This is not a test prep unit (although it could be construed as such). My goal is not to simply expose students to test questions and format (although they will get an amazingly wide variety of exposure and practice). I want to help my students look back on the year and fill in the gaps and build on their foundation through repeated practice. All students need a chance to recall the things they have forgotten (or never mastered in the first place). Our curricula often push forward, build up and look ahead. But we can’t forget to spiral back and give students an opportunity to revisit topics from past units. These lessons represent my attempt designing a rewarding and fulfilling review unit for algebra. It is an attempt to prepare students for the summative exams that we use to asses our curriculum design and mark student progress.
The lessons in this unit are built around a simple format:
The work reserved for the final 45 minutes of class is connected to a series of homework assignments. The materials are designed so that whatever students don’t finish in class, they can do at home. If they finish the current assignment, they can move ahead and complete future assignments.
Each day, the lesson starts with some follow up on their practice work. Based on the previous lesson, I will be prepared to ask specific local and global questions about their work. Local questions can be anything specific to a certain aspect of a problem, like "how did you get that number there?" Global questions can be things like, "how does this example connect to the problem before it?"
The idea of relative error is to quantify how much your error matters. To help students think about this, I ask them to imagine something that is small, at most a foot in length.
"Ok so lets say our ant is about 2 inches long. Yes, it is a big ant. Then lets say you try and measure it, without knowing its actual length of 2 inches, and you tell me that you think it is 5 inches long. What would you say about your accuracy?"
"I'd say I need to try again, because that is way off."
"I agree, how far off is it?"
"How did you do that?"
"I subtracted 2 from 5."
"And if you subtracted 5 from 2, what would you have to do to also get a positive answer of 3?"
"Reverse the sign."
"Right, but do you agree that know you need a positive value here?"
"So how do we do that? How do we get a positive value?"
"The Absolute Value."
"Right so, if we subtract our observed measurement from the actual measurement, or vice versa, and take the absolute value, that is called 'absolute error.'"
Then we reverse the process and talk about something big. The discussion is similar. Perhaps they chose a tall building, like the empire state building. I ask them, "if you tried to measure the empire state building and were 3 inches off, how would you respond?" Here the idea is that this would be incredibly accurate. Being only a few inches off of such a large value is small, relatively speaking. From there we conclude that the way we know an error is small or large, is by first finding the absolute error and then dividing it by the actual value. This is called relative error.
Then we apply this reasoning to the Relative Error Start Up problem. Here we discuss that we first need to find the observed area and the actual area and then make our calculations. The only issues we really encounter with this problem is a misunderstanding of why we divide by the actual value. What often helps is to remind students that your error is out of the actual total. This logical framework matches the calculations in their algorithm.
We give students a set of relative error problems and a helpful Relative Error Template to follow along. The template is set up to help me recognize when a student needs help. They rate how they feel about each problem as they finish and I look at these numbers to figure out if they feel comfortable with the material. I try and keep this rating system simple. Something like, give yourself a 4 if you really understood the question and give yourself a 1 if you felt really overwhelmed.