[Photo by *Irish.]
In my post Elementary Problem Solving: The Tools, I introduced word algebra as a way to help students think their way through a story problem. In the next two posts, I showed how the tool worked with simple word problems.
Now, before I move on to focus exclusively on bar diagrams, I would like to show how word algebra can help a student solve a typical first-year algebra puzzle.
A homeschooling friend who avoided algebra in high school, trying to help her son cope with a subject she never understood, posted: “Help! Our answer is different from the book’s.” Here is the homework problem:
Josh earned $72 less than his sister who earned $93 more than her mom. If they earned a total of $504, how much did Josh earn?
First Step: Translation
In the process of solving a word problem, the student must work his way through three steps:
- Translate the words into a mathematical calculation or algebraic equation.
- Do the calculation or solve the equation.
- Interpret the resulting number in the context of the original problem.
When a student struggles with solving problems, most of the time it is step one that gives him trouble. He does not know how to translate the problem from English into “mathish.” If we want to help our students with their math problems, we need to teach them how to do this sort of translation.
In my friend’s problem, there are three facts that need to be translated. Let’s begin by separating these and isolating them from the rest of the words, so we know where to focus our attention.
“Josh earned $72 less than his sister…”
In word algebra, we use words from our problem to stand for whatever amounts we don’t know. In this case, we will use each person’s name to represent how much money that person earned. So this statement translates into math as follows:
Josh = Sister – $72
Sister = Josh + $72
If you are not sure whether to add or subtract the $72, stop to think: “Who had more money?” Then read the statement again, leaving out the number: “Josh earned less than his sister.” So we have to subtract from the sister’s amount to make it equal Josh’s total, or we need to add to Josh’s amount to find how much the sister had.
By the way, it is a good idea to practice putting the words into math both ways, with addition AND with subtraction. They both say the same thing, but sometimes one form will be more useful to you in solving your problem, and other times the other form will be more helpful.
“…his sister who earned $93 more than her mom.”
Again, we think, “Who had more money?” And again, it’s the sister who earned more. So this fact translates into word algebra as:
Sister = Mom + $93
Mom = Sister – $93
“If they earned a total of $504…”
You can ignore the “if” and just say they DID earn that much. All that the “if” in a math problem means is that, for this problem, we are going to assume the following statement is true. So…
“…they earned a total of $504…”
It translates into word algebra this way:
Josh + Sister + Mom = $504
What Are We Looking For?
“…how much did Josh earn?”
The final part of the word problem is the question. What did the problem ask us to find? It asked for Josh’s amount of money.
Now that we have translated the whole problem into word algebra, we are ready to start solving for the answer. In summary, let me list all the things we know, dropping the dollar signs so it looks more mathy:
Josh = Sister – 72
Sister = Josh + 72
Sister = Mom + 93
Mom = Sister – 93
Josh + Sister + Mom = 504
We have two options for our next step, and both involve a fundamental rule for solving algebra problems:
You cannot solve for two unknown numbers at once, so you have to say everything in terms of ONE unknown amount. That one unknown becomes the X that you will solve for.
Your unknown doesn’t have to be called X, but whatever you call it, it still marks the treasure, the solution to your problem.
First, Find Sister
Since Josh’s money is related to Sister’s amount, if we could find Sister’s amount, then we would know what Josh had. That is the easiest approach, because we have two equations like this:
Josh = (something about Sister)
Mom = (something about Sister)
So if we call the amount that Sister earned X:
Sister = X
Josh = X – 72
Mom = X – 93
(X – 72) + X + (X – 93) = 504
3X – 165 = 504
3X = 669
X = 223
If you do this, remember that you are not done with the problem when you find X. You still have to use X to find the amount of money Josh earned:
Josh = X – 72 = 223 – 72 = 151
Which means that Josh earned $151.
Or Go Directly to Josh
An alternate approach would be to solve directly for the amount we want to find, the money Josh earned. This method will take a little bit longer, but when we finally get to the “X=?” part, we will be done. We don’t have to remember to go back and do that extra step.
This time, let Josh be the X — except that we will use Y, because we just used X in the “Find Sister” section above, and we don’t want to get ourselves confused.
Remember that you can use ANY letter to name your unknown amount.
Josh = Y
Sister = Y + 72
Mom = Sister – 93
Mom = (Y + 72) – 93
Mom = Y – 21
Y + (Y + 72) + (Y – 21) = 504
3Y + 51 = 504
3Y = 453
Y = 151
So again, Josh earned $151.
In the end, it is “real” algebra that solves the problem, the same equation manipulations that the student has been using throughout his algebra 1 textbook. But algebra 1 word problems are primarily an exercise in translation, and the intermediate step of word algebra can help a confused student make the necessary translation from the words of his story problem to the equations he knows how to solve.