Chocolate cake reference

This was Nigella Lawson's recipe, but that page currently doesn't include the ingredient amounts, so I'm reproducing it here. This was a test ahead of J3's birthday, so she helped. Since it seems successful, we will be making it again and I will include more pictures of the process.

Heat oven to 170 deg C

MIX:

• 3/4 cup + 1 tbsp flour
• 1/4 tsp salt
• 1/2 tsp baking soda

1/2 cup water (boiling)

6 tbsp cocoa powder

COMBINE into a thick paste and allow to cool (doesn't need to come fully to room temp)

2 tsp vanilla extract

ADD to cocoa paste and stir

2/3 cup olive oil

1 cup sugar

3 eggs

COMBINE in mixing bowl and mix on high (directions said 5 minutes, we mixed for 2-3)

COMBINE wet ingredients and mix thoroughly

COMBINE dry mix and wet just to incorporate together.

Use small amount of oil to grease a 9 inch baking pan.

Pour in batter and bake for 45 minutes (I rotated the cake every 10 minutes).

Another dessert reference to store, unfortunately dairy: Caramel Sauce

Bath-time math

Who: J3
When: bath before bedtime
Where: bathtub (of course)

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A simple, but classic bath-time investigation: which container holds more water? We've got three, and it is easy to see that the white cup holds less than the other two. However, comparing the silver and blue isn't so easy. Are they the same size? If not, which is larger?

J3 started with the basic technique filling one with water and then using that to try to fill the other. Then, she moved on to usin a common measure, counting how many white cups were required to fill each of the larger containers. Her technique was very splashy, so I wouldn't say every cup of water was equally full or that every drop successfully made the transition from cup to bucket. At this stage, the play was about the spirit of the investigation, the joy of counting, and the fun of splashing water around the bathroom.

More math around the house

Triplet of decorative paper airplanes (courtesy of J1)

A mathematical pizza (courtsey of J2 who made the dough and J1 who snapped the picture)

What's mathematical about pizza? Measuring ingredients, tracking the time to mix the dough, talking different shapes for the base (circle and rectangle are never quite achieved, of course), temperature of the oven, time to cook, ways of cutting, probability that any will be left for mommy or daddy to taste....

Grand Catch-up

Who: everyone in the family
When: over the last week
What material did we use: see below, a lot of different goodies
Where: all over the house

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We haven't posted many activity summaries recently, so this is a review of what we have been doing so that you don't think we've been slacking off.

Calculating and Roman Numerals
Remember my doubts about roman numerals when they were introduced (here)? P has since found that they come up frequently when J1 and J2 are discussing some other calculation (adding 2 digit numbers without paper and pencil was a recent one).  It seems that the kids like the process of conversion or the feeling that they are speaking in a type of code.

Mathsemantically, they appear to appreciate the idea that number names are not the same as number concept. Frankly, I don't know when they developed this appreciation as we've always been a bilingual household (Thai and English) and introduced counting in other languages (Spanish, German, Chinese, Korean, Tagalog, Armenian) when they were very young.

Tip: for kids around 7 years old, try introducing the numbers in a different language and see if they enjoy the disassociation of names and objects as they play with calculations.

Food Math
We've had another round of pizza dough (with J3, our 2 year old) and our first attempt at gougeres (J1 and J2).  The pizza dough with J3 went as usual: a lot of fun measuring, a bit of flour outside the mixing bowl, excessive use of the scale to weigh whatever she saw.

The gougeres were a dairy indulgence, with milk, butter, and cheese as main ingredients. This was my first opportunity to try them since J3, our dairy intolerant one, was out for the day with mommy. We roughly followed this recipe, halved because the remaining munchkins had already decided they wouldn't like them and I couldn't justify eating 20-30 cheese puffs myself (and I'm not generous enough to share them with anyone else). In any case, by the time I thought to snap a picture, we only had 5 left for a nice little pentagonal arrangement.

Mathematically, the activity was interesting because we halved the recipe and we had a debate about how many eggs to use. While the presenter explicitly says 5 eggs, both boys were sure we only saw her use three and we all agreed that she had reserved one for the egg wash. Coincidentally, P brought home some very small eggs later in the day, so we got to have a little (very little) discussion of whether one egg is a proper measure. I'll have to remember to return to that again.

As we've hinted on other occasions, we try to recognize little unplanned opportunities for inserting some numerical discussion into everyday conversation.  Quesadillas at lunch were a chance to talk about fractional parts of a circle based on 1/8.  Below is one serving in the 6/8 (aka 3/4) uneaten state:

Some bites and a cheeky smile later brought us: "daddy, what about this one?"

I offered 1/3rd of an 1/8th for the partially eaten piece and we calculated that this plate still had 7/24th. Maybe I should have rushed to get a clock and show them 3.5 hours?

Catan (Catan, Catan)
Somehow, Settlers of Catan got back on J1's radar and he's been asking to play it everyday for a while. We are still working on a version that fully integrates J2 and it is a bit stale without trading between players.  I tried to introduce a bit of trading today, but J1 was too suspicious of my motives and would rather trade at an unfavorable rate with the bank than trade with me.  For now, we still start him with 2 cities against my 2 settlements, and we also allow all the players to start with the resources from all the hexes associated with both of their developments.  Otherwise, we find the game takes too long to get moving.

You can see J1 (red) in the process of beating me (orange) below:

I can tell these sessions have started to develop his intuition for dice probability.  When allowed to set up a board configuration of his own choosing, he put the 6s, 8s and 9s together on the hexes associated with his developments.

And some geometric designs from J2 while he was watching us play:

Hexaflexagons
I plan to have more discussion of this tomorrow after our math party. For now, suffice it to say that I made some flexagons, left them around the house, and both the boys were really excited to make, decorate, and investigate them.  Almost all of them are trihexaflexagons, so that's what they've come to expect.  I gave J1 a hexahexflexagon with numbered faces and he was delighted to discover the extra faces.  I'm looking forward to seeing how the other first graders respond.

Since it isn't fair to entirely leave you hanging, I'd refer you to this Vi Hart video for a great intro to flexagons: http://www.youtube.com/watch?v=VIVIegSt81k

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: