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.

Round and Round – trig functions

Last week we talked about some basic functions that you will deal with in algebra, calculus and beyond. Today I am going to introduce you to some basic trig functions.

One of the most fascinating applications of trig functions is that of daylight hours. Do you notice how the days get longer in the summer, then shorter in the winter? And then they get longer and shorter and longer and shorter, and the cycle just keeps going on.

Trig functions are just that – cycles.

This is a graph of the function f(x) = sin(x) (the one on the left is one period of the sine function, and the one on the right shows more what the graph does – it just keeps going over and over again – in a cycle.)

 

This is a graph of the function f(x) = cos(x). Cosine is a lot like sine – it just starts in a different place. Where sine starts at zero when x = 0, cosine starts at 1 when x = 0.

This is a graph of the function f(x) = tan(x). Tangent is a ratio of sine and cosine. The reason it is undefined at some places (see how the graph goes up and doesn’t come back down, and then it stars from far below?) is because sometimes cosine is zero, and you can’t divide by zero.

This is just a little introduction to show you what trig functions look like. There are also inverse trig functions, which we’ll talk about later. We’ll also talk about some really interesting applications of trig functions.

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.

Some Tricky 11’s – in each place

Remember that A and I have been talking about patterns in multiplication? Well, the easiest of the multiplication tables (besides 10s) are 11s – just double the number, right? Well, what happens when you get to 11 * 11? It isn’t as hard as you might think – you just have to figure out the right pattern.

Think about 11 * 9:
        We don’t even really have to think about it as “90 + 9” because we can just think of it as “write 9 twice”. But now when we move on to 11 * 10, looking at it this way will help when we do 11 * 11 and 11 * 12:

We can think of the number 11 as “one in the tens place, and one in the ones place” – which when extended to multiplication (for example, 11 * 5) becomes “5 in the tens place and 5 in the ones place” (where you replace “5” with whatever number you are multiplying by.

For 9, that is easy, for 10, it isn’t quite as straightforward – because what does “10 in the tens place” mean? Well, in the illustration above, it means “100” – because a 10 in the tens place means we need to put another zero on the end, and a 10 in the ones place is really just 10. So that’s how we get 11 * 10 = 110.

Now we can move on to 11 * 11. We think the same thing; “11 in the tens place, and 11 in the ones place.” Let’s write it out like we did with 9 and 10:

So you can see that when we write “11 in the tens place” what we get is 110, and when we put “11 in the ones place” we get 11. So the answer is 11 * 11 = 121.

We can do the exact same thing for 11 * 12, and we get 120 + 12 = 132.

Now you can probably figure out 11 * 13, 11 * 14, and 11 * 15! And after a while, you can probably see a new pattern as you go.

So when thinking about the 11’s times tables, you can just think “the number in the tens place, and the number in the ones place.” This rule works for all the times tables – from 1 to infinity!

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.