Thursday, July 14, 2011

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

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

DSCN5158After 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.

DSCN5157Another 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.

Monday, July 11, 2011

Sequences Part II – The Future is in the Past

In Sequences Part I, we talked about what a sequence is, and then we talked about straight up sequences. If you need a refresher, just visit the post by clicking here.

Today, we’re going to talk about recursive functions for sequences. That is, ones that look like this: clip_image002

This is a recursive sequenceor one that uses previous terms to define the next term. This particular sequences is called the Fibonacci Sequence. Mostly it’s just a regular sequence that gets a special name because it’s pretty important to mathematics – but we won’t really get in to much of that in this post.

Remember the cups we talked about? Let’s go back to those and see if we can figure out the Fibonacci sequence. Now, the thing about recursive sequences is that you usually have to take at least one of the cups before the cup you’re working on, and use those numbers in your function. Let’s just take a look at it.

cupsWith this recursive sequence, we have the first two terms defined for us.



So, now that we have our first two cups filled up, we can start with our second cup, which would be clip_image002[3]. The sequence says clip_image002, so basically what that means is that we fill up each cup with everything that was in the last two cups (because we’re adding – and adding is just like putting things together in a mixing bowl when you’re cooking).

So if n=2 (clip_image002[3]), then that means we need to take clip_image002[6] and clip_image002[8] (or the two cups right before cup “2”) and dump them into clip_image002[3]‘s cup. clip_image002[6]=1 and clip_image002[8]=1, so when you dump those into a cup, you get 2, right? So clip_image002[3]=2. (I’ve made some illustrations for what’s happening.

dumpingcups2          dumpingcups3 

Now for the third cup, clip_image002[10], we dump in the last two cups, which are now cups 1 and 2. clip_image002[8]=1, and clip_image002[3]=2, so clip_image002[10]= 2 + 1 = 3. At first the Fibonacci sequences just looks like clip_image002[12], but let’s just keep going. dumpingcups4

We need cups “3” and “2” to fill up cup “4”. clip_image002[3]= 2, and clip_image002[10]= 3, so when we dump in those two cups we get clip_image002[14]= 3 + 2 = 5, so clip_image002[14]=5, and now we don’t have clip_image002[12] - we have Fibonacci!

When you write it out, it’s pretty easy to see what the next term will be:


Now I’m going to give you a few practice problems. I’ll post the answers tomorrow, to give you a chance to figure them out on your own.

Find the first 5 terms of each sequence.
(write your answers in the comment section, or email them, or post them on Twitter or Facebook. Those who come up with the correct answers before I post them tomorrow will be mentioned in the answers post)

1. clip_image002[18]





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.

Thursday, July 7, 2011

Sequences Part I - One After the Other



Say it with me: seee-kwence.

Sequences can look scary, and can make absolutely no sense. What are these lists of numbers that march on toward infinity? Let's break it down a little.

For those who are mathematically challenged, defining math terms in terms of math (especially to someone who doesn't "speak" math fluently) is a lot like using a "circular definition" for example, this one:

Hill - "1: a usually rounded natural elevation of land lower than a mountain" [1]
Mountain - "1a: a landmass that projects conspicuously above its surroundings and is higher than a hill"

So, in an effort not to be circular, I will define the word sequence as you would in your English class before I jump into the math part.

Here are a few definitions from Merriam-Webster to get you "in the mood":

2 : a continuous or connected series: as

b : three or more playing cards usually of the same suit in consecutive order of rank

f (1) : a succession of related shots or scenes developing a single subject or phase of a film story (2) : episode

3 a : order of succession

And here are some examples of some (non-math-related) sequences:

  1. He listened to the telephone messages in sequence.
  2. a chase sequence in a spy movie
  3. I enjoyed the movie's opening sequence.

I hope this gives you a better feeling for the actual word sequence, which I think will help you understand the math variety, too. So, a sequence is just putting things in order. I'm sure you can think of a lot of things that we put "in order" - underwear, then pants; socks, then shoes; turn on the car, then put it in drive. You can probably think of some longer ones, too - like, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. Those are all sequences. Groups of things that go "one after the other" in a certain order.

The place where math sequences get scary is because they are infinite. That means they never end. That's like if you had to start walking and never ever ever stop. You would just keep putting one foot in front of the other, and walk forever.

Well, fortunately, we have a nice sequence we all know (some of us know more of it than others) and we can use it to talk about all the other sequences. This "nice sequence" is called the integers.

We have all been learning about integers since we were little kids. "Jump on three, okay? One, two, three!" Those are positive integers (the integers actually go all the way backward in the negative "direction", too).

You might remember that there is another set called the natural numbers - we also call those the "counting numbers." The only reason we are using integers instead of natural numbers is because the integers include the amazing 0, and we really need that one.

So, let's look at our “nice sequence – the positive integers. We will use this set of numbers to talk about all the other sequences.

0,1,2,3,4,5,6,7,.... you get the picture, right?

Well, now we just use a letter (*gasp!* I know... letters make everything confusing). But before you start hyperventilating, think of our letters as little cups.onecup

We will represent our sequence with image, which is just like the cup on the left. We have “n” cups, which just means for every number (“n”) in the positive integers (0,1,2,3,4,5,6,7,…..), we will have something in our cup. The way we figure out what number goes in our cup is by a function. There are basically two types of functions for sequences. Just your basic function, like image , or a recursive function (which just means each number in the sequence depends on the numbers that came before it), like clip_image002_thumb1[3], which happens to be the Fibonacci sequence.

cups Now, to find out what we put in each up, we take the number that is on the cup and plug it into our “recipe” (the function for the sequence).

Let’s do an example. Say we have the sequence clip_image002[9]. Let’s fill up the first five cups. For the first cup, 0, we plug zero into the function (2n) and we get 0, so the first cup is 0 (in other words, clip_image002[11]= 0). For the second cup, we plug 1 into the function and get 2, so we put 2 inside the second cup (in other words, clip_image002[13]= 2). If you keep doing this for all the cups, you’ll find out that cup #2 = 4, cup #3 = 6, cup #4 = 8 and cup #5 = 10 (in other words, clip_image002[15]= 4, clip_image004= 6, clip_image006= 8, andclip_image008= 10). Now, we can line up the sequence “one after the other” and see 0 , 2 , 4 , 6 , 8 ,10 ,… which I hope you recognize as the even numbers. To write this sequence down in “math” it would look like this:


That is how you would write the sequenceclip_image002[9] in “math.”

Please leave comments on how well you understood this concept.

Tomorrow… Sequences Part II – The Future is in the Past

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.