combining fractions- 1510

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.

In my estimation...1510

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.

Another Art History/Math lesson - 1512

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.

Measurement and the young student- 1512

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.

Number theory and manipulatives - 1510

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.

Homeroom Teacher
Really Good Stuff
Lakeshore Learning
Oriental Trading

Tessellations- 1512

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.