Linear Lesson Plan for 9^{th} grade Algebra 1 Class

This is a good follow-up or extension after students have been taught system of equations.

Objectives: 1. Understand what linear algebra is; 2. Understand the importance (and controversy) of linearity in mathematics

Introduction to new material

Teacher asks the students what linear algebra means and compiles a list on the board.

Teacher then discusses definition from Wikipedia: **Linear algebra** is a branch of mathematics concerned with the study of vectors, with families of vectors called *vector spaces* or *linear spaces*, and with functions that input one vector and output another, according to certain rules. These functions are called *linear maps* or *linear transformations* and are often represented by matrices. Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to the solution of a systems of linear equations in several unknowns.

Of course many students many not know vectors and matrices, but it is a nice teaser for what is to come later in their math lives.

Teacher can then discuss linearity from a less mathematical perspective. Linearity can also mean doing things in a certain order. Progressing from one task to another in an orderly fashion. Implied is that you cannot go on to the next task without completing and understanding the previous task.

At this point, teacher can ask if that is important in mathematics?

If so, can you provide examples?

Teach can show examples of linearity.

To solve multistep equation, you need to know about variables and balancing equations through the inverse properties.

In school, there is linearity of math courses you will take:

Algebra 1 in 9^{th}, Geometry in 10^{th}, Algebra 2 in 11^{th}, Precalculus in 12^{th}.

Assessment Activity

We have learned how to solve systems of equations by graphing their lines and finding the intersecting point. What math principles did you need to know beforehand in order to do this activity?

What do you think you will learn in Algebra 2 as a junior? Do you think it is important to do well in Algebra 1 in order to do well in Algebra 2? Why? There are schools of thought that think sometimes math is taught in a way that is too linear. For example, do you think it might make sense to take Algebra 2 as a sophomore? Why?

Solve y=3x+1 and y=-x+5. Do you think there are other ways to solve a system of equations in addition to graphing?

I think this is a good lesson–it starts with definitions of linearity and some of its important components such as vectors (everytime I see that the word “vector” I think about the line, “What’s your vector, Victor,” from the movie AIRPLANE–classic! Then you move on to the practical side and showing students how to solve a system of equations. In the end you ask them to think about how to solve a system of equations using other methods. To improve this lesson, I would simply flesh it out in the direction that it is already headed. After the students take down the important definitions, I would maybe pull up Geogebra and run through a couple examples of systems of linear equations with the students. I think this would create student buy-in. Maybe the “Big Question” would be, “How can we solve a system of linear equations using our pencils and paper?” Then I would move into the “I do” part of the lesson with a couple of examples using the substitution method. Then the lesson would continue up to the point where students were showing understanding by completing examples on their own. The lesson would then end with an exit slip where students would have to show what they learned. Giddyup.

This lesson touches on a couple of the ideas mentioned in the article about Math sophistication. When you ask them to look for linearity in real life, in a way you are having the students “make analogies by finding the same essential structure in seemingly different mathematical objects.” You have also brought up the idea of “precise language” in your definition section along with the more obvious “value and use precise definitions of objects.”

Very sophisticated lesson Al!

Math anxiety! Ughhh! Many students approach lesson with an “I can’t” attitude becuase their successes have never been reinforced, and thus believe everything up to this point has been a failure. Algebra can be particularly difficult because there are many terms that seem foreign and using letters and numbers together can be very confusing. I think this is a great lesson but some students may feel anxious with the terms to be defined at the beginning of the lesson. I think a great way to reduce some of this anxiety would be to work in a computer lab teaching students to use Geogebra. The teacher can go through several examples of how to enter 2 equations and find the point of intersection on the graph. The lesson could progress so that by the middle of the lesson students are recieve two equations and enter those equations and find the intersection. The teacher could turn it into a competition and see who could give the point of intersection the fastest. Once students see how easy this is, the teacher could point out that the intersection is the solution. Once students know that they have already found the solution, the teacher could then show them another way to find the solution using just pencil and paper, via substitution. I think this progression would allow students to feel successful before moving on to pencil and paper math which may increase their confidence and leave them well prepared for success in finding the solution via substitution.

I’m back! Working on week 7’s task offering suggestion on how to make lessons more sophisticated mathematically, or rather how to create more sophisticated mathematicians. I think one way to do this would be to ask students to come up with a way not to find the solution to a system of equations by graphing two equations on their calculator and guessing at the point of intersection. They could then plug the mistaken solution into either equation to realize the mistake. I think that when students know how “not to do it,” they are closer to internalizing the proper steps for “hot to do it.” Great work Al!