Algebra II : Unit #1 - Modeling with Algebra : Lesson #1

What is Algebra?

Objective: Students will be able to explain why algebra needs definitions and axioms, what some of these first principles are, and what it means to "do algebra".
Subject(s): Math
60 minutes
1 Discussion: What is Algebra? - 10 minutes
What is Algebra, Video Narrative, Discussion What is Algebra.mov
2 Discussion: An Historic Problem - reasoning abstractly and quantitatively - 10 minutes
What is Algebra, Video Narrative, An Historic Problem.mov
A Common Mistake

A Common Mistake on Problem 1:

Students wrote x^2 + 21 = 10*sqrt(x) instead of x^2 + 21 = 10x.  I saw that the class was split about 50-50 on these two equations, so when one student asked me which one was correct, I responded by sending him to the board.  He wrote down both equations, and then explained to us that he wasn’t sure which was correct but that he was inclined toward the former.  He and some other students were helped when someone pointed out that since we used “x^2” for the square number, we should use “x” for its root. 

We also used some numerical examples to help students understand the relationship between the “square” and the “root of the square”.  They understood the concept but had trouble making use of the symbols correctly.  To do this, I built on the class's understanding of the term "square" by asking them for an example of a square number.  They suggested 25, so we tested that against the given criteria.  

First, we added 21 to our square and wrote down "25 + 21".  Next, we set this equal to "10 roots of the square".  Since our square was 25, the class recognized that the "root of the square" must be 5, so we wrote down "25 + 21 = 10(5)".  They immediately saw two things: first, that 25 was not the correct solution, and second that if we replaced our 25 with "x^2" we should also replace our 5 with "x", not "sqrt(x)". 

One student also noticed that our square number would have to end in 9 so that the sum could be a multiple of 10.  This led us to discover one of the solutions (x = 7), but we had to solve the equation explicitly to find the other.  You can see some of these things in the attached copy of student work

Additionally, very few students remembered how to solve a quadratic equation, but when one suggested using the Quadratic Formula they all recognized the formula and remembered how to apply it.  The changes being made to the Geometry curriculum should provide future students with more algebra practice, so I hope this won't be such a problem in the future.

3 Discussion: How do you know? - 15 minutes
What is Algebra, Video Narrative, How do you know.mov
4 Homework - 10 minutes

In the final 10 minutes of class, there are several things to accomplish.

First, I will assign the homework for the night, which is to complete the remaining two problems on the Historic Algebra Problems handout.  This will be due at the beginning of class tomorrow.

Second, I will handout The Weekly Workout general guidelines along with the their first Weekly Workout 1.  Quickly, I will explain the rationale behind the weekly workout - it's about staying strong through constant practice - and ask everyone to write the due date on the top of the assignment.  It won't be easy, but I'll have to avoid getting side-tracked by lots of questions here!

Finally, I will pass out the class syllabus to all of the students, asking them to read it and share it with their parents.  Their parents must send me an email sometime before Friday so that I know they are able to contact me if they need to, and so that I can contact them.  This is another homework assignment!

It's important to write all of the assignments on the whiteboard and to make sure that the students copy the due dates into their planners, but it's also important to send them off with a smile and a big "Welcome to algebra!" at the end of class.

Adjustments to Practice
Grade Book and Data Analysis

As the year has progressed, the Weekly Workout has evolved slightly.

  • I both collect and assign the Weekly Workout on Monday. The students didn't like having to wait 'til Tuesday to get started on the new one!
  • Over the year, I added new problem types and eliminated old ones. I also gradually increased the level of difficulty on the long-standing problem types. In particular, I've been careful to include problems from each Algebra 2 unit that has been begun or completed. Even with increasing difficulty, scores have increased throughout the year!
  • On some short weeks (before or after long weekends), I've produced a shorter workout of just 10 problems.
  • Grading: In the gradebook, I make 18/20 count as 100%. The two extra points are just that: extra credit that stands as a reward for excellent work. I've found that it's not the "smartest" kids who earn these points, but the ones who apply themselves most diligently. The "smart" kids typically race through and make a few careless errors. The diligent ones take the time to check their work and get help when they need it.
  • I made an agreement with my class that we could review one problem from the workout at the beginning of each class. However, I typically refused to completely solve the problem. Rather, I would as students to explain how to get started or what the key strategy was. Once we'd made a good beginning, I leave it up to the individual students to finish. This typically took no more than 5 minutes and gave me the opportunity to clear up many misconceptions.

The feedback I've received from both students and parents has been very encouraging. Many parents have thanked me for providing this continuous practice for their students, and many of my most diligent students have commented the improvement in their skills.

Perhaps unsurprisingly, I've found that the Weekly Workout is an excellent predictor of a student's overall grade. Go figure!