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?"
Since this is a spiral lesson, we are reviewing content that students are already familiar with. They often cover combining like terms extensively in 7th grade and revisit the topic throughout the year in 8th grade. I focus the start up on answer why we are able to combine terms with multiplication and division when we can't with addition or subtraction.
I place a simple problem on the board, something like, simplify the following expressions by combining like terms:
Example 1:2x + 3x^2
After we establish that the first example is already simplified by the second can be written as 6x^3, we begin to discuss why multiplication allows us to combine unlike variables with unlike powers (while addition and subtraction do not).
I start by asking students to plug in values for x to confirm that we can indeed simplify the second example, but not the first. We choose non-zero values for x, take x = 2, and see that example 1 would equal 2(2) + 3(4) = 4 + 12.
However, if we tried to combine like terms in any way, like 2x + 3x^2 = 5x or 5x^2 or 5x^3, 5(2), 5(4) and 5(8) all do not equal 12. So this is a start to confirm that we can not combine unlike powers with addition.
I like to ask, "Are there any x values that would work for addition?" It is important to always look for exceptions to rules. Here, if x = 0, then we can combine the terms for this expression. The identity elements of multiplication and addition are often great candidates for an exception.
To understand why 2x(3x^2) = 6x^3, I remind my students of the Commutative Property. We rearrange the terms in 2x(3x^2) as 2(3)(x)(x^2) = 2(3)(x)(x)(x) = 6x^3.
But to help students really think about why we can multiply unlike terms, we need them to consider what is fundamentally different about multiplication and addition. Even though students often think of multiplication as simply a repeated addition, they need to also think of multiplication as groupings. For example, 2(3) = 6 because we have 2 groups of 3 or 3 groups of 2. In a sense we are not combining these terms when we multiply them, but we are counting groupings.
2(3) = 3 + 3
2(3) = 2 + 2 + 2
In the first example, we have two groups of 3, so we are only adding 3 to itself. 3 is its own like term. In the second example, we have three groups of 2, so we are only adding 2 to itself 3 times. 2 is also its own like term.
This subtle difference seems like semantics, because we could point out the repeated addition that is happening, but here we are identifying that multiplication naturally defines groups of one term by another term.
So x(y) = xy since we could have x groups of y or y groups of x
3x(2y) = 6xy since we have 6 groups of x groups of y and so forth
This conversation is really fun to have with the class. It is important to not just tell them these things, but stop constantly and ask for their feedback. They need to realize that although multiplication and addition are connected, they are different.
The Combining Like Terms template is set up to help me recognize when a student needs help. It asks students to 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.