Double the Number, Double the Fun

For this post we are going to talk about the 2’s Times Tables. The twos are perhaps the easiest times tables to learn.

Multiplying by two is the same as doubling the number (that is, adding the number to itself)! Pretty easy, right? First, let’s practice counting by twos:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24

That wasn’t so hard, right?

Now, let’s think about doubling numbers (adding them to themselves):

2 x 1 = 1 + 1 = 2

2 x 2 = 2 + 2 = 4

2 x 3 = 3 + 3 = 6

2 x 4 = 4 + 4 = 8

2 x 5 = 5 + 5 = 10

2 x 6 = 6 + 6 = 12

2 x 7 = 7 + 7 = 14

2 x 8 = 8 + 8 = 16

2 x 9 = 9 + 9 = 18

2 x 10 = 10 + 10 = 20

2 x 11 = 11 + 11 = 22

2 x 12 = 12 + 12 = 24

That’s pretty straightforward, right? So when you get a “twos” multiplication problem, it’s as easy as adding the number to itself! If you have problems with addition, not to worry! I’ll be posting some tips soon to help you with addition.

Find the rest of the Times Tables here: Times on the Table

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.

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.

A and the Fact Families

I have started tutoring a recent 3rd grade graduate (A) who is struggling in math. Today was our first session, so I tried a few problems out on her to see where the struggle really lies.

Her mom told me she’s had problems with multiplication, which upon investigation was clear – she struggles with multiplication. But I had a feeling that wasn’t where the real problem was, so I talked to her about addition and had her practice some (relatively) simple math facts – like 10 + 20 = 30, and 10 + 2 = 12. Unfortunately, she couldn’t easily and quickly give me the answer – she struggled with these concepts, which are usually understood before a child even begins to learn multiplication.

Addition War

So I decided to play a little game with her I called “addition war.” For those unfamiliar with the card game “war”, it goes like this – each player starts out with a stack of playing cards. At the same time, the two flip over the card, and whoever flips over the higher card wins the “battle” and keeps the two cards. The person who ends up with all the cards is the winner of the war. For “addition war” we took out all the face cards and then we would each flip over a card. Whoever could add the two numbers together first would win the cards. I knew that I would be fast at this game than a third grader (although it’s great practice for me, too!) so I gave her a few seconds head start. I figured that should be all she needed, but it soon was evident that her problem isn’t just with multiplication – she lacks some fundamental concepts for adding in her head, which will cause problems when she tries to do multiplication, and when she tries to memorize times tables (which will have no concrete meaning to her).

Fact Families of 10

With this in mind, we went back to the basics – addition fact families for the number 10. After A masters addition fact families for 10, we’ll move on to what I call “adding to ten” – when you have 7+5, instead of counting up from 7, you add 7 + 3 + 2 = 10 + 2 = 12. This is a much faster way to add numbers, but you have to be really solid on your fact families (especially for 10 – most of the others are relatively easy to remember, and A seems to know most of the smaller fact families, but we will probably need to work on those, too).

To review – the addition fact families for 10 are
10
1,9
2,8
3,7
4,6
5,5

This means that each pair of numbers add up to 10. In order to really drill these “families” in, we used those same face cards (from addition war) as flash cards. I would flip over a card, and A would tell me the other member of the fact family. We wrote all the fact families on a piece of paper for a “cheat sheet” at first. We set a timer for 2 minutes so she would have a reason to be fast, and then I bribed her with chocolate. She gets a fun size candy bar if she can reach our goal of 75 cards in 2 minutes. Today her best score was 52 in 2 minutes, but that was pretty good, since she is still taking a few minutes to name the missing fact family member. We played four or five times today, and it seemed to really help.

Math Manipulatives – Cuisenaire Rods

After we had used the flash cards for a while, we used the Cuisenaire rods to show the fact families. The thing I love about Cuisenaire rods (and really any math manipulative) is that when the student gets a chance to touch math, they are a lot better able to understand it. Math concepts can be abstract, and making them concrete helps students (especially children who developmentally aren’t abstract thinkers) understand. A was able to see the fact families, and get to know them a little better. We talked about how as one of the fact family members get bigger, the other gets smaller (see the stair step pattern?) and that is because we always want to have ten – so we are moving one “unit” at a time from one side of the fact family to the other. This will take a lot of looking at and playing with until A really understands it, but I can tell that she already understands it better than when we started today.

Another game we started to play was “speed” with the fact families. We laid out a bunch of cards and then matched them with their fact families and as she matched them, she turned them over and I would put more cards down for her to match with the fact family. This worked okay, but I think the flash cards was more what she needed today.

 

Goals and Progress

My goal with A is to get her to the point where she can add pairs of one digit numbers without thinking about it for too long. I think we made great progress. She and I will probably be meeting every weekday this summer. Hopefully by the end of the summer we will have multiplication mastered as well. But math concepts have to be built on a strong foundation, so the foundation is what we’re working on right now.

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.