I’m 27 years old, and one of my favorite books of all time is still the children’s book Math Curse. (Actually, I recommend pretty much any book from this author/illustrator combo.) The basic premise is that, inspired by a teacher’s comment, a student starts seeing everything in his life as a math problem. These problems are integrated into the text, while the solutions are printed on the back cover.
When I first discovered the book, I was kind of spooked out by how much I related to it. My dad, who was an accounting major, is really big on math. I don’t even remember how young I was when he started finding opportunities to drill me on my skills at the grocery store (How much does each egg in the carton cost?), in the car (At this speed, how many minutes will it take to get to Seattle?), and pretty much everywhere else we went. Because of this, math has always been integrated (ha!) into my everyday life.
I was lucky. With all that practice and all those applications, the leap from basic arithmetic to story problems wasn’t insurmountable for me. However, I know it wasn’t like that for everyone; I’ve seen countless students who can definitely do the arithmetic get thrown when they have to do the same calculation in the context of a story problem.
I’ve come up with a couple of basic strategies that can help, especially when working with fractions or decimals.
1) Replace the actual numbers with ones that are easier to deal with. When dealing with fractions, kids often get so thrown by the numbers that they lose sight of the situation they’re trying to solve, and it’s frequently a situation they actually do know how to handle. If you replace the fractions with integers (whole numbers), they can figure out what operation they have to do, then put in the actual numbers from the problem.
Let ‘s look at this strategy in the context of this problem:
Ron’s cake recipe requires 2/3 of a cup of flour. He has 8 cups of flour. How many cakes can he bake?
When faced with this kind of problem, kids frequently aren’t sure whether they’re supposed to multiply or divide- and if they choose to divide, they’re not sure which order to put the numbers in. However, if you present the same kids with this problem:
Ron’s cake recipe requires 2 cups of flour. He has 8 cups of flour. How many cakes can he bake?
Most of them will be able to tell you that he can bake 4 cakes. They may have to count it out- 2 cups for 1 cake, 4 cups for 2 cakes…but they’ll get there. Then ask them what operation they can do with the numbers that will get them that answer. They have to divide, so stress that order is important- they had to divide the number of cups of flour Ron had by the number of cups each cake required.
Let’s stick in the original numbers from the problem.
8 cups of flour divided by 2/3 of a cup per cake = 8 * 3/2 = 24/2 = 12. Ron can make 12 cakes.
(Why dividing by a fraction is the same as flipping the fraction and multiplying is a post for another day- and again, this usually isn’t the part of the problem students struggle with.)
2) Do a logic check: should my answer be bigger or smaller than the number I started with? When multiplying and dividing with decimals, it’s really easy to lose track of where the decimal point will ultimately end up. In this situation, putting the calculation in the context of a story problem can actually make it easier, since we can do this logic check.
Apples cost $0.40 per pound. How many pounds of apples can you buy with $7.00?
If a student isn’t sure which operation he needs to do, he can use the integer-replacement strategy to figure out that he needs to divide. Once he’s set up his division problem, it’s time to move the decimal points- but let’s do our logic check first. We’re doing $7.00 divided by $0.40. Should my answer be bigger or smaller than 7? Here’s where the context of the problem helps. I have $7.00 and apples cost less than $1.00 per pound, so I should be able to buy more than 7 pounds. My answer should be bigger than 7.
When done properly, the division looks like this:
$7.00 divided by $0.40 per pound = 700 divided by 40 = 17.5 pounds
It’s always easiest to divide by a whole number, so we multiplied the 0.40 by 100, moving the decimal point 2 places to the right. Then we had to do the same thing to the 7.00, turning it into 700. A student who moves the decimal points incorrectly could end up with something like this:
7.00 divided by 40 = 0.175
This clearly doesn’t pass our logic check.
Thinking about decimals and money reminded me of this classic. Pass it along to anyone who tries to tell you that it isn’t important to learn decimals.