Expressions 2 Spiral Lesson
Lesson 2 of 5
Objective: Students will be able to write expressions that represent a given context.
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:
- A 15-minute start up problem, where we introduce the basic ideas of the concept they are about the review.
- A 45-minute chunk of time devoted to giving students in depth, meaningful practice, where students use very short videos to cover appropriate problems
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 this lesson with this list of Expressions scenarios. These are quick but important exercises, since many tougher algebraic problems assume that students understand the different meanings of variables and coefficients. Often students who can combine like terms and balance equations are stuck on problems because they are not sure why the variable and coefficient could represent. For example, the last statement tells students that q represents the amount of points in one field goal and asks "what does 7q represent?" A common misunderstanding here is that 7 represents the points from a field goal and q represents the number of field goals. However, 7 represents the number of field goals and q represents the points for each field goal.
I try and deepen their understanding around this by asking questions like, "what type of expression could we write to represent an unknown number of field goals that are 3 points each?" For this, we generate the expression 3x, where 3 is the number of points per field goal and x is the number of field goals. I help students understand this by creating a table and picking several inputs for x, like 0,1,2 and 3.
I then like to go further and ask, what if we had an unknown number of field goals for an unknown number of points per field goal?" Here the expression becomes something like y * x, where y is the points per field goal and x is the number of field goals. Again, we choose values for y and x and map out the inputs and outputs in a table. Then we contrast this to our original statement of 7q, where 7 is the number of field goals and q is the number of points awarded per field goal. I encourage students to use the Commutative Property of multiplication to write this term as q7, which matches the format of the other expressions we wrote, where points per field goal comes first.
Each of the statements listed in the Expressions Start Up can be used for a deeper investigation into some of the fundamental misunderstandings that students articulate when working with algebraic expressions.
Expressions 2 Template is a graphic organizer for the students practice work. The template is helps me recognize when a student needs help because students rate how they feel about each problem. 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. There are three primary problems for today's review:
Here is a source for additional problems if your students need more work:
The whole chapter relates to expressions, but I recommend the problems on pages 91 (mild) and page 93 (medium). Page 91 has the most basic types of problems, like "write an expression for 3 less than a number y."
They look something like this, page 91.
Page 93 is a bit more challenging, where problems ask students to attach an expression to a context, like "x represents the kilometers that a bus travels. If a train travels 200 more kilometers than the bus, write an expression representing the distance traveled by the train."
This is challenging for students. They picture two moving vehicles and need to identify that for any distance that the bus travels, they are saying that the train travels precisely 200 km more. This is a tough scenario because it is confusing. It is hard to imagine a scenario where no matter how far a bus travels, the train always goes 200 km further. However, the idea is to help students model any type of absurd situation.
The problems on page 93 look like this, Page 93.