, not a historical record)
what material: the objects to be shared

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This post: http://letsplaymath.net/2014/08/13/fractions-15-110-180-1/#more-28158 gave me the inspiration to create an extended plan to explore ideas around fair sharing. The idea got a further boost from our warm-up discussion about breaking swords and cutting cakes last weekend.

In our house, issues of fairness lurk just below the surface of almost every interaction between the children. Actually, that's not accurate, since fairness is often a visible dark cloud hanging over the proceedings.
Part of the idea for this exploration is to harness their strong feelings on this topic to examine:
- fractions (naturally)
- approximations
- competing theories of fairness/equality
- their own intuition and biases around what is fair and why something should (or shouldn't, or doesn't have to be) fairly distributed, including concepts of ownership ("that is mine!"), earned privileges ("he got X because he did Y"), and private valuations ("you like X more than Y, but she likes Y more than X").

The basic idea is to present different types of sharing problems as thought experiments: talk through or play-act the scenarios, do some analysis of different sharing tactics (maybe using manipulatives, diagrams, etc) and later come back to these in live examples.

Examples of different classes of sharing problems I see:
- cakes/pies: things that can easily be cut into small fractions
- sausages, or apples: something that can be cut reasonably accurately into moderate fractions (maybe down to 1/8th)
- ice cream, or soup/rice/etc: something that needs to be weighed or volume measured
- KEX cookies/small candies: something that can, at best, be cut in half, maybe not cut at all.
- ballons/babies/bicycles/scooters: items that are indivisible objects
- balls/group toys: items that increase in value as more people play (up to a point)

some types of questions are:
1) technically, what tactics can be used to divide, what are the pros and cons
2) strategically, how can you get buy-in that your approach to divide is fair?

Let me know if you have any tips or thoughts on how this will all turn out.

Fair Sharing (warm-up)

who: J1
where: at Kuu, our regular lunch destination at the mall
when: while waiting for ice cream
what did we use: chopsticks

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As we enter this snippet of conversation, J1 was describing/acting out some martial fantasy scene with different elements battling and weapons getting destroyed.

J1: What if I broke his sword into 11 pieces?
J0: 11 equal pieces?
J1: yes, 11ths.  Then he would cry 11 times.
J0: I wonder, is it easy to cut something into 11 equal pieces?
J1: no, hard to get them all the same size.
J0: What is easier, cutting into halves or thirds?
J1: (thinks for a bit) cutting into half is easy. (He then makes a swipping motion with his chopstick and a blade-swooshing sounds.) If you cut it into thirds, you do chop (one smaller slice), chop (another slice about 120 degrees of the previous one), and then you really need to put your back into it (as he makes a third slice toward himself).
J0: (laughing) What did you say?
J1: (laughing, then repeats the last cutting motion) then you really need to put your back into it.
J0: Wow, you were dividing a round cake.  I was just thinking of sticks.  How many cuts do you need for those?

We proceeded to have some discussion of how many cuts were needed, some contemplation of why a it takes n radius cuts to divide the cake into n pieces (for n>2), and a hypothesis about why halves are the easiest to divide (because you only have to compare two pieces and make one cut).

One further comment was worth flagging, related to my n>2 provision above:
J1: hmm, thirds take 3 cuts, but halves only take one cut?

As this was in the middle of something else he was describing, I didn't follow that branch of the conversation, but may return to it later.

*Apology* sorry I didn't include any pictures on this post. I'll see if we can draw a picture of the enemy with  a sword broken into 11ths.