Grade 9 Academic - Day 2

We didn't have class yesterday as the school buses were cancelled, so I'm considering today as day 2.

Before class I handed out small whiteboards and markers and organized bags of algebra tiles (which I won't get to until tomorrow as it turns out).

We started with this warm up which students did on their small whiteboard:

I circulated and then wrote - 10, -5, -2, -6, -4, -7, IDK on the board. Those were some of the answers I had seen. Clearly we had work to do. The reason I wrote all of those answers was for them to see that many students didn't know how to do this correctly - that if they didn't get it right, it was okay. We would work on it together. We then talked about adding and subtracting integers (I have an old blog post about this here).

Then I let them try the warm up question again with much better results.

Next up, today's visual pattern (#129 but I removed the colour as I found it too leading). We used cards to create random groups of three and their task was to figure out the rule for the pattern in as many ways as possible.

I confess that I didn't circulate, listen and engage with my students enough while they were doing this as I was also handing out textbooks. However, I was really pleased with the work they did, the fact that no one went to a table and that they are starting to be able to show their thinking. Here are some samples (some got a little help connecting the representations from me):

I had two possible ways of seeing this pattern ready to go, but added a third to capture some of their thinking - many groups made a constant in the middle of the vertical part. Here are those:

I love that there are so many different ways of seeing these patterns and am always impressed when my students show me a new way of seeing it.

Next, we talked about algebraic models: using algebra tiles to represent expressions.

We had just enough time for them to try a couple of examples. Honestly, I was too rushed doing this and should really have let touch the actual tiles to truly understand. I will fix that tomorrow.

We wrapped up by returning feedback forms (not sure I talked about those - read about them here). 

Completing the Square with Algebra Tiles

There seems to be some interest in how to use algebra tiles in the MTBoS so I thought I would attempt a blog post.  I apologize now if I confuse anyone - I don't claim to be an expert, but I do find they help kids make connections.

For years I avoided algebra tiles. It didn't help that one of my previous colleagues told the story of how one of her students managed to choke on a tile! Thankfully, none of my students have done that.  Our tiles are red and blue so they all feel the need to look through them as if they are 3D glasses (sigh). The kids get to choose which should be positive and which should be negative.  They invariably say red is hot so positive and blue is cold so negative so that is what I am using here.  

We introduce the tiles in grade 9 when we simplify polynomials.  They use them to multiply binomials (it makes a rectangle) and to factor trinomials (find the length and width of the rectangle).  So my students are very familiar with algebra tiles by the time they get to completing the square.

I start with a warm-up to make sure they remember what is special about perfect square trinomials. They work in groups of 4 - each student does one question then they add up their answers.  If the sum is correct, we can move on, if not, they have to find the error.  I stole this from someone at TMC13 (who stole it from someone)^n, who stole it from Kate Nowak.

Then we tackle our first example:

Students understand that instead of making a rectangle, they need to make a square with their tiles.  They each have their own set to work with.  We will place the 7 unit tiles off to the side and work with the rest.

They see that they need to add 1 unit tile to make a square.  In order to do that we have to add a zero pair so a -1 unit tile goes with the 7 off to the side.

We can then write the area of the square a its side length squared and simplify the unit tiles.  And just like that we have vertex form!

We work through a couple more examples in the same way, with students working with their tiles then consolidating with the whole class.  Each time they have to divide the x-tiles - half for the length and half for the width so that they make a square.  Each time they have to add unit tiles.  I get them to notice patterns in what they are doing.   

Next, we use a chart to connect the algebra tiles to the algebra.  The cool thing is that they actually understand why we are dividing 'b' by 2 and squaring it because they have done it with the tiles.

The next day we extend to quadratics where 'a' is not equal to 1.  Again we start with tiles and connect to algebra.  You need identical tile diagrams for each x^2 you have, but it's the same process.  It connects factoring out the 'a' value to the tiles.

I find algebra tiles really help explain why we are doing what we are doing. It helps that my students are asking how to change a quadratic from standard form to vertex form. Well, some of them are anyway!

My SMART Notebook file is here. I left off the extra practice for day 2 as I am not really happy with it, but am not sure how to change it.

If you are looking for on-line algebra tiles, The National Library of Virtual Manipulatives is a great resource.