[Bill Watterson identifies the trouble with math problems, through the eyes of Calvin and Hobbes.]
It’s time to revive and (hopefully!) finish my long-neglected series on solving word problems in elementary mathematics. I’ve been having fun making up the problems, so now I just have to write the posts. Coming up soon:
- 4th grade math inspired by The Lion, the Witch and the Wardrobe.
- 5th grade math inspired by The Hobbit.
- 6th grade math in… I haven’t decided yet. Any suggestions?
Since it has been more than two years since the last entry, however, I wanted to take a few minutes to recap our progress so far and to refer new readers back to the original posts:
- Elementary Problem Solving: The Tools
An introduction to word algebra and bar diagrams. - Penguin Math: Elementary Problem Solving 2nd Grade
Math problems from Mr. Popper’s Penguins. - Ben Franklin Math: Elementary Problem Solving 3rd Grade
Math Problems from Poor Richard.
Bar Diagrams are Visual Algebra
For the rest of the problem-solving series, I will focus on bar diagrams (also called rod models, or simply models) as taught in the Singapore Primary Math series. These diagrams are a pictorial form of algebra, with the length of each bar representing some quantity in the word problem, either known or unknown.
In creating a bar diagram of the word problem, the student must think through the same type of questions that are required in writing an algebra equation:
- What quantities are related to each other, and how are they related?
- How are the known quantities related to the unknowns?
- What is the basic unit (or variable) that I need to find?
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.
When all the parts are the same size, students can use a (number of units) x (size of units) = total diagram to represent both multiplication and division problems, because multiplication and division are also inverse operations. Because the units are all the same, we only need to write a number in the first one.
Practice Makes Perfect
It is important for students to practice applying this method to simple problems, even though they may be able to solve the problems without a diagram. As with any tool, skill with bar diagrams must develop over time, through repeated use.
The 4th and 5th grade problems coming up will require relatively complicated diagrams with multi-step solutions. If you wish to teach this method to your students, be sure to step back to simpler problems (like those in the 2nd grade and 3rd grade posts) at first.
To get more practice creating bar diagrams, your students may enjoy these online tutorials:
Denise Gaskins' Let's Play Math 
What about “Holes” by Luis Sachar for sixth graders?
The Phantom Tollbooth by Norman Juster! Love that book. Or is it too explicitly mathy?
Artemis Fowl by Eoin Colfer or The 39 Clues? The Secret Garden?
Those are great suggestions. I think I would especially enjoy doing a set of word problems from Holes. But for this article, I think I’m due for a non-fiction book. I don’t want Poor Richard to feel lonely! Right now, I am leaning toward D’Aulaire’s Book of Greek Myths. Okay, so myths aren’t exactly non-fiction — at least, not in the same way that a book about stars and planets would be — but they are history, of a sort.
It’s so much fun to adapt these problems that I may continue doing similar posts, even after I finish this series.
Did E. Nesbit have any stories with 6th grade math references? She loved to throw out math ideas into her story, sometimes quiet slyly.