Monday, June 30, 2014

Get your hands in!

I was going to call this "get your hands dirty," but only a tidy mother could consider any of these messy.

We were just playing around with various things on hand.  Mommy did some special RightStart Math with Jin that she will report later, but all I did was capitalize on the good fortune to have toys around.

Jate subtracts a small fraction

While they are playing with the geoboard, we talk about fractions of the circle, angle measures, etc. Another little question we explored: how large did we need the rubber band to be so that it would stretch to make this design?

Colorful Patterns
 Honestly, I don't know what these patterns represent, other than looking nice.  They were variations on a 4x4 which had the inner 2x2 in red, the corners in blue, and the other 8 tiles in green.  See if you can guess what that meant.

Count your eggs before they hatch

Counting, dividing, comparing, adding, spinning, juggling, tossing, laughing, there was no end to the math here!

Another rainbow emerges

Having re-read the manifesto lurking behind this blog, I resolved to make sure there is always one game in play.  This was chinese checkers reimagined.

Primes and patterns

So, what do you see here?

While watching the kids play on Sunday, I amused myself by applying the Sieve of Eratosthenes to our 100 board: blue for 1 (which I'll come to later), red for primes, white tiles for composites.  As always happens, a non-empty subset of the kids got interested and asked what I was doing.  I explained a bit ("I'm marking out the multiples of 2, you know, skip counting by two" and "now I've moved on to 3" then "I don't have to do 4, do you know why?" etc).  At the end, I explained we had a special pattern: for any number covered by a white tile, we could take that many small squares and make it into a rectangle that wasn't a stick, a straight line, a 1xn rectangle.  Yes, I did give them all of those terms . . .

So, let's try an example . . . so Jate picked 54 (maybe with some guidance?) My mind raced, which factorization should I show him: 2x27, 3x 18, 6x9? I chose 6x9 because each dimension is small enough that he could easily see how many squares were on that side and it is aesthetically pleasing.

From the sieve, I had the following tiles to use in my construction:
27 white tiles not used to cover composites: 25 primes + 1 (for 1) + 1 (for 54, just to show which number we were making).
24 blue tiles
Total: 51 tiles, argh! If only the primes were slightly more dense!
As you can see, I had to borrow 3 white tiles from the 98, 99, and 100, to complete my rectangle.

Other factorizations
When you've got a beautiful number like 54 to work with, stopping at one factorizatioin is criminal, even if it is as great as 6x9.  We spent some time rearranging the tiles to make 3x18 and 2x27 (not pictured). I asked if we could make any other shapes and was given a dismissive: "of course, we could just make one 54 squares long, daddy!"

What about 1?
We discussed other numbers and eventually got to 9.  It was deemed a good one because it was a square and then Jate highlighted the smaller 2x2 square inside, so 4 was cool, too. I asked if he could make any "interesting" rectangles with 7, 5, 3, and 2 squares, so we spent some time investigating those.  Didn't quite manage a proof that they are prime, but I think we got some intuition going.

When you are down to 2, then 1 comes next.  We had a good discussion about whether 1 should be in white because it was a square, in red because it could only be made into a 1xn rectangle, or whether it was something else. This led to a compromise (not pictured): we put a white tile under the blue one to show that it could be an interesting rectangle, but mostly it was still blue.

Car trip license plate game(s)

Courtesy of our friend Ko who introduced it on a school trip:

License Plate Games
For times you are stuck driving, especially in the type of dense traffic we have all over Bangkok, you need some games to stay sane and mathematical! There are several you can play, depending on your children:

  1. Spot numbers: each person in the car tries to count up using numbers they see outside.  You can play with each as individuals finding their own numbers and the winner is the one who gets the highest number.  Or, as we usually do, play cooperatively where everyone is searching for the next number. I actually liked playing this when we went walking around London as it usually gave us each time to see the number that the other person had found.
  2. Simple Addition: one spotter calls out the numbers on a license plate and then there is a race to add them up. Works especially well in Thailand where the numbers are usually of the form: LL[xxxx] where the L's are letters and the x's are digits.  For example, AB 1568 gives you 20 =1+5+6+8.
  3. Two digit addition: for slightly older children, break the license plate into two 2-digit numbers, e.g., AB1568 gives you 83 = 15+ 68
  4. Four digit addition: another step more difficult. This time, two spotters call out two plates at the same time and then they get added. An example: AB1568 and CD 3458 gives you 5026 (1568 + 3458)
  5. Simple multiplication:  spotter calls a license plate and then the players multiply the two smallest numbers, e.g., AB1568 gives 5 = 1x5.
  6. Next stage multiplication: multiply the two largest digits. My favourite AB1568 again gives you 48 this time (6x8).
  7. After that, if your players are still having fun, try 3 digits, 4 digits, two 2-digit numbers, etc.  Good going if you get up to 2 spotters calling 2 license plates and multiplying the two 4 digit numbers!
If your kids play these a bit, don't be surprised when they start getting the answer before you do! For our part, we've just moved to stage 3.

Isn't this just computation?
Yes, but . . . computation can be fun, too.  Don't force this game (or any other) if they aren't into it. Also, when we play, we spend a lot of the time talking about what we are doing.  Here are some example topics:

  • "Wow, how did you calculate that?" (to which the answer will be some amount of reordering, breaking numbers up to form common number bonds, etc)
  • "Oh, that is going to be a big one" said when we see a plate with a lot of 8s and 9s
  • "Ooh, this is smaller than the last one"
  • "This is the same as the last number, but just adding 2"
  • "What is the smallest value we could get?"
  • "what is the largest we could get?"
  • etc
This game helped us survive a long drive on Sunday.  Thanks again, Ko!

Can you (or your kids) be good at math?

Parents and teachers: please register for this course and watch at least some of the videos:

How To Learn Math

It is an online course from Stanford, course number EDUC115.

The course leader is Jo Baoler, a name you should get to know in math education. It doesn't have grades or exams, so don't feel anxious about how you will perform in the "class."

Go ahead, register, watch some of the first lessons and think about what attitudes toward math and learning you communicate to the kids.

What if I already know and believe all this?
Fantastic! Share it with friends, parents, teachers and students you know.


As mentioned previously, I find it helpful to have a couple of activities in mind in case someone in the family suddenly asks for a game (even if the request is sometimes signaled indirectly by poking a sibling). Usually, this doesn't mean a full and rich exploration of a topic, but it can be a brief introduction or a continuation of something we've played with before.

Here are two more NRICH activities that fit that:

Jate got to play a little with the chain of changes tonight while I was helping Jin with some homework.  I didn't have time to make or print pieces, so instead we used our mini-wipe board. I drew a collection of "allowed" shapes (square, triangle, circle, pentagon) in four colors (red, blue, green, black) and asked him to make a 4 step chain from blue square to black circle. Then I asked if he could make a one-step chain from the blue square to the black circle.  Then I asked how many of those he could make.  After that, he started adding his own shapes to the selections, so he got to set the tone and practice drawing.


How many of you noticed the mistake in the wagon picture on this post: weekend?

Hint: in this set of pattern blocks: hexagons are yellow, trapezoids are red, 60-120 rhombus are blue, 30-150 rhombus are tan, squares are orange, and triangles are green.

Jate and I had a little discussion, but we both agreed it was a mistake rather than a deliberate choice.

Second question: what did they mean to print?

Bonus questions: if the green triangle has as triangular face that is 1 square unit, what is the area of all the other blocks in this set? Humble brag: I could only come up with 3 distinct ways of finding the answer, though the person who asked me this question didn't like any of my answers.

So far, with the children, I've only posed this question with respect to the easy ones to answer, but now I have an idea for a fourth approach that I can use with the children!

Two more quick games

On weekdays, there isn't enough time to plan a full set of activities.  However, it is still helpful to have a couple of quick games or exercises ready to go.  In some cases, we might find ourselves with surprising free time or, more likely, have a need for some activity to occupy one of them before trouble starts.  Tonight was a bit of the latter.

Jate's Game
We played a couple rounds of another game I got off NRICH.  Here is the link:  The basic rules:
  1. make a number line and label the whole numbers from 1 to 20 (when we started playing, we only went from 1 to 10)
  2. The first player chooses two numbers on the number line, crosses them out and circles either their sum or their difference and circles that number.  The two crossed out numbers are now out of play.
  3. The next player works with the circled number and chooses another number that hasn't been crossed out.  They then circle either the sum or the difference and cross out their numbers.
  4. Each player repeats step 3 until there are no legal moves.
  5. The last player to make a legal move is the winner!
It might help to study the picture below.  It isn't especially clear or tidy, but you should be able to get the point:
Jate is working with 17 (on the right of the paper, circled and not yet crossed out)
He has to choose another number. Since so few options remain, he has two choices, either of which causes him to win the game.  See if you can figure out his move. . .

He chose 12, 17-12 is 5, so he circled 5 and crossed out 12 and 17.  That left me with no remaining legal moves.

Now, this hasn't come up yet, but I have seen the opportunity to argue about a special case: what if he had chosen 5 and circled 12 (= 17-5), crossed out the 17 and 5?  Could I have chosen 6 and circled it, because 12-6 = 6?  Seems to me that you have four options for rules to deal with this situation:
  • 2n - n = n isn't allowed as a legal move
  • 2n - n = n means that you cross out 2n, circle n and the next player uses n for their move
  • 2n - n = n means that you cross out 2n and n, the next player chooses 2 new numbers
  • 2n - n = n means that you circle n, then cross out 2n and n, and this player automatically wins (because the next player can't make a legal move with the circled n, it has already been killed)
In our case, Jate avoided this ambiguity (did he do it intentionally?)  As always, I invite you to explore each variant.

We played 1 and a half games.  The second game, it turned out Jate had set a booby-trap on one of the numbers, so I automatically lost when I circled that number.  How do I know it was pre-planned and not an arbitrary, last minute rule change?  Because he told Jin and they both started laughing as soon as I circled the trick number.

Jin's Game
Writing up the notes on the tidy-up game (see here) reminded me that I hadn't ever fully explored toetactic (inverse tictactoe.)  Jin was amenable to playing, so we drew a couple of boards and played tictactoe as a warm-up.  Then, I explained we were trying to force the other person to get 3 in a row and we played a couple of boards that way. No deep analysis tonight, but it was fun hearing him say "inverse tic-tac-toe" and seeing the plans go through his mind as he worked out his strategy.

Traffic Lights

Something we did before we started this blog: play traffic lights from NRICH:

There is so much I enjoyed about this game, but I mostly want to encourage you to go play it and explore.
This list of ideas is just to get you started and to remind you to pay attention to the kids ideas along the way:

  • what if we had 3 players? this is a natural question for us now as I play these games with the two boys.
  • Jin: what if the traffic lights would cycle (so so yellow follows red, green follows yellow, and red follows green)?
  • Jate: What if we added another color (in our case, purple)?
  • Jin and Jate: what if we had 4 colors and let them cycle?

Some other prompts for an investigation of the game structure and strategy:
  • draw pictures and diagrams
  • try a smaller board,  say 2x2 to consider a simpler case
  • try using 2 colors (or even just one) for a simpler case.

Questions about game structure
  1. Does the game always end?
  2. How many turns are possible?
  3. How many pieces of each color do you need? This is one benefit of making a simple version with paper, it leads you to ask this type of question.
  4. An ending board is where someone has made 3 in a row.How many ending boards are there?
  5. Are there fully colored boards (each square has a light) where no one has won? If so, what is the farthest number of moves such a board can be from a win?
  6. What questions do you have?

Make your own
As I mentioned, making a paper version helped suggest some ideas for investigation, both for me and the boys.  It made it a more fun hands-on activity for them as well (cutting out little pieces and coloring them).

Another route is to write a program to play the game. Here is a pencilcode version I wrote for the basic game: You can play against another person or against the computer, but I hope you also try to make your own version, investigate the variants we mention above, and find your own.


For now, this is a bit of a laundry list, without any special structure.  If there is a theme, perhaps the main idea of the weekend was "games."


Over lunch on Saturday, we played a kind of improv game using sight word cards, these:

In our pack (from FlashKids) there are 86 cards with a word on each side.  I guess the theory of sight words is that these are common words a fluent reader should recognize on sight and read without sounding them out.  We played a game where I would draw two cards and we would each take turns using those words in a sentence, flip the cards, repeat, then draw new ones.

We went through perhaps 10 rounds of cards (about 40 words, roughly 1/4 of the deck) in 10-15 minutes. The boys had no trouble with the words, so as a reading exercise this might be a bit too easy now, but they both were in the mood for this game and had a lot of fun.  Pretty quickly, some themes emerged: Jin used his sentences to tell a story about the Boobaloo characters, while Jate turned all of his sentences into something about bottoms and poop.

We didn't really get up to much writing this weekend, but had some long reading sessions at bedtime.  Jin has started reading to Jane at bedtime, though this weekend his rendition of Spot's Treasure Hunt was a bit grudging and soon gave way to a simple race through lifting the flaps.  Jate read us a Thomas story on his own, Jin read a couple of pages of Asterix, and I filled in with Alexander and the Terrible, Horrible, No Good, Very Bad Day and some sections from one of their children's encyclopedias. Strangely, the Alexander story needed to be jazzed up with some emotive reading (I'm great at imitating a petulant and whiny child, wonder why?) but they enjoy the encyclopedia entries straight.

Math toys
Mommy led two sessions from RightStart Math.  Here's a sample page with the introduction of multiplication:

These are convenient books because they lay out lessons that we can essentially just open and go through with the kids.  If something doesn't seem fun or is too easy, we either skip to the next activity or let them go through it very quickly. If she has time, Mommy will fill in the details of what she did for those sessions.

I think the abacus figured in, or at least it was something with which Jane starting playing, so here's a quick snap.

Both Jate and Jane spent some time working with the pattern blocks.  I usually don't take pictures of them playing so we all can stay in the moment, so here is a non-action shot of the pattern blocks:

The pictures don't really stay the centerpiece of the activity, mostly we spend our time stacking the blocks, tesselating, comparing the blocks, making our own pictures, etc. You know: playing and investigating.

We played two other games over the weekend, both using a nice chinese checker set given to us as a leaving gift by some friends:

As a kid, I don't think I ever got the point of chinese checkers. To spill the secret, it it what you can see in the picture: yellow is about to make a big advance by jumping over the blue, over the next yellow, and over the green to move forward 6 spaces. Jin seemed to get the point quickly and seems about 50/50 playing against either Mommy or Daddy.
With the chinese checker set, tidy-up time also turns into a math game, one of you told us about this as a response to reading an earlier blog post: alternating turns, each player puts 1, 2, or 3 pieces in the box.  The winner is the player that puts the last piece in the box. With a little prompting, the kids can figure out the strategy. Here's how I prompted Jin:
  1. When I got to my move with 4 remaining pieces, I asked what would happen if I took one piece? Two pieces? Three pieces? What does that all mean? 
  2. Jin identifies that it means I had already lost when it was my turn with 4 pieces.  What if it was his turn? Then he would be in a losing position.
  3. So, if it is Jin's turn with 5 pieces, how many should he take? What about 6? What about 7?
  4. Jin identifies that with 5, 6, or 7 on your turn, you have a winning strategy.
  5. What if it is my turn with 8 pieces?
  6. What about 9, 10, 11?
  7. What about 12?
  8. What about 13, 14, 15?
  9. What about 16?
  10. Have we noticed a pattern? At this point, he said that he could tell if someone was in a losing position by counting by 4s. Hmm, so this is where practice skip counting makes some future explorations easier?
  11. Play with 2 colors on the board (leading Jin to say, "how many are there? each color has 10, so that's twenty . . . Daddy, you go first.") We played several times with 20.
  12. Played with 30 and it was more difficult to figure out whether he wanted to go first or second.  Skip counted by 4 up to 28, then he got it.
  13. Played with 10 and let him figure out whether he wants to be first or second and then play.
There are a bunch of extensions of this game. One that Jin suggested was instead of having a simple winner, maybe we should keep track of how many pieces each person collected and that would give them points. We discussed this a bit and then he realized that, alone, wouldn't be a fun game because everyone would just take 3 each turn. I said we could give the person who takes the last piece a bonus, but we didn't really have time to explore that, but here's where you could take it right away:
  • how does the strategy change with a small bonus?
  • what about a large bonus?
  • Are there any critical values of the bonus that cause the strategy to shift?
  • What if you change the game to try make the other player take the last piece (either the inverse of the basic game or the points game with a last-piece penalty, or the inversion of the points game where you want as few points as possible)? 
  • How does all of this work with a 3 player game or even more players?
  • what other variants can you explore?
If I get around to it, I might make a pencilcode version of some of these.  I have a basic AI from  my work on Traffic Lights . . . which I still haven't gotten around to telling you about.

Theoretical digression
Multiplication happens to be a topic I've been puzzling over for a while.  There is a well-established debate about the pros and cons of teaching multiplication as repeated addition, scaling, or something else. Here is part of the internet branch of this debate, a great post from Keith Devlin

I don't know if we are going to get it right, but my approach so far has been to try to present a range of different incarnations for each of the operations. Here is a nice poster that claims to have 12 models for multiplication (I wish we had a copy of that poster!) 

Even for addition, though, there isn't one single mathsematic concept that applies.  For example, adding 2 apples to an existing pile of 3 apples gives you an ending pile of 5 apples, sure.  But traveling 2 km and then going another 3 km is a bit different.  Different again is modular arithmetic (or clock addition, for the under-7s).

Though I think Devlin's essay is helpful, I strongly disagree with his point that it is dangerous to change the rules on children (e.g,. "multiplication is repeated addition" at first, then later "multiplication is something else.") In math, we do this all the time and call it generalizing.  For example, nearly everyone learns linear algebra hearing that a vector space is a real plane and then the concept is expanded. Perhaps this, then, is the real value embedded in how you teach multiplication: don't stress the association by saying "multiplication is repeated addition" but say "one form of multiplication is repeated addition" and ask if they can think of other forms.  Talk about the relationship between them, use examples, draw pictures, stretch rubber bands, etc.  What you really want is for them to appreciate that a concept in one form can be generalized to other areas and applications.

Baking and Kids (and math)

One suggestion we got was to bake with the children.  This is an idea I strongly support for both mathematical and non-mathematical reasons:

  • Baking is full of measuring, so they see and use a lot of numbers. In fact, writing down a recipe for future use or for giving to another person to reproduce is a key motivation for even having numbers in the first place.
  • Fractions appear naturally if you use American measures or recipes
  • You often have to do scaling calculations (halve or double the recipe)
  • The process involves physically interacting with quantities; this is an important step for building comfort and intuition that sets a foundation for picture-based and abstract forms of mathematics.
  • Every product has hundreds of variations and competing recipes, so it encourages a spirit of exploration and investigation (see this brownie fugue, for example)
  • Small children can manage almost all the necessary steps, so it can help reinforce growing habits of independence
  • Of course, baking itself is a skill that is worth learning
  • You get to eat the end products!

What to bake?
I started drafting a treatise outlining the pros and cons of choosing breads or cookies or cakes to bake with the kids. Then I realized that probably no one else will care. For now, let's just say that all of them have their merits and eventually we will end up doing some of each.  Here are recipes in each category that I recommend or have had recommended to me (I'll add to this over time and have included some for which I don't have a link):

Lidia's pizza dough: very few ingredients, relatively easy to mix dough, very short baking time
challah: we make a half recipe
popovers: haven't made this recipe yet, but we will try it this weekend

French Yogurt Cake: also this, but note that we haven't made this!
pancakes: ok, you got me, this isn't baking, but we still love them.

rice crispy treats: again, not really baking, but who cares?
chocolate chip: enjoy this show (videotext) and go wild with variations

Other Cooking Resources for Young Mathematicians
Good Eats/Alton Brown: fantastically entertaining and I love the thematic focus he uses for each show. I've given you an "official" link, but probably better to look for episodes on youtube as above in the chocolate chip section.

America's Test Kitchen brings the philosophy of the scientific method to perfecting each recipe. To complement the above, here's their take on gluten and another with perfect pancake technique.

Bedtime Math: baking IPhones

Tonight, we talked about the daily math story/questions from Bedtime Math. Go ahead, look at the story, read the questions and come back.

Back already? Great.

I want to write about what I think of the questions, the story, and the video, but perhaps the boys' reactions are more important:

  • Video: they loved it, especially the part at the end where he puts his too hot phone in the freezer. They imagined that he would keep over and undershooting the target temperature, moving the phone between freezer and microwave to alternate cooling and heating it.
  • The story: read before the video, this wasn't engaging enough for them to (appear to?) pay attention. Maybe they got more out of it than it seemed as they had no doubts about what was happening in the video
  • The "math" questions: Jin enjoyed them. Jate wanted to be silly instead, but he did offer some digits for the 1/3 of 15 calculation I'll describe below.
I think this is pretty typical of the bedtime math questions.  The story and video are interesting enough for 5 minutes before bed and the math questions seem to interest the boys.  Each one, though, slightly bothers me as the math questions are basically arbitrarily imposed on the story.  For example, why do we care about comparing those temperatures, why do we care about adding one minute to the cooking time, and does batter really shield the phone in the way the last problem assumes? I guess this last one goes onto my curiosity list as item 40.

Maybe I shouldn't be so critical: this story about Big Ben had more related questions.

Field report
Wee and Little Ones
Way too easy for Jin, he answered immediately.  As a result, we'll never know how Jate would have found them, but I think too easy for him, too.

1/3 of 15 minutes: Jin
I had to help explain that 1/3 of 15 minutes involved splitting 15 into 3.  Before calculating, we talked about what halfway would be (he guessed 7 minutes, but knew it wasn't quite enough).  He then asked for half of 7.5 to get 1/4 of the time through as a way to bound the answer to the actual question.  Great strategy, but did he really know what he was doing?

From there, Jin asked us to collectively produce 15 fingers so he could parcel them into 3 groups.  At first, we got a mixed collection of hands and feet (thanks Jate!) By chance, or stealthy guidance from father, eventually we three each contributed one hand, so dividing into 3 groups was obvious.

The sky's the limit
I was surprised that Jin really took note of the initial 2 minutes safety window for the phone and included it in his thinking.  I guess this sort of residual has been emphasized to him in past problems, but I don't recall it figuring prominently in anything we've discussed together.

He eventually got the answer 20 ounces with the strategy of guessing an amount, figuring out if it works to protect the phone for the required cooking time, then adjusting.  I helped him keep track of the pieces of the question, which seemed fair since we were doing this without paper or pencil.

A game to practice number bonds of multiple of 5s

We played a game called Corner game (from the RightStart Math Curriculum).  We dealt out 4 cards for each player.  In this basic version, there are two rules for connecting the cards -- the sides touching each other must have: 1) Same color and 2) Either same number or the numbers have to add up to 5, 10, 15 or 20.  But only the multiples of 5s count towards the total score.  At first, I was concerned that the rule is too complex for a 7 years old.  But Jin got it and immediately looked for pairs of 10s with the same color.  Adding up the total score seems to be a good exercise.

That turtle code

For the last several days, the children have been playing with pencil code. There is so much to love about it:

  1. Cute turtle that makes everything seem silly and fun
  2. Simple, interpreted code that lets even a 2 year old command the turtle (with some spelling help). Save the for-loop that I added later, this is the level of code Jane was typing: Jane's First Code. She would tell me what she wanted the turtle to do ("Get bigger"), then I would tell her the command and which buttons to press to write it.  Of course she can't read the code or remember the commands herself, but she seemed to get the connection between telling the turtle what to do via typed code and then having it move. 
  3. A powerful, "real" programming language that lets you write programs with increasing complexity. This Galton Box animation is one of the nicest  I've seen (and was the example that first got me interested in pencilcode).
  4. Your code is inherently on the net, so you can run your animations on any connected machine, share them with friends, even send them to Grandma and Granpa!
  5. . . . and more
By the way, here's the source for the two animations above: JFC and GBA.

These two were the result of the boys playing: Jin's Gun (source code) and Jate's Star (source code).

I've also shown them cool programs written by other people.  For my own reference, I've been building a repository here: It doesn't even include all the great code that I've already seen (before I had the idea of making a favourites list), so don't feel offended if your awesome work isn't included. Just point to it in the comments!

Now that I think about it, I should also let them browse other people's files and see some simple code that might not quite meet my definition of "cool."

What did they get from this playing, so far:

  • fun and stimulating new game that puts them in control, not because they demanded something, but because they planned what they wanted to happen and worked out the commands to make it work
  • introduction to angles and angle measures, length presented in a new context, coordinate systems, negative numbers, variables, arithmetic calculations, 
  • safe place to experiment
  • a reminder that it is great to learn from other people's work and to credit the original authors

In case you are interested, here's my folder of different pencilcode experiments (at least the ones that were worth saving!):

Note: When I figure out how to do it, I'll embed some of the animations in this page.

Heads and Feet (and multi-variable equations)

NRICH has a fun little problem on its top page for lower primary: heads and feet.

The challenge: A farmer has chickens and sheep. If the animals all together have 8 heads and 22 feet, how many sheep does he have and how many chickens?

My question: Is this low threshold and high ceiling?

The basic task seems to be one unit, but there are a couple of progressions to make it more accessible. My guess is that most children who get stuck will need to be guided toward alternative versions that will let them build strategies for understanding the challenge. Perhaps starting with a version that has fewer animals will be accessible. If necessary, drawing pictures of animals, counting them, counting body parts, etc can help get the exploration started.

Note that drawing realistic animals doesn't necessarily help with the mathematical content (and it can actually get in the way of one solution strategy). However, if you want to practice, here are some instructions that will lead you to a much better animals than I draw!

Plan: have Jane counting objects, Jate drawing pictures of animals and counting body parts, and Jin to get the challenge as originally stated.

The obvious route to a (super) high ceiling is linear equations of multiple variables and the whole world of linear algebra. My idea to open that route is to ask if Jin can write an equation to describe the number of animal heads and another equation to describe the number of feet.

The solution strategy I've seen him use in the past is to draw a picture, starting with the assumption that all the animals are chickens, like this, with heads and feet helpfully labelled:

and then adding feet to make sheep until you get the right total number of feet (the sheep bodies are shaded to make the distinction between sheep and chickens more clear):

I've been wondering about whether this strategy is generalizable, since even three variables seems unclear. My questions, if we get this far:
- can we write an equation or equations that describe the relationships between sheep, chicken and heads? What about sheep, chicken and feet?
- Is always a solution, no matter how many feet are given? Do we notice anything special about the number of feet in this type of problem?
- With the right number of chickens and sheep, can we get all combinations of (heads, feet) = (n, 2m)? Are there any restrictions we can identify?
- Can you think of any problems like this? Maybe chairs (4 legs) and stools (3 legs), bicycles (2 wheels) and tricycles (3 wheels) etc.

This activity wasn't loved and didn't capture Jin's attention during the evening rush.  Maybe something to save for a quieter time or perhaps the lesson is to have various investigations available? Mommy's post mortem:  there wasn't time to do this anyway, so nothing lost.

Well, we ended up with another nice chicken picture, at least:



I started watching some Jesse Schell speeches and eventually came to this one: Learning is Beautiful. His talk is entertaining and he makes some great points, but the one that most hit home was the amazing opportunity open to curious children (here). If they think to ask a question, they can access a huge amount of knowledge, even if they only know to look in Wikipedia or via an internet search.

All children I've encountered are naturally very curious, so I was wondering how best I can help the J's stay curious. My best idea was the curiousity list, to encourage them to ask questions and then take (at least) some of them seriously for further exploration.  Whenever they ask a question, we write it down and, now, it can go on this list.

The List!
  1. What if Josh took a nap mid-day? Jin
  2. What if Josh slept all day and night? How long could he do this? Jin
  3. What causes tummy aches and how can they be treated or avoided? Josh
  4. What is the history of polo shirts, particularly why do they have collars and a couple of buttons? Josh
  5. What causes traffic jams? Jate
  6. Why do all schools start at 8, why is the road crowded at 8? Jate
  7. Why do children go to school? Josh
  8. Why do brothers tease sisters? Josh
  9. Why does stirring ice make a drink colder faster? Josh
  10. How does a freezer work? Josh
  11. How do we log-in to youtube on a Sony TV? Josh
  12. How do the parts of an electronic circuit work? Josh (maybe this is on brainpop?)
  13. Can you spin a turbine with your hands? If you do, what will happen? How does the motion work? Jate
  14. How did the knights' move in chess come about? Jin
  15. Why is so much of Europe devoted to growing crops? Jin
  16. Why do children sometimes not listen to their mothers? Jate (asked with respect to the Madee story การประมาณตน)
  17. Why can egg whites make foam? Josh
  18. How do market sellers weigh/portion sticky rice? Can we trust them? Josh 
  19. How does the price of petrol compare between Thailand, UK, and the US? Josh
  20. What are the fruit seasons in Thailand? Josh
  21. Why are kitkats in pairs? Jin
  22. Why did people make cars if they pollute the air? Jin.  Do trains pollute? What about airplanes? Jate
  23. Why is rice so sticky? Jate.  How do you measure stickiness? Josh
  24. What is stevia and how does it sweeten food ? COul da chocolate cake without sugar be healthy? Jin
  25. Do Pringles have eggs or dairy? Mommy
  26. Why do we need to eat food? Jin
  27. Why do some people sweat out of their noses? Mommy
  28. What are the gift giving traditions in Thailand? Why did our neighbor give us chompoo? Jate
  29. If Mars and Earth were brought closer with their magnetic north poles facing, would the magnetic repulsion or gravitational attraction win? Jin
  30. Does Europe have a desert? Why not, if not? Jin
  31. Where are mangoes grown? Josh
  32. Why do train tracks use/have gravel? some kid on the Heathrow Express (his mother wasn't interested in the question)
  33. When did UCL Academy in Swiss Cottage open? Josh
  34. What do car spoilers do? Jin
  35. Why are exhaust pipes different shapes? Why do some cars have 2? Jate and Jin
  36. When does the rainy season in Thailand start? Josh
  37. How does an Ipad work? Jin
  38. Why do different people's teeth have different spacing patterns? Jate
  39. How did the capital destruction compare with the GDP decline and loss of labour force in WW1 and WW2 (inspired by a discussion of Piketty's new book)? Josh
  40. Is a rackets court the same dimension as a doubles squash court? Josh
  41. How does PV=nRT apply to the coolant cycle in a home freezer? Josh
  42. Why doesn't this ball blow away? Josh
  43. If trees release oxygen gas during the day and absorb oxygen gas at night, do they release or absorb oxygen overall? Jate
  44. How do we make a flashing irridescent light (like the one we saw in the taxi)? Jate
  45. How do they make different flavours of yogurt drink? Jate
  46. Could someone without arms or legs drive? Jate
  47. What is the Higgs boson really? Josh
  48. What are the equations describing a spirograph motion? Josh
  49. What is borax? Josh
  50. Do doctors ever undertake a circumcision as treatment for a condition? Jin
  51. What is the difference between yorkshire pudding and a popover? Josh
  52. Why do we pre-heat the pan for popovers? Josh
  53. What is a winning strategy for the new take-away game (n piles, you can take any number from any single pile, winner is the one who takes the last piece)? Jin
  54. What song has the lyrics: "everything is awesome when you live in a dream/everything is cool when you're part of a team"? Josh
  55. How do they make the little rubber bands for loom band? Jate
  56. Why does eating ส้มโอ make your poop smelly? J0, J1, P
  57. What is a good investment portfolio for higher inflation? J0
  58. How do you buy equities in Thailand? P
  59. Are there any color combinations that are as attractive as a rainbow sequence? J2
  60. What are New Order doing now? J0
  61. What is required to develop an iPhone app? J0
  62. Is the color for green บะหมี่ artificial or natural? P
  63. what type of tree is this? P 
  64. Why doesn't France have an issue with Basque separatism? J0
  65. Why are bathroom electrical plugs an unusual shape in the? J0
  66. What is the relationship between the distance produce is shipped and the associated carbon emissions? J0
  67. What happened to Rachel Mordkowitz (daughter of a trustee of Institute of General Semantics who was mentioned in Mathsemantics)? J0
  68. Do all series an+b (n varies) that have 2 primes also have infinitely many primes? J0
  69. Where is the brain in a crab? J1
  70. What kind of crabs do we eat in Thailand? J0
  71. Does composting attract pests? J0
  72. Does mass matter downhill skiing or cycling without pedalling? J0
  73. Why do some fruits turn brown very quickly when peeled? J0
  74. Why is it cold at higher elevations, doesn't warm air rise? J1
  75. What is the cost of electricity in Thailand, how do they know if you are using electricity? J1
  76. In the turtle night light, why do the stars shift position slightly when we change the light color? J2
  77. Why are the two bathrooms at my school arranged and decorated differently? J1
  78. Who is the best person in the world? J2
  79. why is facebook so popular? J1
  80. How do you know about facebook, J1? J0
  81. Why do leaves fall off of a tree in a tropical country? J0
  82. How do plants sense gravity and respond to that stimulus? J0
  83. Phillips head screws seem more prevalent than flat head, are they and why? Also, why do the heads seem to strip so easily, cheap screws, bad screwdriver sizing, my technique? J0
  84. What is the phase of a chocolate mousse? What is the ideal fat and water ratio to achieve a (relatively) stable version of that state? J0
  85. How are electronic children's rides evaluated for safety? J0
  86. What determines whether something is a hill or a mountain? Does the classification vary by country? J0
  87. How is the cleavage line for a germinating avocado seed determined?  J1 and J2 (see this post).
  88. Conjecture: (2n+1)! is a triangular number. J2
  89. Are there compensating errors or do errors accumulate? This was inspired by the Numberphile video showing the 17-gon construction. J1
  90. What is the story of the orphanage at which Vivaldi taught and why hadn't I heard about it before? J0
  91. What is the non-prodigy interpretation of Mozart? J0
  92. How does the pressure gauge in our bicycle pump work? J1
  93. How is black rice (riceberry) grown? What makes it different from other strains of rice? J2
  94. Does pomelo (ส้มโอ) continue ripening after it has been picked? P
  95. How do crocodiles reproduce? J2
  96. What is the stall speed on a CRJ200? J2
  97. Why do slinky tangle up so easily? J0
  98. What is the best way to untangle a slinky? J1
  99. Who invented straws? J2
  100. Is it possible to find gold to mine? How would we do it? J1
  101. Where did the first person in the world get milk to drink? J2
  102. Are small motors for rotation (cd player, motor for fan rotation) especially susceptible to damage from hot weather? J0
  103. Why is phosphorus a very reactive element? J2
  104. How did Einstein come up with E=mc2 and how do we know it is true? J2
  105. Which countries are the most food independent? which are the least? J0
  106. What do the chinese numbers on the calendar mean? J2
  107. Why did the hippo get stuck in teh door (in the song animals marched 2x2)? J3
  108. Which were more powerful empires: Roman, British, American? J0
  109. Can giraffes swim? J3
  110. Why did Beethoven lose his hearing? J2
  111. Why was Vivaldi's music surpressed? J2
  112. Is π the square root of something? J1. Realized that it is (sqrt of π2, so refined question to: is π the square root of a rational number?
  113. Why were certain animals widely domesticated and others not? In particular, why not elephants? J1
  114. Why does ice sometimes stick to your fingers? J1
  115. Why is it dangerous for children to ride in the front passenger seat of a car? J1
  116. What does an airbag do? Why aren't there airbags for children? Why is an adult airbag dangerous for children? Why are airbags only for the front seat of  a car? J2

What did we do?

Yesterday, we followed the plan very briefly sketched in our previous post.

Playing with the Balance
I started with the boys together playing with balance for a bit. They did a couple of rounds of some of the basics: finding a sum (add one weight to an empty side to balance the other side), a difference (add one weight to the lighter side), and then adding 2 weights (e.g., add weights to balance when you start with 9+6 on one side against 10+4 on the other). 

This was a version Jate got after some playing around adding and removing weights on the left side of the balance:

Jin started asking about multiplication ("how much is 8 groups of 5?") and I challenged them to figure out a way to do these calculations on the balance.  I'm not sure how much progress they made in finding a technique because this devolved into causing the balance to crash. Their technique for a fun chain reaction:
(1) have a moderately heavy right side (maybe around 20 units)
(2) add a lot of weights to the 10 peg on the left side (getting to 40+ units) 
(3) let go, 
The result is that the left side crashes down, those weights fall off, then the right side crashes down and, if you are lucky, those weights also fall off.  If there are any weights still on the balance, it will crash one last time.

I didn't particularly encourage crashing the balance, so they found something else to do.

Percents and fractions

Jin moved off the balance first and went into the box of toys.  He found a collection of fraction and percent cards and set up a game of war (draw random cards and whichever card is larger wins). At first, they played together with just percents, so it was essentially the same as comparing whole numbers.  Mommy got involved and challenged Jin on fractions, accompanied by the fraction chart for reference.  This seems like a promising game for future warm-ups.

Abacus warm-up
Mommy next moved on to the Abacus math exercises.  Jin counted by two from 0 to get a sequence of evens, then again starting at 1 to get a sequence of odds.

Addition table
Warm-up done, Jin then investigated some patterns in add table, starting with highlighting the number bonds for 10 and 5. Jin noticed the SE diagonal was evens, so filled that.  Jate got his own copy and started filling the parts that seemed interesting to him (10 number bonds and the NW corner). 

Some Venn
I didn't see how Mommy transitioned to venn diagrams or all of that discussion.  I came in after the whiteboard had most of this on it: 

I started asking about friends who I knew to be only children (no brothers or sisters). That led to a bit of discussion about where they should go in the diagram, with Jin eventually writing them outside the two strings. I'm not sure he was satisfied with that approach, though, as he scribbled out the names.  I think we should come back again another time to the idea of elements that aren't part of any of the named sets. 

In the meantime, Jate had been playing with the hundred board tiles and came up with this picture. 

Actually, 90 and 99 hadn't been completed, so we asked Jin if he could guess the pattern and finish the picture. He did and we were off to dinner.

The requested book was A Place for Zero. They had read it before and both were interested in telling me how the machine worked to create new numbers from old ones. Both were really excited to discuss what would happen if Count Infinity got sucked into the machine as an addend (they thought another infinity would come out, no matter which other addends got sucked in.) 

Finally, they asked about "that turtle program," so we wrote some pencil code. Jin said what he wanted the turtle to do and Jate helped with the commands that he remembered from 2 weeks ago. I was really surprised how much Jate remembered.  Here is the simple code they wrote: gun

Not sure how much time we will have, but here are some ideas:
- Jate play w balance by himself
- more simple pencil code after dinner
- baking with Jate (probably do this on Wednesday)

Getting Started

Is this just worksheets?
NO! We want to build from and reward their natural curiousity. Games and exploration are the key pillars. At this stage, our thinking is:
- number sense is primary
- guided games and exploration
- communication builds deeper understanding

Number sense
Numbers are all around. Children can be encouraged to attend to the occurrence and use of mathematical concepts. By estimating, measuring and comparing, they build an intuition which leads to comfort. For examples:
- how many marbles in a jar (numbers of objects and volume)
- how tall is their friend (length)
- how long does it take to drive to school (time)
- how much mass does this mango have (mass)
- etc (area, volume, temperature, cost,angle measures, Fermi questions)

For a really intriguing story about the results of one school district using this approach, see:
Take them to interesting places and let them explore. If it necessary, use leading questions to help but this is still best if our attitude is that we are also exploring. This is like a scuba or ski guide (and not a "tell" instructor).


Encourage the kids to communicate what they are doing, what they think about it, why they think it works, etc. They can and should use words, pictures, diagrams, physical objects, and equations. Maybe even musical rhythms and tones?

As we kick-off our plan to add some fun math and reading activities, these are some of the resources from which we are drawing ideas and inspiration:
  1. a lot of fun games and exploratory activities meant to be "low threshold and high ceiling"
  2. RightStart Mathematics books (aka Abacus Math)
  3. US Common Core Standards. We are using this for curriculum benchmarks in both math and language arts. This link has an example for grade 2 math.
  4. Mathematics Mastery:
  5. Khan academy
We have a bigger list that will be useful once we get the rhythm established. *UPDATE* Here is a bigger list of resources we like.

Plan for tomorrow

  1. Venn diagrams of different shapes
  2. Looking for patterns in an addition table
  3. Some number balance games (online balance, other examples). Note: we prefer to use a physical balance.