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?