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:

x + 450 = y

and

\frac{3}{5}  y + \frac{3}{4}\left(y - \frac{3}{5}  y \right) + x = y

…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

Basic bar diagramAll 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.”

Han’s money 1

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:

Han’s money 2

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:

Han’s money 3

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:

Han’s money 4

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.

Han’s money 5

[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:

Han’s money 6

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:

450 \div 9 = 50 \; credits.

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:


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5 thoughts on “Solving Complex Story Problems II

  1. Hi

    The bar diagrams are a good way to teach multi stage word problems. However as the difficulty level increases using bar diagrams to solve word problems would not only become cumbersome but also very chaotic. This will definitely work when we want kids to further brush up their fraction fundamentals. Also this can be used as a method to introduce kids to algebra. Using algebric equation to solve this problem will take 3 to 4 steps but when we use this approach it takes many steps. Also this is very verbose. I would use this for giving kids more practice with fractions and then gradually raise them to higher level. This is purely my understanding.

    regards
    inhome
    http://www.inhomeacademy.com

  2. Of course this is verbose — explanations always are. A student who has grown up with bar diagrams would work through this problem very quickly, since this is a relatively easy one. In fact, my son was often disgusted at problems like this one, because they offered no challenge!

    Bar diagrams need to be taught early and gradually made more complex, as I am showing in my pre-algebra problem solving series of posts. Diagrams should be used regularly as the students are learning fractions. And the true strength of bar diagrams becomes clear when students start working with ratios, when suddenly that most confusing of math monsters seems no more dreadful than a pussycat.

  3. Not “how would I teach this” but “how did I solve it” – which I try to pay attention to:

    He kept 2/5, and then he kept 1/4 of that (so 2/20, 1/10). Means he spent the other 9/10, so 50 must be one tenth.

    Seems to me that the model I use for myself is more like the bar than the algebraic method – but different from both.

  4. Changing the focus from what he spent and thinking about what remains is a very useful technique. Many times, I have slogged my way through difficult calculations, only to realize that if I had changed my perspective slightly, an approach that seemed “backward” to me would have made everything easier.

    For example, this MathCounts problem: “A large water tank loses 1% of its volume to evaporation every day. How much water is left at the end of the month?” My first reaction was to start subtracting the 1%, but that quickly turned into a nightmare. Much better to focus on the 99% of 99% of 99%… that remained.

  5. Hi,
    You and I are exploring the same seminal veins. I have developed this ideal of presenting fractions through division. It’s easy and suggests the natural attack. I have not posted because I have had problem with getting the graphic illustration into my writing. My hat’s off to you.

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