Math for Elementary Teachers II 1512

Permutations

So what is a permutation exactly? Well it is the combination that something occurs, and more importantly the combination and which they occur are important. It's not just the different combinations something occurs, it has a specific order in which they occur; in other words a permutation is a ordered combination.

So what is the difference in a way that is easy to understand? Well basically if you think of it like it is a combination lock it will help you understand. You can only put the exact combination into a combination lock for it to open. When calculating permutation it is basically the same thing. You can only use the different options in a specific order.

Consider the following:

Given 3 people, Joe, Tiffany and Sue, how many different ways can these three people be arranged where order matters?
Let JTS stand for the order of Joe on the left, Tiffany in the middle and Sue on the right.
Since order matters, a different arrangement is JTM. Where Joe is on the left, Sue is in the middle and Tiffany is on the right.
If we find all possible arrangements of Joe, Tiffany and Sue where order matters, we have the following possibilities:

JTS, JST, TSJ, TJS, STJ, SJT

The number of ways to arrange three people three at a time is:
3! = (3)(2)(1) = 6 ways

Math for Elementary Teachers II 1512

What's Probability?



The easiest way to see how probability works is by rolling the dice. When I use this process I am able to clearly see why and how probability works. If I have 2 standard dice. Lets say I want to know the probability of rolling the number 4 or lower when using both dice. Well if I figure that there are the numbers 1-4 on each dice, I know that there are 8 different possible outcomes that could be rolled; another way of figuring this is out would be to make a sample space. A sample place is a listing of all the possible outcomes. In this case {1,2,3,4} and since there are 2 dice I multiply the total of the sample place by 2, which gives us 8.. I already know that each dice has 6 numbers on it, again I multiply by 2 since i have two dice. This tells me I have a possible of 12 numbers. I then know that I have an 8 in 12 chance or the probability of 2/3.

Lets try and figure out the probability of drawing a specific card or cards from a deck of cards. What is the probability of drawing an Ace. Well we know that there are 4 aces in each deck, and 52 cards in a full deck. Which gives us 4/52 or 1/13.
Another example would be finding the probability of drawing a red card. So we know that half of a deck of cards is red, so 26 cards are red. That means that the probability of drawing a red card is 26/52 or 1/2.

To me probability is something that can be used for other situations that are more likely to occur or be more important that know what dice you will roll or what card you will draw. You can also use probability to find out things like the weather based on past weather, or on things like the likely hood that something is going to happen again or things along those lines. Another thing is even used in betting and gambling. What are the odds I will win or what are the odds I will draw an Ace.

Math for Elementary Teachers I 1510

Using the Count-On or Count-back Method

Wouldn't it be nice if all numbers you needed to subtract from another number ended in 0, 5 or even a 1 or 2? It would, but in reality numbers can vary and almost always do; especially when you are trying to do the calculations in your head. Working in a job that deals with money and giving change I have become very fond of the count-on method. It works so well, and is almost fail proof when you do it right! And the best part about it is you always have the necessary equipment needed to use this method; your fingers!

Lets start with an easy example: 729 - 400=? To start we count on our fingers up from 400. 500, 600, 700 so we have 3 hundreds. Then we count up the tens place which would be 10, 20 giving us 2 tens. Finally we count up from the ones place which is 0 up to 9 giving us 9 in the ones place. Then we have our answer of 329.

Another way to describe it would be using an example of giving someone change. Lets say the total bill is $7 and they pay with a $10. So you count back the total change by using your fingers or the count-on method; 8, 9, 10. The change would be $3 dollars. 

There also is another method similar to the count-on method, it is the count-back method. It basically works like the count-on method only you start with the smaller number and add onto it until you reach the total. An example of this is: 293-80=? So to start there are no hundreds in this example so you have 200. Then you see that there 9 tens minus 8 tens which equals 10 tens. Finally there are 3 ones minus 0 ones which equals 3. So your total left is 213.

Math for Elementary Teachers I 1510

Crazy Compensation to Solve Math Problems Mentally

Sometimes when numbers don't end with 0 or 5 they can be a little trickier to figure out in your head.

I enjoy doing things as fast as possible, and if at all possible without the use of a calculator. One way I can figure out a quick math problem, for example how much money I have left to spend at the grocery store or something similar is to use the compensation method.

This method is my favorite because it works well for me and it works fast. I am able to do the math and solve problems pretty quickly using this process. Here is an example: Lets say we have to figure out what 73-59 is. The first thing I do is look at the numbers to see if the end digit is the same, or near the same; in this case it is not. (I will show an example of that next) So I see that 59 is close to 60 and round up, then in my head I subtract 60 from 73 which gives me 13. I then have to remember to add that one back into the equation giving me the final correct answer of  14.

Lets try another example: 48 - 39= ? This time since the ending digit is near the same I just adjust the 48 to 49. Then I calculate 49-39, which is 10. I again have to remember to adjust the total by 1, since this time I added 1 to the first number I now must subtract 1 from my estimated total. I have a final answer of 9!

Math for Elementary Teachers II 1512

Angles and Triangles

What exactly is a triangle? I wonder that myself. I also often confuse the different types of triangles and angles that the include, or don't include for that matter. Hopefully this blog will help you use the tools, or tricks to memorize the different types of triangles and angles.
To begin, a triangle is just that, it has three (tri) sides and three angles, and those angles always equal 180 degrees.

There are three different names given to triangles that tell how many sides (angles) are equal. There can be 3, 2 or even no equal sides or angles in a triangle. These three options give us the three types of triangles possible; equilateral, isosceles and scalene.

The way I remember Equilateral Triangle is that is has three equal sides; Equal = Equilateral (sounds almost the same to me). The three angles of an "equal" triangle are always 60 degrees each.






Next is the Isosceles Triangle, I have no fun way to remember that this triangle only has two equal sides and two equal angles. 

Scalene Triangle has no equal sides and no equal angles. The way I remember this is to me scalene reminds me of scaling, like scaling a mountain. When I think of that I think that no mountain has an exact equal with equal sides. Kind of silly but                                            it works for me.

There are also three different names for different types of angles within a triangle. Those are acute triangle, right triangle and obtuse triangle.

An Acute Triangle all angels are less than 90 degrees.
When I think of acute triangles I think of how cute things
are when they are little, which reminds me that all angles are
little, or less than 90 degrees.

 Right Triangle has a right angle of 90 degrees, the way I remember this is a right angle is 90 degrees so a right triangle has at least one of these.

An Obtuse Triangle has an angle of more than 90 degrees. The strange way I remember this is that Obtuse reminds me of odd, and for some reason when I think of angles I think of the usual 90 degree angle and anything larger than that is odd, or obtuse.


Work Sited: http://www.mathsisfun.com/triangle.html

Math for Elementary Teachers II 1512

You Just Can't Go Wrong With a Manipulative!

Manipulatives in math are just awesome! They are such great learning tools, and I think they are so much fun to use. I had never really gotten into playing around with math things on the computer, but after testing out some super fun sites online I almost think of math manipulatives as games.

There are endless ways of using colorful, fun, exciting objects to make learning math easy and fun. I feel like a spokes person right now, but I have a million ideas in my head of what could be used and why they would work and how students would enjoy them.

Starting at such a young age, babies even use manipulatives. Stacking of bigger objects into smaller objects, the little ball that you put the shapes through the correct cut out and so on. And they aren't even able to talk. With all the possibilities math can be taught at such a young age in a fun way, that children will find it much more enjoyable in school. And hopefully not think of it as such a drag. Math can be fun!


http://www.ixl.com/?gclid=CLj9tr694akCFcbBKgodqwmDbA


There are endless different manipulatives for all different grade levels on this page. You can click on them and find many different options for many different classes! I love it!

Math for Elementary Teachers I 1510

The Lattice Multiplication Method

To start off with here is a video demonstrating both standard multiplication and then the lattice method.



When did they start teaching the lattice multiplication method? And where was I? I definitely do not remember ever learning this process, and to tell you the truth I am glad. It is so strange that I find it more way more difficult than using the standard multiplication algorithm.

Yous tart with the square, then draw the lines and the diagonal lines in the lattice then you have to multiply and try to understand which box which number goes. And as if that is not confusing enough then you have to add the numbers in the right rows and wow I am lost!

Here is a video that starts out a little slow but just watch and it will help with the bigger numbers of multiplication.



I think peoples biggest issue when it comes to multiplication is not knowing the larger numbers. I think most people can do 1-5 relatively quickly and accurately it's all the larger numbers that cause the confusion. So if everyone practiced the larger numbers a little more and found them easier and faster to solve I think long multiplication, or the original multiplication algorithm would be fail proof just about every time, well most of the time anyways given human error!

Math for Elementary Teachers I 1510

Standard Multiplication Algorithm

Seeing as how I was taught multiplication using the standard algorithm, to me it is the easiest and most reliable method; well other than a calculator of course. It makes sense to me. It's simple, step by step process that you use; and as long as you know your basic multiplication and adding you should have no problem with it.

However, some people do have problems following it, and it wonder why. I think one reason is the way they were taught. The way math was taught when I was growing up and in schools compared to the math being taught today is almost another language. Although the information being processed received and processed is basically the same, the way it is being done is completely different. And to me almost more confusing than it needs to be.

Granted, I do understand that people all learn and understand things just a little bit differently. What I don't understand is how to know which way works best and is overall more understood by students. I guess if I knew that I would be a super teacher already and wouldn't even need to be taking math classes for elementary math. I just wish it were easier to know what to teach and how so that everyone could feel like they are on the same page.