Burnt Algebra

Algebra: A Problem in Translation

[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:

  1. Translate the words into a mathematical calculation or algebraic equation.
  2. Do the calculation or solve the equation.
  3. 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.

Fact #1

“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.

Fact #2

“…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

Fact #3

“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.

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15 comments on “Algebra: A Problem in Translation

  1. I like your approach to translate the word problem…this really is the most difficult step for a student in solving word problems. I use the same technique when I teach Statistics. I have the students practice this by only giving them one sentence at a time…to read the problem completely as a first step (usually the way we were taught in school) will sometimes overwhelm a student and discourage them.

    The translation method is a wonderful approach…just remember that when the problem was written down it was translated and placed there one sentence at a time.

    Thanks for sharing your ideas!

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  2. This is a nice and simple way of explaining maths. May you please please send these explanations in pdf format. Unfortunately i do not have the money to buy you books right now. Thanks a lot.

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  3. Solving word problems usually involves translating from words into mathematical symbols, expressions, and equations. This example does that of course. But what I would quibble with is the sloppy way that the variables are defined. For example, Josh doesn’t equal Sister – $72. What would be more correct and helpful for students is to write something more lie, “Let x represent the amount of money Sister has.” Then, write “Let x – 72 equal the amount of money that Josh has.”

    The variable X, or any other variable, is not a person. It is a quantity usually associated with a unit, such as dollars, feet, inches, pounds, etc. Your “let” statements are insufficient, ungrammatical both liguistically and mathematically, and perhaps ultimately confounding for students.

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  4. I had a global “Let” statement near the beginning of the article:
    “In this case, we will use each person’s name to represent how much money that person earned.”

    My experience, both as a student and as a teacher, is that beginning algebra students resist writing “Let” statements with every fiber of their being. To me, it is not worth fighting over, so I allow a global statement like the above to suffice, even if it is only given orally. As they proceed in math, or if they take up computer programming, they will have to learn to be more precise. But at the beginning, they have enough other new concepts they need to keep track of that I think this one can slide.

    I could have been more specific on the units, however, by adding “in dollars” to my global statement. That would have been very important in a problem with mixed units.

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  5. It seems that let statements can be very confusing for beginning algebra students.

    I had a very bright student figure out the answer to a two-variable problem by guess-and-check, and then wrote up the solution as:

    Let x represent 23.
    Let y represent 4.

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  6. “Let x represent the amount of money Sister has.” Then, write “Let x – 72 equal the amount of money that Josh has.”

    I’d also argue that after you write the first sentence, you don’t need a “let” for your second statement. Even in formal algebraic form, I’d think the following is more appropriate:

    Let x represent the amount of money that Sister earned, in dollars.
    Then Josh earned x-72 dollars, and Mother earned x-93 dollars.

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  7. The global “let” with the shorthand (using names to represent the dollars earned) was clearer for me, a mathematically disadvantaged adult.

    I found this blog because I need to bolster my third-grader’s math confidence, and so was searching the Internet for teaching tools. It looks like I might just learn a few things myself finally. THANK YOU for taking the time to maintain this brilliant blog.

    I had no problem with your instruction, but did anyone else get distracted by the fact that both of those kids were earning more money than their mother? :)

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  8. I did notice that, Carolyn. It seemed strange, but then I thought of 4-H fundraisers and stuff like that, where it’s really the kids’ job to earn the money. I suppose it could have been that sort of thing, and Mom was just helping out. (Who knows what goes through the minds of textbook authors?)

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  9. I teach Algebra I and have found the translation technique to be successful overall. I have observed that one of the biggest obstacles for students is writing expressions in terms of another variable – many of my students are experiencing this for the first time (or so they tell me).

    My students are pretty eager to come up with different ways to tackle a problem – for example: “Find three consecutive even integers whose sum is 174.” Whenever I teach this type of problem, I guide the students to assign the variable to the smallest number. Inevitably, a student will ask, “Can’t we make ‘x’ the middle number?”, and I encourage them to try solving the problem that way, compare the results, and justify which approach is easier.

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  10. And which method do your students end up liking the best?
    My natural tendency is to make x the smallest number and then count up, which is probably a holdover from grade school counting. But calculation is easier when we make x the middle number, so the +1 and -1 cancel out.

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  11. In our discussions, my students tell me they think making the middle number “x” is easiest, so that you end up dividing the total by 3 (the natural way to solve the problem without using Algebra). Oddly enough though, on tests I find that they make the smallest number x.

    I really like your blog – My online graduate program required me to visit some blogs and post a comment. It was inspirational for me to see how many people are writing blogs and how much information is out there. Yesterday we had professional development, and I designed my own blog for use with my students:


    Thank you for inspiring me, and I’ll keep checking back in for more ideas!

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  12. i needed help solving the word problem. one leg of a right triangle is 2 inches longer than the other leg. the hypotenuse is 4 inches longer that the shorter leg. find the lenghts of the three sides of the triangle?

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  13. Pingback: Logic Puzzle: Imbalance Problems | Let's Play Math!

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