Loquacious Problems

We’ve talked a little bit this week about how math can sometimes seem like a foreign language. Well, this means that sometimes, we need to “translate” English into “math” – this phenomenon is also known as “solving word problems.”
A “word problem” is really just a math problem written in every day English speak (or German, or Italian, you know, depending on what language you speak). We solve word problems all the time. Like when you tell your mom you want five people at your birthday party, and that means you will need three pizzas, because each pizza has eight slices, and if everyone has three slices (you’re a teenage boy, and your friends are teenage boys), and three slices is 18, but two pieces would only be 16 slices, so you better go with three pizzas and you can save the rest for your little brother and sister’s lunch the next day. That is basically just one really long word problem. Or when you want to run a 5K (3.1 miles) and you know you run a 10 minute mile, so you tell your mom to meet you at the finish in about half an hour (10 minutes/mile * 3.1 miles = 31 minutes).
So really our lives are made up of word problems and they happen all the time, and we even write our own word problems without even being aware that we are doing it! Ha – how’s that for knowing more math than you thought you knew?
But for some reason, when we’re sitting in Mr. Hamlin’s math class staring at our pop quiz that is asking something about people washing cars and how fast can they wash the cars together, our brains freeze up and we feel like we have no clue what these people are talking about.
Well, the first thing you should do is stop and take a deep breath! You know how to do this – you’ve been doing it your whole life. Don’t let the fact that you are sitting in math class and not on the football field running laps freak you out. You can do this!
Let’s put together a sample problem so we can work through it together.

Joe, James, and Jessica are all pro car washers. Joe can wash a car in 40 minutes. James can wash a car in 47 minutes, and Jessica takes about an hour to wash a car. If they all work together washing a car, how long would it take them?

Now, it’s always good to read through the problem all the way before you start working on the problem, just to get a feel for what the situation is. Now, this time when you read it, read it as if you’re actually in the problem, and the situation is your situation. It’s much more motivating to solve a problem when it is your problem. Replace one of the characters with yourself – now it’s you, James, and Jessica, instead of Joe.
After you have read through the problem once (without hyperventilating), I want you to think about what you’re actually looking for. Are you trying to find out how many cars you can wash? How long it would take one person if the other two are helping? What is the question asking you to find?
In this particular “word problem”, the question says, “If they all work together washing a car, how long would it take them?” First off, let’s find out what kind of answer we’re looking for. Are we looking for a rate, a number, a distance? It looks like we are looking for an amount of time: “How long would it take them?” So chances are we’re going to come up with a number of minutes or hours. Since most of the rates given were in minutes (40 minutes, 47 minutes) let’s go with minutes. That means we need to change Jessica’s rate into minutes. This one should be pretty easy. How many minutes are in an hour? Sixty.
The next thing we need to do is write down some kind of relationship between what we know, and what we want to find out.
Well, since we’re talking about rates of washing cars, let’s look at the rates of car washing. Let’s call the rates R, and we’ll give subscripts to each person’s rate.

Now, to find their combined rate of washing the car, we need to add them all together.

So, together, you and your friends can wash 71 cars in 1128 minutes. Well, that’s great, but we don’t really want to know how many cars you can wash in a certain amount of time, we want to know how long it will take you to wash ONE car. So, the easiest way to do that would be to divide 1128 minutes by 71 cars (giving you the amount of time it takes to wash ONE car): 15.89 minutes.
So the basic steps for solving word problems are:
1.) Read the problem all the way through (without freaking out), and putting yourself in the problem, or pretending it is your situation, so you’re more motivated to solve the problem.
2.) Write down what you know (in this case, the rates of car washing).
3.) Figure out what you want to know (in this case, how long it would take to wash one car)
4.) Write an expression, using the information you have.
5.) Solve the expression.
I will have another post next week on word problems. Until then, check out this page over at PurpleMath.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Comparing Decimals

Today, Student A had a worksheet for comparing decimals. She was supposed to say which decimal in each pair was greater than the other, or if they were equal.

Because she originally completed the assignment by guessed (a pretty usual occurrence for A), I decided to give her some steps to follow in comparing decimals before we even began correcting the assignment.

First, we talked about how decimals are just like fractions – meaning they are just part of a number, instead of a whole number. With that in mind, we went on to our steps for comparing decimals:

Step 1. If there is a whole number (the part of the number that is to the left of the decimal point), compare the whole numbers first. If the whole numbers are different, then all you have to do is tell which whole number is greater. (for example, 12.3 and 14.72 – 14.72 is greater because 14 > 12, and it doesn’t matter how much “extra” you tack on to the 12, 12 is still smaller than 14). Otherwise, move on to step 2.
Step 2. Since the whole number parts are the same, we will need to compare the decimals. Remember that decimals are like fractions, so .1 = 1/10, and .01 = 1/100. Well, it’s hard to compare fractions when the bottoms are different (that is, it’s not very intuitive) and it is equally hard to compare decimals when we don’t have the same number of places, such as in 4.35 and 4.3.
To make things easier to “see” (instead of having to think too hard – mathematicians never like to think harder than they absolutely have to – that’s why we have “tricks” for everything) we are going to make sure each decimal has the same number of “places”. To do this, all we have to do is add zeros on the right hand side of the number. Going back to our example of .35 and .3, since .35 has two places and .3 only has one, we will add a zero to .3 and have .30. Now we are comparing .30 and .35 – well, that is just like comparing 30 and 35. It is clear that 35 > 30, so .35 > .30.
If you had .30 and .3, when you add a zero to .3 you will get .30 and .30 – in this case, the numbers are equal.

I was really impressed with Student A’s internalizing this concept. After five or six problems she was able to figure them out quickly, and explain her answers just as fast. She really understood the concept, and that was a huge success for us.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Combining Like Terms – Sorting Letters

Did you ever play sorting games as a little kid? If you fold your own laundry, then you should be a pro at sorting – pants, shirts, socks, sweaters.

Combing like terms is just like sorting laundry, or marbles (as illustrated in my sorry attempt at drawing marbles).

When we have “letters” (also known as variables) in an expression we often want to combine them so that we have an easier expression to look at.

Think of it like having a pile full of different color marbles (like in the illustration above). Let’s look at an example:

2x + 4y – 3xy + 7y – 5x + 2xy

Let’s look at the different “color” marbles we have (that is, the different “terms”)

We have x’s, y’s, and xy’s (the xy’s are their own type of term, because xy is not the same as x or y).

Now we combine the things that are “like” – We have 2x and –5x, so that is –3x. We have 4y and 7y, so that is 11y. Finally, we have –3xy and 2xy, which is –xy.

Our “simplified” expression is –3x +11y –xy.

Because of the commutative property of addition, we can write the expression in any order we want to.

-3x – xy + 11y
11y – 3x – xy

etc, etc

Here are some practice problems for combining like terms (also known as “simplifying expressions”)

Simplify the Expressions by combining like terms

1.

2.

3.

4.

5.

(Pre-Math Tip! Play sorting games with your preschooler. You may not realize it now when you are sorting blocks and dishes and beads, but your preschooler is building the foundation for basic algebra skills like being able to combine like terms!)

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.