Geometry and it's uses

If you've ever considered where to find real life examples to be used in your geometry lessons you have to look no further than the companies who develop Mechanical CAD or Computer Added Design software. These websites are filled with several video and image examples of how geometry is used in design to build devices that are used in everyday life. Simply by going to google and typing in Mechanical CAD provides links to limitless examples.

Here are a few;

 

Examples of their use in a classroom could be at the beginning of a lesson as a hook or breaking down the image to define the various solids, cube, cylinder, sphere, triangular prism, cone, cuboid etc. You could use images and make a game of "finding the solids" and then relate that back how it is used in real life.  
Other opportunities to involve these images into your lessons is to underline how important language is involved in communicating the information presented. Some of the images show dimensions and relationships between parts., units of measure and scale.
With so many various images you can differentiate what is shown by providing more and less complex images.

Number Riddle

I've always been a fan of this riddle. Although not truly math related, the students will assume it is because of the numbers.

Examine carefully the following sequences of numbers:

1

11

21

1211

111221

312211

13112221

1113213211

31131211131221

Although the sequences appear to behave totally at random, in fact, after the first sequence, each sequence is constructed in a precise and methodical way based on the previous one. What is the next sequence?

To solve, simply say the quantity of each number. Ex. in the first line, there is one 1, hence why the second line is 11. Now, the second line has two 1's, hence what the third line is 21.

Get it?

Olympics and Math

Every four years in Canada we are given a wonderful opportunity to mix our Canadian tradition and heritage into the classroom.  Canada has a long and storied history of doing well in the winter version of the Olympics.  In 2010 our class utilized the Vancouver Olympics to study in-depth a variety of math strands.  The activities used also worked during the previous Olympics as well.

Our unit started with looking at Canada's history in past Olympics.  The data management strand offers some very easy connections to this part of the unit.  Students begin by collecting data from a variety of websites (ex. olympic.ca) and creating a variety of graphs.  Data is also used to create and test hypothesis using scatter graphs in grade 8 (ex. Does the money given for a gold medal equate to more hardware for a given country?).  These activities serve to give the student relevant real world data and also create an increase in national pride.

Once the Olympics begin, the students continue to collect data throughout.  In intermediate they analyze the variety of data (medal winners, historical trends, # of athletes, etc.) and determine which type of graph best suits their needs.  This of course leads to the discussion about the purpose of their data and how data can be leading or 'mis'leading.  Students then need to make inferences and convincing arguments base on their data collection.

There are many curriculum connections in the number sense also factors heavily in our discussion throughout the games.  The students can move beyond simple textbook and worksheet questions that may have little connection to their lives by using math they are immersed in throughout the study.  Students use percentages to calculate which countries are doing well (% of medal winners) versus countries that are disappointing.  It is always a lively discussion when we talk about disappointment and how 4th place may seem disappointing, but means you are still fourth in the world.

The students favourite unit involves the unit on percentages and tax.  We try to figure out what a vacation to watch the Olympics would generally cost by making a budget based on data.  Vancouver (yikes!) ended up being very expensive.  Using data from a variety of sources students need to find flights, lodging and of course tickets to watch their favourite events.

Having completed this unit over the past three Olympics, my main reflection is the increased use do technology.  Students are able to follow live Olympic updates, use programs such as Tinkerplots and spreadsheets for graphing, and I-pads or computers for research.  The two seem to go hand in hand in bringing out meaningful math and making the students interested and intrigued in concepts that can get tired when using out of date information.

It also goes without saying that this unit fits quite nicely in other subject areas well.  Teachers can easily incorporate language arts, health and science topics with the Olympics.  The unit also includes completing the events in phys. ed.  class complete with medal ceremonies.  Every time the students complete the unit their reflections always speak highly in regards to fun and interest in math during the Olympic activities.  Every four years I look forward to utilizing the Olympics and in 2010 our Olympians didn't disappoint.

Mind Reader

This is a really cool time-filler anyone could try with their class. The best part, it involves a little bit of math! Take your students through this procedure step-by-step. Tell them to keep it to themselves and do it in their heads, as you can only focus your mind-reading techniques when everything is quiet. Go through the steps quickly, so students don’t have a chance to think up crazy answers!

        

          1.     Think of a number from 1-10

          2.      Multiply that number by 9

          3.      If the number is a 2 digit number, add the digits together

          4.      Now, subtract 5

          5.      Determine which letter in the alphabet corresponds to the number you ended up with 

                (i.e. 1=a, 2=b, 3=c, etc.)

          6.      Think of a country that starts with that letter

          7.      Remember the LAST letter of the name of that country

          8.      Think of the name of an animal that starts with that letter

          9.      Remember the LAST letter in the name of that animal

          10.  Think of a fruit that starts with that letter

Are you ready to have your minds blown??

....

........

............

.........

.....

I’m picturing a kangaroo eating an orange... hanging out in Denmark...
If you have an algebra class, or even just a math class, have them try to explain why this works! 

The Cost of Being Batman!

Hello, everybody! I was thinking of different posts I could create for the blog, and remembered some pretty cool data I came across. I’ve always been a big fan of the Batman: Dark Knight movie trilogy, but have always wondered, could Bruce Wayne really afford all of those cool gadgets? If you haven’t seen the movies, I don’t think there are any real ‘spoilers’ in this post, but if you’d rather wait and watch the movies, do it! It’s worth it.

In the attached photo, you can see some rough estimates on the cost of being Batman. What I did was try to organize the data. I added up all the costs from Batman’s first year, from when he created this superhero and saved Gotham the first time in Batman Begins. This includes his Batsuit, the Tumbler (bat-mobile), his costs of living, gadgets, and training.

In year two, you will notice a huge jump in expenses. According to this article, it would have cost a whopping $600,000,000 to rebuild Wayne Manor after it burnt down. I gave Bruce the benefit of the doubt, and created two sets of data for this reason: One with fire insurance (green), one without (red). Also, I’ve only added the cost of living and his new vehicle, although he did get a new Batsuit and some extra little gadgets.

For the next few years, I’ve only added the cost of living, as Batman went MIA for 8 years after year 2 in The Dark Knight.

In the final movie (year 10), Batman’s biggest expense was his new vehicle, the Bat, which creates another little jump in expenses.

This post is just to kind of put things in perspective! A fun project might be to introduce this data to your students, and have them do a little research on Bruce Wayne’s inheritance and yearly salary, to find out when Bruce Wayne could finally pay-off his expenses of being Batman!

The love of sports to learn Math

Ever since I can remember I was always fond of sports and had a love with the statistics and numbers used in them. One could endlessly review facts and figures of players and teams, wins losses, points talleyed, player stats, dimensions of playing fields one could go on and on. Just look at this site www.atpworldtour.com/ numbers, numbers and more numbers you just have to love it. I can't say how this influenced my ability to understand mathematical concepts but I'm sure it provided great practice in core abilities such as adding, subtracting, multiplying, dividing, percents, fractions, units, decimals, charts, tables, graphs to name a few. Knowing this I watch my eldest son, now in grade 5, his most proficient subject is math, and he like me enjoys sports and all the statistical information that goes along with them. He enjoys spending time reviewing the statistical information and I'm sure doing what I did. So is there a connection between sports and math, absolutely, is there opportunity to use students passion for what fascinates them and tie it to mathematics absolutely. Do we have the opportunities to guide each student to see the connections of math in the real world, absolutley, but will each student use this information to naturally expand their understanding of math it's hard to say. Perhaps knowing some of the students interests and providing more personalized real world connections would provide opportunties to get more students to naturally expand their mathematical abilities and in turn love math.

Big #'s!

This is interesting to look at.  Could be a great overhead, or document camera projection while entering the class.

1 = One

10 = Ten

100 = One hundred

1000 = One thousand

86400 = The number of seconds in a day

1000000 = One million

31556926 = The number of seconds in a year

1000000000 = One billion

7000000000 = The estimated human population on Earth (2011)

1000000000000 = One trillion

1000000000000000 = One quadrillion

1000000000000000000 = One quintillion

1000000000000000000000 = One sextillion

1000000000000000000000000 = One septillion

1000000000000000000000000000 = One octillion

1000000000000000000000000000000 = One nonillion

1000000000000000000000000000000000 = One decillion

1000000000000000000000000000000000000 = One undecillion

1000000000000000000000000000000000000000 = One duodecillion

1000000000000000000000000000000000000000000 = One tredecillion

1000000000000000000000000000000000000000000000 = One quattuordecillion

1000000000000000000000000000000000000000000000000 = One quindecillion

1000000000000000000000000000000000000000000000000000 = One sexdecillion

1000000000000000000000000000000000000000000000000000000 = One septendecillion

1000000000000000000000000000000000000000000000000000000000 = One octodecillion

1000000000000000000000000000000000000000000000000000000000000 = One novemdecillion

1000000000000000000000000000000000000000000000000000000000000000 = One vigintillion

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = One googol (the number 1 followed by 100 zeros)

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = One centillion (the number 1 followed by 303 zeros)

One googolplex = The number 1 followed by a googol zeros (there literally isn't enough room in the universe to write this number).

While not really a number, ‘infinity’ refers to a limitless quantity and is represented by a symbol that looks like the number 8 placed on its side.

Warm Ups

I am currently coaching a number of different running groups (one of which is the high school cross country team).  Lesson number one is always that you don’t work out cool muscles.  A proper warm-up will set the stage for a much more productive, and enjoyable workout.  I would like to submit that the same rule can be applied to any Math lesson.

Math warm ups can come in the form of relevant “real world” or fun videos:

Starting the lesson with a riddle or brain teaser is a great way to get the students participating and get some good discussion going.

There is a basket containing 5 apples, how do you divide the apples among 5 children so that each child has 1 apple while 1 apple remains in the basket? 



Answer: 4 children get 1 apple each while the fifth child gets the basket with the remaining apple still in it.

Two girls were born to the same mother, at the same time, on the same day, in the same month and in the same year and yet somehow they’re not twins. Why not? 



Answer: Because there was a third girl, which makes them triplets!

Lot’s of fun Math jokes, riddles and brain teasers can be found at this website, or many more just like it!

Oh Korea, Korea

   As some of you may know, I'm currently teaching in Korea. For those that don't, I'm currently teaching in Korea. Having done this before, I have eased easily back into the culture and the routine of having long days and hearing "Teacher, Teacher" countless times. I wouldn't be here if I didn't love it though.

   The education system in Korea is drastically different from that in Ontario. Currently, I teach students as young as 3 years old. They come to the English Kindergarden and are seated in front of people, such as myself, whom they don't understand and, I can only imagine, are a little bit scared of. Having seen the children I taught when they were 3, and are now 5, it proves that this "start them off young" mentality works wonders! The children whom once starred up at me with a blank stare, are now forming sentences and asking me questions.

   If the children aren't too scared of us foreigners, and the parents have the funds, the children will stay at the Kindergarten until they go off to Elementary at the age of 6. At this time, I'll continue to see them at the after school program. These students are usually hungry, tired, and irritable. They have spent a long day at school and sometimes the last thing they want to see is me at the front of the classroom. I understand this though and I try and emphasize all the while laying down the law (just kidding). It's with these students that I know I must take a more relaxed approach. If they want a snack, they can eat in class. If they need a second to clear their minds, I'll give them two. Some of my students are at hagwons (private, after school programs) until 8:00. They still then have to go home and complete all the homework assigned at these schools.

   Yes, these kids work and study hard and their test results are evidence of this. Are these kids being pushed too hard? Are our students in Ontario not being pushed hard enough? Having been witness of both education systems, it's clear that the only thing that holds true for both, is that I want to be part of it.

A different Setting, A different challenge - Teaching in a Youth Facility

So, I have kind of mentioned that I teach at a youth facility, but I thought I would "blog" more about my unique setting as many people find it interesting.
First off, I work in a secure facility (yes, it is a jail for young offenders) with some stuff male students, who themselves have gone through a lot, mentally and physically. The majority of the students who come in here are 18 years old and have one or two credits, so they are apprehensive about learning to begin with and don't want to put themselves in any situation that will make them look dumb.  Usually all of them have IEP's and the majority of the students read and write at very low levels.
Now the most difficult part is that in one classroom I can have a max of 8 students, but within that classroom each student could be working on  a different grade and level. So I have to design interesting and engaging lessons, that students can follow independently. The students can not access the internet and we can only use certain manipulatives due to safety reasons. Each one of these students are very needy so I basically have to work one on one with each student, so I try my best to rotate around the room, spending time with each student equally. Even though I only have 8 students, it is exhausting.
I have only taught these guys Science and PE, but I think the biggest challenge with teaching these guys is determining what is the most important and applicable topics they should learn. Especially since the majority of them only stay with me for a couple of weeks, sometimes months and maybe years. Some of the topics I find useless for these guys, but we are technically supposed to follow the Ontario Curriculum. So this is where I use my "professional judgement" and determine what will benefit these guys the most and help them be more successful out in the "real world." The term "professional  judgement" has come up a lot here over the last couple of years and as teachers we are always debating what that means.  I do think that sometimes as teachers we should stray a bit from the curriculum especially if that is the best way to meet the needs of your students.

Re: Kara's comic

Kara's comic reminded me of this tshirt that one of my colleagues was wearing the other day:

Homework Spot

This site contains links to glossaries, games, reference tables, as well as instructions about how to solve different types of Math problems. I find the glossaries particularly useful while working with ESL students.
www.homeworkspot.com

WebMath

The WebMath website is very useful for students having difficulty understanding problems. At first, it seemed to me like a way to help students cheat on their homework, but in reality, could be used as an additional instructional method, as it gives students the steps to completing the problem. Students can choose from a number of categories, such as algebra, fractions, etc., and can plug in the numbers from the problem they are struggling with. The program then generates the correct answer, along with all of the steps necessary to find it. The website is found here: http://www.webmath.com/.

Math in the ESL Classroom

Recently, I had a great experience when I was teaching the students triangle-related vocabulary. I had divided students into groups of 3 and given each group a square piece of white paper. The students were asked to fold their paper to show different terms relating to triangles (such as altitude, orthocentre, etc.) and to present these terms to the class. Even though the students were presenting the same terms as other groups (as a way of practising their English), each group found a unique way to express the words, which was great to see in an ESL classroom.

Multiplication Mania

Hello all,

I thought the blog format would be a great place to share a math teaching ah-hah moment I had on the Friday before Thanksgiving.  Usually a day that the students are a little checked out on with the three day weekend approaching.

As a lot of my colleagues, my students have been struggling mightily with two digit multiplication.  Before getting to multiplying with decimals, this step usually leaves me and the students highly frustrated.  Of course this usually leads to the discussion that today's students just don't know their facts.  "In my day..." Ya, I know you had to walk uphill both ways in your bare feet, is what the students usually tell me.

Back to math, our junior intermediate divisions have decided to have a mini focus on math facts during the month of October.  The students are completing daily activities (times table challenge) and games (multiplication bingo) for prizes and 'bragging rights'.   This is having a real positive effect on our students ability to quickly manipulate single digit multiplication.

However moving to two digit multiplication, there seemed to be no carryover.  Most students were unable to transfer their knowledge of single facts to the standard multiplication algorithm.  Multiplying and carrying at the same time were proving to be a chore.  The students and I had just about given  up, when I happened across Big Ideas from Dr. Small.

She explains in detail, the lattice algorithm (pgs. 37,38) for multiplying multiple digits in a visual sense. 

The lattice system worked exceedingly well for the students.  Each digit is automatically placed in the proper place value column.  In the example above, the pink represents the ones column, the orange the tens column, and so on.

The ah-hah came after ten minutes of dissecting the strategy with the students and their applying the strategy.  We went from two students being able to multiply two digits to 15 students being able to consistently and accurately multiply two digit numbers.  The other beauty is the lattice also works as the numbers get even larger and also work for decimals as well.

I have included a link to a you tube video showing the lattice algorithm in action.  This does not mean that this is the only way to multiply, but if students are struggling with the standard algorithm, this will certainly increase their confidence and lead to increased participation during your lessons.  I actually had students that asked for homework over the holidays.  Even student mentioned that I have never understood multiplying before but this is easy!

Some student before and after samples are shown below.

I can't wait for Monday, to see how this goes with the introduction of the decimal.  Wish me luck!

P. Cornies

Comic - roots, grade 9

Great way to introduce roots in grade 9!

Old Dog, New Tricks

Teaching math has changed drastically since I was employed full-time.  Drastically.  When I graduated teacher’s college and began teaching (Grade 7 - circa 1999), I used chalk, textbooks and overheads in my daily math lessons.

**** 10 years later ****

Enter to the supply-teaching world a 35-year-old mother of 3, “OCT” - who has never used (let alone seen) a Smartboard.  Wowzers.  I have always had a keen interest in technology and taken pride in being able to help my chemical engineer husband overcome minor PC issues (Right click! Sheesh.), but this was way over my head.  Pun intended.  I officially went to teacher’s college in the Dark Ages.

Now, even as a (somewhat) seasoned supply teacher, I find it hard to imagine a day in a classroom without a Smartboard.  What on earth would I do?  The most ridiculous part about this last statement is that I am still learning and am probably still considered a total rookie where this device is concerned.  I wish they would provide Smartboard training for Occasionals.

Thank goodness for the AQ’s.  This is my 3^rd course in the last year and I feel like I have gone to teacher’s college all over again and have gained confidence to get back into the classroom and conquer the curriculum.

I do realize the Smartboard is just the beginning, am learning quickly how to utilize it where math lessons are concerned, and loving every minute.  This old dog is ready for some new tricks.

Frustrating

Today I was called in to supply for a grade 8 class for the afternoon.  When I arrived the teacher was still there and we talked for a few minutes about what my afternoon would be like.  She started off by telling me I would be rotating between her class and the grade 7 class for math.  I was really excited by this, as it would give me more experience for this class.  She started going through the lesson she was going to teach and from what I saw it was going to be really good.  Then she said its a little complicated so you can just watch a movie for both classes.  I told her I was capable to teach the lesson and that I was upgrading to my intermediate in math.  However that did not persuade her and she closed up her notes, handed me the movie and left.  I am really sick and tired of teachers thinking just because I am a supply teacher that I don't know what I'm doing.  Sorry for ranting, just wanted to blow off a little steam.