In my first life as a college student, I was an art major. I know, I know, had I only focused on statistics and probability I would have seen that art majors don't really make the big bucks. Be that as it may, this week's topic on Geometry and more specifically, the centroid of a polygon, reminded me of one of my all time favorite artists, Alexander Calder. I have always loved his mobiles and their mid-century modern look and the colors. Little did I know how much geometry played a part in his creations. That is, until I read about finding the centroid, or balancing point of a triangle. Immediately, I was seeing his amazing mobiles in my brain.
Amazingly enough I was not the first to make this connection. Thanks to Mr. Google I was able to find a really neat math site called Math Cats, that challenges students to virtually balance objects and create their own mobile. I also found a lesson plan that combines Calder's genius and a geometry lesson. There are many more sites that have paired Calder, a mechanical engineer and artist, with the geometry he used to create such wonderful pieces like "The Spinner" that can be seen right here in Minnesota at the Walker Art Museum.
I think that the more we can make real connections for students about how math can be used and interpreted, the less we may have to struggle with enthusiasm and interest. I know that my memories of geometry class consist of hoping not to get called up to the board to write out a lengthy proof and the subsequent explanation of that proof to the class. Had I seen real world applications such as Alexander Calder, perhaps I would have felt more warmly towards it. Let me end with a lovely video that shows his balancing act in motion.
Wow! I have actually enjoyed this topic! I find it fun to draw a factor tree and bring it all down to the lowest low. I found a nice little game at STUDYJAMS that tests your knowledge of prime factorization. What is nice about this site is the tutorial section or Step by Step area that explains the process. I can see using this site in class to supplement the topic. In our district we have a wonderful teacher in 3rd grade who really uses his classroom homepage to offer on-line resources to help his math students to practice their skills at home. Other teachers often link to his classroom page since he's done a great job of finding things like this. I certainly envision using the Internet to connect students with school when they are home and StudyJams would make my list.
The Sieve of Eratosthenes is a great way to help students determine the prime numbers between 1 and 100. Basically, you find a prime number...say 2 and then remove all multiples of 2 like water from spaghetti and move on to the next prime number...3;remove its multiples and so on. Once all the composite numbers have been drained away you are left with nothing but prime noodles numbers. Who was this Eratosthenes? Well, he was a Greek mathematician from long, long ago (say 276 BC to 195 BC) and from the looks of it,
he had a large noodle himself. Thanks to his technique we can use his sieve, or colander, to drain away composite numbers. Efxaristo, Eratosthenes!
The big question remains...what do I do with this prime factorization information? Say for instance you want to figure out the Greatest Common Factor of two numbers. Why would you want to do that? If you had two great big piles of things that needed to be grouped evenly for instance. Then using the prime factorization will entail comparing the two numbers prime factors and combining just those factors that show up in both. On the flip side you could use them to find the Lowest Common Multiple of two numbers (or more if you are so inclined). This would mean taking the factors common to both numbers raised to the highest power of that prime that happens in either number. Then multiply the prime factors out and Viola! You have found the Least Common Multiple. Which can be used to determine when two events might occur simultaneously.
I admit it. This topic confounds me. I had a really hard time with the whole base two idea. The book explained it but not much made it in my head. I'll try and untangle this issue with other sources. One source I found here discusses a "base table" when doing the conversion from a base 10 to any other base. So for instance...we could convert the base 10 number 137 to a base 6 number by making the base chart...
6^3 \ 6^2\ 6^1\ 6^0\
216 \ 36 \ 6 \ 1 \
So the ONLY numbers that can be part of base six are 0, 1, 2, 3, 4, 5, 6 .
The first thing to do is figure out how many groups of 6 will fit into the number 137. We can do this with a standard division algorithm 137/6 = 22.8 (we can ignore the .8 and just focus on the 22)
This means that there are 22 groups of 6 in the number. But since base six doesn't have a 22 we'll have to break that down a little more...
Next we'll figure out how many times 6 goes into 22...well, that would be three, or 3 .
Okay...I just got lost myself. I'm going to explain it this way then...
How many groups of 36 can we break 137 into? why that would be three!! So I can put a three under the 36 in the base chart. 36 x 3 = 108 so we have some left over. That would be 29 and some change left over so we have to determine how may groups of six go in there and my brain says 4! (put that 4 under the 6^1 column) 4 x 6 equals 24 so we have some left over (5 to be exact) and we can add those to the 6^0 column... which makes our base table look something like this...
6^3 \ 6^2\ 6^1\ 6^0\
216 \ 36 \ 6 \ 1 \ \ 3 \ 4 \ 5 \
which makes our final answer 345 six
Whew...I did it! I wasn't sure I could but using a little of the ideas found at this website - namely the base chart- and some other brain figuring I came up with the solution. How do I know it is right? I googled it, of course!
The ideas of other bases and alternate number systems was a new trick for this old dog. I found that the system of the Egyptian numbers seemed the most straight forward, most likely due to it being a base ten system. The downside is that the numbers can get very long. The Mayan system was the next best but the way of reading it top down was certainly a novel idea. The most confusing for me was the Babylonian system which works subtraction into the characters for numbers. It also seemed to figure backwards from how my brain wanted it to go and so I was often quite muddled. There are people who truly understand all this and even chose to blog about it...I found one such person here at hyper mathematics.
Here is a site that discusses Egyptian numbers. Overall, I think this will always be a part of math that makes my head ache and my hands sweat but perseverance is the name of the game.
Seeing as I love videos and we've all had such fun with the conversion of hindu-arabic numbers to Roman numerals, please enjoy one of my favorite catchy tunes by Trout Fishing in America- sing along!
I posted this because it is funny and because it reminded me of the age problems we worked on in our Teaching Math to Elementary Students 1512 class. It also gives a taste of the confusion over proportion and ratio. Sometimes what seems to make sense in our minds may not work out on paper. We do need to teach basic facts along algorithms that work every time in order to avoid raising students who have an Abbott and Costello understanding of math.
So, let's take an example of the age problem from the "other" math class. I will use it to illustrate my take way from Chapter 7. The following story problem is based on actual people, only the names have been changed to protect the aging.
Agnes, Gertrude, and Prudence are three sisters who live in a decrepit Victorian mansion. Agnes is the youngest and Gertrude is one year older than Agnes, Prudence is one year older than Gertrude. Together their ages add up to 294. Find their ages. The first step is to determine what the problem is asking us. And clearly the question asks us to expose their ages. Rude? yes. But that's what they want.
Agnes is the youngest so we will start with her and use X to signify her age.
Gertie is one year older so we can call her X+1
We're going to call Prudence X+ 2 because that is easier than calling her one year older than Gert which would look like this X+ 1+1 which means the same but takes longer to write and may end up being confusing. So Prudence is X+2
We know that altogether the three add to 294 so we'll put that number on the other side of the equal sign, all by itself.
Our plan looks something like this
X + (X+1) + (X+2) = 294
We have a plan- now we should put that plan into action-
We can add all of the X's together which equals 3X and then add the whole numbers together which gives us 3- this changed the left side of our equation this way
3X +3 = 294
The idea is to isolate the variable- or the X so we expose what is means. To do this we have to take three away from each side -
3X = 291
We're almost there (for the first part anyway) - divide each side by the 3 and that will leave us with the value of just one X (or Agnes' age)
X = 97
But wait!! Finding X is great and all but was that all we needed to do? Don't spike the ball just yet...LOOK BACK and see what the question really was....oh yeah, they wanted all the ages, not just Agnes'.
Since we know that Gert is one year older than Agnes we just add one to 97 giving us 98. We can then add one to Gert's age to find out Prudence's age of 99.
The thing I like best in chapter 7 is the chart illustrating how to approach a problem and work towards a solution. I would take this idea and create a poster to hang in the classroom to help students during work and test time.
It goes a little something like this...
I know that using this process during instruction will help give students a structure to rely on when approaching a new problem. If I hang this poster in the room and allow it to remain during testing it may help trigger the students memories from the lessons.
Let me introduce this blog. It is my journey through two classes on Teaching Math to Elementary Students. Now, I'm not saying I'm quite on the level with Ma Kettle but math does have a tendency to make me question myself. I like the idea of learning how to help kids reach an intrinsic understanding of the concepts of math. I did a lot of memorizing in school and not a lot of understanding. Often times I would remember what I needed for a test and then promptly flush it away, only to have to relearn it all over when I needed it.
I plan on teaching the younger grades with Kindergarten being my first choice. I love how the light turns on when they grasp a concept; it's clearly visible on their faces. One of the concepts from this weeks reading has to do with the acquisition of whole numbers and one to one correspondence. "The mathematics curriculum for the young child is built on everyday classroom experiences that exhibit one-to-one correspondence. For example, students record attendance on the attendance board by placing one token for each student present and use one-to-one correspondence to determine if anyone is absent." (O'Daffer et. al. 2008 p. 60) I like this idea of working these basic concepts into the daily routine. It takes math from being a subject students work on during one hour during the day to a language they weave into their experiences.
In my observations in the classroom, I have noticed that teachers have creative ways of "hiding" math in their classroom. One classroom I observed placed a new die cut leaf on the wall each day to measure how many days they had been in school. As the year progressed, the number line grew and they used it to practice skills like counting by 2's and 5's and 10's. The teacher had used a different color marker for the 10's so that they stood out.
Another interesting way to weave math into the daily routine was to use a math game called, "I have...who has...?" As a way to start off morning meeting. Each student received a card with a math equation based on what they were working on such as addition facts. As they read what they have they ask who has a number- everyone has to look at their problem and determine if they have the equation that equals that number. The game is set in order so that everyone will have their turn. A variation I observed during Valentines Day was creating half hearts with an equation that one student would say and the heart with the correct answer would step forward, they would place them together on the floor and give a welcome greeting to the other student.
Works Cited
O'Daffer, C. C. (2008). Mathematics for Elementary School Teachers. Boston: Pearson Education Inc. .
