Lesson 1-4: Problem Solving Strategies

Help! I don’t know how to start!

Sometimes that’s how I’ve felt when confronted with a word/story problem; I just feel lost. Fortunately I’ve learned a few basic problem solving strategies I’d like to share with you. I’m sure you’ve used at least one of them. Check them and see if that’s true!

Algebraic models

Nope, I’m not talking about America’s Next Top Model here.  Do you remember how we described algebra the other day? It is the language of unknown quantities. Anytime we’re confronted with a problem that has unknown quantities, we can use algebra to help us solve the problem.

How do we use algebra to solve a problem? Well, we find an algebra equation that represents the situation and we solve that equation. That equation is what we call an algebraic model of that problem. It is the formula for that problem.

OK, got that…how in the world do we come up with that equation, that model? There are a number of problem solving strategies you can use. We will cover several of them today; the main thing I want you to understand though is the following.

Step 1: What type of problem is this?

Problems generally fall into categories. Especially the problems you will encounter in the book and on SAT tests. Each category of problem has a certain way to solve it, a formula that you use. Put another way, each category of problem has an algebraic model for it.

What if you understood the most common problem categories and knew the basic formula and solution scheme for them? Wouldn’t life be much easier? Then all you’d need to do is determine with category the problem fell into and apply the right formula!

What are some of these common categories?

  1. Cost problems
  2. total cost = cost per item * the number of items (t = cn)
  3. Distance, rate, time problems
  4. distance = rate * time (d = rt)
  5. Labor problems (including material cost)
  6. Net pay = wage per hour * number hours worked + material cost (p = wh + m)

Yes, there are more, but these will suffice for now. 

Read the problem, look for keywords (cost, distance, rate, time, wage, labor, etc) that will help you determine which type of problem it is.

Step 2: Visualize the problem

You’ve heard the saying “a picture is worth a thousand words?” Well, who ever thought it up must have been a mathematician, because the ability to visualize a problem is one of the most critical skills for solving algebraic problems.

Draw a diagram or picture of the problem. Often, just seeing a picture helps you see what needs to be done. In addition, thinking through how to draw the diagram helps you understand the problem better.

Label the picture with the information from the problem. This will help you organize the information and better see how things relate to each other.

Example 2 in the book is certainly one for which a diagram helps. If you look to the left of the example, they have drawn a diagram. It is a straight line that represents the 1590 miles of track. On the left is Sacramento, on the right is Omaha. You know that the Union Pacific is laying more track than the Central Pacific each day. So the place they meet (the Promontory) will be closer to Sacramento than Omaha (since the Union Pacific is coming from Omaha).

This helps you visualize that the total miles of track laid is the sum of the amount the Central Pacific and the Union Pacific each laid.

Step 3: Create a verbal model of the problem

Next, come up with an English sentence that describes the process / formula needed to solve the problem. This is basically the sentence that you will translate into an algebraic equation to solve.

Some examples are actually the description sentences for the problem categories above! Sentences like:

  1. Total cost = cost per item * the number of items
  2. Distance = rate * time
  3. Net pay = wage per hour * number hours worked + material cost

These are the English sentences (verbal models) that you can then translate into an algebraic equation and solve.

For the example 2 problem a good first step verbal model could be:

Total miles track laid = amount laid by Central Pacific + amount laid by Union Pacific

Next, refine by breaking down the amount laid by parts:

Amount laid by Central Pacific = miles per month * number of months

Finally, revise the original with these for each company:

Total miles track laid = miles per month * # months + miles per month * # months

Central PacificUnion Pacific

Step 4: Identify known quantities and plug into the equation

Replace the parts of the equation with the values we know from the problem statement. Once we do this, we will be left with a part of the equation that is unknown.

In the example 2 problem, we know:

1590=8.75* ?+20*?

Total miles track laid = miles per month * # months + miles per month * # months

Central PacificUnion Pacific

We don’t know the number of months. We do know that the Union Pacific started 24 months after the Central Pacific…so if t is the number of months Central Pacific took, then Union Pacific took 24 months less or t -24. This gives us:

1590 = 8.75 * t + 20 (t – 24)

Step 5: Solve for the unknown

Replace the unknown with a variable and solve for that variable. Remember to preserve balance! What you do to one side, you must do to the other!

Finishing our example 2 problem:

1590 = 8.75t + 20(t – 24)

1590 = 8.75t + 20t – 480

2070 = 28.75t

t = 72

And since Central laid 8.75 miles per month, their total is 8.75 * 72 = 630 miles.

Pacific laid 20 miles per month, so their total is 20 * (72 – 24) = 20 * 48 = 960 miles.

Step 6: Check your answer!

There are two easy things you can do here:

  1. Check your equation: plug the answer back into the equation and simplify. If you get something that makes sense (2 = 2) you’re good! If you get a nonsense answer (2 = 1), then you better recheck your answer!

Example 2: 630 + 960 = 1590 which matches.

  1. Apply some common sense: think about your answer and the problem. Does it look like it is in the ball park? Did you some up with something silly like a negative distance or time? Would you really expect to drive that far in that period of time? You get the idea…

Example 2: no negative numbers, both less than the total…

Any other tips?

Yup, see the problem solving strategies page. 

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