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 start off with a problem that reminds students of the importance of being flexible when balancing an equation. A common example of the rigid and automatic approach that students take can be identified with a problem dealing with the distributive property. When students see a problem like:
2(x - 4) = 8
Students seem to always distribute the 2, then add 8 to both sides and then divide by 2. However, it is important for students to realize that they don't have to distribute first. They could divide both sides by 2 and then add 4 to both sides. No only will this solve the problem in two steps (instead of 3) but it will help them on much tougher problems. One can imagine equations where simplification depends on this type of flexibility.
Furthermore, students need to understand that 2(x-4)/2 = x - 4. Dividing the left side by 2 is tricky for students. They often don't understand why we only need to divide the 2 by 2 and not also divide the x and -4.
To help them understand, I ask them to think of a simple numerical example, like 2(5 + 3)/2.
Since 2(5+3) = 2(8) = 16 and 16/2 =8, we can see that we only need to divide 2(5+3) by 2 once. When we have terms connected by multiplication or division, we only need to divide one (unlike addition or subtraction which I often show for contrast). Students understand that 2(5+3)/2 = (5+3), not 5/2 + 3/2.
There is no template for today's problems, since I want students to write out each equation and balance them step by step. Some of the questions are much tougher than others and present a great opportunity for small group talk.
Take Question 14 for example: many students will try and solve by finding the least common denominator of the terms on the right hand side, -(x+11)/4x + 1/2x. A common error is to think that the common denominator is 4x^2 instead of just 4x. For me, this problem presents a teaching moment for canceling out fractions. Having the flexibility to recognize that multiplying each term by 4x will help to quickly unravel the problem is the kind of thinking I want to encourage in all students. Not only do I want them to find ways of plowing through a problem (by plowing I mean solving a problem the hard way) but I want them to compare options and choose the one that presents the clearest path to a solution. Efficiency and precision are effective tools for any problem solver. Notice that if we multiply each term by 4x, the problem simplifies dramatically in one step:
3/4 = -(x+11)/4x + 1/2x
3x = -(x + 11) +2, Here we can distribute the -1 in -(x + 11)
3x = -x - 11 + 2, Combine Like Terms
3x = -x - 9, add x
4x = -9, divide by 2
x = -2.5
Each problem below offers an opportunity to talk about precision and efficiency in balancing equations: