Fractions for Cooks (with Special Bonus Recipe)

For a little weekend fun, here’s another real life math problem you or your kids may enjoy. This one is about fractions. I like it because it’s very concrete, covers a lot of ground, and might have something to interest you regardless of how old you are. Here goes:

  1. I have a recipe that uses 3 eggs and half a cup of flour. (True story; for actual recipe, see below.) I want to use 4 eggs instead of 3, and I need to measure out the right amount of flour for the recipe. My Pyrex measuring cups with fractional increments on the side are in the dishwasher and I don’t feel like cleaning them (I told you, true story). But I do have these:
    IMG_5100That is, four large measuring spoons, with sizes 1/4 cup, 1/3 cup, 1/2 cup, 1 cup. How can I measure out the flour I need?
  2. How can I measure out the flour I need if I’m going to use just one egg?
  3. What’s the smallest unit of flour I can measure out accurately with these measuring spoons?
  4. If I have measuring spoons of sizes 1/n1, 1/n2,…, 1/nk (OK, this is getting kind of fictional), what’s the smallest unit of flour I can measure out accurately?

We talked about the first two of these at the math workshop for 3rd-6th graders that I ran in Montclair this fall (see here). The first is about addition or multiplication of fractions (1/3 + 1/3 = 2/3 or 2 × 1/3 = 2/3). It’s easy enough as long as you take the time to figure out how much flour you actually need (which wasn’t easy for my kids at first). The second one is about subtraction, and I like it because it lets you explain the fractional equation at the heart of the problem (1/2 – 1/3 = 1/6) with something very concrete: 3 eggs – 2 eggs = 1 egg. When we did it at the workshop, that made as much sense to the kids (maybe even more) as cutting up a pie into 6 slices.

The third and fourth problems are about common denominators and least common multiples. You actually need what’s usually thought of as higher math (Euclidean algorithm) to solve the last one fully, but both of them are meant to encourage experimenting. No matter how old you are, you can get started and see how far you get. You might even discover the Euclidean algorithm on your own!

Here’s the recipe, by the way. It’s for our current favorite weekend breakfast item, Russian cheese pancakes, known as syrniki (“syr” is Russian for cheese):

IMG_5181(1)

Ingredients
1½ lbs of farmer cheese (e.g., 3 half-pound packs)
3 eggs
½ cup of flour

Mix ingredients together in a large bowl.

Cover a cutting board with flour. Take out a handful of cheese mixture and roll it in the flour on the cutting board to make a ball-shaped dumpling. Repeat until all the mixture is used up (usually makes 20 or so dumplings). If your cutting board is well-covered with flour, the dumplings shouldn’t stick to it.

Heat a skillet over medium to high heat and cover the bottom with a layer of oil. Flatten each dumpling so it’s about half an inch thick, and place in the skillet (depending on the size of the skillet, you’ll likely be able to fit between 6 and 10 dumplings at once). Cook on each side until light brown (2-4 minutes a side). Serve covered with sour cream and/or sugar. Yum!

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Thanksgiving Recipes I Liked

“When you cook from a recipe book… That’s Mathematics!”
                                                                                 – Tom Lehrer

The trouble with making a particular meal only once a year is that by the next time you make it, you forget everything you did and have to plan from scratch all over again. I fell into that trap this year and never want to again, so I’m writing down this year’s recipes while they’re still fresh in my mind. These have elements of a few different recipes I saw in the Times and in Mark Bittman’s book. The gravy recipe, which we pretty much made up on the fly, is the real winner here.

If you didn’t come here for recipes, the next one will come with a math problem, I promise.

Roasted Turkey

Ingredients

1 turkey, 12-14 lb.
3 onions
4-5 carrots
4-5 sticks of celery
Garlic
Olive oil (for seasoning)
1 oz fresh sage (amount approximate, usually sold as a package)
1 oz fresh rosemary
1 oz fresh tarragon
1 oz fresh thyme
1 pint of hard apple cider
Salt
Pepper (to be freshly ground)

Seasoning

Take half of the fresh herbs you have, rinse, and mince finely. In a bowl, mix 4-5 tbsp of olive oil with the minced herbs. Chop several cloves of garlic very finely and add to the bowl. Add about 1 tbsp salt and 1 tbsp freshly ground pepper. Mix well.

Turkey

Remove innards and wash turkey inside and out. Starting from the breast, reach under the skin with your fingers to loosen it away from the meat. Reach your hands under the skin all the way around the turkey, including the underside, as well as under the thighs and down the legs. You should be able to separate all the skin away from the meat except around the wings. (It’s OK if you have a few holes in the skin.) Then rub seasoning under the skin, all through the turkey meat. Rub any leftover seasoning over the outside of the breast.

Chop about 5 cloves of garlic coarsely and rub in the bowl in which you made your seasoning, so it gets covered with olive oil and any remaining herbs. With a sharp knife, poke holes through the turkey meat and stick a piece of oiled garlic in each hole, so it goes inside the meat.

Place the remaining fresh herbs, and an onion cut up into coarse pieces (4-8) into the turkey cavity.

Place the turkey breast up in a large roasting pan. Sprinkle liberally with salt and freshly ground pepper. Pour the hard cider into the bottom of the pan. Add water so the liquid is about 1/3 inch deep. Add the (washed) turkey innards to the bottom of the pan. Chop the two remaining onions, carrots, and celery coarsely and add to the bottom of the pan. Place a carrot chunk or two in the neck cavity to close it up.

Preheat oven to 500 degrees. Put in turkey, legs first. After 20 minutes or so, or once breast is brown, reduce to 350 degrees. Baste with liquid from the bottom of the pan every half hour or so. Roast until a meat thermometer inserted into the thickest part of the thigh reads 155-165 degrees (likely 2-3 hours total if your stove is like mine). Remove from oven, remove from roasting pan, and cool.

Turkey-Vegetable Gravy

Remove vegetables, turkey innards, and any large chunks of turkey goop from the roasting pan. Heat remaining turkey juice/cider/water mixture in the roasting pan over two burners. Depending on how much gravy you want, you can add broth (we were making cream of mushroom soup at the same time and the extra broth from the mushrooms worked well). Once mixture is boiling, add a small amount of flour to thicken, but not too much (it will get a lot thicker when you add the vegetables).

Take the vegetables that were in the bottom of the turkey pan and puree them well in a food processor. Add to gravy and stir.

Cranberry-Apple Sauce

3 packages fresh cranberries
5 Granny Smith apples (peeled)
1½ cups sugar

Fill a large pot with about 5 cups of water, and add the cranberries and water. Bring to a boil. Add sugar and stir. Simmer for about 2 hours. Let cool (it will continue to thicken as it cools).