What is 8?

I've had a chance to spend more time doing math with the kids again and am hoping to write up our activities more consistently.  Let's see how this works out!

Graham Fletcher created a set of  Progressions videos for various elementary school themes. J3 and I recently went back to his page and found he had a new(er than we knew) progression on early number and counting.  Even for this simple topic, the video highlights some points we hadn't considered explicitly, for example distinguishing producers (of a number) and counters. Also, the cardinality point that smaller natural numbers are nested within larger numbers wasn't something we had talked about, but we soon realized it was part of many examples in how we understand numbers.

With that as inspiration, J3 and I decided to search for a range of examples of a single number, we chose 8, in different forms.  There is at least one obvious version we're missing.

Add a comment (with picture, if you can) to show other forms of the number 8!

Marking 8 on the 100 board, an easy place to start:

8 beads on the abacus shows the relationships 3+5 = 8 and 10-2 = 8 (also 100- 92 = 8)

8 can hide in plain sight. Without labeling the three lengths, it would have been hard to recognize the longer one as 8 cm and, for you at home, impossible to know without reference to show the scale.

It happened that, within the precision of our scale, two chocolate wrapped chocolate bars were 8 oz (2x3.5 oz of chocolate + about half an ounce of wrapping for each):

8 cups of water ended up being a lot, so this version unintentionally revealed a relationship 4 + 2 + 2 = 8

Though I'm not sure I can articulate why or show supporting research, I feel it is very valuable to build experience with physical models of numbers to create familiarity and intuition about what they are/mean. In particular, I hope this helped J3 anchor the importance of units of measure and scale in the interpretation of numbers.

Finally, this construction has nothing to do with the number 8 (or does it???)

100 board Go

In past posts, we've shown some of the make-shift materials we are using to play/learn Go without a proper set. Over the last two days, we had experiences that reinforce the value of this approach.

Exploring an earlier pattern
First, when playing with J3, she noticed that we could complete a repeating blue-green-yellow-red pattern around the boundary of a 5x5 board. In a follow-up conversation with J1 and J2, we explored this:

J1 explained why it would work, grouping the boundary tiles as 5 for each side of the playing square and 4 in the corners, so 5 x 4 + 4. This made it easy to see that the boundary would be a multiple of 4 and also made it easy to extend to any square board: n x 4 + 4.

J2 had a new idea. He thought we were talking about the pattern continuing as an inward spiral.  That gave us this design:

This also led to discussions about symmetry (the blue and yellow have reflectional symmetries that green and red lack) and further investigation on boards of different sizes. Interestingly, we found that, for some boards, none of the colors have a reflection symmetry.

Trying some tsumego
I set J1 and J2 the following challenge (J3 was watching): are the blue cubes alive or dead in each of these two clusters?

Putting aside the interesting Go discussion that resulted, there are two consequences of doing this on the 100 board: extra unnecessary information and a built-in coordinate system. By unnecessary information, I'm talking about the letters on the white tiles and the numbers on the 100 board. This is information that is entirely orthogonal to solving the life-and-death puzzles. This is a simple toy version of one of the key modern challenges in problem solving: identifying which information is useful and which is a distraction.

On the other hand, for talking about the puzzles, we could say things like "what if white plays a tile on 99?" For J3 who was watching, this offered another little example of the idea that numbers are all around.

Some capture fun

For the last example, I set out some white tiles (some alone, some in groups) and asked J3 to capture them with blue cubes. After we did that, we counted the number of cubes we used to capture by moving them to cells in the 100 board (note that we removed one of the lone white tiles). A fun counting exercise, an opportunity to talk about groups of 10, and more familiarization with the layout of the 100 board.

Some sort of number talks with J3

Based on conversations about the dots pictures from Math4Love:

Day 3
I notice

1. there's a number 3, but the number of dots isn't the same (it isn't 3)
2. five over here (pointing to dots) and zero on the down part (the bottom half of the 10 frame)
3. J0: I see some letters...
4. I even noticed that. I noticed there's this plus (points to dash - )
5. J0: I noticed this square
6. I noticed it was a line (bottom row of the 10 frame)
7. I noticed these triangles (the white space in 10 frame sections that have dots)
8. I noticed these are 5 and an extra one (on second page of day 3)
9. I noticed that there are four left (empty cells on second page)
10. J0: you saw 5+1, I see 2 + 4
11. Those two are together. The other ones are lonely.

I wonder

1. Why didn't they make 10 dots?
2. Why did they only cover the middle of the square (points to a dot in the upper left square of teh 10 frame)?
3. I wonder, how do numbers talk? (after I read the title of the slide to her)
4. I wonder, why do they only put 1 on the bottom row?
5. I wonder, can we arrange them so none are lonely

Day 4

I notice

1. Five on the top and five on the bottom
2. Ten
3. five and four, nine

Day 5

I notice

1. This doesn't have a box to go in (a 10 frame)
2. It has a dot in the middle
3. we can count them 2, 2, 2 (pairing them up)
4. if we take 2 away, there will be four
5. the sides are the same (it has a line of symmetry in the middle)
6. it looks like an animals footprint
7. the top four make a diamond
8. If we turn our body to the side, the top four make a rectangle
9. taking out the two in the middle, we have a square
10. it has 8 dots.
11. the number of dots doesn't match the day number

I wonder

1. is it a real footprint?
2. I wonder, if we take the bottom five, it would be 3?

Little one counting

Late last year, Joe Schwartz wrote a very interesting post about the difficulties one student is having with counting and skip counting. I recall hearing a theory that many later math difficulties trace back to when a student missed solidifying the concept of one-to-one correspondence and some related concepts of counting (but don't have a citation or reference). As a result, I was thinking about how our little J's understand these ideas.

In particular, I wanted to try out Marilyn Burn's little game from one of her comments with J3:

Ask him to put out 8 cubes on a paper. [I chose 8 because when I remove one, the child won’t be able to know how many by subitizing.]
Ask: How many cubes did you put on the paper? (8) [Here I look for whether the child has to recount.]
Say: Watch as I take away one cube. Remove one cube and place it on the table.
Ask: How many cubes are there on the paper now? (7) [Does the child have to recount, or does the child just know.]
Say: Watch as I take away another cube. Remove one of the 7 cubes and place it on the table.
Ask: How many cubes are there on the paper now? (6) [This is the same as the previous question, a way to check if the child still needs to recount.]
Say: Watch as I put one cube back on the paper.
Ask: How many cubes are on the paper now? (7) [Similar, but adding 1.]
Sometimes I repeat again removing a cube and asking: Can you tell me how many there are without counting? Some kids shake their heads to indicate they can’t, others say they’ll give a guess, some are able to.

J2

Before I had a chance to try out the question sequence with J3, I had some time alone with J2. He was sorting colored pencils, so we used those as counters. Overall, he breezed through the questions, but there were two amusing points:

1. After separating out 8 colored pencils, I asked how many he had. His response showed that (a) he believes in conservation, so he knew there would be 8, but also (b) he is used to me doing something tricky, so he wanted to verify that there were still 8.
2. His method of verification: split them into two groups of 4, an amount he could recognize by subitizing, not counting.

He asked me why I was asking these questions and I told him it was related to his understanding of hierarchical inclusion. We talked briefly about what that means and he was delighted by the term, so ran off to ask J1, "how is your understanding of hierarchical inclusion?"

J3

My counting time with J3 came during dinner. She was eating cucumber slices, so we used these as counters. This turned out to be a mistake, since conservation doesn't work with edible counters! In other words, whenever I asked her how many slices were in our cluster, she would pop one in her mouth and smile, knowing that she was teasing me.

Mainly, though, I was able to verify that she doesn't yet have the concepts that allow short-cuts to the questions in Marilyn's sequence and needed to recount to get all the answers.

Incidentally, I started the activity by telling her that we were going to count something together. She immediately grabbed this coaster and then accurately counted the circles to 37.

For me, the entire experience was a really interesting illustration that counting actually requires a complex collection of sub-skills.

Surreal numbers and whole body integers

Two unrelated activities to note:
(1) Exploring checker stacks and surreal numbers with J1 and J2
(2) A whole body numbers game with J3

Surreal Numbers and Kids

If, like me from one month ago, you don't know about surreal numbers, I think you'll find they can be a really engaging exploration with kids. The main attraction is the appearance of infinities and infinitesimals, both of which really seem to resonate with young mathematicians. In addition, there's fantastic icing on this cake, too: you can explore by playing a simple (to learn) game with a lot of depth.

Credits: this exploration is strongly inspired by Mike Lawler's recent posts about surreal numbers and the Jim Propp post that inspired him. If you are interested in doing this type of exploration with your kids, I strongly suggest going through all of their posts on the subject.

Note: since we used black and red checkers, while the convention in the other posts is blue and red, I will abbreviate B and R so you can naturally substitute your own preferred color scheme.

How we got started
Using a set of regular, stackable, checkers (black and red), I showed each of the older J's the position RB + BR (a stack with red on the bottom, black on the top and a stack with black on the bottom, red on top) and explained that the basic moves.

This was a good initial example because it let us talk about each of the major scenarios:

• We will investigate cases where B moves first and others where R moves first
• Each one can only take stacks above one of their own color checkers
• If the colors allow, they can take a top checker and leave the rest of a stack undisturbed
• If their color is the bottom of a stack, they can remove the whole stack
• Usually, they will have choices about which stacks to remove

Then, I explained the losing condition: if you don't have any more moves, you lose. They quickly realized this was the same as when they no longer had any checkers of their own color on the board.

Next, we quickly played a set of simple games:

1. Single B checker
2. Single R checker
3. B + R
4. RB + BR
5. RB + B
6. B+B+B+B +R+R (and similar)
7. BBB+R+R (and similar)

Connecting with numbers

I told them that one amazing thing is that we can give each game position a value. Then went through:

1. Single B checker is +1
2. What do they guess a single R checker is? explain -1
3. What about B+R? eventually get to 0
4. Explain the fundamental trichotomy: positive value means winning strategy for B exists, negative for R, what about 0?

Powers of 1/2

The first really juicy bit came when I asked what they thought the value of a BR stack would be. This established a common sequence of investigations:

1. If alone, can we see a winning strategy for either B or R? In this case, obviously B. Thus, the value is positive
2. Compare with 1 by playing the game BR + R. They were really fast about seeing this link, for others this is worth writing out and spending some time discussing. In this case, they saw BR + R must be negative, so the value of BR is between 0 and 1.
3. Guess a value; they've got enough experience to think that often the answers are "nice," so 1/2 was a natural guess.
4. Test. In this case, it meant checking BR + BR + R

Next stop was RB. I wanted to make sure negatives weren't left out and to reinforce the symmetry in the colors, so would ask them to swap the colors and get a value as a quick follow-up.

Next, I asked them to see if they could find a configuration with value 1/4. This took a long time and there were lots of false starts. I didn't think this would be easy and didn't help them shortcut the exploration.

Once they got 1/4, though, they had a fast guess about 1/8. We checked it and then they made a conjecture about further powers of 1/2.

Deep blue and red

Seeing powers of 2 in some form is always fun, but both knew that they had been promised infinity and wanted to see it. Using our plastic dinosaurs, I introduced the deep checkers deep blue (represented by a blue mini) and deep red (represented by an orange mini).

Our sequence for these was similar to the powers of 1/2. The value of deep blue is positive, so they compared to 1 by playing the game BBBB.....+R. Each had a sparkle of insight, but quickly played BBB.....+R+R+R just to check, then announced that the value would be bigger than any integer.

I gave them the name omega, and we checked the deep red: RRRR.....

At this point, one of them put the deep red on a B checker and asked what that would be. Again, followed the previous recipe to realize that it must be positive, but smaller than 1, then smaller than 1/2. At that point, one of them made this arrangement:

Ok, I know that mastadons aren't dinosaurs

If it isn't clear, the realization was that the game sequence B, BR, BRR, BRRR, ... would have values

1, 1/2, 1/4, 1/8, .... and BRRRR..... would be at the end of that sequence with value ... larger than 0, but smaller than every power of 1/2. You can see that they made a similar connection with the negative values.

A little bit about deep purple

The two kids were totally satisfied now, having gotten omega and epsilon (along with omega +1, epsilon - 1, omega + epsilon, etc). To give them something to chew on for later (and because we had purple dinosaurs in our set) I introduced the deep purple, BRBRBRBR.... Immediately, they designated the yellow dinosaurs as the inverse of deep purple (RBRBRBRB.....)

So, what's the value of deep purple? What they've gotten so far:

• positive
• smaller than 1 (by playing BRBRBRBR..... + R)
• Bigger than 1/2 (by playing BRBRBRBR..... + RB)
• guess 2/3. I don't know where this came from, but I confirmed that is the value
• working on finding winning strategies for the second player in R+R+3(BRBRBR.....)

Wrap up videos

As a round-up in the evening, we watched the videos from the first post in Mike Lawler's sequence. I paused frequently to let my two shout out their answers and explanations for where Mike's boys were in their exploration. This seemed to be a very effective way to underline their experience for the day.

Number match on the number stairs

A game for J3. This is a simple game with some variations that makes use of our stairway "number line" and a three year old's natural enthusiasm for running up and jumping down stairs.

As pictured previously, we have labeled the stairs in our house from 0 to 36 (more to come). We have a set of cards with numbers on them. P mixes them up, then gives them, one at a time, to J3 to put on the corresponding step, and they sing count up and down:

Some other variations:

• using playing cards instead of number cards
• Using cards with dot or shape patterns
• Using cards with number words ("one" instead of "1")
• child sends the parent to a particular step, checks if the parent got it right

Pillow forts

In case you missed it, pillow forts have been in fashion recently. Here is an example:

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.
me: what about this one?
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

counting at the pool

who: J3
where: at the pool

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In the spirit of Christopher Danielson's marshmallow post, Tale of Two Conversations, I wanted to flag some simple ways to add some numbers to a toddler's day. None of these are earth-shattering and they aren't hard to do, but they sometimes take a bit of awareness to remember.

Counting up

At the pool recently with J3, I noticed several times that we chose to use numbers when we might have said something else. First, she started jumping in from the side, so we led with a count: "1, 2, 3 jump!"

That's standard and comes easily to most parents (my observation from watching other parents at playgrounds). A math educator once told me that some kids start school thinking that the counting sequence is "1, 2, 3, go" because they hear that formula so often!

One of our additions is to add a little counting song after each jump. I always think of the sesame street count, so we sing after each jump: "1 [later 2, 3, 4, etc] mighty jump, ha ha ha." She got up to 17 before moving on from the jumping game.

Counting down

Later, she did rocket launches from the ladder. Of course, this was a good chance for counting down practice: 5, 4, 3, 2, 1, 0 blast off!

Oh, look at the time!

Wrapping up all of this, we had a series of comments about the time throughout our swimming session. When we first arrived, I read the time to her, with a couple explanatory comments (it was 10:55, so I explained that the long hand on the 11 meant 5*11 = 55 minutes after the hour and the short hand near that 11 meant that it wasn't quite 11 o'clock, so it would be 10). I didn't expect or require her to get all of this detail.

Then, we talked about how long we could stay: 15 or 30 minutes? 30 minutes was the choice, then when would we leave? As the time passed, how much time did we have left? When we got to the agreed leaving time, how much more time did she want?

I found that the conversation about time, in particular, had to be a conscious effort. Since she doesn't have a strong understanding or awareness of time, it would have been just as functional for me to keep all that information to myself rather than explaining it to her. On the other hand, how can the little ones develop their own understanding if they aren't part of these types of discussions?

Some comparisons (two tmwyk transcripts and a puzzle)

who: J1
when: just before bedtime

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the value of being alive

J1: Daddy, now I've got a question for you
J0: Ok?
J1: if I get a new book every time I write 20 pages in my journal, how valuable is each page?
J1: The books are about 150 baht
J0: How much?
J1: let me check, I think the price is on the back cover . . . 169 baht
J0: if you could tell me what I need to calculate, I'll calculate for you
J1: hmm, so 20 pages is 169 baht, I want to know how much one page is, so I need to divide by 20.
J0: do you think it will be more or less than 10?
J1: less than 10
J0; Are you sure? How do you know?
J1: Well, 10 * 20 is 200 which is more than 169
J0: what about 5 baht per page? Is it more or less than that?
J1: More, 5* 20 is half of 10*20, so 100, which is less than 169.
....
<we figure out that the amount per page is 8.45 baht/page>
....
J1: That's not very much!
J0: How much did you get for your birthday?
J1: [x] from grandma, [x] from grandpa
J0: well, how much is that per day. Is it more or less than 10?
J1: More than 10
J0: how <interrupted>
J1: how do I know? well . . .10 * 365 is ...
<some discussion of whether he was right, various other estimates of the amount of money per day>
J0: How does that compare with each page of your journal?
J1: More...but what if I include the [present a] and [present b]?
...
<he estimates how much different presents cost, figures the total, estimates how much that is per day, etc>
...
J3 (who has been listening all this time): wow, J1, that's a lot of money!

J3 explores bricks

Earlier in the evening, J3 has been building sticks with 1x1x1 TRIO cubes. She made four, all the same length, then handed two to me as drumsticks. I counted the cubes in one (I got 11) and then she counted one of hers (she got 12). I put them side-by-side and we saw they were the same length.

J3: but...daddy, I really counted 12, you are wrong
J0: are you sure they should have the same number.
J3: yes, let's count them again, together
<I point at the cubes and she counts them, 11>
J3: Ok, now I'm going to build a shape and you see if you can make a copy. It will be tricky!

A birthday puzzle

With their current ages expressed as whole years (you know, the way everyone talks about ages, except for mothers of very small children):

1. What is a number sentence that relates the ages of J1, J2 and J3? Hint, oldest is 8, middle 5, and youngest 3
2. Will this ever be true again?
3. Was it ever true in the past?
4. When/why not?
5. What about multiplying? Will it ever be the case that AgeY(J1) = AgeY(J2) * AgeY(J3)?
6. Was this ever true in the past?
7. When/why not?

Note that there is a complication since they were not all born on the same day, so the difference in their year ages changes depending on the day of the year we are considering.

J2 wanted to investigate more precisely, so he asked to work things out in months. That meant we had to calculate how many months are between them.

Seasonal math and dice, dice, dice

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

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Queen of mathematical fruits

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

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

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

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

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

Dice Games

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

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

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

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

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

Knights Move

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

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

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

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

Twist on an old puzzle and our number scavenger hunt

who: J1, J2, J3
when: after lunch, before violin lessons
what did we use: mini-white board and things around our neighborhood

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Digit Substitution

You've seen these puzzles before: there are triangles, squares, and circles that represent digits in an arithmetic equation and you need to figure out which digit is represented by each shape. During a recent RightStart Math session, J1 got a couple of these. The puzzles were fun, but standard, each with a unique answer. We decided to add two twists.

How many answers?
The first twist was to find how many answers there are to this puzzle. You can see us mid-work, having already identified two choices for the circle and several options for the other two shapes.
The point is that it invites the kids to think carefully about the abstract relationships between the values that the shapes can have.

Build your own
I always love reversing a puzzle and having the kids build their own puzzles. I opened with the challenge to see if they could create a puzzle with no solutions. Initially, they draw the shapes at random and assume there's no solution. However, once we worked through a couple, they started to notice patterns and to carefully plot their strategies for making the puzzles impossible. Also, it led to discussions about what should be allowed as a solution: can a number have a leading zero, is it ok if two shapes represent the same digit, can we allow a shape to represent a negative number or a number larger than 9, etc.  Really cool thoughts that transcend the specific puzzles we suggested to each other.

Our number scavenger hunt

J3 went looking for numbers around our neighbourhood the other day. This was suggested long ago on Kids Quadrant last September, but I didn't really appreciate how good it would be for us until we started. Here are pictures J3 took of our finds from 1 to 4.

One swimming pool (that is soo big)

Two black metal pieces to open the door

Three plastic bottles (prior to us doing a civic duty and helping clean up)

Four light switches

Except the first, all pictures were taken by J3. We also found an office chair that has 5 wheels, but the picture didn't quite capture it.

Playing this game, J3 was really excited to count everything in sight. I was a bit surprised that she seemed to have very little prior expectation about whether any particular group would meet the target we were seeking, so we will make sure to play this again soon.

Stairway to (number) heaven

Who: J1, J2, J3
Where: our stairs
When: permanently (ha!)

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J2's school has a set of stairs used by the kids with numbers posted on each step. I finally realized we could copy this at home. It has been great fun for each of them, in slightly different ways:

• J3 feels like these are "her" numbers. Everyone else has told her we made them for her and the sense of ownership makes her pay a lot of attention to them.
• J2 really likes having numbers and shapes posted around the house, so he is the one who is encouraging us to continue to the top floor.
• J1 likes taking J3 on a tour of the numbers and admiring the pictures. He also likes coming up with new things to add (like writing a related number expression)
• They all like talking about the ones they made and their contribution to the project

I have three recommendations for you replicating:

1. Start with zero
2. Be open to a lot of interesting number and shape conversations while making the cards.
3. Do it! if you don't have stairs, make it into a horizontal number line on the walls. 

J2 almost finished decorating the 12 card

Cards in situ, ready to be enjoyed

Detail of the 8 Card: such craftsmanship!

Goody bags (fair sharing)

Who: J1
When: before bedtime
Where: bedroom

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Several months ago, I wrote a post (here) about various sharing problems and the difficult question: "What is fair?" I got distracted until I found some ready-made examples from Peter Liljedahl. J1 and I discussed his Goody Bags scenario last night.

Though it seems simple or childish, I found this activity stimulated a rather deep conversation and was really effective. I highly recommend other parents try it.

In a nutshell, you have these 40 gifts to distribute among 5 friends and want to make the distribution fair:

J1 cut out all of the pictures, spent a couple minutes putting them in piles, then we arranged them and talked about whether the allocation was fair.  Each rough column below is a goody bag:

When discussing, we didn't start with them all in a grid, but worked a column at a time, starting from the right.  that means we had two goody bags out and compared them against each other.  J1 naturally came to the arrangement that put common items side-by-side and he was careful to leave spaces where one bag got something that was missing in the other.

As we discussed, he was pretty clear that the common items weren't relevant to deciding whether the two goody bags were fair, we only needed to look at the differences. When talking about comparing unlike objects, he tended to focus on how permanent they would be and compared the difference baskets rather than individual objects within them.  For example, the two rightmost bags differ by +tatoo+sticker - kitkat - gel pen. He judged this fair because the kitkat wouldn't last long, the gel pen was nearly permanent while enjoyment of the the tatoo and sticker would both be moderately lived.

At each step, we compared the new goody bag to the single preceeding goody bag that was most similar.  Eventually, J1 proclaimed that he was satisfied with the distribution and didn't really have a preference between any of the goody bags.

We talked about some of the characteristics of his division:

• when there were 5 of some item, everyone got one
• when there were more than 5 of some item, everyone got at least one
• everyone got the same total number of items.
• no one got more than 1 duplicate

Next, we talked about how he had divided the items. He explained that he had simply done it randomly, by which he meant that he took all the items and dealt them out in a circle. I asked whether it would work well if we tried it again, so he did, but made a very important change.  This time, he dealt the items in a line and gave two items to the bags on the ends in each pass. If that's hard to picture, call the bags A, B, C, D, E and then he is giving goodies to the bags in this order:
A- B - C - D - E - E - D - C - B - A - A - B - C - D - E - E - D - C - B - A (etc).

This process gave us a very different result (not pictured) where most of the previous characteristics were violated:

• A and E got a lot of duplicates
• Often, someone wouldn't get an item, even if there were 5 or more

Everyone still got the same total number, but it was hard to call this split "fair."

We had a good discussion about why the results were different. This included a really interesting idea: if the process used to divide the goodies was "fair," then maybe the ending distribution is fair even if it doesn't look right.  Essentially, he was defending randomness and arguing that everyone had an equal chance to get any particular goody, so the opportunity was fair. This was a pretty sophisticated idea that I hadn't expected to hear. I think our experiment of dealing goodies in a circle vs the line method also demonstrated that apparent randomness can be thwarted with subtle systematic biases, another rather deep idea.

We moved on to discuss whether he had any personal preferences among the items. He ranked them as below, most favored on the left and least on the right. Items that are stacked are considered equal and there is some sense of spacing.

Lastly, he revealed that he had 3 strategies for attacking this challenge, to be employed sequentially, if needed:

1. deal cards randomly
2. swap items if they seemed to be blatantly unfair
3. give up on the whole project and just put all his favorites in one goody bag and day-dream that he got to keep it.

J1's closing comment
When we were all done, I told J1 I was glad we had finally done this sharing challenge. He looked up and said: "oh, this is what you meant! When you said we should do the sharing challenge the other day, I thought you were going to make us share our actual toys, that's why I didn't want to do it."

J2's observation
J2 and I talked a little about this investigation. His solution was simple:

• first row goes to bag A
• second row goes to bag B
• third row goes to bag C
• fourth row goes to bag D
• fifth row goes to bag E
• then the remaining three rows allocated to each bag (in order) by column

He declared this fair because everyone got the same number of items and was entirely unperturbed by the idea that, for example, someone would get 5 potato-head keychains. So, maybe the conversation won't be deep with every child. Or, this indicates stages of maturity in thinking about what fairness is?

Counting challenge (revisited)

Who: J3
When: at breakfast
Where: the dining table

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J3 and I had an opportunity to revisit the advanced counting challenge described in our post here: How many cows?

The first time, I kept turning the cup as she counted: 1 ... 2 ... 3 ... 4 ... 5 ... 6 ... finished!
Somehow, this time she had a sense that there were a finite number and that she didn't need to count any more.

I asked, so, how many cows are there and held out the cup to her.  Her reply: 1 ... 2... buckle my shoe, then giggled and ran away.

Did she actually count and know there were only two, or was she just amused with herself and felt like adding the nursery rhyme line instead of counting further? I don't know.

Monopoly arithmetic

who: J1
when: at bedtime (we were supposed to go to sleep, but stayed up to play)
where: bedroom

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We have recently had an outbreak of Monopoly.

Apologies for blurriness; my shutter speed isn't fast
enough to catch the lightening action of this game

J1 and I have been playing a lot of City Monopoly.  Basic play is similar to the classic version, but on every turn the player uses a little randomization device (not a dice!) to determine how many enhancements they can build on their properties (1-3 building blocks or a railroad). This device can also be used a minute timer for auctions and an hour timer for the overall game. I guess these extra functions justify why they didn't just use a tetrahedron dice.

Monopoly games have been known to last for a very long time, but we aren't allowed to keep a partially-completed game lying around the house.  Instead, we play for about 40 minutes and then tally up the assets to see who has won. One of J1's interesting observations: we usually end up with less money than when we started the game.

Where's the math?
There is a ton of thinly veiled arithmetic in this game as every action requires some type of calculation. This ranges from 2 dice addition when moving the pieces to simple multiplication when calculating the cost of building several blocks to 3 digit subtraction when making change.

There are slightly more subtle points around deciding where to build property enhancements based on which properties give a greater return on capital.  J1 is starting to build an intuitive sense and has made some good observations when comparing between properties.  The most sophisticated analysis he started was looking at the first ten squares and talking about which ones are "easy" to hit.  For example, the very first property can't be hit until players have gone all the way around the board and he was excited to realize that..

Where's the game?
Strategically, there doesn't seem much depth.  The only frequent choice on each turn is where to build the enhancements, including railroads.  Occasionally, there is also a choice about where to build a hazard or a bonus structure.

I said that we usually end up with less money than at the start.  Actually, we have played 10 times and always lost money.  Seems that the winning strategy is just to avoid taking action.  For us, that mostly means not building enhancements, except for the very rare times when they immediately increase the properties rental value as much as the enhancement cost.

Challenges (Programming class 6)

Who: Baan Pathomtham 5th grade class
Where: at home
When: break between terms

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This is a set of challenges meant to test what the kids have learned this term. Initially, I had planned to use these during our last session, but the schedule requires us to leave these for independent work at home during the break.

Instructions

Do as much as you can.
These are fun, but are supposed to be hard.
Work on the challenges in any order you want
Use your computer, old programs, and any references you want.
If you have a question, ask! If you are confused about something, ask!

Challenge 1:
Someone deleted a line of code in two of my programs, can you fix them?

A. Program http://jgplay.pencilcode.net/edit/class/testStar is supposed to make this star

What is missing? Copy the program and add the missing line.

B. Program http://jgplay.pencilcode.net/edit/class/testHi is supposed to write a little message

What is missing? Copy the program and add the missing line.

Challenge 2:
What does the following program draw?

Challenge 3:
Write a program that draws this spiral:

Extra Challenge:
Add the button and animation to the spiral in challenge 3.

Intro to Functions (Programming Lesson 5)

Who: Baan Pathomtham 5th grade class  + Pooh teaching solo (Josh was traveling)
Where: at school
When: 2 hours Monday morning

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Goals:

1) Review for loop

2) Introduction to writing functions

What we did today:

0) Discussing the homework.  They showed the car they modified last time

1) We started with a short program which shows a for loop and and if statement.  This program draws a 20 pointed star:

              speed 100

      pen red

      for x in [1..20]

         fd 80

         rt 100

         if x is 10

            pause 2

2) The children were then asked to create 20 pointed stars in 2 more colors.  Then I showed them how to use function so that they can call the function to draw stars of their color choices.

           

              speed 100

      star = (color) ->

         pen color

         for x in [1..20]

           fd 80

           rt 100

           if x is 10

             pause 2

      star red

3) Then we played with how to make command buttons.  Again, initial program supplied and then asked to extend the program to have more buttons.  I showed that button can do complicated things like drawing the 20 pointed stars as in previous example or even drawing a wheel of a car that was part of their homework last time.

           

            pen sienna

     button 'R', -> rt 10

     button 'F', -> fd 10

     button 'D', -> dot 'green'

4) Last we played with a program that takes user inputs.  It seems effective to have them play around with the program initially and then ask them to extend it in a certain way.  In this case, I ask them to extend the program to also take user input for the size of the polygon.  This one was a bit challenging for them and it was very gratifying to me to see them struggle with it before arriving at the answer.

    

            await read "Color?", defer color

     await read "Sides?", defer sides

     pen color

     for [1..sides]

        fd 30

        rt 360/sides

Homework

Draw a flower garden or a forest.  

How many animals (domino counting challenges)

who: J3, J2, and J1, but the real questions here are for older children
when: morning with J3, evening with the older ones
where: family room
what material did we use: a set of animal dominoes

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When J3 counts, she often says "eleven" after "six" instead of "seven." I set out to show her the difference between 7 and 11.

First, we've got 7 dominoes, I put the blank sides up so that the pictures wouldn't distract us from counting them all as a consistent collection:

Next to those, I arranged a group of 11 (again, all blank sides up):

By this point, she wasn't interested.  I think she found some take-apart cars that needed to be investigated. I, however, was interested in two follow-up questions:

1. how many dominoes are in this set?
2. how many different animal types are there on these dominoes?

The challenge of the first question is coming up with techniques that don't require counting the remaining dominoes.  I have one strategy in mind (other than pure guesstimating) that I think is too advanced for J2 and J1.

For the second question, I think even a direct count requires a thoughtful strategy (though some simple ones are available).  What direct counting strategies can you suggest?

Below, I'll show you how many dominoes are in the set, but I'm still only going to show you the blank faces.  Is that enough information to figure out how many animal types?

---

I'll give you the easy way to figure out how many dominoes are in the set

Now, can you answer the second question: how many animal types are used in this domino set?

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Advanced challenge:
Do the same thing with a pack of Spot It cards.

Loopy loops (programming lesson 4)

Who: Baan Pathomtham 5th grade class (all present)
Where: at school
When: 2 hours Monday morning

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Animations

Again, starting with some code that the kids wrote:

One name in English

The first one done in Thai

Another one in Thai

This wasn't what I was expecting, but it works

What your child has learned

0. Reminder: You can see code that the students are writing through this directory. My own folder with a lot of examples is here: jgplay.pencilcode.net/edit and the class folder is here jgplay.pencilcode.net/class/edit.

1. They each gave a surprise presentation (in English!) of their homework code.  Each explained a bit about how their program worked and then answered questions: which letter was most difficult, was there anything they would change/improve about their code?

2. More nested for loops. We played with their copies of this code (Nesting Catalog) to see (a) what happens if we move the fd 40 code between loops and (b) how to fix our code if we increase the second loop to run from 3 sides to 10 sides (or more).

3. We improved some code that Gan had shown to the class last time.  A friend had written this code (WoWCar) and it was just begging to be rewritten with a for loop.

Remember this reference: http://guide.pencilcode.net/home/

Homework

This week, the challenge is to complete the car that the friend started.  Loops that draw the wheel will make it much easier to add another wheel (or maybe they are going to create a giant truck/lorry with 20 pairs of wheels?)

The following links will help with the geometric commands:

http://pencilcode.net/material/measuring.pdf
http://pencilcode.net/material/arcs.pdf
http://david.pencilcode.net/home/explainer/turns
http://david.pencilcode.net/home/explainer/curves

Please look through these with your children.

  
Some great things they did
As a reminder, this is just a couple of highlights from the class.  Overall, the four are really engaging well and taking advantage of the time to explore and test the programming system (including some little boy/non-traditional variable names).

Titus surprised me today when he was working on the problem of extending the shapes catalog to more sided shapes.  He found a way to shrink the shapes so that they would remain in a grid pattern (at the same time reducing the step between each shape.)  I always love it when someone has an approach that I didn't consider.

Boongie was fun to watch during the nested for loop exploration, particularly as he worked on the problem of getting all of his shapes to appear in the preview panel.

What I learned
The kids just need a little encouragement and a small hint to get going on an exploration.  With a light start, they can be off and running on their own steam.

I think they need a bit of encouragement to ask more questions.  Perhaps the language is an issue (English vs Thai)?

Subitizing and Frustration

Who: J2
Where: at home on the dining room floor
When: just before dinner

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Last night, we practiced counting without counting. Blocked from J2's view, I put xiangqi (chinese chess) pieces on the ground, asked him how many and gave him 3 seconds to see, then covered them again.

I started with the 3 black pieces, then did 4 red (the top two quadrants below). Three seconds was really too long as he was able to count them, so this level was easy.

Next, I combined the two previous groups (bottom left quadrant).

He answered 8, quite confidently.  When I removed the cover and counted them, he was frustrated. Instead of persisting with more examples,  I asked him to make a challenge for me.

Smiling mischievously, he asked if I was ready and then uncovered the configuration in the bottom right. Well, of course, I couldn't count them all and I couldn't work out any subgroups that would help me count clusters.  I said "I'm not sure, but I think there are 30."

Here's my thought process:
- from a visual estimation, I thought there were more than 20
- I guessed that he had used all the pieces and, as with western chess, there are 32 pieces in a full set
- Because he had been frustrated on his last round, I didn't want to get the answer right. I wanted to get close and then have a discussion about getting it a bit wrong.

J2 proceeded to count them and there were 30! Oh no! He was stunned, not the intended outcome.

Turns out that little J3 had stolen two of the pieces earlier in the evening

Counting with 6 hands

Please read for the *questions* below as I'd love to have your thoughts in the comments.

Who: Baan Pathomtham 1st grade class (J1's class)
Where: at school
When: 2 hours Tuesday morning

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We (P and J0) got a chance to spend the morning talking about subtraction with J1 and his classmates. It was an opportunity to see some differences between talking math one-on-one (or one-on-two) and a larger group.  Here are a couple of tidbits from the discussion.

Practice with poker chips
Of course all children need to be familiar with the standard gambling implements: dice, cards, and poker chips. The first two are already well known, so we did an activity with poker chips this time. How would you count all the chips in the picture?  Well, what if "you" were actually a group of 3 first graders?

Here's the strategy one group implemented, spontaneously, as far as I could tell:
(1) divide the chips into equal piles for each child
(2) count the remainder in the center of the pile (in this case, one chip, so the count was trivial and done without an explicit effort)
(3) take turns putting one new chip into the pile
(4) all count together as the new chips are added

Actually, this was their second strategy. At first, it was a free-for-all with all three trying to count all the chips and messing up each others division between counted and uncounted chips. 

Note: counting the chips was just accidental to the activity we were doing, so this shared counting strategy was just a cool thing we noticed along the way. If it had been more central, I would have talked with them about the equal piles at the start (which links with multiplication) and why there was a remainder (which links to the division algorithm).

Enthusiasm
Most importantly, the kids were all really excited and enjoyed working on math. They liked asking mathematical questions about a picture we presented, had fun doing calculations, trying new modeling tasks, and playing the mathematical game.

This confirms, once again, that enthusiasm and curiousity are things we (usually) kill during the educational process.  Not at our school!

Explaining
Given their enthusiasm and apparent facility with the calculations, I was surprised that they struggled to explain their calculating strategies. I can't tell if this is a language issue, if the calculations they were asked to describe are so ingrained that they don't consciously think about them, or if they don't really understand what they are doing.

My key take-away: I will focus a lot more of my discussion time on getting the J's to talk about how they calculated something, see if they can draw a picture, and see if they can explain using a concrete object.

Extensions
As preparation, P and I talked about 3 models of subtraction: taking away, differences, and counting back. P made two comments:
(1) Word problems are harder than straight calculations (said while we were discussing what types of problems to use to have the kids investigate the three models)
(2) "Counting back is such a waste, I always knew the answer through another method and had to artificially demonstrate counting back."

Word problems seemed, to me, the natural way to motivate using a particular model for subtraction. For three quick examples:

• You started with 5 cookies and ate 3, how many are left? This is taking away, obviously.
• Don has 27 poker chips and Tanya has 13. Who has more and how many more do they have? Differences, naturally.
• Walking along a straight line, you go forward 6 meters and then back 2 meters, how far are you from your starting point? Counting back suits this one.

*Question* is this the wrong way to use alternative models? Is it necessary to force them to use "unnatural" models to demonstrate proficiency (for example, using take-away to resolve the differences question)? Does this create difficulties for problems involving alternative missing values in the same types of questions (i.e., you started with 10 cakes and now have 3, how many did you give away?)

Counting back is the same as the movement model, which I called "forward movement" in my post on addition models. This model leads nicely and really easily to emphasizing the role of 0, negative numbers, and subtraction of negatives. Looking a bit farther ahead, it links with vector addition by just extending our operation to more dimensions. Taken along another path (ha, the puns!) it can be used for modular arithmetic (replace directed movement on a straight line with directed movement on a circle).