26 squares

All the squares of side lengths 1 to 26

In looking for an activity to demonstrate "Pythagorean theorem", a colleague and I decided to cut out all the squares of side length 1 to 26 like this. Actually we had the students do it on the first day of class. Once they had them cut out we asked them to make something - we got robots, wedding cakes, smiley faces, guns :( , you name it they made it. Then we asked students to make something with 3 of the squares. Eventually one of them made a triangle with three of the squares. The sides of the 3 squares making the sides of the triangle.

Once we decided to explore these triangles made from the hollow of the squares, this led to a nice discussion about when we would get a triangle or when it was impossible. We generated a list of side lengths of the 3 squares that would create a triangle ( for example 12, 15 and 20 ) and a list of side lengths of the 3 squares that would not create a triangle ( for example 3, 5 and 17 ).
Triangle inequality: The sum of the lengths of the two smaller sides must be greater than the longest side in a triangle.

After this discussion we talked about different classifications of triangles based on angles. So we then generated a list of side lengths that would make an acute triangle ( for example 19, 20 and 21 ), an obtuse triangle ( for example 5, 11 and 15 ) and last but not least the hunt for a right angled triangle. Once we established the rule for right triangles: the area of the two smaller squares must equal the area of the largest square the students were on the hunt for all possible triangles using the squares with side lengths 1 to 26. This is what the board looked like by the end of the period.

 

There were 4 families: FAMILY 1, 3, 4, and 5 and all it's multiples (6, 8, 10; 9, 12, 15; 12, 16, 20; 15, 20, 25), FAMILY 2, 5, 12, 13 and it's multiple (10, 24, 26), FAMILY 3, 8, 15, 17 and FAMILY 4, 7, 24, 25. Nine sum of squares triples amongst 4 families. The students had a fun search for these. We consolidated on how to find an unknown side in a right triangle.

We spent three days doing all this-time well spent.

Next we took the 3, 4, 5 family and explored the relationship between the side lengths amongst the different triangles. Students made connections about the triangles being scaled by some factor. They also decided that the angles in these triangles would be exactly the same. We called these families of triangles similar triangles. With a day of consolidation we spent two days on similar triangles.

Of course I wanted them to tell me what the angles in a 3, 4, 5 triangle were  (or any of the triangles in that family). They decided the only angle they knew was the 90 degree angle and the other two would have to be measured with a protractor. After measuring all the angles in a couple of different triangles in the family the students decided on the smallest angle being 37 degrees and the middle angle being 53 degrees.

I then pulled a trigonometric table and explained the words opposite, adjacent and hypotenuse and we checked out the ratios for a 3, 4 and 5 triangle. Of course the students reaction was who would make this sort of table. We worked through some examples where they found an angle, and some where they found a side length. Trig table only at this point. Again with a couple of days of consolidating we spent three days on this.

Next we made three representations ( table, graph, equation ) for side length of the squares and perimeter of the squares. P=4L. Linear relations and their properties and solving linear equations. Students spent some time with a graphing calculator and learned how to do a linear regression.

Again three representations ( table, graph, equation ) for side length of the square and area of the squares. A=L^2. Quadratic relations and their characteristics. Students spent some time with a graphing calculator and learned how to do a quadratic regression.

The goods are here:
The Goods

Time well spent - took 3 weeks and uncovered 6 of 9 big ideas in MFM 2P math and introduced 3 out of 3 strands. A great activity to start the course  - a little dry and very scaffolded - but a nice start to the course.

Replacing Unit Based teaching with Activity based Teaching

Finally getting back to this blog after a while away. I am going to try and focus and publish a few posts about activity based teaching in mathematics and how I no longer teach compartmentalized units.

Kudos go to all those bloggers and tweeters who I have been discreetly lurking at......but not participating in their conversations. You have inspired me and completely changed my practise in the classroom. Amazing stuff coming from Dan Meyer and others. You are all amazing. Jimmy Pai thanks for inspiring me to get back at this blog. Good advice!

Ok so I have been teaching grade 10 applied math (typically students who don't like and are not engaged in math for various reasons) in Ottawa, Ontario Canada. There are 9 big ideas in the course. Here they are: similar triangles, Pythagorean theorem , Right angled trigonometry, surface area and volume of 3-D figures, lines ( solving linear equations, interpreting linear situations, intersection of two lines ) and quadratics ( quadratic algebra, characteristics of quadratics, interpreting graphs of quadratics ). Here is the official Ontario Curriculum.

I have taught 9 sections of this course in the last 3.5 years ( 7 semesters ). The first three times I taught the course I taught by units. The last 6 times I have taught by integrating the curriculum into activities ( I call this cycling or spiralling the curriculum ). So .......no compartmentalized units. The activities have come from many sources and are scaffolded early in the course and become more student driven and inquiry driven as the course moves on. Here is a prezi of a presentation I did at OAME 2012 in Kingston that describes cycling the curriculum

Characteristics of an activity based classroom:
Cycle / Spiral through the curriculum using activities / tasks / problems / projects as the vehicle.
Students pose the questions for some activities based on a photo, action, video, statement, etc.
Classroom is student centered with the teacher acting as a facilitator.
Conversations about mathematics ( Teacher to student, student to student, student to teacher ).
Curious and creative learners ( students pose questions, students commit to a guess, students determine what they need to know to answer the problem ).
Hands on activities ( Cube-a-links, barbies, cups, ball rolls, marble rolls, squares, catapults, ski jumping, roof trusses, algebra tiles, toothpicks, bridges, ............. ).
Story telling about an image, photograph, etc.

Benefits? You bet..........
Multiple entry points for all students.
Increased student engagement.
Increased student confidence.
Fewer discipline problems.
Math follows the activity / project / task / problem which makes the math relevant for the student.
More connections are made between concepts.
Critical thinking improves and connections to the big ideas in the course develop naturally.
Improved retention by students because of experience with activity. The activity becomes a contextual cue for the student.
Repeated opportunities as the students do activities ( for gap filling, for assessment, for retention of curriculum ).
Cycling or spiralling curriculum allows for repeated opportunities and improves connections between the big ideas.
Meet students where they are at and move them forward ( differentiated instruction ).
More conversations about mathematics, problem solving ( accountable talk). Teacher to student, student to student, student to teacher. Collaborative environment.
Tons of time to get through the course material. ( "uncover the curriculum" versus "cover the curriculum" )

Scary   ;(   not so much. I have been living this for a few years now......and still have a pulse.....barely.

Safe Wheelchair Ramps (inquiry based learning)

Our inquiry based learning group has now planned a lesson on safe wheelchair ramps and we have observed the lesson in two teacher’s classes.
Here is the lesson: https://docs.google.com/a/ocdsb.ca/document/d/1cmBfTzcf53tUxuxGb-Rbl5HBnSUqw7HIlKhBhObtcfc/edit#

In both classes the level of engagement amongst the students has been significant. Both teachers were quite happy with the engagement.
Lots of questioning and inquiry in both classes with students showing mathematical thinking. Definitely some misconceptions in the classes but....I think the students resolved these.

Debriefing the lesson the second time seemed to have a little more structure to it and seemed to be a little more focused. I think we as a group will get better at this process as the semester moves forward. I am so lucky to be able to work with this group of educators who are focused, committed and taking risks. Kudos to all. Looking forward to hosting the next lesson study in early April.

Inquiry Based Learning Professional Learning Community

For this semester I am participating in lesson study with a group of about 18-20 teachers in my board who are interested in creating lessons based on inquiry based learning. All the teachers have classes of either grade 9 or grade 10 applied classes, I am very grateful for the opportunity and am lucky to work in a board that values creativity.
At this point we  have met for one whole day to go over the structure and to develop our first lesson which is safe wheel chair ramps.
This is a great group and should be very exciting-looking forward to the learning. More to come.......

Ottawa Shooting Stars - Small Ball

Ottawa Shooting Stars - Small Ball:

Who: 5-8 year olds (with keen parents)
What: Introduction to Basketball 'fun'damentals
When: Friday, 6:15 to 7:15pm (January 27th to April 27th, 2012)
Where: Glebe High School (Map)

For more information contact Alex Overwijk at alexander.overwijk@ocdsb.ca

Dan Meyer: My Muse

In the spring of 2010 I watched this video of math teacher Dan Meyer blog.mrmeyer.com

His presentation had an immediate impact on me. I have been following dydan's blog for 2 years now. His ideas have influenced how I now teach math.

My Journey as a World Freehand Circle Drawing Champ

Like most math teachers, who have been in the business for a while, I have a repertoire of "stories" that I use with students when I am introducing new concepts. One of my favourite is my story of being a former World Freehand Circle Drawing Champion (this seed was planted one weekend while watching a broadcast of the world bar tending championship - in Vegas).

In June of 2006 (after 10 years of telling the story) a student of mine asked if he could make a video of me and my story. It was posted on our school website where it sat for 6 months. In January of 2007 someone from Fargo, North Dakota found it and posted in YouTube and several other social media sites. The video went viral... and the rest is history.

I will post more about the story over time, but the reason that this is really important is that it has provided me with a vehicle to talk about my true passion, which is engaging students in mathematics.