How to read a word problem

We recommend reading any word problem at least three times. By the conclusion of the third reading, you should be magically whisked away to your home in Kansas.

Read it once…

…to get a general idea of what’s going on. Is the problem about money? Height and width? Distances? Money? Yeah, we already said money, but it’s important. Can you draw a picture in the margin that helps you visualize the problem?

Twice…

…to translate from English into math. Read the problem carefully this time, figuring out which pieces of information are important and which ones aren’t. Then ignore the unnecessary bits. This may sound a bit harsh, but they’re big boys. They’ll get over it.

Three times…

…a lady. Wait, that’s not it.

…to make sure you answered the right question. To be sure, read the question one last time before drawing a box around your final answer. Even if this method of double-checking saves you only once out of every 100 times, it’ll help you in the long run. One out of 100 is better than zero out of 100, but now we’re moving into some advanced mathematics.

Sample Problem

The local department store was having a sale. Holla! Gabe bought a pair of shoes for $21, although they would’ve been cheaper if he’d bought penny loafers. He also bought some shirts that were on sale for 25% off their normal retail price of $18 each. Gabe, always the bargain hunter, spent $75 total. How many shirts did he buy? Also, does he really think any of them will go with those shoes?

Read the problem:

Once…

…for a general idea of what’s going on: Gabe went shopping and spent money. Sounds like an old familiar story. Time for an intervention, friends of Gabe. Preferably before he maxes out his Diner’s Club card.

Now we need to figure out how many shirts he bought, so we read the problem:

Twice…

…to translate from English into math. Remember, we can translate a bit at a time, sort of like how Gabe pays for some of his major purchases when he has them on layaway. Sheesh, Gabe. Get a hold of yourself.

(total amount Gabe spent) = (amount he spent on shoes) + (amount he spent on shirts)

We know he spent $75 total, of which $21 was spent on shoes. This gives us the equation:

$75 = 21 + (amount he spent on shirts)

The problem has gotten smaller. It’s depressing when that happens with cake, but great when it happens to a word problem. Now all we need to do is come up with a symbolic expression for how much Gabe spent on shirts, or the cost per shirt times the number of shirts:

(amount he spent on shirts) = (cost per shirt)(number of shirts)

Since the shirts are 25% off their normal price of $18, they cost $18 – 0.25(18) = $13.50 each. What a deal, and real polyester, too!

We need to introduce a variable for the number of shirts; s will do the trick nicely. 

(amount he spent on shirts) = 13.5s

When we plug this into the earlier equation, we find that:

75 = 21 + 13.5s

Finally, we’ve reduced this thing to a super-simple-looking equation! Good riddance, vestiges of language! Begone, nouns and verbs!

Things look much nicer now, right? No worrying about shirts or shoes or prices or Gabe’s uncontrollable shopping addiction. For the moment, we can forget about the word problem and solve the equation. The answer is s = 4, by the way. In case you were interested.

Three times…

…to make sure we’re answering the right question. We want to know how many shirts Gabe bought. Is that the answer we arrived at? We found that s = 4, and s was the number of shirts Gabe bought, so we’re all done. Four new shirts for Gabe, and three of them feature a Hawaiian pattern. Gabe, if you insist on buying far more clothes than you need, can’t you at least have a decent fashion sense?

When translating from English into math, some information can be ignored. We don’t care that «the local department store was having a sale.» Gabe might, but we certainly don’t. We care about statements that tell us numbers, and statements that tell us what the question is. Any extraneous information has been placed there simply as a decoy. We’re not going to fall for that. Wait a second…two for one? We’ll grab our jacket and meet you there.

Some people find it helpful to underline the important pieces of information in a word problem. You’re like an actor highlighting in a script the lines that are important for him to remember. Unlike an actor, however, you can always look back at the original problem if you draw a blank. Also, you don’t need to wear any stage makeup.

In the problem we just did, the important bits might look something like this:

The local department store was having a sale. Gabe bought a pair of shoes for $21 and some shirts that were on sale for 25% off their normal retail price of $18 each. If Gabe spent $75 total, how many shirts did he buy?

As you practice, you’ll become better at figuring out which parts of the word problem you can ignore and which parts are important.

It’s one thing to solve a math equation when all of the numbers are given to you but with word problems, when you start adding reading to the mix, that’s when it gets especially tricky.

The simple addition of those words ramps up the difficulty (and sometimes the math anxiety) by about 100!

How can you help your students become confident word problem solvers? By teaching your students to solve word problems in a step by step, organized way, you will give them the tools they need to solve word problems in a much more effective way.

Here are the seven strategies I use to help students solve word problems.

1. Read the Entire Word Problem

Before students look for keywords and try to figure out what to do, they need to slow down a bit and read the whole word problem once (and even better, twice). This helps kids get the bigger picture to be able to understand it a little better too.

2. Think About the Word Problem

Students need to ask themselves three questions every time they are faced with a word problem. These questions will help them to set up a plan for solving the problem.

Here are the questions:

A. What exactly is the question?

What is the problem asking? Often times, curriculum writers include extra information in the problem for seemingly no good reason, except maybe to train kids to ignore that extraneous information (grrrr!). Students need to be able to stay focused, ignore those extra details, and find out what the real question is in a particular problem.

B. What do I need in order to find the answer?

Students need to narrow it down, even more, to figure out what is needed to solve the problem, whether it’s adding, subtracting, multiplying, dividing, or some combination of those. They’ll need a general idea of which information will be used (or not used) and what they’ll be doing.

This is where key words become very helpful. When students learn to recognize that certain words mean to add (like in all, altogether, combined), while others mean to subtract, multiply, or to divide, it helps them decide how to proceed a little better

Here’s a Key Words Chart I like to use for teaching word problems. The handout could be copied at a smaller size and glued into interactive math notebooks. It could be placed in math folders or in binders under the math section if your students use binders.

One year I made huge math signs (addition, subtraction, multiplication, and divide symbols) and wrote the keywords around the symbols. These served as a permanent reminder of keywords for word problems in the classroom.

If you’d like to download this FREE Key Words handout, click here:

C. What information do I already have?

This is where students will focus in on the numbers which will be used to solve the problem.

3. Write on the Word Problem

This step reinforces the thinking which took place in step number two. Students use a pencil or colored pencils to notate information on worksheets (not books of course, unless they’re consumable). There are lots of ways to do this, but here’s what I like to do:

• Circle any numbers you’ll use.

• Lightly cross out any information you don’t need.

• Underline the phrase or sentence which tells exactly what you’ll need to find.

4. Draw a Simple Picture and Label It

Drawing pictures using simple shapes like squares, circles, and rectangles help students visualize problems. Adding numbers or names as labels help too.

For example, if the word problem says that there were five boxes and each box had 4 apples in it, kids can draw five squares with the number four in each square. Instantly, kids can see the answer so much more easily!

5. Estimate the Answer Before Solving

Having a general idea of a ballpark answer for the problem lets students know if their actual answer is reasonable or not. This quick, rough estimate is a good math habit to get into. It helps students really think about their answer’s accuracy when the problem is finally solved.

6. Check Your Work When Done

This strategy goes along with the fifth strategy. One of the phrases I constantly use during math time is, Is your answer reasonable? I want students to do more than to be number crunchers but to really think about what those numbers mean.

Also, when students get into the habit of checking work, they are more apt to catch careless mistakes, which are often the root of incorrect answers.

7. Practice Word Problems Often

Just like it takes practice to learn to play the clarinet, to dribble a ball in soccer, and to draw realistically, it takes practice to become a master word problem solver.

When students practice word problems, often several things happen. Word problems become less scary (no, really).

They start to notice similarities in types of problems and are able to more quickly understand how to solve them. They will gain confidence even when dealing with new types of word problems, knowing that they have successfully solved many word problems in the past.

If you’re looking for some word problem task cards, I have quite a few of them for 3rd – 5th graders.

This 3rd Grade Math Task Cards Bundle has word problems in almost every one of its 30 task card sets.

There are also specific sets that are dedicated to word problems and two-step word problems too. I love these because there’s a task card set for every standard.

CLICK HERE to take a look at 3rd grade:

This 4th Grade Math Task Cards Bundle also has lots of word problems in almost every single of its 30 task card sets. These cards are perfect for centers, whole class, and for one on one.

CLICK HERE to see 4th grade:

This 5th Grade Math Task Cards Bundle is also loaded with word problems to give your students focused practice.

CLICK HERE to take a look at 5th grade:

Want to try a FREE set of math task cards to see what you think?

3rd Grade: Rounding Whole Numbers Task Cards

4th Grade: Convert Fractions and Decimals Task Cards

5th Grade: Read, Write, and Compare Decimals Task Cards

Thanks so much for stopping by!

When most people hear “word problems,” they often think of the popular example of trains traveling at different speeds, or unrealistic applications of math. However, word problems represent how most people use math in everyday life, and the SAT includes these problems to test students’ ability to reason logically and solve problems.

For many students, the hardest part of a word problem is figuring out exactly what the word problem is asking of you. Not all word problems are created the same though, so we’re going to break down what you need to know so that you feel confident solving them on the SAT.

Types of Math Word Problems on the SAT

You could divide math word problem in a few different ways. You could do it by topic, for example, such as using the SAT subscores as your guide. Or you could do it by the type of solution the problem requires of you. We’re going to focus on the second way, simply because trying to divide word problems by topic would mean we’d have to cover virtually every math topic the SAT covers, which is way more than we could fit in a single post.

Creating an equation or a formula. In these problems, you’ll be given some information about a scenario and asked to come up with an equation or formula that represents that scenario. In fact, looking for the word “represents” in the question might tip you off that you’re dealing with a problem of this type. These are almost exclusively found in the multiple-choice questions, with different equations or formulas listed as the answer choices.

Solving for a value. This is the most common form of word problem. Based on the information in the word problem, you’ll need to come up with at least one specific value that satisfies the requirements of the problem. Examples include finding “how much” or “how many” of something in the problem, or finding minimums and maximum values.

Defining or interpreting. For these problems, you are often asked to interpret what a variable or a constant means in the situation described, or you might be asked what kinds of conclusions can be drawn from a survey. These types of problems often have no “math” involved, and instead ask you to think logically about a situation.

Tip: Being able to identify problems by type on the SAT is a useful prep exercise and can help you on the day you take the test. You may want to use the Math section from a free practice SAT test and circle all of the word problems. Then, read each word problem and assign it to one of the types. Doing this will help you see the commonalities between different types of questions and will allow you to spend less time on the actual test wondering what type of question you’re dealing with!

How to Approach SAT Math Word Problems

For every SAT Math word problem, there are two things to consider. The first is what we discussed above, which is to figure out what type of problem you’re dealing with to best identify what you need to do. Once you know whether you need to provide a definition, write an equation, or solve for a value, you’ve taken the first step.

The next step will be to do the math that you’ve identified you need to do based on the type of word problem. Let’s take a look at each one:

Write an equation/formula. For problems of this type and for problems that ask you to solve for a specific value, you will often need to understand how to “translate” a word problem into an equation, diagram, or graph that you can work with. Luckily, the SAT is very particular about how it uses words in its word problems, so you can always be sure that certain words are meant to relate to specific mathematical concepts. Here are some of the common phrases used by test makers to clue you in to what kind of math you’re looking for:

Word/phrase	Mathematical meaning/symbol
Is, will be, is equal to, is the same as	Equals, =
Plus, sum, increased by, added to, received	Addition, +
Minus, fewer, difference, decreased by	Subtraction, –
Times, product, multiplied by, of	Multiplication, x
Divided by, quotient, per, for	Division, ÷
More than, greater than	Inequality, >
At least, minimum	Inequality, ≥
Fewer than, less than	Inequality, <
At most, maximum	Inequality, ≤
What, how many	Variable, x (or any other letter)

Solve for a value. With these problems, you might be given an equation as part of your information, but most likely you have to write an equation or formula and solve for the answer. You may benefit from drawing a diagram on some of these questions, especially if the word problem refers to geometric shapes but doesn’t already include one. This can help you visualize the information you’re looking for (the measure of an angle, for example) and help you focus on the right concepts to find the solution.

When translating a word problem into a diagram, make sure you follow the set-up as described in the word problem to the best of your ability, and don’t be fooled by your own diagram. For example, a problem may mention a right triangle, so you draw one. You may end up drawing a diagram that looks like a 45-45-90 triangle, but the problem is actually dealing with a 30-60-90 triangle. By labeling all of the information on your diagram and making a mental note that your drawings aren’t to scale, you can quickly make sense of most problems.

Defining and interpreting. With these types of problems, you often have the equation or formula given to you, and you just need to explain how a part of it works. If you’re not sure, consider whether you’re being asked about a constant or a variable, and eliminate any choices that don’t correspond to the type of value you’re looking at. You can also test the equation with different numbers to see how the value in question influences the equation.

Some problems involve your knowledge of statistical topics, such as understanding how to draw conclusions from surveys, or what factors affect standard deviation. For many of these problems, you’ll need to review these concepts in order to form the correct conclusion.

Examples of SAT Math Word Problems

The following example problems come from the College Board’s free SAT practice tests.

Example of Creating an Equation/Formula

How to solve: We can tell we need to come up with our own formula based on the answer choices and the word “represents” in the problem’s question. We need to find the formula that represents what the original price of a laptop was in terms of p, what Alma paid for it.

Let’s make x the original price of the laptop. In this case, x would be multiplied by .8 to get to the sales price (representing the 20% discount). The sales prices would then be multiplied by 1.08 to account for sales tax. This means that p=(1.08)(0.8)(x). We need to isolate x, so we need to divide both sides by 1.08 and 0.8. That gives us D as the correct answer.

Alternate method: Many problems requiring you to create your own equation are not too difficult, but they can get tricky when there are several steps involved or several variables. You can substitute a number for the variable you’re solving for to find the solution that way. For example, you can set $100 to be the original price of the laptop. Multiply it by .8 to find the sales price, and then by 1.08 to find the price that Alma paid. Then plug the price Alma paid into the answer choices to find the correct answer.

Example of Solving for a Value

How to solve: The word “maximum” tips us off that we are looking for a specific value and that we’ll be dealing with an inequality (see above). Given the information in the problem, we can write this inequality to start: (weight of driver) + (weight of truck) + (14x, or the weight of the boxes) ≤ 6000.

Because we’re told that the weight of the driver and the truck equals 4,500, we can replace that part of the equation: 4500 + 14x ≤ 6000. Now, all we need to do is solve for x. Subtract 4500 from both sides to get 14x ≤ 1500. Then divide both sides by 14 to get x ≤ 107.14… The answer is thus 107.

Note: Pay close attention to how you need to round when working with inequalities. Although this particular decimal probably wouldn’t have made you think to round up, if the result of the division had been 107.9999 you should still round down in this case because the number you’re looking for needs to be less than that value.

Example of Defining/Interpreting

How to solve: Again, the use of the word “interpretation” lets us know what type of problem we’re dealing with. In this case, 12 is a coefficient for the variables n and h, which are the number of landscapers and the number of hours respectively. We can thus rule out choices B and D, since those answers define 12 as a constant in place of these variables.

Since this equation is used to calculate the price of a job, we can be sure 12 represents a dollar amount as reflected in choices A and C. To figure out how things are affected, we can substitute simple numbers for n and h. When both are equal to 1 (there is one landscaper who works for one hour), then 12nh=12(1)(1)=12. When we increase both of these numbers to two, then 12nh=12(2)(2)=48. We can thus rule out C, because the price increased by more than 12 when we added another hour. Choice A is the best interpretation.

How to solve: This is an interpretation problem dealing with statistical concepts rather than interpreting an equation. To be able to “solve” you’d need to know what factors contribute to a reliable conclusion and see what was missing from this survey set-up. In this case, the correct answer is D.

In order for reliable conclusions to be drawn, a sample must be selected at random from the population to ensure that the sample is representative of the population, which in this case is the entire town. By choosing people at the same restaurant on one day, the researcher is possibly biasing their results by narrowing their sample to “people who like this one restaurant.” Because the sample is not representative, choice D identifies the factor that makes the conclusion unreliable. The other choices are factors that, given what we know from the problem, don’t negatively impact the conclusion.

SAT Math Word Problems Tips

As you may have noticed in the last example, you’ll still need to be familiar with math concepts and understand how they might show up in different situations to really feel confident taking any problem on. Here are a couple of other tips worth mentioning:

For most problems, there is more than one way to solve it. We showed this with the first example, where we provided an alternate method for solving a problem. If you find yourself stumped on a problem (we’ve all been there!), then move on to other problems and return to the one that tripped you up later. You may find that you see what you need to do more clearly.

Read carefully. The SAT test makers put a lot of care into how they write math word problems. Be sure to carefully read the question so that you don’t misread something and solve for the wrong thing. (The test makers may even include the answers that result from these common misreadings in the choices!). It’s difficult to read a question correctly once you’ve misread it. If you’re a fast reader who tends to skip over things, try mouthing the words silently as you read. It will force you to slow down just enough that you won’t miss anything, but not so much that you won’t have time to finish the test.

In general, preparing for the SAT Math section requires a multi-faceted approach. You need to be able to identify the types of questions you’re dealing with, know mathematical concepts and how to use them, and deal with other testing issues like rushing through the end or dealing with test anxiety. To help you get started, check out our free SAT Guide that includes 8 of our best test prep tips!

For more information about preparing for the SAT, check out these posts below:

30 SAT Math Formulas You Need to Know

25 Tips and Tricks for the SAT

5 Common SAT Math Mistakes to Avoid

Want to know how your SAT score/ACT score impacts your chances of acceptance to your dream schools? Our free Chancing Engine will not only help you predict your odds, but also let you know how you stack up against other applicants, and which aspects of your profile to improve. Sign up for your free CollegeVine account today to gain access to our Chancing Engine and get a jumpstart on your college strategy!

A Picture is Worth a Thousand Words

“Today, we are going to solve math word problems.” When students hear this from their math teacher, their faces drop, sweat starts to form on their foreheads and they refuse to make eye contact. As a math teacher, I understand. I understand the anxiety and want to make the learning process with word problems more enjoyable. This is where reading, writing and, math collide and the real world of math begins. Let’s talk about how to solve word problems with pictures.

When students struggle with word problems in school, they also have difficulty tackling them for homework. Most students want the “one-size-fits-all” formula for word problems but unfortunately, that does not exist. However, drawing a picture will help students visualize the problem and will start advancing their learning stage from the concrete to the abstract. Let’s take a look at how to solve a word problem using pictures:

Steps for Solving Word Problems using Pictures

1. Read the entire problem: Get all the facts – Underline key word
2. Answer the question: What am I looking for?
3. Draw a picture or diagram: Visualize as a real world situation
4. Solve the problem: Set up the equation and solve
5. Check your solution: Is this answer reasonable?

Word Problem Examples – Example A

Cody has 6 pencils on his desk, Jonah has 4 more than Cody and Vinny has three less pencils than Jonah. How many pencils are there in all?

• Read the entire problem  √
• What am I looking for? How many pencils do Cody, Jonah and Vinny have altogether?  √
• Draw a picture or diagram √

• Solve the problem    √

      6 + 10 + 7 = 23 pencils

• Check your solution  √

      This answer is reasonable

Example B

There are 3 fish tanks labeled   X, Y, and Z. Y weighs 6 times as much as X and twice as much as Z. If Z is 36 lbs. heavier than X, find the total weight of X, Y and Z.

• Read the entire problem  √
• What am I looking for?  What is thetotal weight of fish tanks X, Y and Z?  √
• Draw a picture or diagram √

• Solve the problem    √

18 + 108 + 54 = 180 pounds

• Check your solution  √

      This answer is reasonable

Visualizing a word problem with pictures is a strategy that will help motivate many students to begin the process of solving them.  This works especially for students that may become bored by the excess amount of words instead of numbers or for those students who become overwhelmed by the information and want to break it down into a simpler form.   Another benefit of using pictures when solving word problems involves communicating the results.  The pictures act as justification for answers, make the problems easier to understand and therefore, secure the learning process.

Meet our Guest Blogger: Jan Rowe

Jan is one of Educational Connections’ top tutors. She has twelve years of classroom teaching experience and holds a Virginia and Florida teaching license in middle school mathematics and elementary education. Jan has been with Educational Connections for over a year, working with over twenty students. Her tutoring goal is to help each student understand their learning style so they can improve the speed and quality of that learning. When she is not tutoring or teaching, Jan loves to play scrabble, go hiking and play Frisbee with her new puppy.

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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 51 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	=	51

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.

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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