Thursday, July 21, 2011

I'm so dizzy, my head is spinning!

Today we are going to talk about movement in geometry. Why do we need to know about movement in geometry? It helps us to know how shapes stay the same but can move positions. Just because they move- doesn't mean they necessarily become a new shape. The movements are as follows, with their mathematical term.

Slide: Translation-Imagine opening a window- how it slide up and down

Turn: Rotation- Imagine a Ferris wheel turning in a circle or the turntable inside a microwave.

Flip: Reflection- Looking into a mirror or pressing a inked stamp onto a piece of paper

Please review the following video to gain further understanding on the idea of slide, flip, turn.

These may seem like basic terms to many of us, but to students this concept can be quite difficult to obtain. If you study the below image you will see how these different movements can be easily confused, especially for the movements of slide and flip.

If you have a child who is going through these lessons in school, it would be an excellent idea to draw or even act out these movements. If you draw it out consider using a dot or line paper to show how the shapes are the same size and can even move on the same axis point (turn only on one corner). If you want to act out the movement you can try sliding across the floor or opening a window. You can open a door or pivot on one foot to show flip (from front to back), and for turn you could tell your dog to lay on the floor and then turn him to a 90 degree angle- or just use a pencil or other object to demonstrate this action. Good luck and whatever you do- have fun with it!!

Probability Continued,,, Nonuniform Sample Space

This posting was saved as a draft back from our original probability discussion at the beginning of July and unfortunately did not appear "officially" until now. What are the odds of that happening?! Well, in this case they were pretty likely. So, this is a continuation of our probability discussion. Remembering back we worked with the likelihood that an event would happen equally (uniform sample space). Our example was what our odds were of getting a 6 on a regular die. Today we are going to continue learning about probability- and we are going to find out how to determine the odds of an event happening.

An example of a nonuniform event would be to pick a day of the week using only the first letter of that day (ex. M=Monday, T=Tuesday, S=Saturday, etc.). This becomes trickier since "T" can also stand for Thursday and "S" can also stand for Sunday. This means that "T" or "S" is twice as likely to happen than M, W, F.  We can look into this further by imagining we have a bag with 5 red balls, 2 green balls, and 3 blue balls. What are our following probabilities?
a) Red?     b)Blue?
b) A primary color?

What do we know? Well, we know we have a total of 10 balls. To solve the first question we need to realize- of those 10 balls 5 are red. So, our odds would be 5 out of 10 which we can then reduce down to 1 out of 2. We would draw a red ball 50% of the time. What about blue? There are 3 blue balls out of 10. We can't reduce this down so 3 divided by 10 is .30 which would mean we were drawing a blue ball 30% of the time. What about our last question, if we drew a primary color- red, yellow, or blue. We would have 8 balls that were primary colors and 2 that were green (secondary color). So 8 divided by 10 would make it so we drew balls that were primary colors 80% of the time.

A formula for figuring out Nonuniform Sample Space is as follows:

P(A)=              Measure of the outcomes associated with the event A
                        Measure of all outcomes in the sample space S

To associate this with our last problem we would say:

The Probability of drawing only primary colors is:  8 primary colors
                                                                   10 total colors

So we have 8 over 10 which is equal to .80 or 80% (we can reduce down to 4 over 5 to also get .80)

Included here is a link to a webpage that discusses the odds of an event happening and also the odds of it not happening. It also gives formulas used and great tips.

Additionally, we can extend the idea of pulling more balls out of the bag and probability continued- with the following video:

Monday, July 18, 2011

Making Math Fun with Toondoo

I have yet to meet a person who hates cartoons. Cartoons do many things, they make us happy, they help us think about our world, they make light out of dark or hard situations, they really help us see things in a new perspective. Why not take this concept into other subjects in school or life? Our teacher has actually done this as a project that each student in class must do. We are to create 6 cartoons that incorporate math in some sort of way. I think this idea is clever and fun for students. In a way we are sending a subliminal message to them (about math) and they are enjoying it the whole time.

For class we were supposed to use formulas, theories, or any other idea we could come up with to help students better understand the concepts we as teachers or parents would want them to learn. My example of this is perhaps a little on the stronger end- but I wanted to work with fractions. I thought about the kind of teacher who would be bold and make math fun. Then I thought of a way to do fractions with that teacher involvement. The following cartoon is the one I made for class:

So, although the idea (your teacher being made magically into thirds) is perhaps a little strong- I don't know what student wouldn't want to see that magic trick. The nice part about creating your own cartoon using a program like Toondoo is that you do not actually have to perform the magic trick and your students will still understand the basic concept behind it.

Any way that we can make math fun or school subjects of any sort should be taken advantage of. It is a simple process creating these cartoons and it is very fun. You can even make your own characters that would look similar to your students in your own classroom. So, good luck and get creative!

Tuesday, July 5, 2011

Do you see Geometry?

Why are we interested in Geometry, why does it even matter? What is Geometry even? These are some basic questions we will be looking over in this post... to help us better understand not only geometry- but also they way we comprehend information. First off... what is geometry? Well, Merriam Webster gives us the following definition:

Main Entry: ge·om·e·try
Pronunciation: jemacron-primarystressäm-schwa-tremacron
Function: noun
Inflected Form(s): plural -tries
1 : a branch of mathematics that deals with points, lines, angles, surfaces, and solids
2 : 2SHAPE <the geometry of a crystal>
 ("Definition of - Merriam-Webster's Student Dictionary." Merriam-Webster's Word Central. Web. 05 July 2011. <>.)

I've found geometry to be more than just this simple explanation that talks about lines, angles, surfaces, etc. Geometry connects us to our world. You might say, what does she mean by that? Well, that brings us to our next couple of questions... but first- lets study a picture I took out at the beach yesterday...

The picture may be familiar to many of you... but if you study it closely you can start seeing how shapes form in our minds to create balance and movement when we look at things. Our brains are so well trained to see shapes and put them together... making them blend harmoniously. People have studied geometry for thousands of years... it is a subject that connects us through time as well. Imagine what life would be like without shapes, without repetitions and patterns, without measurements? How would a person be able to tell which room was the biggest in a house? What if we couldn't measure area? How would we ever figure out how much carpeting or tile to put on the floor... how much gallons it would take to paint walls? These are everyday functions we fulfill fairly fluidly for those of us who remember geometry.

Take a look at the above picture again, and see if you can find shapes that form in the image. What shapes have you found? Do you see lines... how about points that might intersect? Below is the same picture... but with the shapes and lines that I saw right away. It took me seconds to see the geometry in this image... but without actually thinking about it and tracing the shapes- I wouldn't have realized there were triangles, circles, a quadrilateral, or even (eventually) intersecting line segments. By taking a closer look I realize how my mind blends all of these things together in a way that makes the image flow and balance.

Granted, there are many scientific and mathematical necessities to geometry... but by taking a quick look at geometry in our everyday lives- we gain greater appreciation for not only what has been taught to us in our past, but the way our brains function... we start to see triangles and guess what degree of angle makes them up. We see lines and wonder how long it will take before they intersect, or we see shapes and wonder how big the area is within them- or how many gallons it would take to fill them. These thoughts connect us with mathematician throughout history... and with our world around us.

I will end this post with a beautiful quote I found quite fitting:

Galileo Galilei. 1564-1642. Italian astronomer, mathematician, and physicist.

The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
Opere Il Saggiatore

Probability in a Uniform Sample Space

Have you ever wondered what the likelihood of rolling a particular number or set of numbers on a dice would be, or a particular card from a deck would be? We can make it even simpler and say, "What is the likelihood that a dice when rolled will land on the number 6?"

I'm going to review over the basics of probability in our post today. Let's take the last question, what is the likelihood we would get a particular number on a dice roll? Now assuming we are using a regular 6 sided die we see the following possibilities: {1, 2, 3, 4, 5, 6} Those are really the only numbers we can roll since they are the only numbers listed on the die. So, if we roll a 1 we have 1 out of 6 chances of rolling that 1. If we roll a 2- that is still only 1 number out of 6 possible and we still have a 1 out of 6 chance of rolling that number. Same thing with 3, 4, 5, and 6. Since each possibility is equal, we call that a uniform sample space- all possibilities are uniform.

What if we wanted to take it a step further to see what the likelihood is that we could roll an even number? This event would mean we could roll {2, 4, 6} out of the possible total of {1, 2, 3, 4, 5, 6}. Since there are 3 possible outcomes from a total of 6 we would say the ratio or likelihood of rolling an even number would be 3 to 6.

We can use the following formula to help us solve probability with uniform sample spaces.
The probability (P), of an even  (A)- appearing as P(A) has the following ratio:

                          Number of outcomes associated with the event A
       P(A)=             Number of outcomes in the sample space S

or if we incorporate this into our problem above we get:

Probability (P) of rolling an even number (A)=     3 {2,4,6}
                                                                       6 {1,2,3,4,5,6} 

You can break this down and get 1 over 2... which would also mean 1/2 (or 50%) of the time we could roll and even number and the other 1/2 an odd number. The following video is included to reinforce this idea. Remember, this is probability at it's basics. If we are working with numbers that do not have equally likely outcomes we use a different formula- I will try to discuss this idea in a later post.

Tuesday, June 28, 2011

Questions & Standards

This blog is similar to Please, Not Math! the original blog page and follows along with a summer college course being taken (Math 1512).

Many of us have various questions about math, like... how do we know if what we have been taught is correct? What are our kids learning now? How do we know if our children are learning all the things they need to know? Well, besides being and involved and informed parent- we can look into something called the state standards for Math.

These standards address what our children are learning, and what teachers are teaching for each grade level.
MN Math Standards
Scroll down the page on the link above and you will find a pdf document of the standards for this year. This is for Minnesota specifically so if you want to you can research national standards as well as your own state standards.

Lastly, I suggest reviewing your child's homework with them. By doing this you can challenge yourself to learn math again- remember from the previous page? Challenging yourself. If you already know math, you can refresh your skills by helping your kids- and they will love it! They will have a buddy to help them with their own challenges they may face in math class!