Solving math word problems and setting up equations
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? I need to be able to break down my own problem solving process into fine details. It is these details and the different 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 and lots of information, but really it is just 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 often manifest the difficulty in this when they read through a simple word problem, and then ask, "Do I go this times this, or do I divide?", just guessing at what to do with the different numbers given in the problem.
The student needs to see the different quantities in the situation and what is the relationship between them. In other words, they need to be able to create a mental representation of the information given in the problem. They need to step out of the 5, 2, 10, 789 or any 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 in the problem 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 make a number sentence for the problem, which solves it:
Relationship: Jenny's marbles + Kenny's marbles = Total marbles
Number sentence: 7 + 5 = __
I wrote __ in the place of total marbles since that is what we don't know; that is what the problem is asking for.
All this may look overly simplified, but helping children to see this 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 sentence/equation.
Relationship: Jenny's marbles + Kenny's marbles = Total marbles
Number sentence: __ + 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 15n + 35 marbles, Kenny has 2n marbles and Penny has 12. How many does Jenny have?
The relationship between the quantities is the same, so it is solved the same way. Just the actual numbers are different.
Relationship: Jenny's marbles + Kenny's marbles + Penny's marbles = Total marbles
Equation: __ + 2n + 12 = 15n + 35
In algebra you many times use the symbol x for the unknown instead of an empty line. So we get the equation x + 2n + 12 = 15n + 35.
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 a missing addend relationship. It leads to an addition sentence where one of the addends is missing. Put an empty line for what is not known.
pages already read + pages still to read = pages total
+ =
This, of course is then solved by subtraction, but initially it is easier to set up an addition equation.
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 hours of the day, some hours already passed, some hours left. What operation do you use? The only quantity we know for this is the total hours for the day so you can put two empty lines.
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. But let's use a variable p for the hours passed to help us along.
hours left = 1/3 of hours passed
= 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
And this can be solved with algebra or by guessing, for example.
Subtraction word problems
One situation in the story problem indicating subtraction is difference or how many/much more.
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 to also understand that difference is the result of the subtraction and tells how much 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 is 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, and 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 about the mass. It's not about how many more so it's not about difference. One thing being greater than other thing implies you add. One thing being less than other thing implies you subtract. And one thing being twice something indicates multiplying by 2.
When I read this problem, I can immediately see that I can write equations from the different sentences of the story problem, but I can't see the answer right away. In my mind, I figure that after writing the equations I will see some way to solve the problem. Probably one equation is solved and gives an answer to the other equation etc.
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?
mass of Great Pyramid = mass of 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 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 (take away) hair length before cutting (equals) 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 get the equations given.
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 x 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 x time = distance AND
distance from A to B + distance from B to C = distance from A to C
