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# How to set up algebraic equations to match word problems

Students often have problems setting up an equation for a word problem in algebra. To do that, they need to see the RELATIONSHIP between the different quantities in the problem. This article explains some of those relationships.

I was asked,

I need an easy and helpful way to teach writing equations.

Example: Helen has 2 inches of hair cut off each time she goes to the hair salon. If h equals the length of hair before she cuts it and c equals the length of hair after she cuts it, which equation would you use to find the length of Helen's hair after she visit the hair salon?

a) h = 2 − c       c) c = h − 2
b) c = 2 − h       d) h = c − 2

Is there a single method to teaching students how to write algebraic equations? I need help.

The first thing I do when trying to figure out how to teach something is to analyze my own thinking. How do I think when solving this problem? What are the steps and fine details? It is these details and steps that I may do automatically that I need to explain to students to help them.

## Seeing the quantities and their relationship instead of numbers

In this problem there is seemingly lots of information, but really it's about recognizing quantities and the simple relationship between them. This is of course the exact same task as translating a situation explained in words into a mathematical expression using symbols.

Children manifest the difficulty in this task when they read a simple word problem and then ask, "Do I go this times this, or do I divide?", just guessing the operation to perform with the different numbers given in the problem.

Students need to see the quantities and the RELATIONSHIP between them. They need to step out of the 5, 2, 10, 789, or any others numbers in the problem, and see the general quantities involved and how those are related to each other. In very simple word problems that relationship usually involves just one of the four basic operations. Then in algebra, there may be more quantities and more operations between them.

## Examples of addition word problems

Example. Jenny has 7 marbles and Kenny has 5. How many do they have together?

The keyword together tells us that ADDITION is probably the operation needed. The quantities here are Jenny's marbles, Kenny's marbles, and total marbles. The relationship between the three is

Jenny's marbles  +  Kenny's marbles  =  Total marbles

From this general relationship between the quantities it is easy to write an equation for the problem, which solves it:

 Relationship: Jenny's marbles + Kenny's marbles = Total marbles Equation: 7 + 5 = _____

I wrote ____ in the place of total marbles since that is what the problem is asking for (the unknown).

All of this may look oversimplified, but helping children to see the underlying relationship between the quantities is important. Consider now this problem:

Example: Jenny and Kenny together have 37 marbles, and Kenny has 15. How many does Jenny have?

Many teachers might try to explain this as a subtraction problem, but in the most fundamental level it is about addition! It still talks about two people having certain amount of marbles together. The relationship between the quantities is the SAME as above, so we still need to write an addition equation.

 Relationship: Jenny's marbles + Kenny's marbles = Total marbles Equation: _____ + 15 = 37

Then, we can solve the equation ____ + 15 = 37  by subtracting. Using this kind of approach in the elementary grades will help children to set up equations in algebra story problems later.

Example: Jenny, Kenny, and Penny together have 50 marbles. Kenny has double as many marbles as Jenny has, and Penny has 12. How many does Jenny have?

The relationship between the quantities is the same, so it is solved the same way: by writing an addition equation. However, we need to denote the number of Jenny's and Kenny's marbles with something. Jenny's marbles are unknown, so we can denote that with the variable n. Then Kenny has 2n marbles.

 Relationship: Jenny's marbles + Kenny's marbles + Penny's marbles = Total marbles Equation: n + 2n + 12 = 50

Example: Jane is on page 79 of her book. The book has 254 pages. How many pages does she still have to read?

This time the word "still" clues us in to an additive relationship where one of the addends is missing. You can initially write an empty line for what is not known, and later replace that with a variable.

 pages already read + pages still to read = pages total + =

This equation is of course is then solved by subtraction, but it is better if you view it as an addition situation and write an addition equation for it.

Example:  The number of hours that were left in the day was one-third of the number of hours already passed. How many hours were left in the day?
(From Grade 5 word problems for kids)

Can you see the general principle governing this problem?  It talks about the hours of the day where some hours already passed and some hours left. This, of course, points to addition once again: we have one part of the day, another part, and a total.

The only quantity we know is the total hours for the day. We don't know the hours already passed nor the hours left, so initially you can use two empty lines in the equation that shows the basic relationship between the quantities:

 hours already passed + hours left = total hours =

Then, the information in the first sentence gives us another relationship:

"The number of hours that were left in the day was one-third of the number of hours already passed."

We don't know the amount of hours passed nor the hours left. So let's use the variable p for the hours passed. Then we can write an expression involving p for the hours left, because "hours left is one-third of the hours passed," or

hours left  =  1/3 p

Then writing 1/3p for the "hours left" in the first equation will give us:

 hours already passed + hours left = total hours p + 1/3p = 24

This can be solved using basic algebra or by guess & check.

## Subtraction word problems

One situation that indicates subtraction is difference or  how many/much more. However, the presence of the word "more" can indicate either an addition or subtraction, so be careful.

Example:  Ted read 17 pages today, and Fred read 28. How many more pages did Fred read?

The solution is of course 28 − 17 = 11, but it's not enough to simply announce that – children need also to understand that difference is the result of a subtraction and tells the answer to how many more.

 Relationship: Pages Fred read − pages Ted read = difference Equation: 28 − 17 = __

Example:  Greg has 17 more marbles than Jack. Jack has 15. How many does Greg have?

Here the word more has a different meaning. This problem is not about the difference. The question asks how many does Greg have – not what is the difference in the amounts of marbles. It simply states Greg has 17 more compared to Jack, so here the word more simply indicates addition: Greg has as many as Jack AND 17 more, so Greg has 15 + 17 marbles.

Example: The mass of the Great Pyramid is 557t greater than that of the Leaning Tower of Pisa. Stone Henge has a mass of 2695t which is 95t less than the Leaning Tower of Pisa. There once was a Greater Pyramid which had a mass twice that of the Great Pyramid. What was the mass of the Greater Pyramid?
(From Grade 5 word problems for kids)

Each of the first three sentences give information that can be translated into an equation. The question is not about how many more so it's not about difference. One thing being greater than another implies you add. One thing being less than another implies you subtract. And one thing being twice something indicates multiplying by 2.

When I read this problem, I could immediately see that I could write equations from the different sentences in the problem, but I couldn't see the answer right away. I figured that after writing the equations I would see some way forwad; probably one equation is solved and gives an answer to another equation.

The first sentence says, "The mass of the Great Pyramid is 557t greater than that of the Leaning Tower of Pisa". What are the quantities and the relationship between them here?

 mass of Great Pyramid = mass of the Leaning Tower of Pisa + 557t

The second sentence says "Stonehenge has a mass of 2695t which is 95t less than the Leaning Tower of Pisa."  Here it gives you a relationship similar to the one above, and it actually spells out the mass of Stonehenge. It's like two separate pieces of information: "Stonehenge weighs 95t less than the tower.  Stonehenge weighs 2695t."  Less means you subtract. If you have trouble deciding which is subtracted from which, you can think in your mind which is heavier: Stonehenge or the tower?

 either mass of Stonehenge = mass of tower − 95t or mass of tower = mass of Stonehenge − 95t

Now since the mass of Stonehenge is given, you can solve this equation, and from that knowledge you can solve the first equation, and from that go on to the mass of the "Greater Pyramid".

If the teacher just jumps directly to the number sentences when solving word problems, then the students won't see the step that happens in the mind before that. The quantities and the relationship between them have to be made clear and written down before fiddling with the actual numbers. Finding this relationship should be the most important part of the word problems.  One could even omit the actual calculations and concentrate just finding the quantities and relationships.

## Problem of Helen's hair length

Problem.  Helen has 2 inches of hair cut off each time she goes to the hair salon. If h equals the length of hair before she cuts it and c equals the length of hair after she cuts it, which equation would you use to find the length of Helen's hair after she visit the hair salon?
a. h = 2 − c      c. c = h − 2
b. c = 2 − h      d. h = c − 2

Solution.  Ignoring the letters c and h for now, what are the quantities?  What principle or relationship is there between them? Which possibility of the ones listed below is right?  Which do you take away from which?

 1 cut hair − hair length before cutting = hair length after cutting 2 cut hair − hair length after cutting = hair length before cutting 3 hair length before cutting − cut hair = hair length after cutting 4 hair length after cutting − cut hair = hair length before cutting

SIMPLE, isn't it??  In the original problem, the equations are given with the help of h and c instead of the long phrases "hair length before cutting" and "hair length after cutting". You can substitute the c, h, and 2 into the relationships above, and then match the equations (1) - (4) with the equations (a) through (d).

## Helping students to write the algebraic equations

One idea that came to mind is to go through the examples above, and more, based on the typical word problems in the math books, and then turn the whole thing around and have students do exercises such as:

• Write 3 different story problems where the solution is based on the relationship

money earned − money spent on this - money spent on that = money left

• Write 3 different story problems where the solution is based on the relationship

original price − discount percent x original price = discounted price

• Write 3 different story problems where the solution is based on the relationships

money earned each month − expenses/taxes each month = money to use each month AND

money to use each month × number of months = money to use over a period of time

• Write 3 different story problems where the solution is based on the relationships

speed × time = distance AND

distance from A to B + distance from B to C = distance from A to C

I'm sure you can think of more similar exercises.

See also:

Why are math word problems SO difficult for elementary school children?
Hint: it has to do with a "recipe" that many math lessons follow.

The do's and don'ts of teaching problem solving in math
General advice on how you can teach problem solving in elementary, middle, and high school math.

How I Teach Word Problems by Andre Toom (PDF)
This article is written by a Russian who immigrated to US and noticed how COLLEGE LEVEL students have difficulties even with the simplest word problems! He describes his ideas on how to fill in the gap formed when students haven't learned how to solve word problems in earlier education.

A list of websites focusing on word problems and problem solving
Use these sites to find good word problems to solve. Most are free!

## Comments

When solving word problems, students must first decide what quantity represents x and then must write all the other quuantities in terms of x. I teach the students to set up arrows according to the language in the problem. All arrows point to x. Example. Harry had 10 less toys than Mark. Sue has twice as many toys as Harry. Set up arrows: Sue--- Harry---Mark Therefore Mark is x, Harry is x-10 and Sue is 2(x-10). Students find this very helpful.

Sandy Denny
My idea is the math teacher might teach and understand the students at the same time and everyone would have a sense of humor. So I think she/he will know if the students are listening or not, when after the class, talk to the student and ask what is wrong. Don't hurt the student's feelings.

lorence

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