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?"
I like to start off with a light question to help students get started on a topic they have seen many times over the years. My thinking is that so many students already have mastery of this topic that I need to identify the few students that are still struggling. The low inference question helps m identify who those struggling students are (they are the ones not able to quickly answer the question). I also like to use the low inference question to gauge my instinct. If I start with a low inference question and almost no one answers, that tells me that many students still have not mastered the content. Either way the light start up question quickly sorts these things out for me:
This is a basic cumulative frequency table question. I ask students to write out a separate table that is non-cumulative. This helps me see if they can deconstruct the cumulative categories. If I see a student rewriting 20-29 and 20-39 as 20-29 and 30-39, I know that they have a sense of how to rewrite the groups into appropriate ten point intervals.
From that point I make sure that they subtract the intervals to find out how to redistribute the frequencies into the ten point intervals that they have created. For example, the 20-29 group has 8 and the 20-39 has 18, this means that the 30 - 39 has 18 - 8 or 10 people. If students can explain this to me they have a strong understanding of the problem.
The class discussion is also structured to help students who need a bit more support. I would ask, "how do the ranges of the interval indicate that we have a cumulative frequency model?" If this question doesn't seem to spark responses, I could be even more specific, "what number does every interval start with? Why would they always return to the beginning?" Then we could follow up with some more flexible questions, like, "I heard someone tell me that the numbers in each group could never decrease. What do you think of that?"
From there we might discuss how students rewrote the cumulative intervals into ten point intervals, emphasizing the use of subtraction.
I also use the problem as an opportunity to ask, "why might we write cumulative frequencies?" Then I begin to drill towards the idea that a cumulative frequency table helps us quickly identify the total number of participants and the interval in which the quartiles live. For example, I know that this group had a total of 35 participants by simply looking at the last interval. I also know that the median(which is the 18th participant) is in the 30-39 interval, since the 18th person fits within the 20-39 range.
For many students these problems are difficult to work with because they have to create graphs for many problems. I point this out to them and encourage to do those in class when I can build their confidence around creating an awesome graph. I also let them use a lot of color. Not that color increases the mathematical analysis directly, but I encourage them to have a bit of fun and pride in their graphs. Adding color makes the graphs that much easier to love.
I also have them use this template to support their graph work: Data Tables Template
Whatever they don't finish in class, they bring home.