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?"
My students can quickly identify the difference between an expression and an equation or inequality. Because the rule is short and easy to memorize (expressions don't relate anything, they have no equal or unequal sign) students are able to quickly answer most questions around this topic. I like to push them a bit on this, and ask, "why do we bother making this distinction?"
A major misconception appears when students try and combine like terms or simplify an expression. This work comes mostly in later lessons, but it is important to help students identify possible misconceptions now.
For example, take the expression x + x/2
Many students want to "cancel out" the 2 in the denominator of the second term. They use a strategy that they often deploy when balancing equations. They start by multiplying each term by 2 to get
2(x + x/2) = 2x + 1x = 2x + x = 3x
To help students understand why you can not use this thinking to simplify the terms, I like to assign a specific value to x. I ask them to each pick "friendly values for x" to confirm that 3x does not equal the original expression of x + x/2. I want my students to experience the realization that they have created a new expression, not a simplified version of the original expression.
One question that sometimes arises is, "How do I pick friendly values for x?" When this occurs, I say:
Let's try an unfriendly value. I think 3 is unfriendly for this problem. Even though 3x = 3(3) = 9. When we plug in 3 to x + x/2 we get 3 + 3/2, which gives a fraction.
I then encourage my students to evaluate 3 + 3/2, since they need practice adding fractions and since they should be able to estimate that this isn't nearly 9, but I want them to pick multiples of 2 to make the fraction easy to work with. In this case, I am teaching both for the current problem and for working with functions in general. I always want my students asking, "what input(s) can I use here to best evaluate the function?"
Once we have plugged in different values to confirm that the expression has changed, I ask them to explain why multiplying by 2 would change the expression. They usually understand that, "You are doubling both terms, so each term becomes twice as large." So I am ready to ask, "if each term became twice as large, what happen to the sum of the two terms?"
This is an opportunity to review out the Distributive Property. Since 2a + 2b = 2 (a + b), we know that our sums must be twice as large. An interesting question at this point is "Why would this work when balancing an equation?"
Here we can bring out any example, say x + x/2 = 6
When we double each term on the left, we also double the term on the right.
2x + x = 12
Here x still equals 4 because we doubled the value of both sides, keeping the equation in balance.
"So how would we simplify x + x/2?"
Here I spiral back to techniques learned in lower grades, if they added 3 + 3/2, they would need to get a common denominator and/or convert to decimal form or even use some type of model. I focus on the common denominator technique, since it brings out the importance of the Multiplicative Identity.
x + x/2 = (2/2)x + x/2 = 2x/2 + x/2 = (2x + x)/2 = 3x/2 = (3/2)x
One key moment is to realize that we multiply x by 2 and also multiply it by 1/2, thus negating any possible change. Then another important moment is to reinforce that 3x/2 = (3/2)x. That is something that stumps many students and this is a great moment to address it.
We give students a set of problems and a helpful Template to follow along. The Expressions 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.
Many of these problems are very basic. So I ask them to write out each expression in words. For example, if they see 3x - 2, I would ask them to write "two less than three times a number, x." Although this is somewhat tedious, it really helps them process each expression represents.