Happy Halloween

Halloween Math Part 2

  Here is the second part of the Halloween math activities.  It's always good to have a variety of possible activities for this time of year.

1. This activity is designed to have students determine which candy has the most calories.  In addition, students are required to create scatter plots.

2.  The American Chemical Society calculated that a 180 pound person would need to eat 262 pieces of fun sized candies or over 1,650 pieces of Candy Corn to reach a point of having a 50% chance of dying due to a sugar overdose.

3.  Here is an activity associating a quadratic function to pumpkin tossing.  There is a huge contest held back east where people go to toss pumpkins as far as they can using some sort of catapult.

4. Provide a pumpkin outline on a coordinate grid to students and have them create a symmetrical face on it.

5. Provide the coordinates for students to graph on a coordinate plane.  The final product could be a ghost, pumpkin, or even a bat so they get a chance to practice graphing.

6. Here is a graphing activity in which the author calculates the amount of candy bought and given out over a two year period and then breaks all the information down into various graphs.  There are enough graphs for students to really practice reading graphs.

7.  This site had a variety of graphs dealing with Halloween from candy given out to retailers who sell candy to factors influencing candy choice to costume choices.

8. From the Infographic Journal comes a wonderful infographic on 100 years of candy beginning in 1910 and lists the top 50 candies.  This is a timeline type infographic which is great for learning to interpret data. 

9. From Full Deck Design, comes this wonderful infographic showing the top 10 favorite Halloween candies. This information could be used to create several different types of graphs and lead to a discussion of which representation is best to use.

Lots of wonderful math activities that do not rely on filling out "Halloween" worksheets designed to replace coconuts with candies or pumpkins.  I believe when students get into high school, they should be doing graphing activities and interpreting data on infographics rather than doing standard work.

Let me know what you think, I'd love to hear.  Have a great Halloween.

Halloween Math Part 1

Halloween is approaching in a couple of days and its time to look at some math our older students can do.  I get tired of the math being mostly for elementary students so I'm providing information students can use to calculate a series of items.

1. In 2012, $2.4 billion in candy was sold for Halloween.

2. Assume there are 30 pieces of snack sized candy in a bag that sells for $6.00 per bag.

3. Figure out how many bags of candy make up the $2.4 billion.

4. Calculate the number of snack sized candies are sold for  $2.4 billion.

5. If there are 41,131,310 between the ages of 5 and 14, how many pieces of snack sized candy would they each get?

If you prefer having worksheets or more traditional activities, Yummy Math has 12 different activities ready to go. The ones that intrigue me the most are the ones as follows:
1.  Calculating the volume of a variety of bags from rectangular prisms to cones and spheres.

2. One on vampire bats dealing with measurements both customary and metric that students are asked to relate to familiar things.

3.  Based upon given information, students decide which bags of candy are the best buy.

4. Calculating the area of three different shapes of chocolate to determine the best.

5.Students examine Chipolte's fundraiser at Halloween designed to raise $1 million and if its possible.

6. Interpreting a pie graph of holiday sales of candy for several different holidays.

7.  The proportion of pumpkins, puree, and pies using the worlds largest pumpkin.

8.  Calculating the sales of costumes based on population, participation, and money spent.

Finally for today, this web site has a graphic and a listing of candies and their popularity around Halloween time so students can read and interpret data.

Have fun checking these out.  Let me know what you think. 

Warm-up

A Good Laugh

Seashells? What Math Is There, There?

 I remember growing up by the beach.  We'd head out to the beach as often as we could.  You couldn't help but see seashells in various stages of being crushed.  Some were whole, some were not.

At that age, I didn't care about the mathematics of them, I just collected them complete with sand.  Its only been recently when I realized that mathematical equations explained everything we see that I wondered about the math of seashells.

Most shells can be described using some elementary mathematics which with a few changes and some programming it is possible to create 3 dimensional computer renderings of a variety of shells. Imagine rotating an expanding semi-circle upwards around a central point.

The most common equation is for an equiangular spiral or a spiral of equal angles also known as gnomonic. These are mostly the spiral shells that grow as the inhabiting animal grows. The shape is always the same as it grows larger and larger. Several mathematicians did an in depth study of shells and concluded shells were formed by expansion, rotation and twisting which are three simple processes in the mantle of the shell.  It is the opening that gets added to and grows in a spiral pattern so the shell looks curved.

Bernoulli described it as the wonder spiral due to the way the widths of the lines ran from the center to the points on the shells but the amplitudes of the angles formed by those lines and the tangents remained constant.  Descartes figured out the mathematical formula is r()=A e ^cot 
where A is the radius of Theta = 0.

If this were done in a cartesian coordinate system, the equations would appear as:
 x(θ) = r(θ) cos θ
 y(θ) = r(θ) sin θ
 with sometimes a third equation for the 3rd dimension.

This particular formula also explains the growth of animal horns, nails, corals, and snails.  Although they use basically the same formula, there are some free parameters which changes the shape so it might be a bivalve instead of a nautilus. 

There are tons of papers out there with all the actual mathematics describing every equation used to describe the growth of seashells along with those used by the computer to recreate shells.  Its quite fascinating and lots of fun to read.

Have a great day and let me now what you think, I'd love to hear.

Alice In Wonderland And Math.

Alice in Wonderland, a book that has been made into multiple movies over the generations.  This famous book, written in 1865 by Lewis Carroll aka The Reverend Charles Lutwidge Dodgson.  Dodgson taught mathematics at Christ Church College in Oxford, England.

Apparently in 1862 Dodgson and the Reverend Robinson Duckworth were rowing three girls up the Thames.  During the five mile trip, Dodgson made up a story about a girl named Alice who wanted adventure because she was bored.  The girls loved the story.

One of the girls, Alice Liddell, asked him to write the story down.  Two years later, he presented her with a handwritten copy titled "Alice's Adventures Under Ground". It was complete with drawings done by him.

He'd based much of the story on actual places in Oxford and in Christ Church such as the stairs in the back of the college's main hall where she began her adventures.  A year later, a greatly expanded version, now titled "Alice's Adventures in Wonder Land", was published by Dodgson under the name "Lewis Carroll". Within a short time, it was a best seller.  This book has never been out of print and has been translated into over 100 languages.  In fact, a sequel "Alice Through the Looking Glass".

Dodgson, being a mathematics professor, included references to math in the story.  In fact, he is the one who is given credit for creating logic puzzles with his Knights and Knaves, the first always tells the truth and the second always tells lies. 

It is said that many of the mathematical references contained in Alice provide a satirical commentary on what was happening to the field at that time.  Dodgson is classified as a conservative mathematician who did not agree with the changes.  To him x times y should equal y times x but others of the time expanded the known number system.

There was an Irish mathematician of the time, William Hamilton, who put forth the idea of "quaternions" which are used to extend the complex plane and used to describe mechanics in a three dimensional space.  He stated one of the four terms in quaternions is time so time had to be involved in these numbers.

Lewis Carroll addresses this topic at the tea party with The Mad Hatter, The March Rabbit, and The Dormouse when he purposely leaves out Time.  He indicated he thinks people should get rid of the complexities and go back to the old fashioned math.  Dodgson actually eleven books on mathematics but very little of it was new. He was considered an excellent mathematics tutor.  He felt that Euclid's "Elements" was the pinnacle of mathematical thought.

He felt that many of the mathematicians of his time did not produce works as rigorous as done by Euclid and strayed from reality into areas such as the complex value for the square root of -1.  To him, i and other imaginary numbers did not represent real quantities. Although he found these new ideas to be absurd, he allowed that they would be of interest to advanced mathematicians they could not be taught to undergraduates. 

 He used his fiction to pull apart the logic of the new ideas, taking the weaknesses to the absurd end so as to show the ideas really were not great. 

The local English teacher told the kids that Dodgson used Hashish as he wrote the book.  She supported this by pointing out the Hookah smoking Caterpillar and demonstrated Dodgson's smoking and writing but I pointed out that  he was not known to use the drug.  She didn't even know about the trip down the river.  It has been said that particular scene along with the mushrooms that shrunk her are actually represent the way the connection between algebra and geometry was cut by Symbolic Algebra.

The whole book "Alice in Wonderland" is filled with all sorts of things like that both warnings and actual mathematics but like most of us, I grew up thinking it was nothing more than a children's story.  I had no idea Lewis Carroll was a mathematician in his everyday life. 

Let me know what you think, I'd love to hear.  Have a great day.

Recreational Math

  Some people read books when they want to relax, some cook, some take long rides but some people enjoy working math based puzzles.  This type of math is referred to recreation math.

Recreational math has been around for many years especially since Scientific American ran Martin Gardeners "Mathematical Games" for over 25 years.

One of his columns in 1975 inspired a woman to challenge his claim there were only eight polygon shapes that could be used to tile a plane.  Within two years, with nothing more than a high school math education, she'd discovered four new tessellations which were published in a mathematical magazine.

Martin Gardeners column introduced millions of people to recreational mathematics.  Today, more and more people are enjoying recreational mathematics in the form of Sudoku, logical puzzles, and so many other fun games.

Recreational math encourages both logical and lateral thinking  rather than advanced thinking skills so the average person can enjoy doing these puzzles without an thorough knowledge of math.  One of the more famous mathematical puzzles is Rubrick's Cube.  Did you realize the cube is so popular that over 350 million of them have sold since their creation in 1977? 

Another popular recreational math item are Sudoku puzzles.  They are logic based puzzles using number placement in a 9 x 9 grid to find the ways you can get the digits 1 to 9 arranged properly.  This modern version was created by Howard Garns, a 74 year old freelance puzzle writer in 1979.  It was first published by Dell Magazines as Number Place.  It is said that millions of people work these puzzles.

I love logic puzzles, the ones where you are given a bunch of clues to sort through using a grid to eventually figure out who did what in what order.  The creation of the first logic puzzle is credited to Charles Lutwidge Dodson, also known as Lewis Carroll author of "Alice in Wonderland".  He is responsible for the knights and knaves puzzles in which the knaves always lie and the knights always tell the truth.

There are schools in some countries that believe recreational mathematics can awaken a joy or curiosity in doing mathematics whereas the Common Core Standards does not look at math that way. Personally, I think if we could fire some sort of joy in our students through recreational math, they might find learning math more fun.

Let me know what you think, I'd love to hear.  Have a great day. 

Apps That Solve Problems.

We know there are now a variety of apps that will show you how to solve equations and other problems and there are calculators for just about anything that will take care of solving a problem.

Many that I've seen show the steps needed to solve the problem or they do give a few words of explanations but nothing in detail.

Yes, I've had students use them to get assignments done but its easy to see they didn't learn anything when I gave a quick quiz with some of the same types of problems.

One way I get around these programs is to have students provide a description of what they are doing each step and why they chose to do each step that way.  This means they have to communicate their thoughts as they solve the problem.  Explaining their thinking helps me figure out where the misunderstanding is in their understanding.  Its more of an assessment, just like quizzes are.

One problem I have with relying on these apps to provide solutions is that many students just copy down the answers without paying attention to the steps.  If they use it to help them learn or check their work, I don't mind.  Unfortunately some of these apps such as Photomath may not provide the correct answer if it misinterprets a x for multiplication rather than a variable or the answer might not be in the correct form. 

Another issue is students may come to rely too much on the app for answers so they never take the time to learn the material.  These apps or websites act like sirens in that they offer to do all the work so all a student has to do is click and the digital device does all the work. 

I realize that the math in many professional jobs are carried out by computer programs designed to make certain things easier but if someone does not know what the answer is supposed to look like, they have no idea if the answer itself is wrong.

Years ago, I worked in a college math lab.  Many students used their TI-82's to graph answers for a class assignment and assumed the answer that came out, was correct.  They were surprised when I looked at the finished product and stated it was wrong.  They honestly thought that it didn't matter how they entered the problem because the calculator would provide the correct answer.

I would love to hear what other people think about the use of these apps in class.  Let me know.  Have a great day.

Animal Watch Vi Suite

Animal Watch Vi Suite is an interesting app I discovered for the iPad.  This particular app is actually created for students with visual impairments but it is also great for use in the math classroom with ELL students or students who are below grade level in reading or math.

The app was created by people at the University of Arizona to help build their algebra readiness skills while learning more about animals.

The first unit introduces students to the Giant Panda.  They learn about how one looks, its weight, what it eats, and it's life span.  The unit provides a description and drawings on the creature before discussing its diet and facts about its life such as they are good swimmers but do not have a permanent den.  At the end of the unit, there are three word problems centered on the Giant Panda. 

What is nice is there is a digital pad to enter the answer.  If the answer is wrong, it provides immediate feedback  while telling you to try again.  There are two hints available should the student need it and there is a scratch pad should a student want to use their finger as a pencil to find the answer. 

Each problem allows the student three attempts to solve each problem before giving the answer.  The problems on the panda include one where they have to read a graph to find the answer so its not just finding the correct information in a word problem but it also uses word problems to introduce graphical information.

Some of the skills covered are multiplication and division via the Black Rhino, converting between systems of measurement using the Burmese Python, the California Condor provides experience in statistics while the Cheetah provides practice on fractions, percentages, and rate-time-distance problems.

Furthermore, Geometry is experienced by learning about the Gray Wolf, Hippos cover solving expressions with variables, Honey Bees let students experience probability while Poison Frogs cover adding and subtracting fractions or working with mixed numbers.

On the other hand, learning about Polar Bears help students become better at adding and subtracting integers while the Sea Turtle and Snow Leopard Units help students become better at forming fractions or adding and subtracting fractions with different denominators.  The last unit or White Shark has students practice multiplying and dividing fractions.

I love the way the developers have created an app which integrates reading, science, and math into a cross curricular situation.  This is an app, I want on my classroom set of iPads because my students need practice reading in addition to working word problems.

Check it out if you get a chance and best of all, its free.  Let me know what you think, I'd love to hear.

Warm-up

If a blueberry is about 1 cm across, estimate the length and width of the pile in inches.

Warm-up

Estimate the number of peas in the photo

Body Mass Index or BMI

  I grew up with a mother who was always watching her weight.  By the time, I hit 13, I understood all the math involved in counting calories so one could loose weight.  Later on, after I graduated from college, I learned calories was no longer the thing.  People now checked their Body Mass Index or BMI.

The formula used to calculate a person's Body Mass Index is their weight in pounds times 704.7  all divided by their height in inches squared.

So if a person weighs 145 lbs and is 5 ft 6 inches tall the formula would look like (145 * 704.7)/66^2 which gives a BMI of 23.46.  Anything between 19 and 24.9 is considered healthy.  One problem with this formula is that it does not measure overall fat or muscle. Its pretty much just a ratio.

I thought it was was something new but the formula has been around since 1830 when it was developed Lambert Adolphe Jacques Quetelet, a Belgian mathematician and scientist.  The formula was created to give an idea if the person weighed a healthy amount.

At the time he developed this formula, there were no calculators, computers, or other such devices so the formula had to be calculated by hand. Some people argue that the formula says you are lighter if you are shorter and heavier if you are taller.  In 2013, a mathematician argued the formula needed refinement.

He recommended the formula now read ( the weight in pounds * 5735)/height in Inches^2.5.  I tried it out and for me, the BMI produced by this formula wasn't that different from the one produced by the original formula.  As stated earlier, the formula does not take into account the difference between fat and muscle.  It is known that muscle is 18 percent denser than fat.

This means that a person who is 6 feet tall, weighs 203 pounds and does not exercise will have a BMI of 27 while a person who is 6 feet tall, weighs 211 pounds and is an Olympic athlete will have a BMI of 28.  Two people in very different shape but with almost the same BMI.

It has been suggested that the Waist to Height ratio is a better indicator of heath than the BMI  because it is more accurate. The idea is that the waist circumference should be no more than half your height.  So if you are 60 inches tall, your waist measurement should be no more than 30 inches.  So if you are 60 inches tall with a waist of 25 inches, you are in good shape.

There is research indicating that over 54 million people has a BMI indicating they are over weight while their waist to height ratio indicates they are within a normal range.  Which is right? Who knows.

I decided to write on BMI and waist to height ratio's because they are both used in the health industry.  Let me know what you think, I'd love to hear.  Have a great day.

The Mathematics of Hair Braiding.

I live in a village in Alaska where so many kids wear their hair braided in intricate pattern or can braid hair beautifully. There have been days when I fixed my hair in a side braid and one of my students told me, I'd done it wrong so she redid it.

Other times, a girl asked to braid my hair and fixed it so it formed a french braid into a braided bun.  Some moms go crazy with their creative braiding.

In general, hair braiding is made up of repeated geometric patterns.  In mathematics, a braid is any collection of strands that cross each other. Braids also have a beginning and end which as static but the strands themselves can be moved around by pushing or pulling so one braid changes into another braid.

African American culture is known for it cornrows which have been studied.  Cornrows use translation, rotation, reflection, and dilation to create styles from simple linear rows to complex curves and spirals. In addition, the process involves fractals because if you've ever watched anyone braiding hair, you'll notice the same shape is repeated over and over but in the process it shifts and becomes smaller and smaller.

If you are interested in playing around with braids, you can check this program out.  In the lower left corner, you can click on the picture of a woman with braids.  Every time you click, you see a new picture with a different braid pattern done in the hair.  some are quite simple, while others are quite complex.  In addition, you can play with the program to see how changing numbers change the final product.

In mathematics, standard braids are made up of three strands but they can be made of two, four, five, or more strands but think back to the hair braids you've seen.  Most are made of three strands either up and over or under and up to create the pattern but sometimes you see two strands twisted together to create a pattern.  I've done something called a fishtail braid using four strands in an almost woven pattern and I know some professional hair braiding people have used five strands.

If you get a chance to look at pictures of braided hair, especially the kind that uses multiple rows, you might notice the rows run parallel to each other and sometimes, they might run into each other at perpendicular angles.  The next time you see someone with braided hair, remember you are observing math at work.

Let me know what you think, have a great day.

I See Math All Around Me!

  If you mention the word "math" around most people, they throw up their hands in the form of a cross, hoping you disappear as if a vampire faced with a cross.  Or they mutter something like "I've never been good at it". Yet they miss the math found everywhere.

Look at the fantastic curves found at the Sydney Opera House.  Curve after curve gracefully standing in beauty.  As you drive towards your destination, you find curves as part of the road or if you head out fishing, you'll see curves as the river twists and turns its way across the landscape.

Most people when they sit down to watch a baseball or basket ball game pay more attention to the players than the ball but both sports utilize are perfect spherical shapes.  Other sports use spherical shapes such as soccer, golf, etc yet some use more of a oval shape such as in football.  If one looks carefully, patterns can be detected in the plays, patterns that remind people of the Pythagorean Theorem.

Oh and roads, roads, roads, are parallel and perpendicular.  Some are set so two parallel roads are crossed by another road which acts as a traversal so instead of seeing a busy intersection, I see so many different angles such as corresponding, alternate interior and exterior or consecutive. They are all there hidden by businesses and people.

How many times have teachers required students to calculate the slope in a purely theoretical basis rather than exploring a real life application on the roads and roofs across the country.  Each has a slope yet the slope is called a grade or pitch yet its the same.

Of course architecture uses so many geometric shapes both two and three dimensional such as cylinders, pyramids, prisms, found in the Leaning Tower of Piza, the Great Pyramid of Giza, or a skyscraper.  Many building exhibit a series of parallel and perpendicular lines in addition to having some beautiful example of zero and undefined slopes.

The more modern architecture often offers striking angles playing against each other in a composition similar to the music at a concert. Furthermore, the skyline of a city resembles a wonderful bar graph with its different heights spread across the horizon.  Every city has a different graph of its skyline.

Of course each map has a special coordinate system with longitude and latitude in degrees pinpointing a location.  Maps of streets use a alpha numeric system to divide the city up into areas so D3 covers several streets.  I remember as child finding maps in the phone book and looking up the street I lived on.  This is before Google maps.

Have you ever gone through a hardware store where each can of paint tells you how many square feet it covers, how large each tile is, or how many square feet the wallpaper covers.  Beautiful numbers combined to discuss area, real area not the theoretical weird shapes you seldom see in your neighborhood.

This is just a short list of math you see everywhere you look.  It doesn't include all the equations used to explain how your air conditioner or heater works, how the business sector determines profit and loss, the Dow Jones index and so much more.  I love the world and all the math swirling through our lives.

Let me know what you think, I'd love to hear.  Have a great day.

Science Fiction Math Became Real

 Up until I read an article in the Atlantic, I didn't realize how much Science Fiction contributed to the space program.  Yes, you read that right.

Imagine, you are watching a movie released back in 1929.  You see someone board a rocket ship and fly to the moon.  Their trip follows a figure eight trajectory to the moon. You shake your head because you know it could never happen.

If you were to compare the flight of Apollo 11 to the one from that movie, it would be the same.  The trajectory was calculated by Hermann Oberth one of the first scientific advisors to films and known as the father of rocketry.  In addition, he designed a rocket ship for the movie that was so realistic, the Gestapo confiscated all his drawings in the 1930's.

Although he was paid to create a real rocket ship, he was unable to do so at that time but a year or two later, he ended up working with Wernher Von Braun to create the well known V-2 rockets.  Eventually they came to the United States to help with their space program.

This is not the only case of real life imitating Science Fiction.  Scientists have used Hollywood to foster support for possible or developing technologies such as driverless cars or targeted advertising, in addition to testing new theories that could end up as real world such as Kip Thorne who worked as a scientific advisor for the film Interstellar.  

He struggled figuring out the mathematical way black holes and worm holes worked for the film and managed to get them to use CGI to create visualizations of his theories.  This lead to his writing several papers on the topic.  Another scientific advisor involved with the Stargate series came up with a binary system to explain the radio active event needed for the story line.  A few years later, such a system was found.

Sometimes, scientists have a theory they get Hollywood interested in using as a way of promoting it.  For instance, the man who consulted for Jurassic Park, integrated his theory that dinosaurs were warm-blooded into the story. At the time, the theory was controversial but now its accepted.

The one thing all of these people had in common was their ability to use mathematics to explain events prior to them actually happening.  All of these scientific advisors had strong enough mathematics or knew a mathematician who could verify the calculations.

So often we see the results of mathematics but not the mathematics themselves.  I'll be talking about that another day.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Warm-up

White Boards, Let Me Count The Ways.

  I have a set of whiteboards in my class that I use regularly.  I love whiteboards because they are an empty surface waiting to be filled with mathematical ideas.  They also start conversations between partners as they strive to find the correct answer.

I keep a set on a shelf and I usually have some sort of whiteboard application on the iPads but this year they are late getting the iPads out so I can't use those.

Honestly most of my students would much rather work the problem out on a white board than show their work on paper.  In addition, if they make a mistake it is much easier for them to make corrections. I don't know when the desire to write only answers happened but its a habit by the time they arrive in high school.  Here are some of the ways students use physical or digital whiteboards in my math class.

1.  During jeopardy, they have to write their answers on the whiteboard.  I ask them to place their calculations on one half while the answer is on the other half. This way I can do a quick assessment to see where they made a mistake if their answer is wrong.  The version of jeopardy I play is that all the students receive points if they have the correct answer. 

I've found that if you play it the same way they do on television, many of the struggling students give up in a couple of rounds because they are unable to be the first with the correct answer.  In addition, I am able to check every student for understanding of certain concepts.  I use a lot of premade games at Jeopardy Labs that makes life easier.

2.  In Pre-Algebra, I have students draw tiles on the white boards to answer questions like 5 - 8 to find the answer or I have them illustrate in some way basic math for integers so I can see if they understand what is going on.  Yesterday, we did it for simple addition and subtraction of integers and a few had a light bulb go off when they saw that - 8 - 5 gave them 13 negative blocks rather than 3 positive blocks.  Up to this point they didn't see the signs.

3.  White boards, especially ones with a coordinate plane, are wonderful during both guided and  individual practice because I can check work or correct problems more easily.  The coordinate planes allow students to practice graphing and I can spot check for problems.  The other thing, is I can place a problem on the board and students hold their boards or iPads up so I can check their work.  Its quick and easy.

4.  When I have them write word problems using two numbers and a term, I like having them create a drawing of the situation for the word problem first before they begin writing the word problem. As they say a "picture is worth a thousand words" and it clarifies their ideas when they sketch the situation.

5.  If they have questions, they can write it down, signal and I can go check the board so they don't feel embarrassed to ask.  Furthermore, I write the answer down so they have something they can read rather than relying only on their listening skills in a room that sometimes gets a bit noisy.

These are some of the ways I use whiteboards in my classroom to  encourage all students to work.  Let me know what you think, I'd love to hear.  Have a great weekend.

The Pythagorean Theorem and Sports.

How many times have you taught the Pythagorean Theorem using the standard triangle, drew the squares to show it all works out, or you've had them draw a standard triangle like a 3-4-5 or 12-5-13 on a piece of paper.  Next, they've drawn square off the legs, color them before cutting them out and placing them inside the drawn square for the hypotenuse?

Instead of talking about ladders stuck up against a multi story building, lets apply it to football, a topic most males in your classroom can relate to.

The National Science Foundation created a wonderful three to four minute video designed to show people how the Pythagorean Theorem is used to calculate the angle of pursuit taken by a football player who wants to tackle the guy with the ball.  Sorry, I don't watch football so I really don't know all the names of the positions.

Picture this is you will.  The receiver catches the ball and starts running in as straight a line as he can down the field.  The person who needs to tackle him is about 30 fyards to the right.  He runs in a diagonal so he can tackle the receiver at around the 40 yard mark.  Using the Pythagorean theorem, he'll have to run about 50 yards to intercept the receiver and tackle him.  The video is well done and shows students how this works.

This site provides a written explanation of the material on the video. The nice thing about a written explanation, is students have a chance to practice reading for information which meets a reading standard.

In addition,  a researcher discovered that this same theorem could be used to predict a teams expected wins based on the number of runs scored and the number of runs they allowed.  Think of it this way:
Expected wins = (Number of runs scored)^2/(number of runs scored)^2 +(number of runs allowed)^2.  This is referred to as the Pythagorean Expectation.

It works this way.  Say the Dodgers scored 720 runs while allowing 640 runs.  You square the 720 for 518,400, then square the 640 for 409,600 and add the two numbers together for 928000.  Now we have 518,400/(518,400 + 409,600)  or 518,400/928,000 = .558.  So the Dodgers should win 55.8% of their games over the season.  This means if they played 170 games, they should win 95 of their games.

Furthermore, the application of the Pythagorean Theorem in Baseball is used to determine how far the second baseman has to throw to home or third base has to throw to third.  The baseball diamond is 90 feet from base to base so that makes it easy to calculate the distances.  If you'd like to do an activity on Baseball, this site has the worksheet and answers one could easily use in the classroom.

Basic Statistic Information

Math is Fun is a great website for introducing students to basic statistical information.  It has topics from "What is Data" to different ways to show data, to setting up surveys, to measures of central distribution, to measures of spread, probabilities, to types of distributions.  Basically everything you need in one place.

I really enjoy the organization of the topic and the way its handled.  For instance the lesson "What is Data?" begins straight off explaining qualitative vs quantitative complete with examples.  It further defines the two types of quantitative data with examples so students have a better idea of the difference.  The lesson takes time to talk about collecting the data including information on sampling.  The sentences are short with lots of pictures, highlighted words and at the end, there is 7 question quiz on the material presented.

Each question is multiple choice with a help button available. The help button takes the student back to the material.  Once the question is answered, the program gives immediate feedback telling you why its right or wrong and tells you the percent of people who got the question right.  I love the immediate feedback.  At the end of the quiz, you find out your overall percentage and which questions you missed.  Unfortunately, the questions remain static so if you retake the quiz, the questions are the same.  At the bottom of the page are links to more information.

In addition, there are four hands on activities spread throughout the unit.  One of the activities is on asking questions followed by a second activity on writing better questions which can be used to collect data.   These have lots of examples but still ask the students to create their own questions based on certain criteria.

Under measures of spread, you'll find the usual range but the topic also covers  quartiles and the interquartile ranges, percentiles, the difference between mean and statistical deviation with access to statistical deviation formulas and calculators.

Comparing data covers univariate and bivariate data, scatter plots, outliers, etc.   Furthermore, the site has advanced information sprinkled throughout the list, so students are able to explore topics in even more detail.  

I love the way each topic is set up with definitions, examples to support the definitions, links to important vocabulary and ideas.  The lessons are fairly short and are easy to understand.  I will be the first to admit that I only took one statistics class in college and that was quite a few years ago.  So when I have to teach a topic, I'm rusty on, I go here to get a quick review.

I want to use this site instead of the book because the material is easier to take notes on since most of my students are ELL and need the site's organization to take notes easier than trying to get them out of a book. 

Check it out and let me know what you think.

How The Brain is Changing

  I've been hearing from more and more people that the use of digital devices is changing the way we think and the way our brain works.  Last spring,  the local high school ran a documentary on this exact topic.  I wish I'd been able to go but they ran it when I was off at a conference.

The current research indicates that the growing use of technology can both benefit and harm the way children think.

Remember, the brain continues to develop to about the age of 26, when it has reached its adulthood.  In young children, the brain is malleable and  growing, the wiring of their brains is being changed by frequent exposure to technology.  Every age has had technology that changed the brain.  For instance, when reading became more wide spread, the brain became more focused and imaginative because we pictured the words in our minds.  One good thing out of the growth of current technology is that brains are able to scan through information more efficiently.

There are four main areas effected by the rise of technology: attention, information overload, decision making, and memory/learning.  Attention is what allows us to think and leads to decision making, problem solving, creativity and other things.  It is highly malleable and influenced by its environment. So children's type of attention is formed directly by its environment. Unfortunately, constant exposure to the internet has lead to distraction being considered normal, focused attention impossible, imagination is not needed and memory doesn't develop normally.

I've seen how imagination is inhibited because most of my students go to the internet to find pictures to recreate rather than developing their own ideas.  They'll spend hours finding quotes they like to turn into posters.  They look for apps to solve their math problems rather than developing the ability to approach the problem in a variety of ways.

In fact, studies indicate when a person reads text without interruption is able to do it faster, have better understanding, recall, and have learned it better than those who read the same material filled with ads, hyperlinks, and videos.  Furthermore, there is a body of evidence to show that students who are on the internet during a lecture remember less of the lecture and scored fewer points on the same material as those who didn't have internet access.

Technology is good in terms of  improving visual spacial abilities, the ability to identify only important pieces of information, and increase both reaction times and attentional abilities.  Today's students are less likely to remember the information they find on the internet but they are more likely to know where to find it.  In addition, if the brain doesn't have to remember as much information, it is thought that it might open the brain to participate in higher order thinking skills but they are not sure.

So if the attention span of your students is shortening, this might be the reason for it.  There is now a generation of students attending school who have spent their whole lives being entertained by their mobile devices. 

Let me know what you think, I'd love to hear.  Have a great idea.

Statistics and The Real World.

Over the next couple of days I'm going to be exploring sources for statistics we can use in the classroom when we hit that part of the school year where we have to include touch on it.  Most schools I've taught at do not have a class on probability and statistics.  We are expected to include it in our regular classes which means, I have to figure out where it fits.

There is also the problem of finding statistics that have been done in a reasonable way rather than adjusted to fit the presenters point of view. In addition, most students see the problems in textbooks to be stupid.  They see no relevance but today, I'm sharing math lessons created by the United States Census Bureau for grades K to 12.  If a child is old enough, they might remember the census workers who visited in 2010.  The next time they are due to visit homes is in 2020. 

The link will take you to the Math page where it lists all the exercises with a name and the type of analysis used for the activity.  For instance, one says "Fitting a line to data - earnings and educational data.  This lesson is suggested for grade 8 but it could be used in high school easily.  It lists the learning objectives, materials needed, time required, and an activity description along with both teacher and student printed material.

The teacher packet is 15 pages long. It has everything listed on the main page, plus lists the common core and NCTM standards the activity meets.  Furthermore, it lists which of Bloom's Taxonomy adjectives it meets and provides information  for before and the activity before including all of the student pages complete with suggested answers.

All data is at the end of the three part activity.  Part one requires the student to plot the number of years of schooling with the average earnings, draw a line of best fit before finding the equation of the line based on y intercept and slope or two points.  Students are asked to compare the equations they came up with and discuss them.

The second part has them graph the same data broken down by sex.  They find the equation of the line for men and women before comparing their lines with the one from part one.  The final part of the activity requires students to write a one paragraph summary based on the data they plotted in parts 1 and 2.

If you are looking for an activity for a specific grade level, there are tabs listing K-5, 6-8, and 9-12 so it is easy to find the right one.  They have activities designed to help students learn more about frequency distribution, applying correlation coefficients, interpreting dots and box plots, creating and interpreting histograms, sampling means and viability, and looking at what is a statistical question.

In addition, the site provides teaching resource including data access tools which has three lessons designed for students to use some of these tools in class, games, warm-ups, and so many other resources, some are for math, some aren't. 

Check it out, see if you can use any.  I've seen several including the one I explored,  I can integrate into my math classes this week.  Have a great day.  Let me know what you think, I'd love to hear.

Warm-up

Warm-up

National Geographic and Math

  When I think of National Geographic, I never thought of lesson plans using math but thanks to a tweet on Twitter, I had a chance to explore their site.

National Geographic has some wonderful things that can be used in Math classes.  I did a general search for materials that are science based but can be used in math and the search provided me with 59 different possibilities.

The majority of items that came up are actually articles which discuss topics that use math such as engineering.  In addition, there are some historical articles, and reports on people.

I clicked on an article about Foreign Finances written by someone from Visa Credit Card Company.  In the article, the author who discusses things students should think about if they go overseas for a semester or year of schooling.  He reminds students they have to arrange to have bills paid while gone or it could effect their credit scores.   He also recommends mail be forwarded to parents rather than having an acquaintance pick it up.

Furthermore, he discusses what documents should be photocopied, which things should be left home and even takes time to discuss credit cards themselves in terms of foreign exchange rates and use fees.  He tells people to remind the credit card companies you are going overseas so they do not refuse certain charges.

I discovered this happened when I ordered a rail pass that I never got.  I found I need to talk to the company before ordering the rail pass so they would have accepted the charge.  He spends the rest of the article continuing his discussion on everything connected with traveling overseas. The article even comes with a vocabulary list so students can check vocabulary if they do not understand it.  In addition, there is a link to Khan Academy for math related to the topic.

Some of the articles are very short such as telling you when Sesame Street began or the Electoral college. Some come with questions but the questions may be more Social Studies focused however, its not that hard to create mathematically based questions so students can analyze information.

The main plus I see on the articles is that students practice reading real world writings to synthesize or pull out information in order to answer questions.  This requires they practice their "reading for information" skills which shows them cross curricular applications. 

Many of the articles expose students to job possibilities by spotlighting a person and discussing their jobs in details.  I read one on a woman who helps provide natural disaster preparedness plans for various cities such as one for a place in Indonesia.  She uses GIS to create a plan in case of tsunami.

I admit, some are not directly focused on math but the job does require it as part of doing the job.  The article on this woman could be used to introduce students to GIS, how it works, and perhaps even arrange a few exercises using it so they can see the role math plays in the overall picture.  I see this site as a way of showing the role math plays in the real world.

Check the site out, let me know what you think. I would love to hear.  Have a great day.

Increasing Mathematical Discussion

  According to both math standards and research, mathematical discussion is important to students learning mathematics because if they can clearly explain their thinking, it means they understand the concept.

Unfortunately, many of us do not include as much discussion as we'd like because of all the things we are mandated to accomplish in a 45 to 55 minute period.

When we listen in on various discussions, we are able to assess student understanding of the material.  Furthermore, we can tell how well they listen to others while being able to express their ideas, something they will need in the future.  In addition, it gives students a chance to develop the ability to construct arguments designed to defend their point of view.

It is important to create safe atmosphere in class so students feel as if they are creating a community.  It is a place where students are recognized for their questions while being encouraged to comment on ideas.  One part of this is for students to become active listeners so they are able to follow someone's argument and be able to counter it, should it become necessary.

Furthermore, it is important to encourage student interaction because it is the interaction which helps develop discussion.  This discussion should happen equally well in pairs, small groups, and whole class.  Unfortunately, one cannot jump directly to whole class discussions without allowing students the chance to become comfortable discussing math topics in pairs and small groups.

One way to encourage pairs discussion is to provide conversation starters such as:
#1 What answer did you get for problem______?
#2  I got____________.  Did you get the same answer?
#1 Yes.  How did you do it?
#2 Explains how he or she got it.
Continue with #2 taking the lead.

Another way is to set up prompts which hold students accountable for listening to what is being said.  For instance, a student might restate the point another student made when speaking.  A way to keep things interesting is to use Popsicle sticks with names or dice to select the next student.  It might also be necessary to ask the student speaking to repeat what they just said if they are too quiet.  In order to comment on someone's ideas, they have to hear them.

Finally, once students have learned to be active listeners and are able to restate the points, they can then agree or disagree with what was said by providing support for their stand. Although it can be a slow process, it is worth the time investment to help students become comfortable in expressing their ideas.

Let me know what you think, I'd love to hear.  Have a great day.

A Vocabulary Game For The Math Classroom

We all know that learning vocabulary in math for many students is like learning a foreign language due to the three levels of words.  They are either words that have a general meaning such as "hello" or "What's up?" or they might have both a specific and general meaning such as product or the math only meaning like Torus.

We all have that one section in the book where students have to change algebraic expressions to words and vice versa.  Most of the time we rely on those worksheets or some similar activity but what if we could incorporate games.

I got this game from the  Sadlier school blog but I had to modify them since they were made for English classes.

1.  Vocabulary Yatzee - requires students to be divided up in to groups of three to five students.  Give each group five dice to roll.  Each player has three chances to get the best dice they can.  Students have a list of math words they will use during this game.

Two of a kind are worth the total sum of the top three dice.  If they roll two dice, they must use two of the vocabulary words in a sentence.  Example might be something like addition and subtraction are related to each other.

Three of a kind are worth the sum of the top four dice.  If they roll three dice, they need to use three vocabulary words in a sentence.

Four of a kind are worth 25 points and requires students to use four vocabulary words in a sentence.

Full house is worth 30 points.  The student must compare a vocabulary word to another vocabulary words while contrasting it with a different word.  An example might be plus means the same as add but minus means the opposite.

Small straight is worth 35 points. The student must compare a vocabulary word with two others and contrasts it with a different vocabulary word.

Large straight is worth 40 points.  The student must compare a vocabulary word with two others and contract it with two different vocabulary words.

Yahtzee or 5 of a kind is worth 50 points.  A student is required to use five words in a sentence which must make sense and have the proper context.

Students have one minute after rolling the dice to come up with the sentence or the comparisons.  If they cannot, they must forfeit their turn and it moves to the next player.  This gives students, especially ELL students, a chance to become more proficient with the language of mathematics in a safe environment.  This is a good step to use before having students begin discussions.  Tomorrow, I have something on increasing mathematical discussion in class.

Let me know what you think, I'd love to hear.  Have a great day.

12 Interesting Math Facts

I love math and I love the way it makes way more sense than the English Language.  I also love certain facts I've learned about math that I want to share with you.

1. The Hairy Ball Theorem which boils down to "You can't comb all the hairs on a tennis ball without getting a cowlick."

2.  If you divide any number by 7 that does not have the result in integer form, you will see the numbers 142857 repeat in some way in the answer.

3. International paper sized A4 has a ratio of 1:square root 2.  If you cut the paper in half, the ratio remains constant and doesn't change.

4. There are 52 cards in a deck and 52! ways to shuffle the same deck giving you 80658175170943878571660636856403766975289505440883277824000000000000 different ways.  Such a large number and hard to conceptualize.

5. If you count in binary, you can count to 31 using one hand while if you use two hands you can count to 1023.

6.  73 is an interesting number because its the 21st prime number while its reverse 37 is the 12th prime number.  In addition, 21 = 7 * 3 and 73 when written in binary forms a palindrome of 1001001.

7. Although there are 3 dimensional objects such as Gabrielle's horn with an infinite surface area, they have limited volume.

8. There is a paradox which states you can cut a sphere up and reassemble it into two identical spheres which are exactly the same as the first.

9. If you randomly select 23 people and put them in the room, there is about a 50% chance 2 people in the room will have the same birthday.

10. If it were possible to fold a piece of paper 103 times, its thickness would be larger than the universe we can see.

11. The universe is not big enough to contain the written digits in a googolplex, even if each digit is the size of an atom. 

12. Since the ancient Babylonians did their math in base 60, they gave us 60 seconds in a minute, 360 degrees in a circle, etc.

I hope you found these facts as interesting as I did.  Let me know what you think, I'd love to hear.  Have a great day.

Addition and Subtraction of Integers

This past Friday, I introduced the addition and subtraction of integers.  I tried teaching it a different way than I usually do.  Ordinarily, I would use number lines to show how it works, talk about adding if you have two negatives or two positives and subtract if the signs are different.

In fact, one of the kids contributed that to the discussion but they are in this particular math class because they struggle with the basic concepts.  From previous experience, I know they have issues translating the visual when done with number lines to applying it to numbers.

One issue is the idea that on a number line, the positive and negative represent directions when they are used to thinking of those signs as indicating addition or subtraction.  This time, I tried introducing it a different way.

I told them that addition of integers is actually combining numbers with the same signs.  I didn't talk about adding, I used combining.  Then we talked about a can of soda where one student drank two ounces then drank another three ounces.  The students were able to see the situation could be represented by the equation -2 + -3 = -5 because each time they drank something it was removing the liquid from the soda can.

Another situation we discussed was giving money to friends. First one student gave two dollars to a friend and three dollars to another friend so he gave away $5.00 total.  They said it was minus $5.00 because he no longer had the $5.00.

When I introduced subtractions, I explained with the number line that it was finding the difference between two numbers for instance 5 - 7 = -2.  The example I used was the high temperature of the day was 5 degrees but that night it dropped seven degrees so the temperature was -2 degrees.  The second example used was you had $5.00 in your account and used your debit card to buy $7.00 worth of stuff at the store but you'd overspent by $2.00 or you were -$2.00 in the hole.

Of course when we hit the minus a negative number, I reverted to explaining 3 - (-4) is actually 3 + -1*-4 or 3 + 4 = 7.  Its hard to explain in any other terms because my students are ELL and I do not believe the science they get allows me to explain any scientific situations on this type of topic.

What I loved most was the mathematical discussion it evoked.  Most of the time, I get total silence when trying to involve them in a discussion but this one, they volunteered ideas on situations that matched the problems.  I would say this is one of the best lessons I had.

Let me know what you think, I'd love to hear.  Have a great day.