Sunday, August 31, 2014

Triangles (programming lesson 3)

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


Again, starting with some code that the kids wrote:
A golden star for their first for loop

More complex 3 levels of nested for loops

What your child has learned

This is just a quick list; you can ask them to show you and, if anything seems unclear, I've got some links that can help or just send me a Line message.

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

1. Basics of for loops: this is a way they can run commands repeatedly without having to write repeated lines of code. There are some spacing and punctuation issues to get this right, so encourage them to use the blocks to get things lined up correctly.

2. How to use the command line to run single lines of code. Easier for them to show you than for me to explain it.  This is something they can use to test an idea or in debugging.

3. How do direct a named object. Using this program, they got to test their knowledge of the command line and movement commands by shifting around the named triangles:

Remember this reference:


This week, the challenge is to draw their name in Thai and English.  These links are really useful if they need help understanding the turning and arc commands lt (left turn) and rt (right turn)

Please look through these with your children.

Here is a version of my name in Thai:

Some great things they did
I'm going to start flagging things the students showed in class that are really good and we should encourage.
For loops: when asked to match code with output, Win made some good observations that helped him form a hypothesis without tracing through every line of the code.  In the golden star code, he saw that the for loop would run 5 iterations, so he was looking for a picture where 5 was a prominent feature. In addition, he saw that the pen was going to be gold, so he could match up the color to add confidence to his prediction. Over time, as they encounter more complex code, they will need to use features like this to get a sense of the program, even if there are commands they don't understand.

At the end of the class, Kan started showing programs his brother had written.  Really fantastic to see that Kan enjoyed what we were doing enough to share with his family, that he had learned enough to help get someone started.  This also got the whole class excited about exploring code written by other users. We want to encourage them to learn from other examples they see, which means:
  • searching for examples
  • thinking about what other programs do
  • testing them
  • changing them and seeing the effect

What I learned
Make the tasks concrete and explicit, at least for one example, before letting them pursue a more general/less structured investigation.

Have a back-up for when the network is down.

Friday, August 29, 2014

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

This post: 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.

Mathematical distraction (argh!)

who: J0
when: J3's naptime
what material: scale, bottle, water, formula

I looked down at my hands and saw a partially prepared bottle of baby formula.  Hmm, did I add the right amount of each ingredient?!

Stepping back, how I had I gotten into this (minor) predicament? By day-dreaming about some other math (which I hope to post in the near future), I had managed to mess up this simple recipe:
1. add 4 ounces of water to the bottle
2. add 2 scoops of formula powder to the bottle
3. close and shake the bottle
4. dispense to the consumer

My solution: use a little more (very simple) math to figure out what I had done.

I got out my trusty scale and made the following measurements:
(1) empty bottle: 60 grams
(2) 2 scoops of formula: 20 grams (pictured below is one scoop, I had tared out the lid and the scooper)
(3) 4 ounces of water is approximately 120 mL/approximately 120 grams
(4) my partially made bottle: 155 grams

Hmm, 155 grams? My target mass was supposed to be 200 grams. If I had forgotten one scoop of powder, it should have been 190 grams. So...

Looking more carefully, I had only put in about 3 ounces of water (mass about 90 grams).  I concluded that I needed an extra 30-40 gr of water and another scoop of powder.

What lesson did I learn?
Maybe I need to be cautious thinking about engaging math problems when I'm driving?

Monday, August 25, 2014

Flags (programming lesson 2)

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

Apologies that this post is a little bit of a muddle. I am including material for the school teachers and other parents so that  you know what we are doing and can help your child if they want. For the more general audience, I want to explain what we did and share my thoughts about what I learned and teaching more generally.


Let's start with the really fun bit: some code that the kids wrote:
A Thai flag

Colorful yin-yang (inspired by the Korean flag)

What your child has learned

This is just a quick list; you can ask them to show you and, if anything seems unclear, I've got some links that can help or just send me a Line message.

0. You can see code that the students are writing through this directory. My own folder with a lot of examples is here: and the class folder is here
1. Basics of setting up an account, logging in to our accounts, opening a new file for programming, saving our work, and changing the name of the file. All of these are summarized quickly on this page: Making a Pencilcode Account.

2. How to test the program and run it in full screen mode, how to use blocks to build a program and how to toggle between code and blocks.  Play around clicking on the basic screen and these are all pretty easy to see.

3. Basics of moving the turtle (fd, rt, lt, jump) and drawing (pen, dot, pen path, fill). This handout has enough to get started and is what we worked on for the first lesson.

Important there is a lot of helpful material here: Feel free to make use of any of this, even code from which I have drawn the in-class activities and homework assignments.

The challenge this week is to replicate this turtle drawing (the last figure on the hand-out from last week):

What are they really learning?
This course isn't really about learning how to make a little turtle move and maybe it isn't even about programming. Here are things we did and what I wanted to show them.

To start the lesson, we looked at the programs each student wrote to draw a Thai flag. I've given one example above, but suffice it to say that all five of the programs were different! I was a little surprised we got so much diversity from this simple flag, but that was good for the kids to see: there isn't a single right answer.

Some students were only partially successful.  That's also fine! We got to talk about where they were stuck and learn another pencilcode command (pen path . . . fill <color>), which we later spent most of the session investigating.

I showed them my flag programs: South Korea (more complex shapes and animation) and Thai (simple animation and music). Was I just showing off to the beginners? No, letting them know that there isn't a limit to what they could build (it isn't just moving a little turtle around).

The main meat of the day, though, was spent playing with filled paths, drawing arcs, and drawing simple shapes. This was really a geometry lesson and I was delighted to hear them asking each other: "how many degrees? how many sides, how long should this be relative to that?" They also did a lot of experimenting with new commands, trying different angles and sizes to get things to fit together. Overall, a lot of great math.

And. . . once again we saw that simple things can be attacked from many different directions.  For the task of drawing a simple equilateral triangle, the kids came up with 3 different approaches (though they couldn't get all of them to be equilateral).

What I learned
First message I got was to ratchet down my expectations for how quickly we can go through the programming material.  My intention had been to introduce for-loops today (and that was even a step back from my original plan to go through for-loops and nested loops!) We need give them a chance to really get used to the basic operations and elements around programming.

The second lesson is to make the math part more explicit. Inherent in all their drawing programs will be distances, angles, circles, arithmetic, and some algebra. For students this age, I need to respect that these are also important concepts and big challenges.

Saturday, August 23, 2014

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

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.

Wednesday, August 20, 2014

Addition, it's just making a bigger collection, right?

Who: J1 (also guest appearance from J2)
When: at dinner
What materials did we use: our mouths (just talkin') and then a bunch of miscellaneous items later

J1: [Friend X] isn't good at multiplication.
M: That's not surprising. Has he started learning multiplication in school?
J1: Actually, he's not so good at addition either. [Friends Y and Z] also don't really understand addition.
D: Great, so maybe you can help them. Is there a picture you can draw that shows how you think about addition? For example, 4+3?
J2: (looks at his fingers for a moment) that's 7! (exclamation, not factorial)
D: So what does that mean, when you add?
J1: (making a big gesture with his arms) Gathering things together, collecting them.
D: Are there other models of addition that you know? Other things you do where you need to add?
J2: counting, then counting some more
. . . tbc

So, how many models of addition have your kids seen? How many are there? Let me know what you think in the comments.

Repeated Counting

For a hobbyist counter, like J3, this activity is an excuse to count up to 20.  Twenty? Well
5 fingers + 5 almonds + 5 fingers + 5 almonds = 10 fingers + 10 almonds = 20 etwas.

My argument is that the fingers and almonds aren't really being combined, they are all serving as objects for counting, then counting some more.

Grouping Together
On the 4 different stems, we see 1 longan, 2 longan, 3 longan, and 4 longan. How many ลําไย are there all together?

I'm calling this distinct from repeated counting because the 10 ลําไย do make a coherent collection all together in a fruit bowl, while their separation into 4 subsets probably won't be relevant for their future destiny and, in fact wasn't relevant to why they came to our table (they were all bundled together in a cluster).

Forward Movement
You've seen this game before: roll the dice and move your fierce dinosaur closer to the finish. A number line gives you a similar model, with the benefit of fractional steps, but I liked the fact that the motion is only sequential by convention here.

Combining Mass

A lucky one as the measurement error doesn't destroy our perfect whole number addition here.

Combining Volume
Yes, it was necessary to pour out 1/4 cup + 1/2 cup = 3/4 cup of chocolate milk in the course of this investigation!

As with the mass model, note that this is imprecise and the experimenter must decide from the context what gap can be tolerated between the theoretical and empirical results. This is especially critical for valued substances like chocolate milk!

(Musical) Time
One quarter note plus four eighth notes get played for a time lasting one measure, in 3/4 time.  Or, if you want to be measure independent, four eighth notes together last the same time one half note.

This is a bluebird of happiness. It is not a model of addition . . .  or is it?

How many cows?

Who: J3 (2 yr old)
When: at breakfast and again at lunch
Where: the dining table
What did we use: a cup with a repeated picture on it

Look closely at the cow (cows?) on the cup below. I've taken the picture from 4 orientations, so you can see all around the cup. How many cows are there?

I discussed this with J3 the other day.  She pointed to the cow and counted, then I turned the cup until she saw another cow, pointed at it and counted.  We kept doing this until she had counted up to 5.  I asked her to try counting them again, and again I stopped turning when we got to 5. I asked how many cows are on the cup and she said 5. We did it again, but this time I kept turning until she counted 6.  We did it again up to 10. Finally, I kept turning until her counting got very unorthodox (mid-teens) and with large skips and also going back to smaller numbers.

Later in the day, I put a piece of tape next to one of the cows (this is actually when I took the pictures above). I repeated the same turn-and-count with J3 (I think up to 10).  Then I drew her attention to the tape and repeated the counting.  Same result.

As I typed this, she looked over my shoulder, pointed to the pictures and proceeded to count up to 5 cows.

We do a lot of counting objects, moving them between piles, associating them with fingers, comparing which is larger, so I already knew that her one-to-one association of objects and the count is shaky. Still, I was surprised that she was willing to accept an arbitrary number of cows on the cup.

Did you notice the white plus signs on the cup? How many are there?

If you didn't notice them, perhaps this was an invisible gorilla experience.

If you did notice and also counted them, particularly if you distinguished large and small ones for a more detailed analysis, then you can be an honorary member of our family!

Sunday, August 17, 2014

Turtle Houses (Programming Course Lesson 1)

Who: Baan Pathomtham 5th grade class (minus one student, plus one friend)
Where: at school
When: 2 hours in the morning

We had our first lesson today and it was a lot of fun.  Here's a quick synopsis:

Robot Turtles
We started with the basics of robot turtles.  Everyone got a pack of command cards (forward, left turn, right turn) and then I executed the moves as they played. We cycled through a couple of times with them giving commands in turn, then each student took a turn as the moving turtle. By chance (?) each turn ended when we crashed the turtle into something and I exclaimed "Crash! Next player."

We were having trouble with the network connection, so we then played three rounds of robot turtle on the board:

  1. no obstacles, each player plays one card at a time and I moved the turtles
  2. Add walls that they need to negotiate, each player plays one card at a time
  3. With walls, each player plays 3 cards at a time for a batch move.

This was all easy, but seemed engaging.

In the future, I will plan robot turtle activities either as a warm-up or when we have to take a break from the computers (or network is giving us trouble).

Pencilcode Intro
I showed them how to create accounts and open their first program.  Then, we worked through the first two activities on this worksheet:

Periodically, I would stop them to point out actions or utilities:
- the run triangle to play their program
- how to save
- run in full screen
- open a new program file
- shift to block mode and shift back to code mode
- move through folders to explore other users' code

They did a good job adding a door to the house from the worksheet and immediately started experimenting with other things they could draw.  Here's an example:

I think it worked really well to show them the blocks after they had written a decent program first. They were able to drag in new commands to see what effect it had and those effects were often surprising, especially when they put a new line in the middle of their existing program.

Finally, I explained their homework and gave them some time to work on the challenges, think about them, and ask questions.

This week, the kids have 3 homework activities and 2 open activities:

  1. Show their parents how to draw a Thai flag using pencilcode
  2. Figure out the code to draw each of the designs on the left (from the first "chapter" of the pencilcode book)
  3.  Draw what they think each of code blocks on the right do and then test it (this is from the second "chapter" of the pencilcode book)
  4. Look through other programs and highlight ones they find interesting.  I'm not expecting anything concrete on this, so it is a half assignment.
  5. Think about something they would like to program as a project.  Again, want them to start thinking, but I'm not looking for anything concrete yet.

What did I learn?

  1. Important to have an offline, back-up activity when the network connection isn't performing
  2. Most of my priors were confirmed: the kids were eager to experiment, they benefited from a little guidance and some prompting to try particular things, they were good at explaining to each other how to do something, and the intro activities were really easy to blend into a fun lesson.

Matific Games (quick review)

Who: J2 (with some help from J1 at times)
Where: dining table
When: after lunch (was everyone finished eating? Not clear.)
What we used: on our laptop

J2 saw I had an open website and started playing some of the games: Bees and Flowers, Spy-a-Meerkat, Monster Shop I, and Tile-a-Shape. For a free (it seems) site, I was impressed: the animation was cute and the activities were engaging, even if these four were a bit easy for J2.

Two points of discussion:

  • Spy-a-Meerkat: J2 was fine up to 5, but then had difficulty when there were 6 or more.  J1 had no trouble and I should have had him talk about his strategy for figuring out the answer
  • Since all of the questions were pretty easy, the two together didn't get any wrong, until J1 said: "I wonder what will happen if we get one wrong. Let's try it." I love that.

They got stuck finally on the last shape tiling.  See if you can figure out the answer.  You can only translate and rotate the purple shapes, no scaling:

Saturday, August 16, 2014

21 with uno cards

Who: J1 and J2
When: after bath and before bedtime
Where: bedroom
What material did we use: pack of UNO cards

I'm working through the games and activities listed on this page of ideas for J1's school. Tonight, three of us played a version of 21 with the UNO cards.  I considered removing the non-numerical cards, but instead made up the following rules:

  • everyone is dealt 7 cards at the start, you draw a card every time you play one, so should stay with a 7 card hand
  • numerical cards played are all added, up to 21, with the last player able to play collecting the stack
  • reverse and skip cards can be played at any time (perhaps I should have restricted it to play on a card of the same color?)
  • wild automatically take the stack to 21 if the player can say what value is required
  • draw 2 and draw 4 lead the next player to draw extra cards (so it helps an opponent)
Overall, I think the game was fairly enjoyable.  The most interesting point was asking J2 the value of a wild card he had played onto a stack with value 5 and he replied "5+5+5+1." He did this again several times, showing a couple of things about his thinking: first, he wasn't really subtracting, he was filling in a gap, he is comfortable with some number bonds (often his first step is to get to 20 or 10, and he likes adding by 5s if there is room.

Extension discussions
For several reasons, we just played the game and didn't talk a lot about the structure.  Here are two areas of discussion we could have pursued:

  1. Are there strategic plays? Given the numbers available are 0 through 9, how many turns are required before we get to 21? Generally, if you have a choice of two cards, should you play the smaller or larger one?
  2. Hey, this is a game I just made up.  What do you think about the rules? What is fun, what's not? What else could we try?

Conclusions about the game
J1 was generally pretty fast with all of the little calculations required, so it felt like playing a game rather than drilling arithmetic. Based on this test run, I'd say it is a decent game for J1's class, or a similar group of students.

I really liked the rule about wild cards and will use that again when we play next. However, J1 didn't like the treatment of draw 2 and draw 4; he suggested that the player should get to draw the extra cards for them self.

Skip and reverse also seem like duds. As noted above, I may restrict their use to when they can be played on a like color. Another idea is to assign them a numerical value (10 and -5 both appeal to me). Another possibility is that they reset the pile value to 0.

Just play UNO? 

On the other hand, playing with the UNO cards made them feel like we should just be playing UNO.  We ended up playing two rounds of 21 and then one game of UNO.  It has become accepted that J2 will always win UNO when we play as a family. Of course, this isn't likely to be true over time, but tonight was no exception. In fact, his last card was a lucky play: after a string of reds, J1 played a red 5 allowing J2 to drop his last card, a yellow 5!

Thursday, August 14, 2014


Major international mathematics prizes were awarded yesterday at the International Congress of Mathematicians. For the Fields Medalists and Nevanlinna Prize winner, Quanta magazine has really nicely written profiles.

I just want to offer a personal note of congratulations to Manjul Bhargava (Fields Medal) and Jacob Lurie (Breakthrough Prize).  We were undergraduates together and it was a lot of fun to work on mathematics together so long ago!

Since this post doesn't have any actual math in it, here's an animation to make up for it:

Wednesday, August 13, 2014

Currency conversion and 1001 nights

who: J1 (and a bit of J2)
where: bedroom
when: bedtime (particularly after lights out)
what materials: internet connection (not really necessary)

This is another Talking Math With Your Kids -style conversation.

  • J1: Daddy, what's 100 pounds (British pounds) in Baht (Thai currency)?
some discussion about which way he wanted to convert as I hadn't really been listening
  • D: Well, there are about 50 baht per pound.
  • J1: So, I need 50 groups of .  . .can you use the computer to calculate it?
  • D: Yes, I can, but we don't need to.  First, though, we need to figure out what calculation to do. You said you have 100 pounds and there are about 50 baht per pound
some mumbling, not really getting anywhere; I'm tempted to comment and guide, but hold back.
  • J1: I have 100 groups of 50.  What's that?
  • D: What about 10 pounds in baht?
  • J1: (pause) that's 500
  • D: how did you calculate that?
I was assuming some strategy for directly calculating 10 x 50, either just adding a zero or building from 10 x 5 (which he would calculate as 10-20-30-40-50), or 50-100-150-200-250-300-350-400-450-500.
  • J1: well, I know 20 pounds is 1000 baht.  Then 10 is half of 20 and 500 is half of 1000.
  • D: Interesting.  How many 20s are in 100?
  • J1: (thinking) 5000 baht in 1000 pounds!
some conversation about what  you can buy with 5000 baht.
  • D: what about 15,000 baht.  How many 5,000s are in 15,000?
  • J1: 3, so 300 pounds.  Wow, that's a lot

Why did I find this so interesting?
First, I was really surprised by the strategy to calculate 10 x 50.  This reinforces the magical phrase "how did you think of that?" Sometimes my own preferred approach seems so obvious that I feel there won't be anything interesting gained by hearing the child's approach and, in this case, asking the question was just because I wanted to build a good habit.  I was so surprised by the answer that I forgot to tell him about an alternative strategy.

Second, there had been several other times earlier in the day when I tried to lead him into a mathematical conversation and he wasn't taking the bait. I guess I should relax and see where chances arise instead of controlling it.

Ok, but 1001 Arabian Nights?
We've started reading the Project Gutenberg version of 1001 nights. Nice mathematical title, no? Well, tonight we started The Story of the Husband and the Parrot. What you need to know is:

I am reading the story of Sheherezade to J1 and J2.  In this story . . .
. . . Sheherezade is telling King Shahriar The story of the Fisherman, in which . . .
. . .the fisherman is telling a genie The Story of the Greek King and the Physician Douban, in which . . .
. . .the King tells his Vizir about a story told by another vizir to King Sinbad, in which . . .
. . . we get The story of the Husband and the Parrot (which involves the Parrot telling the Husband a short story).

Counted generously, that's 6 stories-within-a-story.  And now I've told you, so that's 7 layers.

Tuesday, August 12, 2014

Dots and Boxes (and adding and multiplying)

who: J1 and J2
when: while waiting for lunch during a Chinese religious holiday
what material did we use: paper and pencil

Games, Games, Games!

So, I've been (binge) reading the posts over at Talking Math with Your Kids. I suggest you go over there an do the same (binge reading).  Try to absorb as much as you can, then go back from time to time in order to reinforce the new habits.

Here, I want to offer you a cheat: games. One key objective of TMWYK is to achieve the moment of reflective wonder ("I wonder why/what/if/etc") and games have the wonder built in: ("I wonder what I need to do to win?")

To provoke a rich conversation, the games can have pretty simple rules.  We've written about others in past posts; recently we've played several rounds recently of dots-and-boxes.

On a piece of paper, you start with an array of dots. Each player takes turns drawing a horizontal or vertical line connecting adjacent dots. If one player completes a box by drawing the fourth boundary line, they take possession of that box and get to take another turn. Here's the wikipedia page, from which I'm borrowing this helpful illustration of a simple 3(dot)x 3(dot) game:

Sample topics we've discussed:

  • How many dots are there?
  • How many boxes will there be? 
  • Is there a relationship between the number of dots and boxes?
  • Can this game (for a particular grid) end in a tie?
  • Simple strategy: filling or avoiding long chains.
  • More advanced strategy: double cross
  • Symmetrical play: what happens if the 2nd player copies the moves of the first player (reflected through a line or point of symmetry)? This is interesting for other games as well.
Other than the mathematical content, these games have been a welcome distraction during the dead time waiting around family obligations.

Oh, and for the other Thai readers here, Happy Mother's Day!

Thursday, August 7, 2014

Adding and Subtracting Games

Who: J1's first grade class
When: in class time or extra time around school
Where: at school
Why: to accompany lessons on adding and subtracting
What will we use: cards (playing cards or number cards), dice, possibly some packaged games

These are ideas for enrichment and deepening activities, to be discussed with the teacher. Do you have any thoughts on which are the best games? Please comment below. 

Future posts will repeat this exercise looking for activities that support time+calendar work and another around money.

(1) Blackjack:
Game A: Use number cards instead of playing cards, no double down
Game B: With playing cards, can use simplified traditional rules (no double-down).
Game C: To focus on adding by a particular number, redefine the face cards to that value, e.g., 9, to make the frequent additions more challenging.

Note:  B or C can be modified to fix the value of ace as 1 instead of allowing the 1 or 11 option.

(2) 21 (as with blackjack, these can be played with traditional playing cards or number cards)
Game A: deal out n cards to all players (n should be between 4 and 7, depending on number of players). Players take turns putting down a card and adding to the sum until the next player can't stay under 21. The last player to play collects the cards which have been played.  Count number of cards played at the end as points.

Game B: Same as A, but each player can add or subtract the value of your card from the accumulating value of the pile. The value of the pile has no bounds and gets collected by the player to land exactly on 21.

Game C: modify the value of the face cards (if using traditional playing cards).

(3) Strike it out: described on this NRICH site and blogged about here

(4) Number bonds of multiples of 5
We blogged about this game here:

(5) Sum swamp/Chutes and Ladders
For those who don't know Sum Swamp, players roll three dice, two with number values and one with addition/subtraction operators. They perform the indicated calculation and move that number of spaces along a path toward the finish line. In other words, it is chutes and ladders, but with the operation die added.

Game A: play with 2d6
Game B: play with other pairs of dice. I think Sum Swamp will work up to 2d10, but the path is probably too short for larger dice. Chutes and Ladders can probably work up to 2d20, but I would impose a rule that the winning player has to land exactly on the final square (or maybe within 5?)

(6) Dotty Six
Game A: describe on this NRICH page
Game B: change the target to complete each box to 10 and use tally marks instead of dots
Game C: change the target to another higher number (I suggest either 15 or 20) and use tally marks.

(7) Pass the peas, again, NRICH
Though the kids will probably end up throwing dried food at each other?
Game A: as described
Game B: use 2 dice and then multiply the die value with the value of the number square it lands on, then subtract those from your residual figure.

(8) Subtraction squares

(1) Eggs in a basket: NRICH
(2) 5 steps to 50: NRICH. Their version only uses 10s and 1s, but the step sizes can be changed. Also, they don't specify what dice to use, so I would prefer 2d10 to get 0 to 99 as the range of starting values. I made a pencilcode program to explore this (here) but now see that I missed the point about using dice to start, so maybe an interesting constraint is formed by using 2d6?

(1) Play with a number balance (as pictured on this page)
(2) using a digital scale: weigh pairs of objects separately.  Figure out how much you think they will weigh in together and then check.
(3) magic boxes: another NRICH activity

note on materials:
Dice: [n]d[m] = use m-sided dice and we need n of them. For example, 1d6 is a single six sided die (usually with standard 1-6 markings), 3d8 means three octahedral (8-sided) dice.
Traditional playing cards: 52 card deck, two through 10, jack, queen, king, ace
TPC numbers: 40 card deck using ace (1) through 10
number cards: special card deck using numbers, we got ours through Abacus Math

Wednesday, August 6, 2014

Zero...One...Two...Three... (Counting)

Who: J3 (and a little bit of J2)
When: at times you want to help teach counting (e.g., all the time)
Where: anywhere there's stuff to count
What we use: whatever is available, fingers if that's all we've got

Many of you have heard me say this before: "there are three kinds of mathematicians, those who can count and those who can't."

I never get tired of that (why?), so I should take the opportunity to publicly apologize to everyone who has (or will hear) me say it many times. Part of what lies behind the joke is how fundamental the basic counting skill seems to be for everything else in our standard math education curriculum. I've heard a high school-focused educator claim that many kids struggling at her level are really dealing with the lack of a firm grasp of one-to-one correspondence (a sub-skill for counting). If you really want to, you can see this skill highlighted as a foundation block in the US Common Core Standards: base skill level in counting and cardinality,

So, what do we do? Basically, we play these 4 simple games (from Amy at Kids Quadrant) almost all the time until the kids are counting fluently. There are only some small points I can add to Amy's great post:
  1. start with 0. I make two balled fists and wiggle them when I say zero, unless I'm counting hands in which case I just say 0. The point is to make sure they realize 0 is also a number.
  2. be silly: this is a game for them kids, so feel free to make silly sounds and gestures.
  3. try to find things they can grab and move around as they count. My intuition is that the more of their body involved and the bigger the motion, the more they will remember.
  4. (optional) try counting in other languages. If you don't care which language, Chinese and Thai are good choices because of the logical naming system they employ (I think other Asian languages are similar).
Here's another really interesting post from KidsQuadrant outlining the skills behind simple counting: here. What I want you to take away: even though counting seems easy, even obvious to you, be relaxed about how long it takes to click for your kids and keep enjoying it as a repeated game.


When I tell the joke and say "count," I'm internally thinking about combinatorics.  I have long felt a bit weak in this area, with anxiety that my counts were either leaving out cases or double counting somewhere. That's the real reason I like this joke so much, because the self-deprecation has a meaningful kernel of truth.

Tuesday, August 5, 2014

Pattern replication

who: J1 and J2
when: just after breakfast and mid-afternoon, in between rounds of Sleeping Queens
where: in our play room and in my office at home
what did we use: pattern blocks and an oversize book (and play Sleeping Queens)

With J1 out of school all week, I've plenty of opportunity to use the activities that I'd listed in advance of our math party. I particularly wanted to try the second of my pattern blocks challenges:
Pattern Blocks Activity B: needs pattern blocks and a large pillow to block the view.  With a friend, take turns making a secret design and verbally describe it to the friend to tell them how to replicate it.
J1 was reluctant to humor me, since he wanted to play Sleeping Queens (all day long, as it turned out). However, when I explained what I wanted to do, he got more interested, saying, "ah, we've done this at school before, it's easy." I was about to set up half a pillow fort to use as  screen to block his design when I realized we had inherited a gigantic (roughly 1 meter tall) book about mummies that was perfect.

J1 set up a design and then started describing it to me and I asked questions when I didn't understand.  Here are the two patterns,

So we got pretty close, thanks primarily to his use of the word "tessellate" to describe shapes that fit snugly together. You can see that we each took for granted the orientation of the top green triangle. J1 didn't tell me which way to point it and I didn't think to ask.

Then, I went behind the screen and put together a fish about to chomp some square orange food, but our communication was really poor and J1 made the design on the bottom:

We got interrupted, otherwise I would have liked to try a couple more times in each role.  In the afternoon, I got a chance to do it again with J3. Not surprisingly, he went for a more complex design than he could describe, including a slightly innovative orientation for the orange squares. He was having trouble giving me instructions, so we ended up with him just rebuilding the original in front of me. In any case, he'd used more than half our orange squares, so we couldn't do a faithful reproduction.

Incidentally, the second design above is what he put together when he was playing with the blocks on his own afterward.

My lessons:
I really liked this pattern activity.  First, it gives a semi-natural reason why they might care about some basic elements of math such as number and shape. Those form some of the basics for explaining what pattern they've created.  Similarly, it is a natural place to start introducing angle measures and concepts like parallel and perpendicular.  When I am describing my designs, I try to use a full range of vocabulary and encourage them to ask about any words they don't know.

Second, like the flexagons, it is a mixed math, art, and language activity.

Third, this was something the kids found engaging but challenging, so it seems like the right level for their current abilities.  In general, it is hard to find things that fit like this.

Sleeping Queens

Finally, the game Sleeping Queens loomed over our day.  One of the friends had brought and left it yesterday for the math party. J1 had played it at school and was really excited; left to his own devices, he would have played continuously and it really was the last thing he did before going to sleep. By the end of the day, J2 was also playing along (and winning). I have to give it a positive endorsement on their enthusiasm alone, but I do have two caveats: for our small children, they feel they are being attacked when someone steals their points (a standard part of the game play) but the game seems to simple to suit older children (and certainly won't be much of a mathematical experience for them).

Baby pictures
Just for amusement, here are some snaps by J3:
Does the Pharaoh approve our pattern design?

Our future coconut crop?

Monday, August 4, 2014

Math party (and flexagons!)

Who: Baan Pathomtham First Grade Class+J3
When: 9am - 2pm
Where: our house
What will we used: see below
Why: oh why, oh why? (actually, it was fun!)

First, thanks to all the kids for being so friendly and polite.  Thank you for leaving everything tidy when you left, though that was under a mother's supervision and probably not surprising.  What did amaze me was when Tanya stopped everyone from rushing to eat a snack and you all went right away to collect the toys/games/crafts.

Second, did any of our plan survive reality? Yes, actually some of the things we prepared went well:

  • Passports: the kids enjoyed having their activities noted and getting stamps in the passports
  • Flexagons: everyone got to play a bit and learned about flexing the hexaflexagons and tetraflexagons. A couple even made their own hexaflexagons and someone decorated a blank I'd left around.
  • Pizza: as usual, kids enjoyed assembling their pizzas and were astoundingly patient while they baked.

  • Origami stars: some kids were interested in making these
In truth, though, this was simply a group that wanted to play games and could nearly have been left alone the whole day with a selection.  As it was, they played a lot of Uno, managed part of a game of Settlers of Catan, and played some assorted other games: Sum Swamp, Walk the Plank, Spot It!, and Squares

Some lessons (for me):
  • Though fairly small, our play room is large enough to host two distinct stationary activities, but they have to be child-selected to be sufficiently engaging for the two groups to remain intact
  • Origami (including flexagons) for this age probably needs a smaller group (one-on-two likely works). The kids have the skills required, but either they found it to concentrate, I found it too hard to concentrate, or a friend would suddenly come over and try to take control of the project.
  • Activities requiring a meaningful amount of preparatory instructions either need to make sure the whole group is listening first, or have the instructions delivered by one of the children.
  • Kids need more encouragement taking things apart (see Flexagons, below)
Below are a sample of the passports, the flexagons, and folded stars from today.

I promised to write up our flexagon experience. After the party today, I'm even more enthusiastic about this activity. There are three reasons why I strongly suggest you start playing with them today:

1. Cheap and easy to make
This is great because, if you break them, then just make more! Since even learning how to flex the shape takes some investigation and practice, there's a danger of tearing a flexagon. That should be encouraged! Cut them open to see how they are folded, force them if you can't see how to flex it. Taking things apart is a great habit/skill.

I think the kids today were overly anxious about their investigation. Perhaps this was the other side to the coin of them being so polite? In any case, this is something I would actively seek to encourage in the future.

2. Mix Art and Math
A blank flexagon isn't much fun, it has to be decorated to make the mystery really come out. For people with a traditional conception of mathematics, this may seem odd: art as a tool to explore the mathematical structure? Yes, yes!

So, if your child doesn't like art, this is a back-door into drawing some patterns or pictures.  If they love art, then this is a back-door into equilateral triangles, angles, rhombus, hexagon, how many sides a piece of paper has, state diagrams, etc.

3. You probably don't know much about flexagons
That means you can let the child lead this activity and just let it develop. That's great because they can be the teacher or you can be equal partners.

Alternatively, you might get excited and have your own questions about flexagons. In this case, this is a chance for you to investigate and for your kids to see you investigating. Do you try to figure it out on your own, draw diagrams, build other models, dissect specimens, experiment, watch videos on youtube, look at wiki pages, ask a friend, talk with your child, all of these things? A great opportunity to show your child how you explore something that interests you.

My flexagon recipe:
Only steps 1 through 6 are really required
1. Watch the Vi Hart video:
2. Make a trihexaflexagon by using tape to complete the loop
3. Color it
4. Flex it and think about how it works
5. Take it apart (this is why we used tape)
6. Make a bunch of additional trihexaflexagons, use different ways to make the series of triangles (folding, compass, 30-60-90 drafting triangle, printed template, etc)
7. draw more pictures and patterns
8. Try making a hexahexaflexagon (again, using the basic comments from the video)
9. Look at some templates online, including Tri-tetraflexagon, Flexagon Portal, More templates and more
10. watch the second part of the Vi Hart flexagon series:
11. watch this third Vi Hart flexagon video:

Sunday, August 3, 2014

Grand Catch-up

Who: everyone in the family
When: over the last week
What material did we use: see below, a lot of different goodies
Where: all over the house

We haven't posted many activity summaries recently, so this is a review of what we have been doing so that you don't think we've been slacking off.

Calculating and Roman Numerals
Remember my doubts about roman numerals when they were introduced (here)? P has since found that they come up frequently when J1 and J2 are discussing some other calculation (adding 2 digit numbers without paper and pencil was a recent one).  It seems that the kids like the process of conversion or the feeling that they are speaking in a type of code.

Mathsemantically, they appear to appreciate the idea that number names are not the same as number concept. Frankly, I don't know when they developed this appreciation as we've always been a bilingual household (Thai and English) and introduced counting in other languages (Spanish, German, Chinese, Korean, Tagalog, Armenian) when they were very young.

Tip: for kids around 7 years old, try introducing the numbers in a different language and see if they enjoy the disassociation of names and objects as they play with calculations.

Food Math
We've had another round of pizza dough (with J3, our 2 year old) and our first attempt at gougeres (J1 and J2).  The pizza dough with J3 went as usual: a lot of fun measuring, a bit of flour outside the mixing bowl, excessive use of the scale to weigh whatever she saw.

The gougeres were a dairy indulgence, with milk, butter, and cheese as main ingredients. This was my first opportunity to try them since J3, our dairy intolerant one, was out for the day with mommy. We roughly followed this recipe, halved because the remaining munchkins had already decided they wouldn't like them and I couldn't justify eating 20-30 cheese puffs myself (and I'm not generous enough to share them with anyone else). In any case, by the time I thought to snap a picture, we only had 5 left for a nice little pentagonal arrangement.

Mathematically, the activity was interesting because we halved the recipe and we had a debate about how many eggs to use. While the presenter explicitly says 5 eggs, both boys were sure we only saw her use three and we all agreed that she had reserved one for the egg wash. Coincidentally, P brought home some very small eggs later in the day, so we got to have a little (very little) discussion of whether one egg is a proper measure. I'll have to remember to return to that again.

As we've hinted on other occasions, we try to recognize little unplanned opportunities for inserting some numerical discussion into everyday conversation.  Quesadillas at lunch were a chance to talk about fractional parts of a circle based on 1/8.  Below is one serving in the 6/8 (aka 3/4) uneaten state:

Some bites and a cheeky smile later brought us: "daddy, what about this one?"

I offered 1/3rd of an 1/8th for the partially eaten piece and we calculated that this plate still had 7/24th. Maybe I should have rushed to get a clock and show them 3.5 hours?

Catan (Catan, Catan)
Somehow, Settlers of Catan got back on J1's radar and he's been asking to play it everyday for a while. We are still working on a version that fully integrates J2 and it is a bit stale without trading between players.  I tried to introduce a bit of trading today, but J1 was too suspicious of my motives and would rather trade at an unfavorable rate with the bank than trade with me.  For now, we still start him with 2 cities against my 2 settlements, and we also allow all the players to start with the resources from all the hexes associated with both of their developments.  Otherwise, we find the game takes too long to get moving.

You can see J1 (red) in the process of beating me (orange) below:

I can tell these sessions have started to develop his intuition for dice probability.  When allowed to set up a board configuration of his own choosing, he put the 6s, 8s and 9s together on the hexes associated with his developments.

And some geometric designs from J2 while he was watching us play:

I plan to have more discussion of this tomorrow after our math party. For now, suffice it to say that I made some flexagons, left them around the house, and both the boys were really excited to make, decorate, and investigate them.  Almost all of them are trihexaflexagons, so that's what they've come to expect.  I gave J1 a hexahexflexagon with numbered faces and he was delighted to discover the extra faces.  I'm looking forward to seeing how the other first graders respond.

Since it isn't fair to entirely leave you hanging, I'd refer you to this Vi Hart video for a great intro to flexagons:

Friday, August 1, 2014

activities for a math party

Who: 6 classmates of J1 + J1 + J3
When: all morning + lunch
Where: our house
What will we be using: see below

Some planning notes for a math (and English language) party at our house next week.

- Brief explanation of basic rules of the house as everyone arrives: be kind, be respectful, be safe, persevere
- Explain schedule, particularly if there are going to be full group activities (reading a story, making food, eating food, change to 100 board pattern)
- have water, fruits and cut vegetables available for snacking throughout the day
- activities set up in stations around the house with some printed guidelines. Tidy up after you finish each activity (to make it appealing for the next people)!
- Kids will go through the activities in pairs, may depend on attendance.  
- Maybe J1 allowed to join whichever pair he wants? Otherwise, create group of 3 to deal with odd number?
- Do we need a map so that the kids have some idea of what is available? How many activities do we really need for 4 hours (which will include lunch)?

To start, everyone gets a passport book and decorates the back cover with their name, partner's name and own design. The book is to record which activities the children have done through the day.

100 Board
Guess my pattern: needs 100 board and 2 colours of tiles (5-10 minutes total time)
3 or 4x through the event, J0 will set up some patterns on the 100 board.  Kids guess what the pattern is each time. Pattern ideas:
- odds and evens
- multiples of 5
- squares
- Fibonacci sequence
- Primes

Pattern Blocks
Activity A: needs pattern blocks, design cards, and a camera
- Copy some pre-made designs
- Make your own design and take a picture (adult will help with the picture?)
Activity B: needs pattern blocks and a large pillow to block the view
With a friend, take turns making a secret design and verbally describe it to the friend to tell them how to replicate it.

Stacking blocks 
Tallest tower: needs collection of building materials (we will use duplo, stacking blocks, trio blocks, polydrons and magnatiles), measuring tape and camera.
- build the tallest tower you can
- estimate the height
- measure the height
- take a picture

Ninja stars: needs paper cut into squares (or even better, 1x2 rectangles)
- take apart some pre-made stars and investigate
- think about how to make them and ask questions
- try to make your own
- decorate
- throw them at designated targets (maybe this is too risky to encourage?)

Hexaflexagons: needs paper cut into strips
- with help, make a trihexaflexagon
- decorate the sides
- explore
- make more if interested
- make a hexahexaflexagon (advanced, gets an extra stamp)

Other origami: can be added according to interest of children
- animals
- polyhedra
- etc

Reading: needs a selection of books
- Select a book with your friend
- read and discuss with an adult (depending on preference, kids can read themselves, adult can read to them, take turns, etc)
- Should this be done all together or pairs?

Writing: needs paper, pencils, some coloring pencils/crayons
- modelled on the story we have read
- insert yourself and make some additional change to the structure
- write your story
- illustrate/decorate

Pick a game and play with your friend
- Sum swamp
- Chess
- Chinese checkers
- Chinese chess
- Uno

Pencilcode intro
- Adult shows them basic intro to pencilcode set-up
- let them explore the gym, other people's code and play

Food: I'm inclined to drop these ideas for the first trial run of this event; mini-pizza is the strongest contender to keep
Mini-pizzas: needs pizza dough (shaped), tomato paste, cheese, cut veg
 each pair gets a blank dough to top and decorate as they desire
- bake
- eat

Coconut faces: needs coconuts and craft material
- Ask an adult to open a coconut for you
- Make a face on the top
- drink it and pretend the face is making silly noises at your friends

Mini-bread experiments: needs water, flour, yeast, salt, eggs, sugar, milk, oil, honey, seeds, cheese
to be done with J0 supervision, have everyone do this all at one time?
- each pair gets to measure out 150 gr flour, 2.5 ml salt (1/2 tsp)
- measure optional wet ingredients (egg, milk, honey, sugar, oil) up to 80 ml total
- add water to top up wet ingredients to 80 ml
- mix wet and dry
- knead and shape
- rise
- bake
Note: generic ingredient ratio should be checked against some sample recipes