Introduction to Equations

Objective: When done with this lesson, you will have demonstrated how to solve simple problems using words to explain your thinking.

How long will it take? Will we have enough? When will we be there? How much will it cost?

When we answers to questions, we want SOLUTIONS! We'll be studying different techniques in this unit to find solutions to problems. Throughout this unit you'll become an expert 'solution sleuth'--a real Sherlock Holmes!

So what makes a good sleuth? Here are some of my thoughts. I think a good sleuth would:

Know what the problem is.
Be able to find the solution to the problem.
Be able to clearly communicate the solution to others.

If a person doesn't have one of these key skills, then he or she is not a good sleuth. Think about it...

Suppose he or she didn't know what the problem is. Then he or she will know where to start or when he or she is done!
Suppose he or she wasn't able to find the solution to the problem. Then he or she would wander around aimlessly hoping to stumble on the solution.
Suppose he or she wasn't able to clearly communicate the solution to others. Then nobody would know who the culprit was or where the money was buried or anything!

In each lesson in this unit we will work on each of those three skills. In this lesson in particular, we will emphasize the need to clearly communicate understanding.

Before you try your hand at sleuthing, let's look at a sample problem and some sample solutions people might offer.

Sample 1 Problem
"Mable bought a bagful of marbles for $4.50. When she got home she counted out 75 marbles. She was wondering how much each marble would cost individually."

Sleuth

Solution Offered

6 cents

4.5 ÷ 75 = .06

Since there are 75 total marbles and if each one cost the same amount, then each of the 75 would share the total cost of $4.50. Therefore, dividing the $4.50 equally amongst 75 would give the price per marble.

We can use a calculator to find that 4.50 ÷ 75 = .06. So we know that the price per marble is $.06 or, as most of us like to say, "six cents per marble".

Sample 1 Problem
"Mable bought a bagful of marbles for $4.50. When she got home she counted out 75 marbles. She was wondering how much each marble would cost individually."
Sleuth	Solution Offered
	6 cents
	4.5 ÷ 75 = .06
	Since there are 75 total marbles and if each one cost the same amount, then each of the 75 would share the total cost of $4.50. Therefore, dividing the $4.50 equally amongst 75 would give the price per marble. We can use a calculator to find that 4.50 ÷ 75 = .06. So we know that the price per marble is $.06 or, as most of us like to say, "six cents per marble".

What do you think about their "solutions"? Let's evaluate them.

Sample 1 Solution Critiques
"Mable bought a bagful of marbles for $4.50. When she got home she counted out 75 marbles. She was wondering how much each marble would cost individually."

Sleuth

Evaluation:

Correct in the "number".

No explanation.

You don't know what the shark is answering.

You don't know if you can really trust in the answer since non of it was explained.

Correct in the "number".

No units. Is that cents or dollars or crackers?

Correct in the "math work" shown.

There wasn't much of an explanation.

You have to guess from where the work and numbers come.

Correct in every regard.

Easy to read.

Easy to follow.

Easy to know that it is the correct solution.

Your don't have to guess about anything.

Sample 1 Solution Critiques
"Mable bought a bagful of marbles for $4.50. When she got home she counted out 75 marbles. She was wondering how much each marble would cost individually."
Sleuth	Evaluation:
	Correct in the "number". No explanation. You don't know what the shark is answering. You don't know if you can really trust in the answer since non of it was explained.
	Correct in the "number". No units. Is that cents or dollars or crackers? Correct in the "math work" shown. There wasn't much of an explanation. You have to guess from where the work and numbers come.
	Correct in every regard. Easy to read. Easy to follow. Easy to know that it is the correct solution. Your don't have to guess about anything.

Remember, a good solution always has these parts:

the problem situation explained
clear, easy to follow reasoning
the correct answer in correct units

All these are part of being a good sleuth! If you're not so sure they are all needed, take a look at this situation:

Sample 2 Problem
"XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX."

Sleuth

Solution Offered

Crazy Carl
"It'll be $150."

Sharp Sam
"It'll take a while to do, but I can get it done for you for about $150."

Wacky Ward

"Let's see...Your bathroom floor is 5 feet by 9 feet which makes 45 square feet.

The carpet you like is $1.50 per square foot, plus the padding which is $.35 per square foot, and labor is $1.20 per square foot. That's a total of $3.05 per square foot.

So, $3.05 per square foot for each of your 45 square feet makes $3.05 · 45 = $137.25.

Of course, there will be tax. That's 8.5% of $137.25 which is $11.67.

Your total would then be $148.92."

Did you notice the problem was all X'd out? Hopefully you know what the situation was from one of the solutions offered. Do you see how much more thorough Wacky Ward's solution was? Notice that his solution has all three important parts:

the problem situation explained

clear, easy to follow reasoning

the correct answer in correct units

Great job, Wacky Ward!

As for the others, well.....they leave a lot to be desired.

Sample 2 Problem
"XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX."
Sleuth	Solution Offered
Crazy Carl	"It'll be $150."
Sharp Sam	"It'll take a while to do, but I can get it done for you for about $150."
Wacky Ward	"Let's see...Your bathroom floor is 5 feet by 9 feet which makes 45 square feet. The carpet you like is $1.50 per square foot, plus the padding which is $.35 per square foot, and labor is $1.20 per square foot. That's a total of $3.05 per square foot. So, $3.05 per square foot for each of your 45 square feet makes $3.05 · 45 = $137.25. Of course, there will be tax. That's 8.5% of $137.25 which is $11.67. Your total would then be $148.92."

So remember that a good solution has much more than just a quick "answer."