Uno

Who: J1 eagerly, J2 reluctantly
When: before bedtime
What we used: Uno card deck
Where: bedroom floor

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The card game Uno has recently become popular at Baan Pathomtaam and J1 caught the bug. I picked up a pack last week and we've already played 20 (or 30, or 50?) times.  Apparently, there was even a solitaire game played at 5 am this morning.

I guess . . . that I don't really get the point of the game, so I'd welcome any comments pointing out interesting things that I'm missing.  Here's what I've considered so far:

• For very young children, this reinforces recognition of colors and numbers.
• Generating "uno" patterns: sequences of objects with two (or more attributes) where only one attribute changes for each step of the sequence.
• A bit of counting when you decide what color to call for a wild, executing a draw 2 or wild draw 4, and noticing that the other player has gotten to one card.
• Some strategic thinking to determine what cards to hold toward the end, or more advanced tactics if you can remember enough information to figure out what the other player is holding.
• You can modify the rules to introduce some arithmetic practice (some ideas are in the links below).

I've cheated a little.  Not at the game, but just in introducing a bit of extra math artificially.  When I deal out the starting 7 cards for each player, I put them in a grid so we get a quick reminder of one of our multiplication pictures:

Other ideas
That trusty friend, Google search, came up with some ideas for math games using Uno cards:

• place value war and 10s memory (both explained here: http://living.weelife.com/2012/08/two-uno-math-games.html) and 
• an uno progression (http://www.squidoo.com/how-to-teach-math-with-uno)
• a fun paper studying the structure of UNO by Erik and Martin Demaine et al: http://arxiv.org/pdf/1003.2851.pdf.

Baking math

Who: J3 (2 year old)
When: late morning, after J1 and J2 had gone to school
What did we use: water, yeast, flour, sugar, salt, oil, measuring cup, measuring spoon, scale, bread machine
Where: kitchen

See how little of the finished product was left before I remembered to take a picture:

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I'd posted before (here!) about some of the (mathematical) reasons to bake with kids.  Here is the math we did today while making bread:

1. counting
2. measuring volume and mass
3. estimating
4. next steps: ideas I forgot, but you can include them!

Counting
Maybe I go overboard, but my habit is to count everything around the smaller children.  Based on the recipe alone, there wasn't much scope for counting (2 eggs, 2 tsp salt, 2 tsp yeast) but we also counted the number of measuring spoons in our set and how many scoops of flour we needed to get our target mass. Along the way, we probably counted fingers and maybe even toes, too.

Measuring volume and mass
This is the obvious place to develop number sense while cooking.  We end up doing a lot of extra measuring during the process, playing with the different tools, and talking about comparisons between them.  For example, flour (21 ounces, about 600 grams) was the only thing we needed to weigh. What we actually weighed: flour for the bread, flour not used for the bread, the mixing bowl, the measuring cups (full and empty), a water bottle and box of crackers near us on the counter, the measuring spoons, and the sugar jar.
That allowed us to explore a range of weights from about 20 grams up to 1.5 kg.

We talked about metric and imperial units and looked at all the quantities in each. In particular, I use the imperial units as a good place to talk about fractions.  For today, I just read out the fractions close to what J3 measured, and included those in our chatter about comparisons.  My goal is to head off any (distant) future anxiety that comes from seeing fractions as a different "type" of number.

Estimating
I try to encourage them to estimate measures before they see the result.  This is easiest for mass, I just say "I think that will be 100 grams (or whatever I guess)" before they put the object on the scale. For the older kids, they will almost always respond "I think it will be less/more/[x grams]." J3 will usually just copy me, but she makes a delightful sound when we read the measurement together and comment on how close our estimate was.

Next steps
In truth, I didn't really go through every step with J3.  We measured the ingredients into a bread machine pan and then let the machine do the mixing and kneading.  Of course that saves a lot of time and effort, but we didn't see the exciting phase transitions where the dough goes from  separate liquid+dry flour to lumpy/sticky to smooth/elastic.  For older kids, it is interesting to ask them why they think the dough changes. Also, I did the shaping myself and that could be an opening to talk about braids/twists/knots.

As I wrote up these notes, however, I realized that I missed an opportunity with J3 to talk about temperature and time.  I think time was the big one: time for mixing/kneading the dough, time for a rise (actually there are 3 rises in this recipe) and time to bake.  With that in mind, I'll leave you with a picture of our kid-friendly timer:

patterns, estimation, communication (ideas for a grade 1 math course) (draft)

*UPDATE* this is not a class we are going to teach, but the ideas behind it and spirit inform the activities we do with the children.

Spirit of the class 1: "It is good to be confused."
Quote from Glenn Stevens (and Arnold Ross?) My initial thought on a Thai version is:

 ความสับสนเป็นสิ่งที่ดี

The goal is to operate within a realm of difficulty where the students are making some mistakes. From a music teacher, I got the heuristic that roughly 10-20% of the time is a reasonable target.

Students should be praised and encouraged when they say they don't understand something, when they ask questions.  Typical cues:
- show excitement (smile, eyes wider, raised voice pitch)
- emphasize this is where we get to learn
- ask them to explain how they are thinking, use words, symbols, numbers, draw a picture, etc
- ask other students for their thoughts
- ask everyone (including the original student) approaches for how they think we could figure out an answer

Spirit of the class 2: Always looking for patterns.

Communication
Throughout, emphasis is placed on having the students talk to each other about what they are finding, organize some of their thoughts, share with the class, write down what they think.  Written material will include:
- equations
- diagrams/pictures
- words (Thai and English)

Practice comparisons:
- X is similar to Y because A, B, C
- X is different from Y because A, B, C

Later, ask for things they know that are similar (and how those things might be different).  When appropriate ask for things that are close but different.

Patterns
Starting discussion:
- What is a pattern?
- What patterns do they know?
- What do they do with patterns?

using single trio blocks hidden in a paper/cardboard tube. For all, questions are

1. can they guess a pattern.  
2. how many patterns they can guess that fit the existing data. 
3. how much more data they need to see to test their hypothesis (confirm or reject).
4. For the pattern they guess, what color comes 30th in the sequence? What about 191st? or some other numbers that are moderately and then very large.

- repeated AAA etc pattern
- repeated ABAB pattern: what color is the 30th in the sequence?  What about the 191st?
- repeated (ABC)^n pattern: what color is the 30th in the sequence? What about the 191st?
-  prod_i=0...inf (A(B^i))
- prod_i=0...inf (A(B^(2i))
- random
- (AB)^8C: to show that there could be something else going on

Using a piano or violin, play patterns like:
- aural versions of the patterns above
- jumping up an octave
- CDCECFCGCACBCcCdCeCfCgCaCb
- CEDFEGFA etc
- CDEDEFEFGFGA etc
- forte, piano, forte, piano (single note or chord)
- different rhythms
- play a simple piece (e.g., twinkle) and stop before the last note.  Is there a pattern.  How do they know that the song doesn't feel finished?  This is just for discussion, no real expectation to have deep answers.

Movement patterns
- movement versions of some of the block patterns
- follow the leader patterns (everyone assigned to follow someone else, someone assigned to follow the teaacher, then teacher slowly walks a loop to allow the snake to form up)
- 2 separate snakes
- wall and monster: everyone assigned another person who is a monster (to them) and another person who is a wall (to them).  They try to walk so that their monster is on the other side of their wall.  Discussion: did it look like there was a pattern?  We had a rule about what was going on, but it was very complicated.  If another teacher came to watch, could they guess our rule?

Ending discussion:
- What other places can they find patterns?
- What are their favourite patterns?  What do they find appealing about those patterns?
- Are there any patterns they don't like?

Homework:

1. Find patterns in new places: movement, colors, shapes, in language
2. Find new patterns: for something they thought they already knew the pattern, can they find another rule that fits the data?  Maybe this one is too hard

Follow-up activity: 

Here are 4 things, find the one that doesn't belong.  

• Why doesn't it belong? What is the pattern or rule that the other three fit?
• Now, can you find another rule or pattern that kicks out a different object?
• For each of the four, can you find a rule that excludes it from the other three?

Estimation/Measuring
Note: other quantities to measure are listed here: http://3jlearneng.blogspot.com/2014/06/getting-started.html

How many are in my jar/box each day at the beginning of the day (can be something left out for the whole school to enter a figure.)

Temperature using the infrared thermometer gun
Distance using the tool

School run: give the students a gps to measure the following
- Time: how long does it take to get to school? Does it vary from day to day? How does it look over the course of a week?
- Distance: how far do they travel? Again, does it vary from day to day and how does it look over the course of a week?
- Speed: average and max speed as measured by the gps.  Which varies more over the course of a week?
Each day, ask for the data of the day and then ask for an estimate for the following day. Rotate through the students who has the GPS.

These will also be good for introducing graphs. I would just show the picture and explain what it meant, but not dwell on the graph.

Chains (part 2)

Who: J1

When: before bedtime

what: figuring out the rule for the 60-chain and finding other chains

why: because J1 wanted to discuss the chains exploration

J1 was getting more and more concerned as he looked for his homework folder and didn't find it.  Suddenly, his eyes lit up and he held up a crumpled piece of paper: "Daddy, look what I've got!"

It was our 60 chain from earlier in the week.

I suggested we try some other starting points: 0 2, which he then wrote down until we got back to 0 2.
Next, he asked what would happen if we started with 2 3.  I gave him time to think about it and then pointed to 2 3 in the 60-chain and asked what he thought that meant.

Next, I pointed out that our 0 2 chain had a 2 2, 4 4, 6 6, and 8 8. What about other doubles? He found the 60 chain had 1 1, 3 3, 7 7, and 9 9.  So, what was missing? Eventually, we got to this:

I left him with this challenge: there are two other chains, what are they?

Before he would go to sleep, he wanted to discuss a little whether it was easier to find long chains or short ones.  Eventually, he came to the conclusion that long chains might seem more complicated, but they are easier to find. Can you see why he thought that?

Scaling (part 2)

Here is an example of an activity that didn't really work out.  I'm curious if you have ideas why and I'll explain my take on the situation.

Who: J1 and J2
When: mid-afternoon when I am working
Where: in my home office
What: drawing pictures in larger scale using a grid copying technique
Why: build intuition for scaling as multiplication and distract them while I was busy

Background
I wrote about one plan to have the J's play with scaling through a collection of grid drawings here:

http://3jlearneng.blogspot.com/2014/07/multiplication-scaling-and-groups.html

After getting that inspiration, I prepared several pages using pencilcode and some pre-grided pictures I found on the web.  Then, I even went a step further and got an Asterix action shot and made it into a grid drawing:

Here's my pencilcode directory.

How could the older J's not love it?

In play
 At first, both J1 and J2 seemed interested. They asked me to print particular pictures, compared how many gridlines each had (some are 4x4 and others 5x5) and then asked me if I could find other pictures (leading me to put together the Asterix one).

From there, J1 made an attempt on the butterfly pattern, but I don't think J2 even made any progress on his.

Post mortem
Here are my ideas for why they didn't get into the activity:
(1) it was too hard
(2) they were distracted by something else
(3) I wasn't doing it along with them.

Sometimes, it is frustrating to spend time planning and organizing an activity for it to fizzle out. When it happens, though, I think it is important not to keep pushing the activity.  Sometimes, it can be introduced again at a different time or place. Sometimes, in fact, they will ask about the activity on their own, helping to prove that often something goes in, even when they don't seem to be listening.

Pattern Blocks (mini follow-up)

Who: J1

When: 5 minutes in between other play

What: finding an area relationship and learning a surprising technique

How: I made a design and then we admired it together

A pretty little picture, no?

Last month (specifically, here: http://3jlearning.blogspot.com/2014/06/tease.html) I asked you to calculate the areas of the different pattern blocks. One of the trickier ones is the thin, tan parallelogram. This arrangement gives you one way to see a relationship that might be hard to spot.

I set out the blocks and arranged them as in the picture.  At some point in his play, J1 paused and started looking at the blocks.  I moved the top two (orange square and green triangle) away from the others and said "hmm." That was enough to launch us into a discussion of
(a) are the two clusters the same shape
(b) are they the same area
(c) are the green triangles the same
(d) can we draw any conclusions?
(e) if they are the same area, why can't we make a square from the two parallelograms?
(f) what equations can we write that describe our picture?

Do you know other examples that are like this (you add something to a problem/picture/system that suddenly makes it more clear rather than more complex)?

One example that recently came to my mind was completing the square for a quadratic equation. A more sophisticated example is moving from affine to projective space to look at curve intersections in algebraic geometry.

Number sense, calculating, and stuff

Apologies, this post is a bit of a grab-bag of different things we have been doing recently. They are a mix of activities across each of the age groups, done at different times of the day and different levels of engagement, though most of the one-on-one time with J1 comes just before bedtime.

Number Sense: J3
We have made a conscious effort to surround the children with numbers and are often looking for good ways to present concepts in a different format, especially one that has physical objects and some relationship to their body.  Meals are often rich with opportunity, especially if you don't mind a little bit of playing with the food.  Here, J3 has matched up 10 almonds with each of her fingers, counting as we made the layout , then counting again as the almonds got put in a little cup, then counting again as she ate them.

Similar ideas:

• counting stairs going up and down
• counting musical beats as you listen to a song
• weighing food, weighing each other
• estimations (amount, length, weight)

Snakes and Ladders, err, Dinosaurs and Dinosaurs
I've mentioned before and will repeat frequently that I enjoy playing games, but snakes-and-ladders format aren't really games since you can't actually do anything. For now, though, the kids like them and the provide an opportunity for two things: (1) asking interesting questions about structure and (2) introduction to probability.  Along the way, there is practice calculating, but this is just small addition problems and not especially rich.

Tonight, J1 and I played a dinosaur version, you go down if you land on a land-dinosaur head square and up if you land on a designated square associated with a flying reptile:

We made one big modification to the game: playing with two dice instead of one.

The main point of interest is hearing J1 calculate dice sums, which square he is jumping to, and analyzing the size of the boost (or drop) from hitting the bonus squares and the penalty squares. It fun to talk about the probabilities of hitting the bonus and penalty squares, particularly because he asked all the questions and did most of the talking.

Simple starter cues, if you want them:

• is it possible to get to x from where you are now?
• how many squares away are you from y?
• How many ways are there to roll a 5?

Toys

Really, all of this post is just leading up to praise for some toys the kids really like and that can soothe parental anxiety about "stimulating development."  Frankly, I don't really care, since I enjoy playing with these, too!

Rainbow Loom

Extremely popular in J1's school group and even J2 can weave a nice design on his own.

They also discovered that youtube has useful how-to videos and they even worked together!

So, what are the (future) mathematical benefits:

- pattern building and recognition

- building a mental model of how the structure fits together

- understanding algorithms (especially with repeated loops)

Snap Circuits

A fantastic kit with electronic components that all snap into a grid-circuit board for easy assemble and disassembly.

 There is a book of projects which show circuit layouts and give some commentary about what you should be examining to understand the operation of the circuit.  Usually, they introduce each new component in a show piece of its own so the kids can build a sense of function.

I do have one complaint about the kits, though: a lot of the projects depend on using an integrated circuit.  These are black-boxes, so it misses a bit of the fun of building everything up from truly elementary components.  For now, that's certainly fine for our kids level of sophistication since it is fast to build a fully functional circuit. We do miss out on really understanding what is going on inside the box, though.

Trains

Really just an excuse to post a picture that reminds me how much fun J3 had playing with one of our toy train sets: