How many ways can we combine or mix up some fractions? A few. We can add them, multiply them, divide them and subtract them. I like this kind of math, it feels like a fancy occasion. Really? You doubt? What does one wear to a fancy schmancy occasion...if you're male and like to dress up? A bow tie!! See? Fancy. We use the bow tie when we ADD or SUBTRACT fractions when the denominators are not the same. A bow tie fixes it all up and neatly, I might add. See? I made this little demonstration for you...
Works like a charm because bow ties are just that, charming. Even Pee Wee knows that bow ties work.
Bow ties are a great way to show kids how to get matching denominators in a snap. Of course, they can also use a factor tree to find the least common multiple and certainly we can teach them that method as well but the bow-tie is just so neat and tidy.
We can also multiply fractions and in my humble opinion, this is easier than adding them. Multiplying fractions is so easy that you just multiply the numerators and multiply the denominators. It is really that easy. Don't believe me? Then google it. This rule is everywhere out there on the internet, not just in Wikipedia either. I even found a game that could be used in a classroom setting. It is based on the Millionaire game and the teacher can use it with a SMART board since it would be fun for a touch screen application. There is a solo option or a team option. Try it here at Math-play.com
We know that division is the opposite of multiplication so dividing fractions just involves a little twist or flip. When we have two fractions to work with, invert the second and multiply. Something like this...
Very simple process. Why does it work? Because division is the inverse operation of multiplication. And where do I go to make all these lovely visual aides? Picnik, of course! It is a great resource to modify photos and create pictures with some personality.
Estimation is not always for jelly beans guessing contests. No, no. Estimation is an important skill to have when doing addition, subtraction, multiplication and division. There are many strategies to help students estimate. Why do this? It can be a useful first step in order to get us on the right track as well as check our final answer.
What is important in estimation is knowing what you want to estimate...amount,lentgth, volume, an answer to a multiplication problem...once you know what you're looking for you can employ one of many strategies to determine your estimated answer.
Let us begin with answers to multiplication or addition problems...a superior strategy is rounding. If you are given a set of funky numbers to add or multiply you can use rounding to make them more friendly. So, say you've been given 23, 31, 19 and 15 to add. They are snarly numbers that don't add real nice at first glance. So you round them to the nearest 10...like 23 is nearest to 20 and 31 is nearest to 30. Those were easy but what about that mean looking 15? The general rule is to round UP. If we use that rule then the 15 would be thought of as 20. The 19 is snuggled right up next to 20 so that would also be 20 and we could then easily add 20+30+ 20+ 20 which equals 90.
In some cases, the estimated answer is good enough- like when you're at a big box store and throwing items willy-nilly in your cart and you have 100 bucks in cash and need to estimate if you have exceeded your wallet's capacity. If those numbers were prices I'd be able to stop because I'd know that I had enough money but I'd better high-tail it to the check out because one more clearance bargain might push me over the edge.
If they were numbers that represented diamonds, rubies, emeralds and opals, I'd want to know exactly how many precious gems I had in my possession. So I'd check it with my calculator and see that all told I had 88 gems and if I lend them to the Queen of England, I'd better get all 88 back.
So say you want to estimate the number of donuts to bring into work each day for a week and you remember than the last time that donuts were brought in there were 34, 27, 29,36 and 31 donuts consumed. You could use the clustering technique to estimate how many you should bring- all those numbers sort of cluster around a number...which looks like 30 to me. And since the donuts were brought in for five days you could multiply 30 times 5 and determine that bringing in 150 donuts will probably see you through.
Estimation, as I said, is not just for jelly bean guessing games, it is for donuts too.
Now you may ask,"Just how can she tie the surface area of a cylinder to Art History?" Of course, that would be strange that you would have actually asked that, unprompted, but since I told you to, I bet you are wondering...
Andy Warhol is just the crazy, modern artist type that can help me out. Not that he was really into formulas to determine surface area, he wasn't. He was more into the mod scene and icons as muse. He used many very familiar objects as the subject of his art and really who else could come close to Marilyn Monroe but the Campbell's Soup Can. Not really the same appealing curves but for mathematicians, the surface area is what really gets them going.
So the question remains...how do we find the surface area of a cylinder and learn about Andy Warhol in the process...lets ask Andy himself.
Thanks so much, Andy! I'd shake your hand but I know that you absolutely hated to be touched. (See, I told you that you'd learn something about him!)
Okay- I'm going to go and grab an actual can of soup and give you the measurements and then we'll discover the surface area together!! Okay, I have in my hands, well that's not exactly true since I'm using my hands to type. I have next to me on my desk...a can of Campbell's Tomato Bisque Soup (how fancy!) and it measures 4 inches tall (h) and the diameter is 2 and 1 half inches or 2.5 which means that the radius would be half that or 1.25 (r). We will use 3.14 as Pi and proceed according to Andy's instructions...
SA= 2(3.14)(1.25sq.) + 2(3.14)(1.25)(4)
which equals...41.21 square inches
Which then means that you can find the total amount of tin represented by Warhol's 32 soup cans from 1962
Since one can equals 41.21 sq. inches then 32 cans would be 1318.72 sq. inches. If you'd like to learn more about the soup cans you can check them out here at the MOMA. One of the basic tenets of Andy's pop art ideal was that he took mass produced images and had us look at them in a different way. He repeated images and that can be mathematical, after all, he had to know that 32 would end up in an 8x4 square instead of creating 41 cans which would have been hard to square.
Measuring marigolds might just be a great idea when introducing the concept of measurement to young students. I really like the idea of using non-standard units of measure so the marigold could be the ruler for that matter! But seriously,folks, I do agree that young students will grasp the concept of measuring much better if you begin with things they know. Have them use their feet or their hands to measure their desk or chair. In my observations I have seen teachers use things like ice-cream scoops to measure the height of the kids in the class, marking off each student on a wall chart made of a super stack of construction paper scoops. I liked that one a lot. The kids will all get it and will probably tell their parents for the next year how many scoops high they are.
Teaching measurement skills will work so well with estimation skills that I can already see a load of activities to try out...take a guess how many spaghetti noodles tall you are! Guess how many Ritz crackers are in a stack as high as your chair! Take a stab at how many juice boxes it will take to fill an aquarium! The best part would be actually carrying out these crazy stunts to see how close they came to the correct answer.
The idea is to get the making reference points for size, weight and volume before they begin to practice with standard units of measure. So having access to various containers to see how many of this fit into that will be important. They can also begin to compare their own measurement with that of other students and adults. It would be interesting to have the students estimate how tall older students might be in 5th grade, 8th grade and 12 grade and then go and actually measure a group of students to find out if they came close. I think I'm getting an overload of ideas for this topic and that, as Martha Stewart would say, "is a good thing." Yes, it is.
By the way, if you estimated that Martha is tall you'd have been right- she is in fact, 5'9" tall or you could estimate her at around 23 McDonald's french fries (which in turn I'm estimating at 3 inches...)
Once students are getting comfortable with standard units of measure like feet, yards, pounds (or grams, centimeters and liters if you are so inclined), you could use these estimating skills in other subjects as well. Make a treasure hunt for social studies and have them find things that measure certain amounts...like find an object that would have been on the Great Plains that would have weighed 2000 lbs or something that would have been a foot tall standing or half that walking...the possibilities are endless.
Can you guess which animal I was looking for in the questions above? Thanks to all the references you've made over the years it should be easy to figure that the Buffalo is taller than 12 inches so it must have been the one that is around 2000 pounds.
Of course, I can't leave without giving you another video that should appeal to young students as they grapple with the concept of THE FOOT. Enjoy.
I wanted to take a post and consider what is out there for teachers to use in their classrooms. Mainly I'm looking at different teacher supply sites and what they offer for mathematics support; things like manipulatives and other hands-on materials to help students grasp (literally) the concepts of numbers. The go-to site for me has always been Discount School Supply. It is an online store that I made use of when I did daycare. They have a great selection of counters and other math supplies.
Learning that important one-to-one number correspondence is part of the kindergarten world. Counters are certainly a great way to help the students learn that the written number 4 is equal to four of something. There are counters that resemble fruit, pets, dinosaurs, cars and trucks and teddy bears. Counters often come in different colors and can then be used to help sorting skills. Having a variety of these counters in your class will help engage students in number theory. The various shapes can also connect math to other subjects- say you are reading a story about pets during circle time- have the household pet counters during math and make that connection back to the story. Eventually they can be used to discuss sets- you can set up a station that would illustrate the intersection of blue animals by having a few different pets in a few different colors- you can ask the kids to set up the intersection of dogs or cats or yellow animals.
I think one of the most important considerations to make when selecting manipulatives for your class is making sure you have open-ended, multi-use objects. One of the best would be the interlocking one inch square colored blocks. Not only can they be used as the counters above, for making one-to-one correlations but can also be used as simple base 10 blocks and for measuring and estimation. In my job at school we often have a basket of these available to kids when they are pulled out to work on math individually. They can estimate the number of blocks an object will measure and then use the blocks to check their estimation. Students have them available to help with addition and subtraction. They also use them to show "counting on" concepts. For instance, you want to student to add five and four. You give them the already connected block of five and have them count on, or add on four more blocks. These little cubes are really a wonderful item to have around.
As students progress and begin to learn about addition and subtraction a number line is an important tool to make use of. There are plenty of ways to make cheap and easy number lines to have at each desk or table spot but this floor model caught my eye. Since we have learned in our text about helping students to visualize walking up and down a number line as they add or subtract, it makes sense to have them really do this. I think you could easily have two of them and just add negative signs to one line once students are learning about negative numbers. Older students may benefit from getting up out of their desks to actively engage themselves in the concept. Of course you can take this idea and create it yourself out of masking tape and construction paper numbers.
Here are some other sites that sell classroom supplies. Since money is always an issue in stocking the classroom I think checking out these sites a great way to spark inspiration for something you could create yourself or a new way to use something you already have.
This may be one of the most engaging topics you could hope to cover with students. What are tessellations? Well, they are a pattern made up of geometric figures that act like a jigsaw puzzle only the pieces are all the same or are made up of a few interlocking shapes that cover the plane they are on. Now doesn't that sound like it will fascinate students? Yeah, I agree. The definition hardly does justice to what they can be in real life. I find that is often the case with mathematical terminology; very dry but when interpreted into the 2 and 3 dimensional world, very cool. Tessellations are just like that.
Of course, one of the most famous interpreters of this concept is M.C. Escher. He was a master at putting this geometry concept down on paper and allowing us all to step inside. His brain operated within a geometric framework. So often we see math as a single subject, an hour out of the day but when we are confronted with Escher's work we can see that it can be part of ourselves and not separate as many students and teachers see it. In this piece we can see that Escher shows the way to move from a simple tessellated polygon and evolve it into an image of a lizard- something we all are familiar with in our 3-D world. He bends the frame and slowly changes the tessellation. This is certainly a great example to show to students.
If you think Escher will amaze students you "ain't seen nothin' yet!" Tessellations can transform a 2 dimensional piece of paper into a three dimensional object. Nobody does this better than the creators of paper folding, or origami. In the following video from Japan you can see how this very intense paper folding can create something amazing out of something very ordinary.
So we can go beyond this one more step. The kinetic tessellated sculpture. Theo Jansen creates wind propelled sculptures that make use of tessellation concepts in their design. You can see how he repeats interlocking polygons to build the frame for these astounding moving creations. In teaching shape geometry it would benefit students to see how a simple concept can evolve. They are limitless opportunities to express the concepts they are learning about in their classroom.
We would all like to know just what our risks are when going through life. While these commercials are funny they also show that probability is a part of the economics of our lives. We select where we live, looking at the risk of floods, mudslides or massive amounts of snow. When we become parents, the idea of the risks of how a car will or won't keep us safe becomes important. Some people like to deny probability when they buy and believe in the lottery. While being able to predict acts of god may not be part of our everyday lives, we can use the formulas for probability to help us make smaller decisions and work out necessary issues.
This topic was a tough one for me; especially combinations and permutations. What is the probability given a set of 10 questions on permutations and combinations that Kirstin would select the incorrect formula for each question? I would hazard a guess that it may be the inverse probability of either of the commercials above. Since I didn't fair so well on mastering the concept that ORDER MATTERS for Permutations while a combination says, "whatever;" I will try and make it clear for all and my own self, here and now.
So say we are watching a race...a horse race. Now we all know that CHARLEY HORSE wins the race each and every time you play against your brother...but in this race we want to know out of the 8 horses running, how many different orders of win, place and show can occur. So since the ORDER MATTERS we use that handy dandy permutation formula.
Once we've entered all the pertinent information such as n = 8 and k= 3 and we've done the factorial for each we get ...336.
On the other hand, combinations are less detail oriented. Order doesn't matter. Just pick already! The difference between the permutation formula and the combination formula is found in the denominator.
nCr = n!
k!(n-k!)
Often it is those little details that make all the difference. So for a combination the formula is basically the same but we multiply the denominator by k (or the number of selections). If we look at the same numbers but change the story - say we have a bag with 8 different kinds of chocolate truffles inside and we're going to pick three out we don't care about the order, we just want the candy. How many different COMBINATIONS could we have? 56. Isn't that a lovely thing to imagine? 56 different combinations of truffles. sigh. Oh, where was I? Combinations...for when you don't care who comes first.
Father Knows best- my math skills were busted by my very smart Math Major Dad who let me see that I'd made a little mistake in the above example- the answer was not 120 as I had first figured but 56- I even made it happen on my calculator so I see the error of my ways- Thanks, Dad! "Sure thing, Kitten!"
This seems like a lovely topic to introduce one of my favorite all time math sites ever. It is called Brightstorm and it has thousands of videos of master teachers explaining topics in math and science. This resource would have proved invaluable for me in junior high and high school but alas, my 64K PC jr. didn't even exist until I was a senior in highschool and just wasn't meant for the internet which as far as that goes, didn't even exist either! But now it does and Brightstorm is amazing! They have recently begun to work with the current texts being used across the country. I think this service is one that all students and teachers should use. What is the probability that I will use Brightstorm again and again? 100%.
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. .
