The previous lesson on solving equations and inequalities graphically was very difficult for the students, and I ended up having to do an entire extra lesson of just practice. They worked very hard and I heard quite a few "my head hurts!" comments. Even with that effort, they were taking much longer than I had predicted to do the work, so I ended up pushing the quiz off until the next lesson (this afternoon).
It's interesting: my students are easily able to solve equations such as 2|3x - 4| + 5 = 9 this way, by identifying points of intersection of the functions f(x) = 2|3x - 4| + 5 and g(x) = 9, and looking at their x-values.
As soon as the problem changes to an inequality like 2|3x - 4| + 5 > 9 , they get lost. There seems to be a disconnect between knowing where the f(x) values are greater than the g(x) values (which they understand), and being able to identify the set of x-values that produce those y-values. Many of the students seem to flip back and forth between the x- and y-values and get confused with what they are doing.
In the review lesson, I had them start again with writing the solution in inequality and interval notations when given a number line. This seems to have helped some of them make the connection. I think that many more students were getting the idea by the end of class. I'll see how much they have actually learned when I grade the quizzes today.
After the quiz, they will be working in study groups, which is one of my goals for this year. Part of their homework for today was to look at the different sections that will be on the test (which I listed for them), to flip through notes and old quizzes to remember what types of problems are in each section, and then to rank those sections based on how well they think they are prepared. I resisted creating a review packet for them for when they will be working in their study groups today. Instead, they must compare their rankings and decide as a group what topics to study. Then, they need to find example problems in their notes, homework, and old quizzes. This is risky, because students (even my lovely honors kids) without a concrete assignment (like a worksheet) tend to lose focus and not use their time well. But that is what I want to coach them on, so we'll see. I did make a short review sheet with some key problems for them to complete as homework. I hope to take away more of this scaffolding by the end of the year.
Here are the sections on the Unit 2 Test:
1) Functions and relations
2) Function notation & function composition
3) Representational fluency (using equations, tables, arrow maps, and graphs)
4) Operations on functions (adding, subtracting, multiplying)
5) Domain and range
6) Piecewise functions
7) Translating functions
8) Transforming functions
9) Solving equations and inequalities graphically
Quiz 3 scores:
10| 0 0
9| 2 6 8 8 8
8| 0 2 6
7| 6 6 8
6| 0 4 4 6 8 8
5| 0 4
The class seems to have split into those that really understood and those who didn't. There's something about inequality signs that just drives some kids bonkers.