Patterned

A very significant Pre-math concept for kids to understand is patterns. Patterns are easy to make with anything you have on hand (fork, spoon, spoon, fork, spoon, spoon, etc), and can provide hours of entertainment for you and your child.

Work with your child on creating an extending patterns. Take turns starting a pattern and finishing it. For example, maybe you start a pattern first, then have your child extend it, then let your child start a pattern, and you extend it.

You should also help your child be able to start a pattern that you state. For example, you say “Can you make me a pattern that goes fork, knife, fork, knife?” And your child should be able to display that pattern for you.

I am working on creating some fun use-at-home math games and manipulatives that should be available soon.

Are there any concepts you would specifically like to have manipulatives for?

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.

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:

This is a recursive sequence – or 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.

With 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 . The sequence says , 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 (), then that means we need to take and (or the two cups right before cup “2”) and dump them into ‘s cup. =1 and =1, so when you dump those into a cup, you get 2, right? So =2. (I’ve made some illustrations for what’s happening.

           

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

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

2.

3.

4.

5.

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.

Sequences Part I - One After the Other

 

Sequence.

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.

We will represent our sequence with , 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 , or a recursive function (which just means each number in the sequence depends on the numbers that came before it), like , which happens to be the Fibonacci sequence.

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 . 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, = 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, = 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, = 4, = 6, = 8, and= 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 sequence 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.