Exploring Consecutive Whole Numbers

This year with my grade 8 students I am trying to spend one class every so often giving my students time to work collaboratively on mathematical tasks. The first one I did this year went really well. For this task, I had my students find the sum of the numbers from 1-10 and then from 1-20. For these, they just used their calculators and computed the answers. Then I asked them to find 1-50 and 1-100. Of course, some thought that using a calculator was still the best method, but there were others who knew that there had to be an "easier" way and worked to find it (especially once I asked them - what about from 1-1000?). This task led to some great thinking and some great ideas. We eventually got to the point where the students were able to come up with a formula to find the sums of consecutive numbers. I told them about Gauss and how when he was younger he did a similar exploration in his math class and came up with some amazing ideas in the area of number theory. For those that were interested, we extended this problem to think about what happens when you don't start at 1 (eg. find the sum of all the whole numbers from 10-90) or what happens if you are adding consecutive even or odd numbers.

The second task that I presented to my students comes from the NRICH website. Here is the link to the video that I used to introduce the problem to my students:
http://nrich.maths.org/507

They really got a kick out of the British accents!

Once they watched the video, I had my students work in groups of 4 to answer the two questions the video posed (see below) and any other questions that came to mind as they watched the video.

What is special about the numbers that can be written as two consecutive numbers or three consecutive numbers or 4 or 5, etc?
Can all numbers be written as the sum of consecutive numbers?
Once again, I was so impressed with the level of thinking that my students showed. Most groups started with thinking about what types of numbers can written as the sum of 2 consecutive numbers. They quickly realized that all odd numbers can be written in this way. When I asked them why this was, they remembered that when you add one odd number and one even number, you always get an odd number. Some groups were also strategic in their approach and had some students looking at sums of three numbers, some looking at sums of 4 numbers, etc. The number of patterns that the students discovered were quite amazing. Some were even able to see the connection between this task and the last because, as an example, they realized that the sum of 5 consecutive numbers is always a multiple of 5 and the smallest one that will work is 15, which is the sum of the numbers from 1-5. When challenged by me to think about what numbers wouldn't work, they realized that all powers of 2 are not able to be written as sums of consecutive numbers. I also threw random numbers out at them (like 81 for example) and challenged them to think of all the ways that they could write this number as the sum of consecutive numbers.

For anyone interested in trying problem solving tasks in their classroom, I would highly recommend this problem, as it has a low entry point (all of my students realized that all odd numbers can be written as the sum of consecutive numbers) but a very high ceiling.

Mission #3: Pieces of Eight

Misson #3 for Explore the MTBoS was to choose from a variety of collaborative websites. The possibilities were amazing, and I am hoping to find the time to go back to each and every site. The one that intrigued me the most right now, was the Collaborative Mathematics site. This intrigued me the most because Jason Ermer (@CollaboMath) (the creator of the site) had posted a comment on my blog post about my 1-100 Assignment (see the post entitled A Few of My Favourite Things) that I should show my students his Pieces of Eight Challenge. This challenged related very well to what we have been doing in class, which was looking at the 4 4s puzzle and then tackling the 1-100 Assignment.

The first thing that was really interesting was that a bunch of my students said, "Done!" I've got it!". They quickly found a solution that worked and were satisfied that the solved the "challenge".

Here is an example of student work giving me only one solution

It surprised me that for many of my students, their thinking ended with one solution. Most ignored the latter part of the challenge which asked them to think beyond one solution. It asked them to find another, and another, and then, figure out how many solutions there might be. There are many students I sent back to the drawing board.

Examples of students being asked to go back to the drawing board. Now I was getting more than one solution.

Some of my students came up with more than one solution, but once again, I asked them to continue to think about it - to look for patterns and relationships between some of the numbers they are finding that work - namely 8s, 88s, and 888s. A few of my keen students really started thinking about this. One of my students immediately got on board and was VERY motivated to exhaust all possibilities. I was very pleased with his thinking. Here is what he came up with, which we think is the final solution.

Another student kept trying. She would email me one solution then I would ask her if she could think of more and I got her thinking about looking for patterns. After a few back and forth tries, she emailed me a solution with all 14 possibilities worked out. I was so proud of her for persevering and for the excellent thinking that she showed.

Many of my other students are still working on it, and I am hopeful that they will come up with all the possible solutions.

The second thing that surpised me was that many of my students immediately asked me, "If I do this challenge, will I get any bonus marks?". You see, I showed them this challenge as an additional thinking task to try, it was not the focus of a lesson. I was shocked that my students were willing to do the challenge, but only if it counted for something. What have I (or we - the school system) done to students to make them think things are only worth doing if they are "worth" something. What about the love of learning? What about curiousity and intrigue? Are those not "worth" something? Where has the internal motivation to learn gone? It probably has something to do with the emphasis we put on grades and marks. My school is beginning go down the path of examining our assessment practices so that hopefully soon we can get rid of numerical grades. Perhaps this will make a difference. But, in the meantime, it is disheartening that many students have lost their internal drive to learn. Maybe I will just keep doing more of these challenges, and hopefully they will jump in and reignite their love of learning!

Mission #2...How I "met" Fawn Nguyen

A while back I thought twitter was just for movie stars and famous (or not so famous) people who have too much time on their hands. I mean, who really wants to know what some random person ate for lunch that day? Back in the winter when my colleague, John, introducted me to twitter, I was skeptical. I remained open minded, however, and tried out twitter. After a few weeks, I was still a little unsure. I mean, how would I have the time to keep up with posts, some of which were useful, and some which were not. Then, I must have made some smart decisions and picked the "right" people to follow. These people led me to the #msmathchat, which in turn, led me to #MTBoS. Now, I can't imagine not being on twitter.

Teaching can feel somewhat lonely at times. This seems ironic seeing as you are constantly surrounded by people, but it is true. I also get frustrated sometimes when I see people that have taught for years and are doing nothing to grow professionally. Time is always the number one barrier to professional development, and I still haven't figured out the answer to that. I suppose the answer may simply be: if it's important to you, you can somehow find the time. Twitter has eased my frustrations and loneliness as I can always find inpiration there. If I feel lost or stuck for ideas I can also reach out and get ideas from others. My favourite cheerleader on twitter is definitely none other than Justin Aion.

This week's mission was to go outside of my comfort zone and become a little more active on twitter. One of the first things I did was to introduce myself to Fawn Nguyen, who is just amazingly inspiring! In doing so, I may have started some sort of twitter battle (just fun ribbing, really) between her and @mr_stadel. Then, a whole bunch of others got in on it, including THE Dan Meyer (wow!!), @MTChirps, @Mythagon, @lbburke, @nathankraft1. Although they threatened me with duct taping and hazing of some sort, I think they are all nice and just joking. Or, so Fawn tells me........look for my next post: Attack of the "Nice" MTBoS Gang!

The Wheel of Theodorus

In this week's #msmathchat the topic of irrational numbers came up. I talked about an assignment that I used to give my grade 8 students called the Wheel of Theodorous. I adapted this from an assignment that Leslie Lewis wrote about in an article in the April 2007 Mathematics Teaching in the Middle School.

This project highlighted the connection between math and art and gave my students a chance to blow me away with their creativity. Julie Reulbach encouraged me to blog about this assignment, so here goes.

This assignment was introduced to my students after they had been doing some work exploring irrational numbers and square roots. It was also a great way to review and practice the Pythagorean Theorem. The assignment outline can be found here: https://www.dropbox.com/s/fty0rgxmqccfsa9/The%20Wheel%20of%20Theodorus.doc.

Essentially, all students started with a right angle triangle that had legs that were each 1 unit (most used cm). They then had to show how to find the measurement of the hypotenuse. In this case it was sqrt 2. This then became the base for the next triangle and they made a rotation of 90 degrees and made the adjacent leg 1 unit. Once again, they had to find the measurement of the hypotenuse. In this case it was sqrt 3. When repeated again, the next hypoteuse was sqrt 4, which they had to write as 2. This allowed them to see the difference between perfect squares and non perfect squares and reinforced which numbers were irrational and which numbers were rational. They kept repeating this process, labelling all sides. They ended up creating a sprial, or the Wheel of Theodorus. They then had to make this into some sort of creative art work. Here is a link to more specific instructions in case that didn't make sense. I had the students hand in this worksheet along with their completed piece of art work: https://www.dropbox.com/s/9smr0e3gsmb986b/Wheel%20of%20Theodorus%20Work%20Sheet.doc?m

Here are some pictures of my students' work. I was blown away by their creativity.

Giving Thanks

Today is Thanksgiving in Canada so I thought it would be appropriate to write about how thankful I am that I found the MTBoS community. I wouldn't have this blog without the MTBoS community and without Twitter. I am so thankful that my colleague @cya_outside introduced me to Twitter and I am so glad that I have found some amazing educators to follow. My mission this week for Explore MTBoS is to spend some more time on Twitter. So, I guess I am about to put myself out there further and interact with new people or with people I follow but have never contacted. There are some people that seem to me like Twitter "Rock Stars" who I can't imagine would be interested in hearing from me, but they are just teachers like me! I think that's the point of the Explore MTBoS missions....they are meant to stretch you beyond your comfort zone. I think this will also help me to expand my PLN. There is so much going on that is so inspiring. So, thanks to the MTBoS for helping me to become a better teacher. Hopefully, one day soon I will have more to add that will help others become better teachers as well.

Who Am I?

I guess I should have created this post first, but oh well!

Welcome to my blog!

Here is a little bit about me:

My name is Carrie Annable. I live in Hamilton, ON. I just started my 12th year of teaching (11th at my current school). I have taught anywhere from grade 6- grade 11 math, but am currently teaching grade 8 math. My main goal is to convince my students that math can be awesome! I try to make math as fun as I can, although this is always a continual work in progress. A few years back, I completed my M.Ed. from Nipissing University. My thesis examined what it was like to teach grade 6 students math using  problem solving based approach. Last June, I received my Ph.D. from University of Toronto. My thesis examined how laptops were changing the landscape of mathematics teaching and learning.

I recently joined twitter and can be found there by the name @CarrieAnnable.

At my school I am in charge of the Middle School Math Program. I also co-chair a committee that I co-created with my colleague, Ryan Baker. This committee gives teachers at my school a place to learn from each other, discuss current trends in education, take part in action research and ultimately become leaders in pushing the boundaries of education.

Me at my graduation

Look at the cake my awesome colleagues got for me!

Me and my awesome dog Mollie! She loves math too :)

A few of my favourite things....

Since @samjshah referenced the Sound of Music (one of my all time favourite movies) in his intro to Mission #1, I figured I can reference it too!

Rain drops on roses and whiskers on kittens,
Bright copper kettles and warm woolen mittens,
Brown paper packages tied up with string,
These are a few of my favourite things....

Hmmm..... not too mathematical...how about:

Calculators clicking and my students thinking,
Groups loudly working and problems a-shrinking,
Solutions to problems all tied up with string,
These are a few of my favourite things....

Okay so that was pretty bad. I'm a math teacher. Not a poet or a musician. Give me a break! On a more serious note, one of my most favourite tasks for my students is a few tasks in one.

First, we start with a game called Math Dice, essentially students work in groups and roll 2 twelve sided dice to get a target number (the two numbers are multiplied together to get the target number). They then roll three regular dice and use these three numbers any way they wish to get as close as possible to the target number. For example, if a 5 and a 7 were rolled on the twelve sided dice, the target number is 35. Then, let's say a 2, 3 and 4 were rolled on the other dice. Students use these numbers to create an equations as close as possible to 35. They might do 34 + 2, or 2^3 x 4, etc. I usually give them about 5 min for each game and have 2 winners. First, who ever gets closest and second, who ever can make the most equations that are mathematically correct (any equations). This way, students will all try and students will keep working even if someone quickly gets the target number. Here is the handout I use: https://www.dropbox.com/s/cvyo1xwimaspphu/Math%20Dice%20Activity.doc Let me know if it doesn't work. I am new to drop box!

After we play the game. I show my students how they can use factorials and summations to make larger numbers. Then, I introduce them to the Four Fours Puzzle. The students work on 0-10 in their table groups and we have a mini competition to see who can get these 11 solutions the fastest. Then, I have them work together as a class to try to get 11-50. They work together and compete against the other classes I teach. This year, the prize was homemade chocolate chip cookies. Yummy!

All this is a lead up to the first big assignment that I give my students. I call it From One to a Hundred. It works just like the Four Fours Puzzle, but each student is given 4 diferent numbers from 1-9 and then use these numbers to create as many equations as they can from 1-100. Here is the handout I use: https://www.dropbox.com/s/izdnlvwkxksysf2/FROM%20ONE%20TO%20ONE%20HUNDRED.doc