Organizing a Word Problem

Date: 03/21/2000 at 13:28:00
From: G Wimbley
Subject: Re: Math - subtracting/trading/borrowing

How can I get my 3rd grade son to understand and know where to start 
on word problems? Sometimes he doesn't know whether to add, subtract 
or multiply. For example:

Backstreet Boys tickets are on sale. Front-row tickets cost $35, 
center stage $25, balcony $15, T-shirts $10, and posters $8. John 
bought 2 front-row tickets for himself and his brother, 2 T-shirts, 
and 2 posters. How much did John spend?

How can I get him to understand where to start and how to figure out 
what to do first? Do you have any sample word problems and 
explanations of how to do them?

Date: 03/21/2000 at 16:57:17
From: Doctor Peterson
Subject: Re: Math - subtracting/trading/borrowing

Hi, Gloria.

Our FAQ includes a section on Word Problems, with some general hints 
and links to other places to look for examples:

  http://mathforum.org/dr.math/faq/faq.word.problems.html   

The archives on Elementary Word Problems may also be useful:

  http://mathforum.org/dr.math/tocs/wordproblem.elem.html   

Some people like to give lists of keywords to look for, like "each" 
which suggests you'll want to multiply, or "less than" which suggests 
subtraction. I prefer to focus not on words but on concepts; if a 
child knows well what each operation does, and can visualize what is 
being done in the problem, it should be easy enough (though sometimes 
tricky) to match up the right operation with each part of the problem. 
If he is not good at visualizing and imagining (or maybe even if he 
is), he might want to draw or act out the situation; if he has 
trouble organizing his thoughts, he might want to use a table or chart 
to list what he knows and see how it fits together. Whatever fits his 
own style of thinking will be best for him.

The first thing to do is to look at the goal: what question do you 
have to answer? Sometimes that can save work if there's a lot of 
extraneous information that might lead you in the wrong direction. 
Once you understand the goal, you can take a look around and see what 
resources you can use to get you there. Let's try your problem as an 
example:

>Backstreet Boys tickets are on sale...

We can make a chart of prices, or just imagine one; if nothing else. 
This helps to make sense of the wording of the problem, namely the 
idea that there are three different kinds of tickets in the list:

     Tickets
        Front row      $35
        Center stage   $25
        Balcony        $15
     T-shirts          $10
     Posters            $8

If we see what's coming, and have a little experience with this sort 
of problem, we can make up a shopping list, which combines our goal 
and our givens:

     Item                Count   Price   Cost
     ----                -----   -----   ----
     Front row tickets     2      $35
     T-shirts              2      $10
     Posters               2       $8

(Note that some of the paperwork adults take for granted, invoices and 
forms and so on, are really problem-solving tools that save a lot of 
thinking, by helping to organize data; just getting used to some of 
these real-world devices can be a useful experience. Do some shopping 
together, and look at bills and invoices and receipts.)

Now we can go through the problem one step at a time, as if we were 
really shopping. If we buy two tickets for $35 each, what will they 
cost? This should trigger the word "multiplication" -- not so much 
because of the word "each," but because of what is happening, two 
identical things being added together. We can multiply to find how 
much is being spent on each item, and then add the results together. 
If even the idea of adding isn't obvious, we can picture actually 
spending the money: put down $35, then another $35, then two $10 
bills, then two $8 piles. Eventually the idea of addition will pop 
into our minds. To sum up: the goal is to associate an action with the 
corresponding arithmetic operation: putting together adds, separation 
subtracts, repetition multiplies, splitting into groups divides, and 
so on.

There are other tricks for problem solving that help with more 
complicated problems that have multiple steps, but it sounds as if 
we're dealing with fairly basic problems in which the goal is simply 
to learn when to use each operation. If making subtraction concrete 
helped your son, then probably acting out these problems will help, 
too. Many children seem to just see the numbers and ask "What can I do 
to them?" rather than seeing the actions in the problem and asking 
"What is being done?" By focusing on the concrete problem rather than 
the abstract numbers, you can make the connection more easily.

Let me know if there are any specific kinds of mistakes he makes that 
might suggest a more specific remedy.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/