6th Grade Math—Semester B

Sample Lesson - Introduction

Lesson 11.05: Setting Up Real-World Equations

There's a whole bunch of equations buried in here, along with who knows what else.

(Source)

Your Mom may see a revolting mess in this picture. You may see your room after what you call a cleaning. We see a lot of real world equations just waiting to be discovered. Are they hiding under the magazines? Desperately trying to get out from under the pile of dirty clothes? Hiding under the bed? Yes to all of those questions.

Can't see them? We'll give you some help. Here are some examples:

There are twelve dirty shirts on the floor. If there are twenty shirts total, how many are clean? Assume all the other shirts are clean and not crammed under the bed.

If you have 60 pounds of magazines to take to recycling and each magazine weighs ½ pound, how many magazines are you recycling?

If your room isn't cleaned at least once every thirty days, Mom will crack from the smell. If this picture was taken at the two-week point, how many more days will pass until Mom cracks? (Because you aren't cleaning your room without being told, that's for sure.)

When you finally have to clean your room for real, it takes you 1 hour to clean one-quarter of your room. How many hours will it take to clean your whole room?

What's that moving under that pile of clothes? That's not a math problem—we are just a little frightened. Do you have any missing pets, or has one of your socks come to life?

Now can you see the possibilities? There are equations to be found everywhere, even in this mess. We'll show you how to set them up. However, we are not helping clean your room, regardless of what is crawling around in your dirty clothes.

Sample Lesson - Reading

Reading 11.11.05: Solving Equations: Stuff Just Got Real

Setting up real world equations is easy, once you understand what the problem is asking for. Sometimes it helps to go over what you know, in order to figure out what you want to know. That way you know what you know, and you know what you don't know…you know? Ow, that made our heads hurt.

Let's use the examples in the unit intro to begin.

How about the shirts? We know that we have twenty shirts total. We know that twelve shirts are dirty. What we want to know is how many shirts are clean.

What else do we know, but didn't say up above? We know that our shirts are either clean or dirty—so our total number of shirts must equal the number of dirty ones plus the number of clean ones. Here's how this looks as an equation:

Total Shirts = Clean shirts + dirty shirts.

We'll call our unknown value(s) x, just because that's what we like to do. You can call yours Jason or Ashley or whatever you want. It's a free country. We just think x is a bit more efficient. In this case, the unit of measurement of our unknown value is "clean shirts". We know there are 20 total shirts and 12 dirty shirts. Plugging all of this info into our equation, we get:

20 = x + 12

Our answer will be expressed in clean shirts.

The magazine problem is just as easy. We know how much each magazine weighs in pounds, and we know how many pounds of magazines we have. What we want to know is how many magazines we have.

What else do we know? We know that the weights of all the magazines add up to make the total weight—but since each magazine weighs the same amount, we can multiply them instead of adding them. So the weight of each magazine times the number magazines equals the total weight of all the magazines. Our equation looks like this:

Total weight of magazines = weight of each magazine × total number of magazines.

Our unknown x is the number of magazines. Substituting, we get:

60 = ½(x)

If we included the units, it would look like:

60 pounds = ½ pound per magazine × x magazines

We can double-check our units to make sure they match on both sides of the equation. We can write pounds per magazine as pounds/magazine, and see if the units cancel out in the numerator and the denominator. Let's check.

60 pounds = ½ pound/magazine × x magazines

It works! We have pounds on both sides of the equation.

For the room-cleaning problem, we know that Mom will crack in 30 days if you don't clean your room and two weeks have already passed. What we don't know is the number of days until Mom cracks, so we'll call that x. Using the same methods we can set up the equation

30 = 2 + x

Wait, that doesn't look right. Let's check the units.

30 days = 2 weeks + x days until Mom cracks.

Aha! We need to change weeks to days. We know there are 7 days in a week so we could add a term:

30 days = (2 weeks × 7 days/week) + x days until Mom cracks.

Simplifying, we get

30 days = 14 days + x days until Mom cracks.

If that isn't a handy real-world equation, we don't know what is. There are some more examples of equations here for you to check out as well. Now that we've set all of these equations up, how about you solve each of them and check to see if your solutions are correct? Come on, you're quickly becoming a pro at this solving equations thing. We have faith in you!

Recap

To set up real world equations, think about what the unknown value is—in other words, the answer you want to find out. It helps to list what you know as well as what you want to know. You will use those things to make your equation.

Make sure you know what the units are, and that they match up on both sides of the equation. You can cancel out units that appear in both the numerator and the denominator on the same side of the equation.

Sample Lesson - Activity