Solving Complex Story Problems II

[Oops! I found one more post from my old blog. It apparently slipped off the back of my metaphorical desk and has been sitting with the dust bunnies.]

Here is a math problem in honor of one of our family’s favorite movies…

Han Solo was doing some needed maintenance on the Millennium Falcon. He spent 3/5 of his money upgrading the hyperspace motivator. He spent 3/4 of the remainder to install a new blaster cannon. If he spent 450 credits altogether, how much money did he have left?

[Modified from a word problem in Singapore Primary Math 5B. Stop and think about how you would solve it before reading further.]

Algebra: Substitute and Solve for x?

How can we teach our students to solve complex, multi-step word problems? Depending on how one counts, our problem would take four or five steps to solve.

One might approach it with algebra, writing a two-equation system like:

and

…and then simplify the equations and solve for x.

Of course, elementary students have not learned algebra yet, and many adults have forgotten most of what we ever knew. Instead, Singapore Primary Math teaches students to draw pictures which I call bar diagrams. These pictures model the word problem in a way that makes the solution appear almost like magic.

It is a trick well worth learning, no matter which math program you use.

The Whole Is the Sum of its Parts

All bar diagrams descend from one very basic diagram showing the inverse relationship between addition and subtraction. The whole is the sum of its parts. If you know the value of both parts, you can add them up to get the whole. If you know the whole total and one of the parts, you subtract the part you know in order to find the other part.

As word problems get more complex, the whole may be split into more than two parts. Also, the parts may be related to each other in ways that require a more involved bar diagram. No matter how complicated the story, however, one usually begins by drawing a bar to represent one whole thing and then dividing it into parts.

Han Solo’s Spaceship Repairs

We start with a bar representing all the money Han Solo had to start with. If I am working with students who are new to bar diagrams, I tell them, “Imagine all the money spread out in a row on the table.”

The first fact we are given is that 3/5 of the money went to upgrade the hyperspace motivator. This is easy to show by dividing the bar into five sections and marking three of them as spent:

A Part Becomes a “Whole”

Next we are told that Han used 3/4 of the remainder to install a blaster cannon. The words “of the remainder” are easy to overlook, so be careful! These words indicate that we have a new “whole thing.” The money Han has left is now going to be treated as if it were the only money in the story.

The easiest way to show this is to draw a new bar below the original:

On this new bar, 3/4 went for the blaster cannon, and the rest is still in Han’s pocket. We show this by dividing the bar into four pieces and mark three of the pieces as used:

Simplify to a Single Unknown Unit

We are almost done. The part of our “remaining money” bar that is left unmarked stands for the money Han has at the end of the story, which is exactly what we are trying to find. We know that all the money Han spent adds up to 450 credits. There are six sections (called “units”) of spent money in our diagram, but we have a problem:

• The units are not the same size.

The units on the original bar are much larger than the units on the “remaining money” bar. If the units were all the same size, we would be able to divide the total amount spent by the number of units, and that would tell us how many credits were in each unit. Therefore, we must ask ourselves:

• Is there any way we can adjust our diagram to make the units the same size?

Can you see that each small unit is exactly 1/2 the size of the large, original units? That means we could divide each of those original units in half, and then they would match the small units on the second bar. This is the bar diagram equivalent of finding a common denominator for fractions.

[Note: An experienced student would probably have noticed the relationship between the units before drawing the “remaining money” bar. We could have divided the larger units in half at that point, making it possible to mark off the money spent for the blaster cannon without drawing a second bar. There is often more than one way to work through a story problem diagram.]

The Arithmetic Is Easy

In all, nine units of money were spent, which made a total of 450 credits. If we merge our two bars back into one, this is easy to see:

As soon as we can connect a unit (or a set of same-size units) with a number, the difficult task of thinking through the problem is almost over. From here on, all we need is simple arithmetic. Nine units are 450 credits. That means one unit must be:

And one unit is all the money left after finishing the repairs. So the answer to our story problem is: Han has only 50 credits to his name.

For Further Study

If you would like to teach bar diagrams to your students, you may want to explore the Thinking Blocks website. These interactive lessons offer plenty of story problems for online practice:

• Thinking Blocks Addition and Subtraction Word Problems
• Thinking Blocks Multiplication and Division Word Problems

Update: My New Book

You can help prevent math anxiety by giving your children the mental tools they need to conquer the toughest story problems.

Read an expanded explanation of the Han Solo word problem—and many more!—in Word Problems from Literature: An Introduction to Bar Model Diagrams. And there’s a paperback Student Workbook, too. Now available at Rainbow Resource Center and other online bookstores.