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.

Quickies

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:

• 5 steps to 50
• Chain of Changes

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.

Tease

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: http://nrich.maths.org/6589.  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.