Sunday, December 28, 2014

Sprouting! (observational botany 2)

Who: All Js
Where: at the dining table and in front of our house
When: after lunch (observations from 19 December/day 18)


Our avocado pit has made some progress.

J1's observations
  • the root has sprouted
  • maybe it is actually upside down and the avocado is confused?
  • the pit split in the middle.
  • there is a sprout in the middle
  • some of the pit has peeled off
  • the exposed pit is a bit more rough than when we examined it last week. it feels like bumpy wax.
J2's observations
  1. Maybe it is actually upside down and the avocado is confused?
  2. the avocado pit looks like it is going to poop on us (referring to the emerging root)
  3. the exposed pit is smooth
  4. the calculator is 9.5 cm long
J3's observations <made at dinner, had been napping while the older two discussed>
  • There's a plant!
  • These are floss, we use it to clean our teeth (gesturing to show how)
  • Oh, some of the *this* fell off (noticing that dried skin from the pit was in the bottom of the water bowl from when one of the older two peeled it off and dropped it in).  
We also had a discussion about the division of the pit into two halves. When we started, there wasn't any clear indication that it would cleave along this line. We were wondering if there was some mechanism to prevent it from cleaving where we placed a toothpick, if the splitting location is random, or if there is a clear place it will split. I guess we add these to our curiosity list.

3 toothpicks: 0 gr on our scale, indicating that they are less than 1/2 gram
The avocado pit and 3 toothpicks together was 64 grams (J2 noted this is 8 x 8)

The sprout in the middle of the pit and extending down was 4.5 cm long.
J0's nose is 6 cm long
J1's nose is 4.5 cm long, so the same as the sprout.
From end to end, the pit and root sprout are 7 cm.
The root protrusion is 2 cm.

When we started, we estimated the pit's mass 56 grams, (J2 remarks, 56 = 7.4833147 x 7.4833147)

J1 measured me with the tape measure and proclaimed that I have 101 kg of fat. His reasoning: some 
distance around was 101 cm and he assumed I must be 1 kg per cm of perimeter along that slice.

J2 added, the following.
6 = 2.4494897
4.5 = 2.1213203
I asked J2 if these square roots had any meaning, in the context of our seed. He said, "no, it is just for fun."

Introducing: Orange seeds

Last sunday (13 December) we planted some orange seeds. Vaguely following these instructions, we used two methods:
  1. Planting in soil, keeping the soil covered and most: no developments yet
  2. Planting in a pool of water with some soil and dried leaves: interesting developments this week
Our "interesting" case this week

So, what happened with the soaking seeds? Here are our observation notes:
  • Oh, the seeds are sprouting!
  • Hmm, the sprouts seem to be wiggling!
  • Those aren't sprouts, they are larvae, probably mosquito larvae
  • Ooh, it smells like cows. It stinks
We poured it out on the street in front of our house, in the sun, and watched the water dry out. We observed the larvae moving around in the small puddle of mud as it dried and talked about what they needed to survive. J2 noticed that there was a storm drain a meter away from the puddle and asked what would happen if they went down the drain. Then we talked about whether the could get to the drain (having to cross a meter of dry ground) and how they could know that there was a safe destination on the other side. We made conjectures about their senses and ability to communicate:
  • probably cannot see/no eyes
  • probably cannot talk, but we guess the do have a mouth to eat
  • not sure about ears
  • cannot read or write
  • no ability to communicate with adult mosquitos, ants, humans, or other creatures
In summary, we concluded that their knowledge is restricted to the limited part of the universe that their limited senses can observe directly. Can you see the editorializing?

As you might expect, fire was introduced at some point in the conversation, we ended up burning a handful of dry leaves and some paper scraps. As you do, you know.

Wednesday, December 24, 2014

A christmas eve mystery

Who: J3
Where: at school
When: over 2 weeks

This is actually something being done at school, but it matches our seed growing at home very nicely.
They did a couple of experiments to test the effect of different conditions on plant growth:
  • with and without water
  • in different potting media (soil, sand, rocks)
  • with and without sunlight.
Here is the picture of the plants without (left) and with sunlight:
Sorry about the blur, this was my only shot through a swarm of excited toddlers

Thus, the mystery: why did the plants grown in the dark grow so much taller? Add your hypothesis in the comments!

This is a special day for our family: Grandpa G's birthday.  So, in the spirit of celebration and birthday wishes, we send some powers of 2 (and square relationships):

Sometimes 6s got to get a bit crazy, right?

Sunday, December 21, 2014

Dice Farmer (game)

who: J1 and J2
where: reception floor (having been deported from the kitchen doorway)
when: after lunch

We recently got a pound of dice:
From the toy category: "dad uses kids as an excuse to buy something for himself"
Today was our first round of dice farmer (from Leftover Soup, rules below) and the three of us had a lot of fun. Actually, J3 stole a handful of dice from our reserves and had fun, too.

This is a fast game that both J1 and J2 found pretty compelling. In fact, J1 and I were in the middle of playing Munchkin, J2 was getting antsy, so I told him the rules and started playing on the side. J1 got so interested, he abandoned Munchkin mid-combat!

J2 kept winning, so that helped make sure everything was fun. They were both surprisingly tolerant when their dice "died" and pretty friendly about sharing favorite dice (as you can see, almost all have a unique color). One parent warning: on a hard surface floor, dice will bounce everywhere.

The basic rules are easy enough so this game is a low threshold activity, but the number of combinations made it computationally challenging for our players. During play, we started having conversations about the shapes, probabilities, and how to assess who was ahead. All of these will take some time for us to really develop.

Just before going to sleep, J1 asked me to promise to add this to the blog tonight.  That's how much he enjoyed it!


Equipment: A "herd" of dice of different shapes. We used standard D&D platonic solids + 10-sided.
Set-up: all players start with three 6-sided dice
Play: on each turn, roll all your dice. Any dice that come up 1 are dead and go back to the reserve. From the remaining dice collect sets that add up to sizes of dice shapes. You then add these to your stable.
Who wins: the first person to collect three 20-sided dice as part of their herd
  • Change the starting number and/or composition of dice. For example, I often started with 3d4 to reduce my chances of winning. Usually, though, I suggest all players start with the same configuration or establish a budget for # of sides.
  • Change the winning condition. J1 got excited about requiring the winner to have 1d8 + 1d10 + 2d12 + 3d20. I wasn't around to see how this worked out.
  • Use different interpretations of what it means to form sets that add up to a target number of sides. For example, you could require hitting the target exactly or, as we did today, set that as a minimum. Perhaps you could also get some amount of excess back as a rebate, though I usually considered those lost.
  • Change the condition to die or add other scenarios (e.g., 1 = that one dies, 2 = asleep, so doesn't count for that round). 
  • Add unusual dice configurations

Saturday, December 20, 2014

A simple geometry puzzle

who: you
where: online
when: right now

This shows a toy swiss roll in vivo and in the mirror. What is wrong with this picture?

Here is a detail

Friday, December 19, 2014

How do you know? (talking math with your kids)

who: J2
where: in the car
when: this evening, driving home from a picnic

Yep, in this part of the world it is perfect picnic weather. I would have taken a picture, but

I was driving, so I didn't hear the full conversation, but what I caught was:
J2: Yes, 384 is a multiple of 12
P (mommy): How do you know?
J2: Because 384 is 192 + 192 and 192 is a multiple of 12.
P: How do you know?
J2: Becuase 192 is 96 + 96 and 96 is a multiple of 12.
P: How do you know?
J2: Well, it is 8 x 12 . . . also, it is 48 + 48 and 48 is a multiple of 12.
P: How do you know?
J2: 48 is 24 + 24 and 24 is a multiple of 12
J0: how do you know?
J2 (exasperated): Daddy, everyone knows 24 is equal to 12 x 2!
So, I'm still not sure how he knew that 384 was a multiple of 12. Surely it wasn't really through the decomposition in this conversation . . .

While "How do you know" is probably overused in this conversation, I think it is a good question to have as a parenting habit. You can see we fall back on it when very tired. Also, the kids expect to hear it, so they are immediately ready to enter into that type of discussion.  I'm looking forward to the times when (1) they turn it around and ask us "how do you know?" and (2) when they use it on other people (each other, friends, teachers).

Wednesday, December 17, 2014

Observational botany (step 1)

Who: All Js
Where: at the dining table (for future reference, this post has notes from 1 december 2014)
When: 5 minutes a day, before dinner

The author of Five Triangles made a suggestion somewhere (maybe his/her other blog?) that a great science activity is to plant a seed and make observations of the developing plant for a year. We are starting this with an avocado pit.

The pencil is a stand-in until our dental hygiene
catches up to our scientific zeal

J1's observations

  • The pencil smells like okra
  • It's red gray
  • It feels like my hair
  • I think it is 5 cm long
  • I estimate the mass is 1trn grams

J2's observations

  • The avocado pit smells like okra
  • It feels like your poop (J0:"My poop or your's?" "Your's daddy")
  • It is 36 feet (J0:"Long, tall, or wide?" "Every dimension")
  • It weighs 1000 pooplizes (J0: "what is 1 pooplize?" "The mass of all humans on the earth put together.")
As you can see, someone wasn't really taking this seriously

J3's observations

  • This is floss (pointing) and this is floss (pointing again) and a pencil (pointing for a third time).
  • It is not symmetrical
  • One side is round and the other is pointy (indicating the side down in the water as round and the end pointing up as pointy)
  • It is smooth
  • it has no smell
J3 also asked for the pencil I was using to take notes, then drew some scribbles on the page and said she was drawing avocado pits.

We made the following measures of mass:

  • Empty bowl: 107 gr or 3 3/4 oz
  • Pencil: 4 gr or 1/8 oz
  • Dry pit+bowl+2 toothpicks+pencil: 167 gr or 5 7/8 oz
  • water added: 133gr 

By implication, the pit and 2 toothpicks is 56 grams.

A general procedure for growing your own avocado tree (don't expect to eat the fruit, though);

As usual, wikipedia has something useful to say (my emphasis added):

Usually, avocados are grown from pits indoors. This is often done by removing the pit from a ripe, unrefrigerated avocado. The pit is then stabbed with three or four toothpicks, about one-third of the way up. The pit is placed in a jar or vase containing tepid water. It should split in four to six weeks and yield roots and a sprout. If there is no change by this time, the avocado pit is discarded. Once the stem has grown a few inches, it is placed in a pot with soil. It should be watered every few days. Avocados have been known to grow large, so owners must be ready to repot the plant several times.

In other news
Doesn't the icosidodecahedron look oddly, asymmetrically misshapen?

Tuesday, December 16, 2014

Math games 7

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

Skip counting warm-up

We've talked about warm-ups in several previous posts. Today, it proved to be a more substantial discussion than expected for a warm-up.

For skip counting, we go around the room with each child saying the next number in the sequence. This time, we started with some easy skip counting (2 and 3, based on requests of the kids in the class) and then something a bit more difficult (6 or 7).

Along the way, we saw some people getting stuck and knew that they would benefit from seeing some new strategies. When one child appeared to do a calculation very quickly, we asked them to explain their thinking. It turned out to involve splitting and regrouping:

We asked why they decided to split 6 into 4 + 2 and this was the explanation:

This discussion happened in each class and, in each class, we gave it the name of the student who explained it (Minnie and Jiping). For the rest of the day, when someone got stuck, their friends would offer encouragement and say "try X's method."

By the end of this warm-up, the second graders were excited enough that wanted to show off their technique for multiples of 9, so they spontaneously launched into that.

Parents at home: you can do the skip-counting warm-up driving together or at a meal time. Ask your child to show you how to get started.

If they are having difficulty with a calculation, first give them time to think through it.  Next, suggest tools they can use: write it down on paper, draw a diagram, use some objects. Finally, ask if they can break it into pieces that they know.

Ring your neck

Since some kids weren't in class last week, we began our games with a review of the new game from last week. It was a bit messy in second grade, but in first grade we got them to take turns explaining rules (one child explained one rule) and then they split into two teams to play a demonstration round.

Our intention had been to discuss strategy for this game, but the group dynamics didn't work well. These are the types of leading questions that encourage them to think about the structure and strategy of the game:

  • Do you want to go first or second?
  • If there is only one card left, do you want it to be your turn or your opponent? What about 2 cards? What about 3 cards? What about 4 cards? ....
  • Is it "good" to take a card (do you expect to get points or lose points when you take a card)? What is the most points you could get? What is the least? What are all the possibilities?  What is the average?
  • Do these things change as the game is played?
Parents at home: when playing games with the kids, ask them to explain the rules. As they are explaining, encourage them to show examples.  When you are playing this game, encourage them to add up their score each time they choose cards. Be patient when they need to take time doing the calculations and use the suggestions above when they get stuck.  When they aren't stuck, ask them to explain how they were thinking.

Finally, ask them the strategy questions.  It can be a fun discussion, especially when you don't know the answers!

Strike it Out

Players: 2
Material: paper and pencil
Set-up: Draw a number line and tick for each whole number. We gave the kids printed pages with number lines up to 20, but you should feel free to make longer or shorter number lines.
Start of play: the first player chooses a number and circles it
Each round: using the latest circled number, the player chooses another unused number, and forms a number sentence where the result is a second unused number. Circle this result and cross out the other two numbers.
Winning: the last player with a legal move wins.

This is an NRICH game and these pages show examples: student page and teachers' note.

Parents at home: This appears to be a more complex game as there are so many choices at each step.  When there are a small number of choices left, ask them to predict whether they can win. Also, ask them how many ways there are to form the number sentence and whether it makes a difference.


  1. Play strike it out with parents, friends, and siblings
  2. Play at least one of the card games we have taught this term.


First, we had a lot of fun playing these games with the kids this term. I think we found a collection of games that were fun for the kids and reinforced mathematical concepts appropriate for their current understanding. When playing the games, the kids were focused and engaged. Also, at least one of these (ring your neck) also has potential as a more subtle strategy game, though you have to reduce the value of the final bonus card.
When we weren't playing the games, we often found it difficult to keep everyone together for a discussion. This is something we will discuss with the teachers and think about strategies for next term.

Sunday, December 14, 2014

Our function machines (programming class 13)

who: Baan Pathomtham Grade 5 class
where: at school
when: Monday morning (bright and early!)

Ah, lucky number 13.

Homework discussion

We started by trying to figure out Titus's function machine.  This was a good continuation of our function game last time as he didn't have a chance to present a mystery function. Why don't you have a go:

Here is a link to his function machine so you can test more inputs.

While examining the output of the function S, we noticed that the results often included a repeating decimal.  For example, putting 5 into the machine gives us 12.142857142857142 (which should continue, up to the precision of the computer's calculation). This gave us a chance for a short conversation about repeating decimals and rational numbers.

I'd note that figuring out the underlying function was quite hard for the kids.

We then talked about where the other three got stuck on the homework and then spent the rest of the class helping them work through different associated issues.

The four function machines

Eventually, everyone got their function machines working, at least to the level of taking an input and giving us an output. Of course, Titus's is linked above. He is working on extensions, particularly making a loop.

Here is Gun's:

and Win's:

My function machine

As an example with some extra functionality, I showed them my function machine. At first, we entered numbers as input and it seemed pretty silly.  Then, they got a surprise when they tried something else:


The kids have two homework assignments:
  1. Extend their function machines to incorporate functionality from my program fctMchn_2.  Example extensions: add a loop so the user can try multiple inputs, animate the input and output, keep a list of the input-output pairs that have already been tried. They should feel free to make changes as this code isn't necessarily as clean or simple as it could be.
  2. Think of a project for next term. Perhaps they want to build a game, an animated presentation, something to demonstrate more mathematical concepts, a beautiful picture or they have other ideas? Encourage them to explore the following to spark some thoughts:,, other pencilcode user accounts, or have them do an online search.

Friday, December 12, 2014

Calculating and computers

Another post that isn't about the kids, oh well.

I got a bit carried away in a comment over on Dan Burfiend's blog: Quadrant Dan. As an opener for his geometry class, he asks about some large numbers. I suggested a couple of follow-up questions, for those who wanted to pursue the opener further:

Exponent Investigation
Are there any numbers (feel free to restrict to integers) where a < b but ab < ba? What are they?

I will leave you to play with this one.

Approximating big powers
A rough approximation that can be really helpful is 210 is close to 1000, aka 103. For 4234, you could approximate:
42 is approximately 40 = 4 * 10
So 4234 could be close to 268 * 10 34
Using our approximation of 1000 for 1024, replace 268 by 28 * 10006, so we get
256 * 1052 or 2.56 * 1054

Of course, that's still only about 1/6 the precise value calculated by worlfram alpha, but seems pretty good for such simple calculation.

Approximating compound interest
Let's say you want to do better than the previous approximation (we do, we do!) Can we make a useful adjustment to correct for replacing 42 by 40? Well, 40 = 40 * 1.05, so 4234 = 40^34 * 1.0534.

That second term looks like a calculation for compound interest, right? One rule of thumb (the rule of 70) is that a compounding process will double in approximately (70/rate) periods. In other words, the time it takes your money to double at interest rate r% is about 70/r years. At 5%, about how many doubling periods do we get when we compound 34 times? About 34/70 * 5 which is about 2.5. So, we can approximate 1.0534 by 22.5

Depending on your love for √2 , you ignore that bit and end up with a final estimate of 1055 (approximately 2.56*1054 * 4). However, for those playing along who want to say √2 is close to 1.5, then we get a final approximation of 1.5*1055.

Exercises for the reader
Try to approximate 3442. Is your approximate result larger or smaller than the approximation we got above? How confident are you that this allows you to determine which is larger, 3442 or 4234?

What did we learn?

Well, in cleaning up this post, I learned how to do exponents and square roots in html, so that's cool.  More seriously I feel this example shows something important about the roles of manual calculations and computer based math.

First, this wasn't blind calculation following an algorithm. At each step, we were thinking about relationships, albeit approximate ones, and ways to short-cut the direct calculation.

On the other hand, the sequences of approximations could easily have taken Dan's whole class. Would it have been fun for the students? The use of the calculation engine brings this into scope as a 5 minute class opener for a class that will eventually be about something else entirely (I guess).

Even if you wanted to talk about the approximations in class, I think seeing the answer from Wolfram Alpha actually makes the hand calculations a lot more fun. The kids would be thinking something analogous to this: "sure he can fly over that building in an airplane, but can he really jump over it?!"

Thursday, December 11, 2014

Math Games Class 6

Who: Baan Pathomtham 1st and 2nd grade classes
Where: Bangkok, Thailand
When: after science and before lunch

Pattern discussion

Homework from last week was to investigate the following patterns:
Pattern 1: Red - Green- Red - Red - Green - Red - Red - Red - Green .... (+1 increasing sequence of red blocks punctuated by single green blocks)
Pattern 2: Pink - Red - Pink - Pink - Red - Red - Pink - Pink - Pink - Red - Red - Red ...
(+1 increasing sequence of pinks followed by an equal number of reds)
As a specific  target in the investigation, we asked which color appears as the 100th term of the sequence and what is the closest position to 100 for the color that isn't the 100th term.

In class, we talked about how the kids approached this challenge.  Almost everyone took a brute force approach and wrote out the entire sequence up to 100. What else could they have done? Let's start with Pattern 1.

Most kids noticed in class that Pattern 1 has far more red blocks than green.  Given any large number, then, their best guess is that the block will be red.  They were also good about recognizing that their confidence in this prediction would be higher if the number was bigger. For older kids, this could be an interesting path for further exploration.

In class, we talked about some other ways to explore these patterns. After completing this post, I will write up an outline of what we discussed.  Part of the homework this week is for the kids to share with their parents what they learned from this exploration.

Ring Your Neck (New Card Game)

Materials: 2 players, standard pack of playing cards (all 52) and a piece of paper to keep score.
Set-up: Deal out 13 cards face down in a circle in front of the players
Play: Players alternate picking up either 1 or 2 cards (their choice) from whatever remains on the
table 13 cards
Scoring: After all 13 cards have been collected, players add up the cards they have collected according to the following values:

  • card 7 has value -7
  • card J has value -11
  • card Q has value +12
  • card K has value +13
  • card A has value 1
  • all other numbers have their face value

Most importantly, the person who collected the last card (or two cards) gets + 50 bonus!

To play the next round: Collect the first 13 cards in a discard pile, then deal face down the next 13 from the deck.


  1. Talk with your parents about what you learned in the pattern exploration
  2. Play Ring Your Neck 4 (or more rounds) with your parents and show us the scoresheet

Sunday, December 7, 2014

The function game (programming Class 12)

who: Baan Pathomtham Grade 5 class
where: at school
when: Monday morning (bright and early!)


As with loops, we are spending time focused on functions to make sure that the students master these concepts.  Today, we started with a function game. I drew a function machine on the board and gave it a name. Each student took turns giving me the value of an input, then I would tell them the output. Their objective was to figure out the function rule. Early on, I added a table showing our input and output values, to keep all the information organized.

The picture shows our starting function: constant 5.  I was pleased that, even on this simple function, we got to see an example where an invalid input was tried, so we could talk briefly about the domain of definition.

Other functions we tested were LIN(x) = 20 - x and Pyth(z) = z^2 + 1.  The students were pretty fast about guessing all of these.

Next, Boongie and Win got to lead the game.  This was a lot of fun, though calculation time slowed us down a little bit, especially when students input large numbers.  I'd note that there was little systematic thinking about what inputs to use, but this was the first time we've played.

Next time we play, I will use a non-numerical function to remind them that functions aren't only about arithmetic.

Homework review

Homework links: TitusBoongieWin, and Kan (not done!)

Our usual opportunity to talk about the code that the kids wrote.  There wasn't much to report here, though Win got a bit stuck (he didn't finish his code) and Kan didn't write the program. We did talk a bit more about Kan's boxbox program which nicely illustrated three things about pencilcode functions:
  1. They can have multiple inputs
  2. Numbers, words, arrays can all be inputs (indeed, the possibilities are much broader than this)
  3. Inputs can be left out and the program will still run.
Of course, we already knew most of these things, but it is nice to have rediscovered them by accident and it makes a break from the number-centric functions that were used in our number game.

There's a link above to the program, this is what it produces (press the button to run):

I showed them a couple of programs I wrote.  This one piggy-backs off their use of switch-when-then-else to build a long list of people and birthdays: long list. This other one was an attempt to add some additional functionality and include an example of composition of functions: bells-and-whistles.

spike and starburst sequence

The main challenges of the day were to replicate these animations: SpikeSB0SB1

We were lucky that Kan chose to write all the steps explicitly, while Titus used a for loop. This allowed us to compare and contrast and we saw that the for loop wins out because:
  • it takes fewer lines of code (9 vs 27)
  • it is easier to debug (consequence of the fewer lines of code)
  • it can be changed easily. For example, sending all the spikes out 10x farther required 6 key strokes, while the other program required a lot of changes (nearly every line of code)
These were a bit of a challenge and there is still more work to do for most of the students. Next time, we will continue with these and move on to the third starburst version (passing a function as an argument to another function.)


The homework this week is very challenging: to create an animated function machine. The program should draw a picture of their machine, ask the user for an input into the function, use animation to show that input going into the function and animation to show the output coming out of the function machine.

For user input, they can use the requester( ) function and program phrase from this code:

Monday, December 1, 2014

Math Games Class 5

Who: Baan Pathomtham 1st and 2nd grade classes
Where: Bangkok, Thailand
When: after science and before lunch

Special note for class parents:
Thanks to all the parents who have been playing the math games with the kids. They were really excited to talk about it and it shows how much they appreciated working on the challenge with you.

The homework this week is to investigate some patterns. We haven't found a great translation of this word into Thai as there seem to be different words for each of the slightly different meanings. As you help the kids with their work, you can encourage them with the following questions:

  • What do we notice? 
  • Why is this happening? 
  • What do we think is happening next? 
  • What can we do to test our prediction? 

It will be great if you can spend some time talking about these issues, even if you don't feel that they got to a particular "answer."

Today, we had a slightly different lesson plan:
  • Skip counting
  • Talking about the snugglenumbers game from last time
  • Exploring some patterns
  • Playing indian addition poker

Skip counting
For each class, we gave them a number to skip count and P said an initial number.  For example, 
skip count by 2, starting with 0. We would go around the class at least twice. Today, we skip counted by 2, 3, 4, and 1/2.

This was intended as a fast warm-up, so we didn't choose anything that was particularly challenging. Even so, it proved to be good practice for nearly every student.  Also, there was a moment of deeper interest in the 1st grade class when we were skip counting by 4 starting with 1 and someone asked: "will we get exactly 100?"  That led to a good discussion of the pattern and then the class was really interested to keep going to see if they were correct.

Snugglenumbers game
The kids were all enthusiastic about this game which, frankly, has surprised us a bit since the game play is very limited.  They wanted to talk about whether they won or their parents.  For grade 2, we posed the following questions:
  • Are there any numbers that are very helpful? if so, can you say which are the most helpful and why?
  • What is the maximum possible score? Does it matter whether you are playing with one deck or two?
We didn't have an exhaustive discussion for either set of questions. For parents at home, I would encourage you to discuss these questions with your children.  Full disclosure: J1 and I had already discussed the question of the maximum possible score last night, so I had sworn him to secrecy for the class today.  As it turned out, we didn't really discuss this in grade 1.

This activity was shown to us by David Ott, the K-12 math coordinator at ASL, one of whose recent exploits can be seen here: Family Math Night.

Set up: using construction blocks, I set up 5 patterns, the hide them in rolled-up paper, then hide those in a small bag.

The big reveal:
One pattern at a time, one cube at a time, I reveal the colors. At most steps, I will ask the kids for a show of hands to vote which color they think comes next.  Rotating around the room, we then ask for some explanation of the choice. I will usually also tease them a bit to encourage them to think more broadly. When they have all settled on a particular pattern and agree (which I don't force or encourage) then I whose the whole stick and ask them for a continuation, for example, what color is the 20th or 40th or 100th cube in the sequence.

The reaction
The kids absolutely loved this activity.  They were really excited to predict what would come next, eager to argue their view in the voting and then really emotional when the next cube came out and either matched their expectation or disproved their prediction.

Here were the patterns we used today:
Simple tools to cause mathematical delight

Your challenges:
What is the color of the next cube for the pattern on the far left?  If the glare makes it hard to see, it is pink-red-pink-pink-red showing. What is your reasoning?  How many other possibilities can you think of for that color and what is your justification?

If the pattern in the middle stick is repeated, what is the color of the 100th cube? What if I tell you that the pattern I want to extend is increasing whole number red series with a single green separating the runs of red?

Indian addition poker
P introduced this game two weeks ago and blogged about it here: As she wrote at the time, this is a really good game.  We played two rounds at the end of each class and I was really struck by how much respect the kids showed to themselves and each other.  They took their time calculating, didn't get frustrated, and those who discovered their number faster were encouraging for the other students. To recap: everyone is dealt a card that they hold on their forehead and the teacher reveals the sum of all the cards.  Students then try to figure out which card they have.

In second grade, we played with a full deck, making J= 11, Q= 12 and K= 13 (A=1 in both classes). In first grade, we took out the face cards so we had A through 10.

One other modification is once there are 3 or fewer students left and they seem to be stuck, we tell them the sum of the cards from the remaining players.

I would encourage families with several kids to try this at home.  An alternative version is to have everyone take a hidden card, then reveal sums of 3 cards.  Feel free to experiment and see if you can find another version you enjoy.

We asked students to work on the two more complicated patterns, the increasing reds with green punctuation and the increasing pink and red stick.  The questions they were asked were:

  1. What is the color of the 100th cube? How do they know?
  2. If the red/green pattern has a red cube in the 100th place (a strong consensus among the classes) then what is the place of the closest green cube to 100?
Note: these questions can be answered by brute force, building the sequence or writing it down. What we would like to encourage is, of course, looking for patterns. For example, for the red-green pattern, do they notice anything about where the green blocks appear?  Can they make predictions about when another green will occur?  In the pink-red pattern, can they say when a block of reds has ended, do they notice anything about those places?

Saturday, November 29, 2014

Blog tools

We've gotten feedback that it is hard to find particular content and navigate our blog. Today, I added two utilities that should help.

Search bar: pretty standard for all of us who use Google search everyday.  I haven't experimented much to see how the pages get ordered, so I'm interested to hear whether this is helpful.

Tag cloud: we try to tag all of our posts with key words.  These are shown with the most frequent tags at the top in a larger size, while the least used are small at the bottom. Clicking on a tag will take you to the collection of posts with that label. Make sure to check out some of the little-used tags as they are references to fun games or explorations.

What do you think? Is this enough structure to get you to the content you want to see?

Flowers, Squares, and Functions (programming class 11)

who: Baan Pathomtham Grade 5
when: Monday morning
where: at school

Homework review
Forests/flower gardens (some of these were old?): 

Boongie made two versions: 
The first led very nicely into functions, the topic of the day as he had put the petal drawing into a function.  The second was a really nice cap to our discussions about for loops. In his original version, it drew one lime flower and three blue ones. Working together, the class modified the code so that all the flowers were a different color.

Win showed us his flower program, cryptically named untitled5:

This is a good candidate for streamlining using a function.  Something for him to consider over the week.

To round out the discussion, I showed two related programs that I had found from other users and modified:

We didn't discuss this program as Titus thought it had gotten lost:

Focus for the day: Functions and vectors
Today, our plan was to build up a more complex function structure based around drawing some simple squares.

The first task was just to create a function that would draw a single red square. This code is an example.

Once completed, we added an input variable that would allow them to control the size of the square. Finally, we wanted them to make nested squares that would show why it is useful to capture a block of generic code in a function.  Here's our example.

At this point, Kan had two interesting ideas: First, he wanted to add the color as a parameter to the function.  That was pretty easy, but it was a nice way to see that functions can take numbers and words (in this case a color keyword) as arguments and that they can take multiple arguments.  His next idea was to include a variable that would tell the turtle to move after drawing the square. I've copied his code here

He wasn't quite sure how to properly call the function.  Notice what he did: three arguments to the function, one of which is a vector!  Good stuff . . . and we will discuss this more next time.

We also talked about the idea of functions in mathematics and worked through three examples:

  • f(x) = x+2
  • goal(y) = 2y
  • birthday(name) = {the day of that person's birthday}

The challenge this week is to write the birthday function that will work whenever a member of their immediate family is entered as the input variable.  This is not going to be easy, so we expect them to experiment, explore, and ask questions if they get stuck.

Some other interesting code
In preparation for the class today, I noticed Kan had created this program:

Both Win and Kan had these programs

At some point, we will return to these programs to talk about what they do and why.  I know the two were experimenting with them at the end of class and I'm eager to see what they can discover.

Wednesday, November 26, 2014

เกมคณิตศาสตร์ ป.1 - ป.2 (25/11/2014)


Source: Snugglenumber by Anna Weltman 

สิ่งที่ต้องเตรียม: ไพ่ 1 สำรับเอา 10, J, K ออก
     ๑. แจกไพ่คนละ ๑๑ ใบ
     ๒. ให้ A แทนเลข ๑ และ Q แทนเลข 0
     ๓. เด็กๆจะต้องนำไพ่ทั้ง ๑๑ ใบมาใช้ โดยที่ไพ่แต่ละใบใช้ได้ครั้งเดียว ให้เอาตัวเลขตามไพ่ที่ได้มา             ใส่ในช่องตามแผ่นเกมที่แจก (ดูตัวอย่างข้างล่าง)

              0     ____
              5     ____
             10    ____  ____
             25    ____  ____
             50    ____  ____
            100   ____  ____  ____
            - ได้ ๓ คะแนนถ้าไพ่เท่ากับจำนวนพอดี
            - ได้ ๑ คะแนนถ้าค่าต่างไพ่กับจำนวนที่ต้องการไม่เกิน ๖
            - ไม่ได้คะแนนถ้าค่าต่างไพ่กับจำนวนที่ต้องการเกิน ๖
            มีไพ่ในมือดังต่อไปนี้  2, 2, 3, 5, 6, 6, 7, 8, 9, Q, Q  นำมาเรียง

              0     _Q__
              5     _6__
             10    _2__  _7__
             25    _2__  _8__
             50    _5__  _3__
            100   _Q__  _9__  _6__
         คะแนนที่ได้คือ 3, 1, 0, 1, 1, 1 รวมกันได้ 7
Update: some pictures of players in action

Monday, November 24, 2014

Nested Loops Review (Programming Class 10)

who: Baan Pathomtham Grade 5
when: Monday morning
where: at school

Reviewing last week assignment

Last week, students were asked to draw this shape using double for loops if possible.  Only 1 in 4 students managed to use double for loops but none of them see this as drawing a triangle 3 times.


Nested for loops revisit

Using the homework as a starting point, I simplified the task by having them imagine the triangle to be just a line.  After they wrote the simplified program, I asked them to change the line into a triangle.  Eventually, we made our program to take any number of petals.

    num_petals = 3
    speed 10
    pen blueviolet
    for [1..num_petals]
      for [1..3]
        fd 50
        rt 120
      rt 360/num_petals

The next task is to get the students to understand how the use the counter in the for loops and the importance of placement of commands and indentation inside the for loops.  I actually had them do a simplified program before we get to the one below which creates 3 rows of colored dots (I first asked for one row of all the colored dots.)  Even these few lines of codes, it's still useful to breakdown the task into smaller, more simplified programs to learn the concepts.

    rt 90
    for x in [blue,green,pink]
      jump 25,-75
      for y in [yellow,orange,red]
        dot x, 21
        dot y, 7
        fd 25


Draw a flower garden using at least double loops.  I encouraged students to use triple loops or as many as they fancy.  I also mentioned that I didn't get to review functions but that could be handy for them to use for this homework.  Some of them seemed intrigued and I am hopeful that they will do their own review of functions to do this homework.

Fun with Polygons (Programming Class 9)

who: Baan Pathomtham Grade 5
when: Monday morning
where: at school

Reviewing last week assignment

As usual, I started with homework review which is to draw a scary ghost with a bonus point if they can make their ghosts disappear and reappear.  Turned out to be a nice segue into my main agenda this week which is to review the for loop and a little bit of geometry.  I pointed out that their codes could be shorten by using for loops.

Fun with Polygons

Although my main goal for this session was to review for loops, I wanted to throw in a bit of geometry review (have been noticing that the kids were using too much trial and error strategy when they drew non-quadrilateral shapes).  The structure of this session was I asked them to draw various polygons using for loops: 
  - we started off with the basics: square and equilateral triangle   
  - Then I asked them to generalize their program to make any polygon.  This requires two things:
       1) Understand how to change the number of iterations in the for loops
       2) Recognize that the angle for the turtle to turn is 360/n where n is the sides of the polygon.  I showed them Josh's little program to illustrate this fact (
  - Next I asked them to use for loops to draw a rectangle.  Here they have to recognize that the input into for loops can be other things than a sequence of natural numbers.

Things to Note (Debate?) for Future Lessons
Up til now, our main teaching approach has been to show them codes, run it to see what happens, and then ask the students to extend the codes.  While this show-don't-tell approach is great at stimulating their creativity and interest in class, my opinion is that our G5 students need a bit more explicit blackboard instructions to get a strong grasp of the concept.  Thus, I think we need a balance between learning by exploration vs. with explicit instructions.

Math Games (Class 3)

Who: Baan Pathomtham First and Second grade classes

Where: at school
When: mid-morning

Here is the third installment of our math games series.  Despite shortage of staff (Josh is traveling), it's a successful session -- the kids loved it so much they begged for more after they finished their first round.

Indian Poker Addition Game

This game is a twist to the indian poker.  Although it's more fun and challenging with more players, for younger kids, I find that 3-4 players work out best.

Equipment: A pack of cards with face cards (J,Q,K) removed.
Here's what we do:
- Deal one card face down to each player.  When given the signal, they hold the card outward on their forehead so they cannot see their own cards but only see their friends' cards.
- The card dealer announces the sum of all the cards. 
- Each player guess his or her number.

We have 7 children in each class so they have to add up 6 cards and then subtract from the announced sum.  For children who struggled, I gave additional hints -- I told them the sum of his or her card plus two other friends' cards so they only have to add up 2 cards.  

Close to 100

Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.
- Deal out 6 cards to each player
- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers.  The goal is to get the sum of the two numbers as close to 100 as possible but cannot exceed 100.

Wednesday, November 12, 2014

Multiply my love

Who: J1 and J2
When: while not hacking minecraft on raspberry pi
where: in my home office

Recently, we found a couple of nice games and puzzles built around multiplication that I wanted to share with y'all.  The important point is that these aren't drill-in-disguise where the primary objective is reciting multiplication facts, they are games with their own goal that is facilitated by multiplication.

In this puzzle, you have to cover a grid with rectangles. The trick is that the grid has numbers sprinkled throughout and every rectangle you draw has to contain exactly one number that is equal to the area of the rectangle.

Solving (left) and make-your-own (right)

Here's the game and a collection of puzzles:

For credit, the game was built by David Radcliffe (@daveinstpaul) and I originally read about this game on Moebius Noodles.

Why do I love it?
(1) Fun and challenging puzzle worth doing on its own
(2) Great reinforcement of the area model for multiplication
(3) Tremendous scope for further investigations

The last two points are related. Creating a puzzle helps stimulate thoughts about the structure of the puzzle.

Examples of things you might want to explore
a. Is there always a single solution or could there be several?
b. How many ways are there to partition a rectangle into sub-rectangles?
c. If there can sometimes be multiple solutions, can we recognize this or recognize that the solution will be unique in advance (before we solve the puzzle)?
d. If you put numbers into the square grid, will they always form a puzzle that can be solved? If not, what conditions are necessary? What conditions are sufficient?
e. Is there an algorithm that will always find the solution (when one exists)?
f. Are any game versions fun (and what mathematical structure do they have?

Times Square
This game comes from Calculation Nation. They have several good games, so it is worth taking a look at their collection. Other than this one, I particularly enjoy Nextu.

The objective here is to get 4 in a row before your opponent does. On each turn, you move one of the sliders below the playing square and capture the square that is the product of those two.

Why do I love it?
(1) The game is fun and challenging (this is the sine qua non of games, no?)
(2) Players have to use multiplication and division when planning their next move
(3) The interaction with  your opponent is not straightforward
(4) J1, J2 and I had a delightful conversation about which numbers are in the playing board, which are missing, and why.

In this case, I don't see as much scope for further investigation, but  you might have more ideas than I do. At least I will leave you with one:

Why should we have expected that the playing board would be a square, thus justifying the name?

Some other things related to multiplication
Prime Climb
A beautiful multiplication table emphasizing prime factorization, from Math4Love:

This is linked with their game Prime Climb (which I don't have, so insert unhappy smileys here!)

Factor Game
As the name suggests, a game related to factorization.  Rules here.  I will blog about this when we play the game at home or with one of the school classes.  Also, I noticed a pencilcode user that started building a program related to implement this game (Introbot's FactorGame).

An alternative algorithm
A video (here) and picture (below) are getting pushed around the web. My take:
thoughtlessly teaching/learning/applying any algorithm isn't very useful, but playfully investigating and thoughtfully considering why it works is always worthwhile.

Tuesday, November 11, 2014

Subtraction Games (math class 2)

Who: Baan Pathomtham First and Second grade classes
Where: at school
When: mid-morning

Today was our second session playing mathematical games with the younger kids at J1's school. Another fun session with a new game the kids can teach to their parents.

As usual, we were too occupied to take any pictures.

We are settling into a standard agenda:

  1. Warm-up questions: quick questions that are intended to require 10-20 seconds of thought, but are not very challenging for the kids.
  2. Mathematical observations: based on a picture, the kids practice making observations about number, measure, shape, sequence, etc.
  3. Something from an old game
  4. A new game
  5. Homework

Old Game: Euclid
As a reminder, the rules for this game are here: Euclid's Game.
Last time, we played on a 100 grid with the second graders and a 60 grid with the first graders. This time, we had them all on a 100 grid.

To start out, we had a refresh by playing the game once: the kids on one team and I was on the other team. As we played, I did the following things to spark thoughts about the game:
(1) always explained full equations to justify my moves. Example: 40 (pointing to the 40 that has already been crossed) minus 5 (pointing to the 5 that has already been crossed) is 35 (crossing out the 35)
(2) I got them to say full equations and had one student per turn cross out the number for their move.
(3) I asked them to confirm my calculation.  Sometimes I said the wrong answer, so this wasn't an empty activity.
(4) I asked them what was the largest number we could cross out in our game (once the first two numbers had been chosen)
(5) I asked them to look for emerging patterns in the numbers we were crossing out.
(6) I asked them to look ahead in the game to see who was going to win.
(7) I asked what would happen if we somehow manage to cross out 1 in our game.

After playing the game to make sure we all got the rules, I posed the following:
- last week, I saw two students play a game where the starting choices were 60 and then 36.
- What pattern of crossed out squares do you think emerged by the end of the game?
- Do you think you can start with 60 and 36 but get a different pattern?
- Who do you think won?

Look: another 100 grid for you to cherish!

Note: even though this game seems to only involve subtraction, analyzing the game and finding a winning strategy is more related to multiplication, division, and factoring.

New Game: Card subtraction
Source: Motion Math

Another subtraction game with a nice feature that many people can play at one time.
Prep: take a standard deck of playing cards and remove all 10s and face cards.

Each round:
- Deal 4 cards to every player
- Deal 2 cards in the center, the first of these becomes the 10's digit and the second the ones digit in a 2-digit number that is the Target
- players make two 2-digit numbers with their 4 cards and find the difference.
- Players then find how far their difference is from the Target and this gives their number of points.
- Lowest points win

For example, say we deal 5 and 3 as the central cards, making the target 53. Now, let's say one player gets an 8, 7, 2, 3.  One way they could play is making 38 and 27 for a difference of 11.  That is 42 away from the target for the round (remember 53), so they could claim a score of 42 points. Think about it and you will see that there are other combinations that get much closer to the target and score far fewer points.

Observation: the two digit subtraction problems that come up in this game were challenging for the kids.  When playing this game with your children, encourage them to use the following tools to help with their calculations:
(1) write down the calculations on a piece of paper
(2) use concrete objects to work on the subtraction questions
(3) use the 100 grid from the Euclid game.  Ask them to show you how they can use this grid to help them analyze a subtraction problem.
(4) use a number line.

Let us know what other strategies you use.

Finally, encourage them to look for patterns and relationships.  What pair of 2 digit numbers could they form that has the smallest difference? What about the largest difference? Which grouping gets closest to the target? which gets farthest? Are there several combinations that are the same difference from the target?

- play a game with your parents and show the grid to us at the next class
- try the game starting with 60 and 36.  See if you get the same pattern we got in class?

Card subtraction:
- play this game with your parents or siblings for 5 rounds (or more!) and bring us the scoresheet.

Photo Math, computer based math, and hand calculations

Who: J2
Where: all over the house
when: after school almost every day last week and all weekend

Recently, there has been a lot of excitement about the photo math app (on-line community at-large) and hand calculations (just within our house). Is there a place for doing hand calculations and learning standard calculating algorithms when technology has already automated so many mathematical operations and is attacking problems of increasing complexity?

I'm not going to attempt to answer comprehensively or theoretically, I'd just like to make some observations based on J2's explorations this past week.

It started with squares

Who knows why, but J2 was building a sequence of squares one afternoon with our colored tiles. I think he had seen something when we were doing another investigation or had heard me make a remark and wanted to investigate square numbers himself.  Making these squares was a fun and colorful way to do the calculations.

At some point, he realized that he wasn't going to have enough tiles to keep making separate squares and he consolidated into one square which he kept growing. I think this 13x13 is where he stopped that day.

What was he thinking?
He was absorbing the numbers and looking for patterns.  Early on, he realized that it was annoying to keep counting all the tiles to calculate the new square, so he wanted a faster way.  At one point he came to report his progress and explained: "I have 100 tiles in my 10x10 square.  When I make the 11x11, I know it will have 100 + 10 + 11 tiles."

Symbolically, he was recognizing (n+1)^2 = n^2 + n + (n+ 1)

After that, he kept using this relationship to check his results. We also played around with doing the multiplication directly. Whenever we did a multiplication, I would find a way to illustrate the distributive property and usually invoked some number bonds.  Here is one illustrative example, though mostly I just drew diagrams like this on a paper:

Incidentally, I got this design from Mike Lawler's video giving a physical illustration of why the product of two negatives is positive.

Another tool:
For several of the calculations, J2 was using a 100 chart or our 100 board.  He spent a bit of time looking at the board thinking about whether the squares were easy to see on this board.  his intuition was that, somehow, it would be nice to see them as the vertices of growing squares within the grid.

I suggested he build his own 100 spiral and look for patterns along the way.  This is slightly less than halfway (well 40% of the way, to be exact):

While he did this, he noticed three things:
(1) the squares are appearing along diagonals
(2) the even squares are rise moving northeast and the odd squares increase going southwest.
(3) we also get non-square rectangles at 1x2, 2x3, 3x4, 4x5, etc

I have it on good authority that  you can see something interesting with the primes in this configuration (see Ulam's Spiral) but that will have to come later for our J's.

Some further adventures
Along with his hand calculations, J2 started entering his squares into a spreadsheet. This let him explore larger squares than he could multiple right now and well beyond our tile collection. Also, we could explore first and second differences, seeing his old recursive relationship in a new way.

Beyond the squares, he has since done similar things with cubes (constructing physical cubes out of trio blocks, building a table in the spreadsheet, looking at differences), powers of 2, and quartics. Looking at these all together allowed him to start seeing connections around more advanced questions:
- which powers of 2 are squares?
- which cubes are also squares?
- which quartics are squares?
- how fast do the different sequences grow?

A hint of what is to come: before going to sleep last night, J2 mentioned that he wants to talk to me about triangular numbers next . . .

What do I conclude
For J2, the hands-on manipulations and associated hand-calculations are helping him see number patterns more closely and become familiar with a lot of interesting relationships.  This work has provided him with a platform to then engage with more computationally powerful tools. Importantly, he doesn't see it as a binary choice between manual and automated calculation, but is very happy to alternate between the two.

Further reading
For a more thoughtful and comprehensive discussion of the app and the impact on teaching, see Dy/Dan.