Wednesday, August 4, 2010

Linearity in Algebra: A Lesson Plan

In teaching Algebra 2, one of the areas where we spend a fair bit of time is solving linear systems of equations using three different methods: 1) graphing; 2) substitution; and, 3) elimination. This mini-unit will be for solving linear systems using substitution.

Design a student task or mini-unit centered on understanding linearity.

Essential Question: How do I solve a linear system using substitution?

Lesson Steps:

1. "Please Do Now (PDN)" - Solve equation 1 for y: 2x - y = 2

2. Review PDN with students.

3. Add equation 2 to the equation 1 to make a linear system: x + 6y = 27

4. Walk through the Algebra steps to substitute the PDN solution into equation 2 to solve for x.

5. Walk through the Algebra steps to substitute x into equation 1 and solve for y.

6. Walk through the Algebra steps to check the solution in both equations.


Homework assignment: For each linear system given, identify the equation and variable for which you will solve as step 1 in substitution. Walk through the steps to solve for that variable in the equation.

Closure / Ticket Out:

At end of this multiple day lesson, students will complete and hand in a solved system of linear equations.


Note that this is a multiple day assignment, so I would start this lesson by giving an overview of linear systems based on what we did in solving systems by graphing. We will also discuss that some linear systems have no solutions (parallel lines) and some have an infinite number of solutions (same line). Homework assignments will break the steps into pieces until the students are able to fully solve a linear system using substitution.


  1. Doug,
    I thought I had posted a comment earlier but perhaps that comment did not find its way to your blog? Anyway the comment I had hope to make refers to the way in which we teach the concept of linear systems. My experience last year was that students who struggled with the idea of a linear equation in two variables had considerable difficulty when we got to systems of linear equations. I believe that many of my students needed to grasp the basic concept of a linear function and that the graphic representation provided the best format to achieve that goal. Once students understand how geometric representations are connected to algebraic equations they can make the connection to understanding linear systems. I found that the visual model had the greatest potential for success in teaching the linear systems part of the curriculum. The interactive applications that are provided in the on-line textbook give us some nifty supplemental material for this lesson. Once students understand the underlying concept of a linear system the various methods of finding the solution may make more sense.

  2. Week 3- Linearity in Algebra
    Response to Doug Snyder's linearity assignment:
    My suggestions for minimizing the anxiety in this assignment are as follows;
    Step 1. Anxiety and frustration walk hand in hand. When making up the Please Do Now for the lesson be sure to develop problems that introduce the new material by building upon skills your students have already mastered.
    Step 2. I believe that many students will not ask questions fearing that they may appear foolish or stupid when they don't get an important idea. Try to encourage student interaction and reward it. A little candy can be a great way to draw your students into the discourse and once involved their self-confidence is bound to improve.
    Step 3. Time is always a factor in the classroom but it makes no sense to forge ahead when students have not demonstrated their understanding of a topic. Sometimes that extra time invested on a problem can make a big difference. This is important when we are providing guided instruction as in tasks 2 through 6 in the lesson plan.

  3. This comment if for tasks 7.1 & 7.2


    One idea used to combat math anxiety is to give more students a chance to “connect” with the lesson. One possible approach is to prompt them to develop an intuitive feeling for what the equations mean.

    1. For instance 2x – y = 2 can be restated
    as y = 2(x – 1). This equation can be interpreted verbally: get y by taking x, subtracting 1 and multiplying the result by 2. You could plot a few sample points:
    x y
    0 -2
    1 0
    2 2
    6 10

    There’s now some context in the problem for students who are more verbal and visual.

    With the second equation x + 6y = 27, do the analogous thing. Rewrite as
    y = 4.5 – x/6. Verbally, take one sixth of x, and subtract it from 4.5.

    Plot a few sample points
    x y
    0 4.5
    1 4.33
    2 4.17
    6 3.5

    Also, conjecturing and connecting to real life are considered antidotes to math anxiety.

    2. With the visual information above, there is an opportunity for the students to stop and think about the situation.
    Could be some point that the solution sets have in common?
    Is so, where would it be likely to be?
    Is there a way to find it exactly?

    3. Real life applications: Give an example of a real life situation where knowing the intersection of two lines is helpful (though I’m at a loss right now to think of one).


    Conjecturing is also considered a point of mathematical sophistication (point #3), so suggestion 2 above might be helpful with sophistication as well.

    Leading the kids to conjecture a general rule about when pairs of linear equations have solution sets might also promote mathematical sophistication.

  4. Linearity in Algebra- Math Sophistication 7.2
    Clearly the concept of linearity is one of the fundamental ideas in any algebra course. The level of math sophistication in this area can vary dramatically from student to student. It is my belief that one of the keys to developing higher levels of math sophistication is to build solid vocabulary skills. Definitions in mathematics are precise and must be clearly understood before students can move to higher order thinking activities. Meaning in this area revolves around the ability of students to recognize the relationship that exists between variables in a linear equation. Once students are able to make the connection between the abstract definitions of linear equations and the applications for this concept in real world problems the level of sophistication can increase to a point where students can synthesize their own mathematical interpretations of this concept. This process is best implemented by providing students with an ample supply of examples to play with.