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.

Math Fun

This lady, Vi Hart, makes some really fun videos about math and some of the less useful, but seriously fascinating applications!

Here is a video for your enjoyment. Be sure to check out her YouTube page, and her website, for more fascinating math fun!

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.

Prove It – how (and why) to write proofs

For some reason, when geometry students get to the proof writing segments of their classes, they freak out. “Why do we have to learn how to write proofs?” “This is useless.” Etc, etc.

Well, first off, let me tell you that writing proofs is not useless, and you do it every day (whether effectively or ineffectively) without even realizing it.

Remember how you wanted that new video game that just came out and your parents said “No” and you tried to convince them to let you buy the video game? Did you just say “I need the video game.”? Probably not. That’s not very convincing. What you probably did was talk about all the reasons you “needed” the video game (or car, movie ticket, date, you name it – whatever it is that you had to convince your parents you “needed”)

Well, little did you know – but you were (verbally) writing a proof! Who would’ve thought! You already knew how to write proofs!

The tricky part about writing proofs in geometry is that there are so many rules, and most of them don’t usually make sense (if you have a bad attitude about it, like so many kids do). The thing is, you know more math than you think, and most of the “reasons” for geometric things being the way they are come pretty naturally to us as human beings.

The way I like to think about all those rules we use in Geometry proofs are as tools that I keep in my Proof Toolbox. There are a lot of rules. Such as, when two parallel lines are cut by a transversal, the alternate exterior angles are congruent. (WHAT!?)

The fascinating thing about using words in Math is that we usually use words that make sense. Don’t think about all these crazy terms like “alternate exterior angle” and freak out – just calm down and look at each word in the term as an individual word, and check out what it means.

First let’s take the word “alternate” – what does the word “alternate” mean? Well, in a selection process, such as when someone gets chosen for a play or for a spot on a team, an “alternate” would be someone that would take that person’s place if for some reason they weren’t able to perform. But that’s not really what we’re looking for. What about when you skateboard through some cones in the road “alternating” sides of the cones? Now that’s more like it. So “alternate” just means on opposite sides of the transversal (the line that cuts the parallel lines).

Now let’s look at “exterior” – well, that one is pretty easy. It just means “on the outside”. If we think of our parallel lines like a tunnel, then the “exterior” angles are on the outside of the tunnel.

We’ll talk more about how to figure out what all these crazy math words mean in another post.

Now let’s talk about how to write a proof. Since we have been talking about parallel lines, let’s start there.

Usually a proof problem has two parts – the “Givens” (i.e., the things you know for sure) and the “Prove” (what you are supposed to show is true based on the things you know for sure)

Given: and are congruent

Prove: lines l and m are parallel.

Now there are a lot of theorems and properties (i.e. tools) that will help us write this proof, but a tool won’t help you unless you actually have it. In our case, that means that a theorem or a property isn’t going to help us unless we know it. So if you have been studying your tools, you will know right away that angles 1 and 2 are “alternate exterior angles.” We know that If two parallel lines are cut by a transversal, then their alternate exterior angles are congruent. The converse (which just means switching the parts of the if-then statement) is also true: If the alternate exterior angles of two lines cut by a transversal are congruent, then the two lines are parallel. Which means that this proof is easy – we basically just state the converse of the alternate exterior angles property.

I will be starting a series on proof writing soon that will have useful information for writing proofs all the way from beginning geometry into college analysis. If you have specific questions about writing proofs, let me know in a comment, via email or Facebook or twitter, and I will try to incorporate your answer into the posts for the series.

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.

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.

Functional Graphs

There are some graphs of basic functions that every math student should know. If you can recognize what a function is going to look like when it is graphed on the xy plane, you’ll be a lot more likely to be able to figure out problems about that function, because you’ll be familiar with is.

Think of this – when someone is talking to you about a person, it’s hard to understand exactly what they are taking about, unless you have some kind of mental picture of the person they are talking about. Otherwise, you’ll be saying “Huh?”

So, to help you avoid saying “huh”, I am going to help you learn to recognize the graphs of basic functions.

First, , the most basic function. And probably the most boring. Blah.

And the next one, , this is a “parabola” – think “bowl.”

 

After the parabola, we must have a “cubic” (think, a “cube” has three dimensions – height, width, length)

Next, , also known as the “absolute value function”.

The slightly odd function, - a tricky function, because it is not defined at x=0!

Now for one of my very favorite functions,(an “exponential” with the famous number “e” – yes, it is a number.)

 

 

And we have have the exponential function without it’s inverse,(which is also not defined at x=0)

So there you have it – the seven basic functions (that are not trig functions – we’ll deal with those in another post). If you can memorize what these look like, you’ll be set through almost all of your math years. But I won’t let you get away with just memorizing them. Eventually we’ll study exactly why these functions look the way they do, and we’ll learn a lot of really amazing stuff about them.

Which is your favorite graph? Why?

(ps – I made these graphs using an amazing FREE graph drawing tool for your computer called WinPlot – if you don’t have a graphing calculator, this will be very useful for you. You can do a lot of the same things you can do on a graphing calculator using this program)

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.

Core Standards – what your child should know

What should a child know?

You can always use your state’s curriculum guidelines for a starting point of what your child should know at what grade level, but if you are homeschooling, or you just want a general sense of what math topics are out there, you can try some national cores that have been developed by different organizations.

First, there are the Standards and Focal Points developed by the National Council of Teachers of Math.

There are also Mathematics standards published by the Common Core State Standards Initiative.

Both look pretty useful and in depth, and I hope to use them myself in coming up with topics to write about on this blog, as well as in teaching my own children.

What standards do you use in your math teaching? Do you look at your state core? If you homeschool, which standards do you use?

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.

you know more math than you think

Do you ever feel like math is a foreign language?

Think about how it feels to learn another language, or to be in a place where people are speaking a language you don’t understand.

Now think about how a child learns to speak their mother tongue (the language their parents speak). If a child is exposed to a language from the time they are born, and if someone interacts with them on a regular basis in that language, the child will eventually learn to speak that language. Are children born speaking a language? Maybe baby talk – but mostly they just cry.

Turns out, the ability to learn math is programmed in us similar to the way the ability to learn a language is programmed in us.

You can read about a study that was done on this subject here.

In the article, they mentioned that a lot of Euclidean Geometry (lines, space, shapes – all that basic stuff) concepts were “innate” in members of a remote tribal community in the Amazon who had no formal training in geometry. They got as many answers right about geometry as formally educated Americans.

Something that was interesting, however, was that younger children (5 and 6 years old) didn’t know the answers. The researchers weren’t sure why, but I have a hypothesis.

Since math is similar to a foreign language, and young children don’t often understand the concepts of grammar, it make sense that they also wouldn’t understand the concepts of math especially if math has not been “spoken” to them regularly. This goes back to the principle of teaching “pre-math” skills by exposing young children to concepts like classifying, sorting, number conservation, counting, etc.

If you want your child to understand math concepts better, speak math with them! Try to make it a point to “speak math” with your child every day. Whether it is by sorting and classifying, or by talking about shapes around you, or multiplying a recipe, or talking about how “steep” the stairs are, speak math with the children in your life!

How do you “speak math” with the children in your life?

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.

Pre-Math: One of These Things

Sometimes as parents, and as teachers, we discount the things that children do as they play that help them learn math skills. We can encourage children to do things that develop good “math brains” without making them sit down and look at flash cards or do worksheets (I know a lot of kids love flash cards and worksheets, but we need to expand our horizons a little!)

One very important math concept and skill is that of sorting. In a previous post about combining like terms, I talked about how combining like terms was a lot like sorting laundry or marbles, or anything else than can be sorted and grouped.

Children like to sort and group naturally. My son will sort his toy cars by color, my daughter loves to help put the dishes in their correct places.

When our children practice sorting during play, or while helping mom and dad around the house, they are learning valuable math skills that will play a part in whether or not they are easily able to grasp the concept of things like combining like terms later in school.

Another part of sorting is being able to classify objects. In the picture above where my children are sorting laundry, they wouldn’t be able to sort the laundry unless they knew what set of classifications we were using to sort. Are we sorting kid’s clothes and grown-up’s clothes? Are we sorting every color? In our case, we were sorting according to “light” colors, “dark” and “ bright” colors, and “whites” – which meant that I had to explain what each of those classifiers meant.

When my children understood the classifiers, they were able to sort. Being able to classify objects (especially in different ways – like I said above) is also very significant and an important ability for children to develop so they will be successful in math skills later on. They have to be able to classify math “objects” such as numbers, coefficients, variables, exponents, etc.

Another game to play is “One of these things is not like the others” where you have a group of objects, and one object is not like the others. In this game, you are classifying objects, but you are also teaching the concept of “one” (which might seem like an easy concept, but trust me – you probably don’t actually understand “one”, even as an adult. I didn’t fully understand what “one” meant until I was finished with my undergraduate studies).

Try to find ways to help your children classify and sort objects. Try to make it a fun game that has to do with your life (sorting silverware or laundry, classifying cars when you are driving “Can you see a red car? How about a red truck? What about a blue truck?”). As your children learn to classify and sort objects, and as they do it on a daily basis, their brains will be gearing up to understand much more complex mathematical situations as they grow up.

What ways do you teach your children how to classify and sort? Do you feel like you are able to teach your children math concepts?

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.