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.

Math for Elementary Teachers II 1512

Fundamental Counting Principle

When two events occur sequentially, and you want to know how many ways for these events to occur together, you can use fundamental counting principle. And it's something that is very easy to visualize and figure out! Two things I love when doing math.

For example if you have event X that has 2 different outcomes and event Y that has 5 different outcomes, then there are 2*5 different ways for events X and Y to occur together. It doesn't get much easier than that!

You can also use fundamental counting principle when you have more than just two events as well. You just continue the multiplication process. An example would Be event A has 3 outcomes, event B has 6 and event C has 3. The total different ways for outcomes to occur would be 3*6*3.

I think this process is so easy to use and follow. It is very quick too!

Math for Elementary Teachers II 1512

Tim Bedley and His Teaching Techniques

So Tim Bedley likes to use a process of teaching by letting students figure out the answer without being told the answer, even when they have questions. Instead of giving them the answer, he turns there question into another question that they can understand and are able to figure out the answer. I think this is awesome. What better way to teach than to actually show students how to figure things out, on the own.

Without a doubt I think this process not only teaches students things, but teaches them in a way that they will remember and feel confident in using. They can actually think of how they previously solved the problem and how it can be used to solve other problems! Brilliant!

In his video he uses different outfits to engage his classroom and how many possible outfits they could make with 3 shirts and 3 pants. Here is the link if you would like to check out the video https://clc.ims.mnscu.edu
/d2l/lms/content/viewer/main_frame.d2l?ou=1389676&tId=11007266.

An example I think that would work great with students would be using healthy snacks, how many different healthy snacks could the students make with 2 fruit choices, a choice of milk or juice, and a choice of yogurt or nuts. I think the children will like seeing what is possible and they always love food so if they could eat some of their results I think it would help the process stick even better!

Here is a short video of a student talking about how much she enjoyed Tim Bedley's teaching style and how she had fun and learned more!

Math for Elementary Teachers I 1510

Estimation

Where would we be in this world without estimation? About this much, about this far, almost, close to and so on. I would be lost! And I think that students would also be missing something if they are not taught estimation and how or when to use it and how it helps.

There are so many different benefits of estimation. Lets start with the simple ability to quickly add up or subtract a group of numbers for fast results; an example would be at the grocery store and trying to figure out your total, or how much change you will have.

I cannot imagine not having this skill, I'm pretty sure it is something I use on a daily basis. It isn't only helpful when I am grocery shopping, or giving directions but also when trying to figure out an array of other things. When students are learning to add it can help them come up with the answer faster than figuring out exact figures. They can estimate, then if needed go back and figure out the exact answer and see how close their results were. It is a great learning process and tool.

If you have difficulties estimating here is a helpful site: http://www.aaamath.com/est.htm 

Math for Elementary Teachers I 1510

Reaching a Consensus

After viewing a video watching a classroom figure out an answer to a problem, without too much help fromt he teacher was an eye opener. There is a teacher in the classroom who is there to help the students, but not give away the answer, after seeing how well it worked and how much the students helped each other solve the problem when one was wrong, showed me a new way of teaching. This is the video that I watched demonstrating this technique.



I think reaching a consensus is a great way of teaching, instead of just asking for the answer right away when they don't know it. This way they are hearing several different ways other students have found the answer and seeing different ways of the thinking process. Although, I can see how it might be a little confusing if they are getting the wrong answers for themselves and other students, this is where the teachers presence and leading them in the right direction without giving away the answer is so much more beneficial.

The students in this video had so many different ideas on how to  figure out how many dogs each person had, almost every student had a different way of thinking and showing a different way to come up with their answer, whether it was or wrong it was how they thought. Then when talking with other students they were able to see the correct way to solve the problem and agree on one set answer.

Math for Elementary Teachers II 1512

Mean, Median, Mode and Mid Range.

Sounds like a bunch of words that have nothing to do with math to me, well at first they do anyways. When I think of figuring out the Mean of a group of numbers I usually call it the average. It is the same procedure and gives you the same answer yet it has another name, why? To try and confuse me? Or is there a real reason that has logic behind it?

Median to me automatically makes me think of middle, which helps me remember what step this is; the middle number in a list of numbers. Nice and easy, especially if the list of numbers is an odd amount!

Mode, why not just say most? That would be easier for me to think of when I am trying to find a mode in a group of numbers. No mode, would mean there are no numbers that repeat themselves, so again most would still make sense (to me anyhow) because if there are no numbers that appear MORE than any other number then the answer is no.

Finally, the word Mid-Range, which for some odd reason makes me think of home on the range every time I see it. Kind of makes me chuckle, but of all the terms listed in this blog I would say this one is closet to what you are actually trying to figure out. So whomever made up these terms for these procedures gets a thumbs up on mid-range :)

Here is a catchy little tune for remembering these procedures: http://www.youtube.com/watch?v=oNdVynH6hcY

Math for Elementary Teachers I 1510

How fun are Egyptian numerals, Babylonian numerals Mayan numerals. I can only imagine how long it would take to actually work with these graphics, but they do look cool. I've never actually worked with numerals other than numbers and I found this part of the chapter pretty interesting.

When using Babylonian numerals, for example, I did first have to learn how to multiply by the base number which at first was a little challenging. but after doing a few problems I caught on pretty quickly. It also took me a few minutes to see where the position of a digit is in relation to its value. It does take a little longer and a somewhat drawn out process of finding what each graphic equals then multiplying then adding, but none the less it looks cool so I like it!

I wonder what it might have been like to live in a time where I would have actually used these graphics for math problems. I also wonder if what I am think about them is what an ancient Egyptian might be thinking about our numbers and how we do math. A question I will never have the answer to, such is life!

Here is a short YouTube video that talks about Babylonian numerals. http://www.youtube.com/watch?v=fPHBeYtp1Tw&NR=1

Math for Elementary Teachers I 1510

So it is the second week of class and I am already starting to feel lost. Numbers can be very overwhelming sometimes. I usually enjoy math and find it fun to solve a problem and get the right answer. But somethings in math I don't think I will every fully understand.

For example, this week we have been learning about sets, subsets, intersections, unions and a lot of other stuff I one minute I think I am understanding things and the next I find myself at a loss and very confused. Does anyone else seem to have this problem? If so what or how do you figure things out? Or if you understand these mathematical words and procedures that go with them can you please explain them to me in a light that I will possible understand! Please :)

A website that I sometimes use to help me through some questions I have is http://www.onlinemathtutor.org/help/math/sets-intersection-union-subsets-disjoints/ it gives a few definitions of things, and shows what symbols mean, and even has a spot where you can do math problems.

Math for Elementary Teachers II 1512

Analyzing data, where to begin right? After going over this weeks topics I feel like I am seeing things for the very first time. Ways of graphing and plotting and the what have you's on how to put information into a readable and usable form.

But if it's supposed to be so readable why am I some what if not more confused by the data shown in a certain ways than I am just looking at the numbers? For example a back to back stem  graph; am I the only person who has never seen or heard of this before? When or why would anyone use this or how would  this be an easier way to look at numbers?  When I see a list of numbers all missing the first digit which I then have to reference back to to the stem I am like what? To me it just seems more difficult or easier to misread a line and have a wrong number.

However, a way to show data that I do like is in a graphical way. Like a dot plot or a circle graph. It seems so much simpler to just look and see what you are looking for. Now I know not everything can be put into a graphical display for but wouldn't that be nice? It would be for me anyways :)

Here is a fun little website that you can make all sorts of graphs in easily, it's actually kinda fun too!
http://nces.ed.gov/nceskids/createagraph/

My First Blog, EVER!

Hello! This is my first blog and I am very excited to be learning something new! My future blogs are going to be math and teaching related and I hope to learn new things and even teach you a few new things with these future blogs!