I want students to solve systems out of necessity.

I want them to feel the interconnectedness of the two (or three) equations. In the past, I’ve asked small groups to build a functional 4×4 magic square. Soon they realize that changing a single number has multiple effects; this is the nature of the system. Unfortunately, abstracting the connections results in more than two variables. This year, I wanted to create the same feeling with only two variables. (The familiar *x* &* y*).

Enter: Alex Overwijk.

# Category: linear functions

Let it be known that Sadie Estrella is a Hawaiian treasure.

She made her way north for SUM2015 in Saskatoon and I got the opportunity to learn from her about counting circles (as well as share an eventful dinner).

It is probably good to understand her work on counting circles before reading a couple of ideas I had during her session.

I went to her blog and searched for #countingcircle, and the results can be read here.

This semester, I’ve been attempting to infuse my courses with more opportunities for students to collaborate while solving problems. This post is designed to examine the shift in student disposition throughout the process.

I have noticed an increased conceptual understanding almost across the board and this is reflected in the differing solutions on summative assessments. It is also nice to see their marks grow on these unit tests. I do not believe that paper-and-pencil tests are the best venues for displaying conceptual understanding, but it is awesome when the two intertwine.

## Animating Patterns

There is a very strong emphasis on linear relations and functions in the junior maths in my province. In Grade 9, students begin by analyzing patterns and making sense of bivariate situations. The unit–which I love–concludes with writing rules to describe patterns and working with these equations to interpolate and extrapolate.

Grade 10 students continue along this path in the light of functions. There is a large degree of abstraction that occurs in a short amount of time, and droves of students abandon the conceptual background (pattern making) in favour of memorizing numerous formulas. (Slope formula, slope-point, 2-point-slope, slope-intercept, etc.)

**Some (much prettier) quadratic patterns, which are introduced in 11th Grade, are posted here**

## Relation Stations

This semester I desperately wanted to improve how I taught linear relations to Grade 9 students. I had tried some interesting activities in the past, but lost patience and ended up drilling them with notation and algorithms. I wanted to find a way to show the students that equations were just explanations of patterns. I began compiling different linear patterns and dug in for the long haul.

I stumbled upon a collection of abandoned, square tiles and decided to use them to put students in the center of the pattern making.

## A Discussion on Slope

I have taught Grade 10 math more than any other class. I still have lessons that I created during internship that I use. Other sections of the curriculum I have perfected over the years. Today, I added another lesson to the list of those that I will do for a long time. This is my desperate attempt to describe and catalogue it. If I don’t do it now, it will filed as a good, but vague, memory.

My goal was to introduce the idea of slope and be able to get numerical values for slopes from graphs. I also wanted to introduce the four types of slope: positive, negative, zero, and undefined.

The class began with a quick discussion on how rate of change relates to slope. I handed out student whiteboards at the beginning and drew four lines GeoGebra.

Many teachers tell me that it is their creativity that limits their ability to be adaptive in the classroom. Somehow the “reform” movement (or should I say re-movement) has pigeon-holed itself into a connotation where high-energy teachers give vague tasks to groups of interested students. Out of all this, curricular outcomes explode in no particular order. This can’t be further from the truth. In my view, the biggest steps toward changing student learning is changing teacher perception.

The grocery store is a brain workout for the mathematically inclined. Not only do the varying metric and imperial conversions tease out the micro-savings of bulk, but neon yellow discount signs encourage percentages and good ole’ multiplication tables. Often you find adults transfixed in a complex division trying to figure out which ham will be cheaper. Once that calculation is complete, they turn their attention to making sure the portion will be enough to feed their whole family. The sheer volume of available estimations overloads me; coupons just complicate the matter–significantly.

Maybe you have seen the Burger King Stacker commercial where “Meat Scientists” work on an interesting problem. Needless to say, it piqued my curiosity the second I saw it; it was not long until I was trying to suck every ounce of mathematical value from the video. I am sure that I did not accomplish this goal, but I did manage to find some interesting problems and questions.