The kitchen lends itself to endless mathematical opportunities, and today I want to share an activity that you can do with your own children that illustrates the properties of fractions. Understanding fractions is a key step in a child's mathematical development. This is the last "big idea" before algebra, and it is not uncommon for difficulty in algebra to be caused by difficulty with fractions and proportional reasoning. The activity below can be adapted and extended in many ways. If you try it, or a variation, tell us about your experience!
The key idea behind this activity is called the "area model" for explaining fractions, and is especially useful in explaining multiplication and addition of fractions. Do you remember the "pie charts" that we all group up with for representing fractions? Well, it turns out that instead of pieces of pie or pizza or something else that comes in a circle, it is much better to illustrate fractions with something that comes in a rectangle, and the tastiest rectangle that I can think of is a pan of chocolate brownies!
To explore the relationship between two fractions, one third and one fourth, say, go ahead and bake a pan of brownies, if you want to get really decadent, you can use the recipe below, which is adapted slightly from the Ad Hoc at Home Cookbook by Thomas Keller. While the brownies are baking, have your child think about how they would answer the following questions:
What is 1/3-1/4?
If we had taken two pans of brownies to a potluck and had one third of one pan left and one fourth of the other, how much of one pan do we have leftover?
If I take one quarter of the pan of brownies and you take one third of what is left, who gets more?
The important thing is not that they get the right answer right away, but that they think about the relationship between the question and they brownies in front of them. Once the brownies are cooled, cut them into thirds in one direction and fourths in the other. This will give you twelve very rich brownies. Demonstrate the answer to the questions above with the brownies! Try to speak in terms of what fraction of the whole you have as well as how many brownies. Notice that if you cut the pan in to sixths instead of thirds, the number of brownies changes, but the fraction of the pan does not.
Preheat the oven to 350oF and line your rectangular pan with parchment paper. Melt half of the butter and pour it over the other half so that you have a lovely creamy blend with a some small unmelted bits. Whisk or sift together the flour, cocoa, and salt. Beat together the eggs and sugar until pale and fluffy. Alternate adding the dry ingredients and butter (1/3 of each, then 1/2 of what is left, then the remainder...another math problem,! How much in each addition?), mixing on low speed until the batter is just homogenous. Stir in the chocolate chips. Spread the batter into the prepared pan and bake until a cake tester comes out with a only a few moist crumbs. For a 9x9 pan, this is about 40 minutes.
The recipe's proportions are given below. Triple the amounts for a 9x9 pan of brownies and enjoy more kitchen math!
1 stick (1/4 lb) unsalted butter
1/4 cup flour
1/3 cup cocoa
1/3 tsp salt
1/2 cup sugar
1/2 tsp vanilla
2 oz bittersweet chocolate chips