Saturday, March 14, 2009

Algebra 1: Introduction to Inequalities

I'm not planning our Algebra 1 classes this year, so I have not been producing much for it. But I did put together a scaffolded introduction to inequalities. The objectives are for students to:

  • Compare numbers using a number line (i.e. "<" means "to the left of")
  • Understand the difference between open and closed circles
  • Graph the solutions of a statement like "x < 3"
  • Understand graphically why adding/subtracting by any number or multiplying/dividing by a positive number does not change the relative position of two numbers, while multiplying/dividing by a negative number does. In other words, students should understand when and why to "flip the inequality sign" when solving inequalities.
  • Solve and graph linear inequalities
Here is the file.

Friday, March 13, 2009

Algebra 2: Reducing Polynomial Fractions

I started this lesson with some theatrics. I asked them to simplify the fraction shown in the picture, and of course they all wanted to cancel the terms (as expected). I let them do it, and then changed the pretty pink heart into the fiery eruption you see here. I told them that those red slashes are like daggers through a math teacher's heart. I also told them that, when they go to college, I never ever want them to make the mistake of canceling out terms. Cancel factors, not terms! We spent a lot of time talking about the difference between factors and terms, and why this rule is true. We talked about why you can't add 5 and 5x, but you can cancel the 5's in 5/5x. I think this was time well spent, because this canceling problem is a persistent weed. From there, we practiced factoring and canceling. Pretty straightforward. In the following lesson, we multiplied and reduced products of polynomial fractions. There really were no new skills to learn, so after modeling one problem, I had them do independent practice work.

And now, I am caught up on postings!

Lesson 11 (Reducing Polynomial Fractions) doc / keynote / quicktime

Lesson 12 (Multiplying Polynomial Fractions) doc

Algebra 2: Factoring Difference of Squares

Continuing with the lessons, we learned to factor difference of squares expressions. I used a geometric approach to help make sense out of the pattern, and it has really helped some students figure out how to more easily factor the nasty ones like 25x^2 - 16y^4. A quick sketch of the squares, labeled with their side lengths, has proven quite useful.

Lesson 9 (Difference of Squares) doc / keynote / quicktime

Lesson 10 (Review and Practice) doc / keynote / quicktime

Algebra 2: Factoring, and More Factoring

It's been a while since I posted. The last week of February was our Junior Trip, in which we take all of our junior class on a 4-day-long trip around California to visit various CSU campuses. It's an incredibly important part of our program, because it is the time when our juniors really start to imagine themselves as college students. The tours, the student panels, seeing the dorms and classrooms, the admissions directors, and the DCP alumni all bring things into sharper focus for the 11th graders. We moved the trip earlier this year (it used to be in April) because kids come back inspired and ready to make positive changes, and so we wanted them to have more time to improve their grades before the end of the semester. It's also a great time for students and staff to bond and get to know each other in different ways. Needless to say, a 4-day, 3-night field trip with 80 high schoolers is tiring. We're all pretty much recovered now, and it's been back to business as usual. Time to catch up on some lesson postings.

In Algebra 2, we're nearing the end of the polynomials and factoring unit. I've been focusing on basic factoring techniques (look for the GCF first, then either use trinomial factoring or difference of squares, if possible). I'm still deciding whether to throw sum/difference of cubes into the mix this time around. I decided to bring simplifying and multiplying rational expressions into this unit (instead of waiting for the rationals unit) because it seemed like a good way to have them get more practice with factoring without repeating the same exact problems again and again. Plus, these questions are prominently featured on the STAR test.

One thing that has been helping students deal with factoring out the GCF is teaching them to write the prime factorization of each term in the polynomial, every time (including a -1 factor when there is a minus sign). Though it takes longer, this is pretty much a foolproof way of factoring out the GCF - many students have a lot of difficulty with the "what's the largest expression that divides into both" method.

Lesson 6 (Factoring the GCF and Trinomials) doc / keynote / quicktime

Lesson 7 (we used Algeblocks to get a better understanding of factoring trinomials) doc

Lesson 8 (Factoring Trinomials by Grouping) doc / keynote / quicktime