Algebra 1: Representations of Linear Equations

Increasing my students' representational fluency has been something I've been working on for a while.  Our second semester started last Monday, and to start my Algebra 1 students off easy, I had them do a four-fold poster of a linear relationship to review what we did last semester: situation, equation, table, and graph.  They did the work fine overall, but quite a few students had more troubling questions than I had expected (i.e. "how do you make a table?").   I guess it just shows that we have to keep going through these different representations and their connections again and again.

We have started the new unit - working with linear equations - in which students have to write the equation of a line given its slope and a point, or two points, or a point and a parallel line.  In the past, I have done this only algebraically (except for the initial explanation of concepts); this time around, the students will have to practice the problems both algebraically and graphically.  And, more importantly, the skills tests will require them to show mastery with both methods.  Let's build those connections!

Here are the first few lessons in the unit.

Lesson 1 (Representations of Linear Functions)
Lesson 2 (Graphing Practice)
Lesson 3 (Write the Equation of a Line)
Lesson 3: Keynote / Powerpoint

And some snippets from the worksheets to illustrate what I am talking about:

 

Introducing Linear Inequalities

To show that a line is a representation of an infinite number of points, I like to give my algebra 1 classes an equation, like y = 2x + 3, and then give each student a couple of different ordered pairs - some that are solutions and some that aren't.  I have them each work out their points, and then go to the board to plot an open or closed circle, depending.  Once all the students sit back down, we look for patterns and see that all the closed circles fell on a straight line.  Discuss, and voila.

This extends nicely to linear inequalities (and systems of equations and inequalities).  On Tuesday, my algebra 2 students were reviewing linear inequalities so I did this activity with them.  I really like it, because it is engaging, and it helps build a mental picture that they can rely on later on when they are struggling through graphing problems on their own.  My students often get stuck on the "pick a test point" part of the process; but now, I ask them if they would have plotted a closed or open circle based on their result, and to think about what the picture on the board looked like.  This usually helps them see which side of the boundary line to shade, and to be able to explain why.

Here is what the board looks like after students plotted their points:

Then, we looked for patterns.  Usually, a student will come to the board and draw some sort of line after getting frustrated with trying to explain it in words.  Then I reveal the shading:

And there are usually some audible "ahhs" and such.  Another great benefit of this is that the string of open circles on the boundary helps students see what the dotted line is all about, and why changing the inequality to include an equals sign would create a solid line - a string of closed circles.

Here is my lesson that goes with this.  And the keynote.

Algebra 1: Graphing Lines Practice

I just used this worksheet from Mr. K for the first time the other day.  I thought it had a pretty cool setup, but I didn't realize just how effective it would be until I used it in my first class.  The "solve the joke" aspect of it helps draw them in, but the hidden beauty is in its self-checking properties.  Since each line must pass through exactly one number and one letter, a line that doesn't do this must be graphed incorrectly.  Students started realizing this and would go back and find mistakes without having to check with an answer key.  The only bad part (sorry to say) is that they had absolutely no idea what the answer was supposed to mean (see earlier post).

I made up a "balloon pop" homework to go with this that was inspired by Green Globs.  I wish I had the tech access for my students play that game.

Algebra 1: Skills List - Spring Semester

I spent a good deal of time right before break trying to figure out exactly how far I can push my students for the second semester of Algebra 1.   These skill items will be broken down into chunks for the skills tests, and MC-ized for the benchmarks and final exam.  I regret how many concepts I had to leave out due to time pressures; and still, the list seems daunting and endless.

If you're interested, this is what my students will be doing over the coming months.

doc / pdf