Third grade just wound up its exploration of single-step and multi-step word problems. These types of problems are challenging for some students. If yours is one of them, the crucial first step in any word problem is to visualize what’s happening in the “story”; I tell the children to “make a movie in their minds” of what’s happening. Then, write an equation that “translates” what’s happening into a math sentence, with an X to represent what you don’t know. We’ve been calling this a Plan.
A very simple example (much simpler than the ones we’ve solved in class): “Matt baked 18 chocolate chip cookies, and then his dog ate 12 of them. How many cookies does Matt have left?” The action starts with 18 (cookies), then 12 are taken away (eaten), and we don’t know how many are left: 18 – 12 = X. I strongly encourage the children to write underneath each number and unknown what it represents, because when word problems get complex, it can be easy to lose track of that, and then you end up with a number as an answer, but you’re not sure what the number actually means, in terms of the story. So this equation, or Plan, would look like this (see image below).
The Unknown doesn’t always appear at the end of the story, of course. “The next day, Matt baked some more cookies. His dog still managed to get to them, and she gobbled down 15! Now Matt has only 5 cookies left. How many cookies had he baked?” The equation that translates this story would look like this (see image below).
The next step, after writing down an equation (or Plan) that describes the story, is to choose a strategy and solve it. Some children might use the standard algorithm to solve 18 – 12; some might think (and write) “18 – 10 = 8, and 8 – 2 = 6.” Some might think, “Well, the difference between 12 and 18 is 6” and just show that fact on an open number line.
A Plan that starts with an unknown can seem puzzling. How can you take away 15 when you don’t know how many you’re starting with? But if you put that Plan on an open number line, look at what it shows you:
This (see image at left) shows X – 15 = 5, but it also shows 5 + 15 = X, which suggests a very easy strategy for solving it! And it makes sense, logically: the number of cookies left added to the number of cookies eaten should equal the original number of cookies.
What’s the point of writing a Plan? Why can’t children just solve the problem? Yes, the most important thing is that children be able to solve these problems, and come up with an accurate answer. But when word problems become complex and require several steps to solve, students can feel confused about what to do and how to solve them. Writing an equation that represents the story, and putting that Plan on an open number line, can often show them what they can do to solve it.
How can students tell when a problem requires more than one step to solve? When I asked them this, most of their answers
focused on the fact that there’s something you need to solve first, before you can solve the rest of the problem. We write those different steps as separate Plans, although as children gain in mathematical sophistication, we start to combine them into one equation, using parentheses to show what should be done first.
Here’s an example of a multi-step problem from class: Danny’s reading goal is to read 250 pages this week. He reads a fiction book that has 90 pages. He reads a non-fiction book that has 75 pages. How many more pages does Danny need to read before he reaches his goal?
Most students thought about this problem this way:
Step 1: 90 (fiction) + 75 (NF) = X (pages read)
Step 2: X (pages read) + Y (pages he still needs to read) = 250 (his goal)
Then they solved it. Here’s an example from one Math Notebook:
And here’s another, where the student combined the two steps into one Plan. It’s a beautiful Plan, but you’ve probably noticed that this student made an error in subtraction. This really emphasizes why it’s so important to show your thinking! If you solve something in your head, and you make an error, it’s easy to overlook it. If you put your thinking down on paper, you’re more likely to notice your mistake.
We’ll continue practicing complex word problems through the year, in class and for occasional homework, as they’re such an important skill to master.