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 love consecutive integer problems, so I start off with one that helps students see just how flexible the algebraic modeling process can be:
Lets say you add four consecutive integers and get 130, how can we find those numbers quickly?
First I remind students that integers are positive or negative whole numbers and that consecutive whole numbers are two whole whole number right next to each other, with no whole number in between. Then I walk them through the modeling process in a conventional way:
We don't know what the first number is, but we do know that there is some first number that fits. We can call this number x.
Students consistently want to name the next number as y, since it must be different from the first, but adding more variables complicates our algebraic work, I ask them "how can we represent y in terms of x?" This phrasing is difficult but important for parametric equations, since students need to solve for one variable in terms of another, I explain that writing y in terms of x means representing our second number by only using the variable x. This helps students realize that the second number should be x + 1 and the third number should be x + 2 and the fourth number x + 3.
Here's a question that often challenges my students:
"Using this approach could we write the nth consecutive number as x + (n-1)?"
This is another tough question for them, but if we list out the pattern from our example, students will see that the first consecutive number is x + 0 or x + (1-1) and the second consecutive number is x + 1 or x + (2-1) and so forth. It is not something at all to memorize, but it is something to observe. They need to be comfortable with the idea of modeling the sum and look at the relationship between terms and term numbers.
Once we discuss these ideas, students can quickly solve for x by combining like terms and then substitute the value of x to find each of the four consecutive numbers. What surprises students is that we can model this scenario in a variety of ways. I ask them if we can pick any expression for our first term. "What would happen if our first time was not x, but x - 1?"
Here students write out:
(x-1) + x + (x + 1) + (x + 2)
It is important to observe that -1 and +1 cancel out and reduce the amount of work needed to solve this problem.
I love it when a student says, "So could we pick any expression to represent out first term?" The answer is yes, as long as each subsequent term is consecutive.
We could have something like:
(x + 899) + (x + 900) + (x + 901) + (x + 902) = 130
It surprises students that this still works. I make sure to walk through the steps and show them that we still get the same value for x. The only difference from the first problem is that x no longer represents the first number in the sequence, but the value that we can substitute into x + 899 to find the first number in the sequence.
Although I would not encourage students to model a consecutive problem with these values, it important that they are extremely flexible in the modeling process. Consider the question "you have 7 consecutive numbers that add to 28."
Instead of writing:
x + (x + 1) + ... + (x + 6) = 28
They could write:
(x - 3) + (x -2) + (x - 1) + x + (x+1) + (x+2) + (x+3) = 28
In this equation, all of the constant terms cancel out. We get:
7x = 28
x = 4
So the numbers are 1,2,3,4,5,6 and 7.
This approach is much quicker than adding all the constant terms in:
x + (x + 1) + ... + (x + 6) = 28
For practice I give students a set of problems and a helpful template to organize their work: Consecutive Integers Template. The 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.
Here are the four practice problems for today: