Comparing Linear Combinations in Ax +By= C to y=mx +b

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SWBAT write linear combinations in the form Ax + By= C, then rewrite the equation in slope-intercept form using appropriate methods.

Big Idea

Recognizing the mathematical structure of a context involving a linear model helps students to write equations whose form is consistent with their thinking.

Warm Up

10 minutes

In this Warm up I introduce students to a problem that involves a linear combination of quantities described by two variables. It is a problem that is much easier to write in Standard Form (Ax + By = C) than slope-intercept form (y = mx + b). I plan to use about 10 minutes  of class time for my students to complete the problem and to review it with the class.

In recent lessons, my students worked several problems in which a constant rate of increase (or decrease) was described and they modeled these problems in slope intercept form. I chose today's problem because students will be able to reason with the numbers in the problem, but they will not find it easy to apply the same process used in recent days. I expect my students will start by thinking of different combinations of $2 loaves of bread and $3 cakes that can be purchased with exactly $20.

In this video I demonstrate how I will discuss the problem with the class: 

It will not be a surprise if none of my students write a correct equation for today's Warmup problem. We may work with a table of values. If so, I will call on students to offer combinations that work and we will discuss whether or not there are patterns. In the next section of the lesson we will complete Guided Notes. As we complete the notes students can record a process for writing a linear equation in standard form.                                                   



Guided Notes

10 minutes

Towards the end of today's Warmup, I like to introduce the Standard Form for the Equation of a Line. Once this model is introduced, I also demonstrate how to solve the standard form for y, using inverse operations to undo the equation and convert the equation to slope-intercept form.  

Today's Guided Notes scaffold students' work as they compare two different methods to graph a linear function. In Method 1, students use the x- and y-intercept of a linear equation written in Standard Form to graph the line. As I show students this method, I emphasize the importance of writing the coordinates of the point, immediately after solving. I like to show my students this video from Khan Academy to re-enforce how to find the x- and the y-intercepts. My students appreciate hearing things explained in a different voice and having a resource that they can look to outside of class.  

Teacher's Note: Khan Academy has a set of Grade 8 Lessons called Two-Variable Linear Equations and Functions. I found these resources really helpful for my students. I will explain more about this in my reflection. 

Guided Practice

30 minutes

During this section of the lesson I give my students a set amount of time to complete each of the Guided Practice problems.  In general, it is difficult for my students to write equations when given a word problems. I prefer to give them about five minutes to persevere with each problem. Then, I will ask students to explain what they have done so far and we will work together to find a correct answer, if necessary.  This allows me to model a few different types of problems before the end of class. It also helps me to understand which problems are most difficult for my students. For today's set of Guided Practice problems, the problems the students do not complete in class will be assigned as homework.

At this stage of the lesson I will allow my students to work these problems with their table partner. In my class, partners are typically in homogeneous pairings.

I expect Problem 4 to be the most challenging task. Therefore, I encourage students to attempt it in class. We may review the problem as a class, and, I will ask students to try to complete it again on their own at home. In the video below, I provide a demonstration of how I review the problem with my class. 



Exit Slip

5 minutes

With about five minutes left in class I will distribute today's Exit Ticket. It reviews two of the skills that I wanted students to learn in this unit:

  • Write and graph a linear equation in Slope Intercept Form
  • Write and graph a linear equation in Standard Form

Overall, I want my students to be able to plot the y-intercept and another point on the line using the slope. And I want them to be comfortable using the x- and y-intercepts to graph a line.

As shown below, my students need to be more precise in their work when working with changing from Standard Form to Slope-Intercept Form and with graphing using the x and y-intercepts.