Different Forms of Linear Equations

The purpose of this warm-up is to give students a chance to review all of the key concepts of the unit so far, and to give you a chance to assess their understanding.

As you circulate, constantly ask students to explain their answers and conclusions. Avoid telling them whether or not you agree with their answers. Rather, ask them to listen to each other’s answers and determine whether or not they agree (MP3). The more you make a habit of this in the beginning of the year, the more likely they are to do this on their own throughout the year.

Problem (2) is the most computationally heavy task, so whenever you talk about this problem with students, make sure not to get bogged down on the computations. Instead, talk about the concepts behind the computation (MP1). Students may have developed their own method, but it is always important to ask them what their computations actually mean and why they make sense.

Students may not want to write down actual functions for the designs in problem (3). This is fine—they shouldn’t be asked to do something that feels tedious. They can describe the relationship between the lines and explain how this shows up in the equations. In fact, this approach actually gets at big ideas more powerfully than writing the equations. For students to do this, they will need to think abstractly about the functions and the relationships between them (MP2).

Problem (4) is another problem that students can solve in many ways. Even though the entire lesson yesterday revolved around this type of problem, students may need to discuss the concepts again. It is important not to rush students or to get frustrated when they don’t seem to remember things. The only way for students to feel safe trying to make sense of problems is if they don’t have to memorize algorithms meaninglessly. Instead, I find it is more successful give students the chance to think through problems again and again until they establish ownership over the ideas (MP1).

This lesson gives students the chance to make more connections between graphical and algebraic representations of linear functions and to examine the structure of each form of a linear function (MP7). By graphing any form of the function, students can discern information about the other forms.

A good question to ask is: How is graphing the slope-intercept form accomplished most easily? From an initial graph, students can easily find any other point to create point-slope form. They can also find the x- and y-intercepts which will help them find standard form. They may or may not have made these connections during their work on the lesson, so the closing provides them with the chance to do this.

Ask students to explain any graphical or algebraic methods that they have developed and emphasize the idea that they can think about these problems either way. This helps them get at the deeper structure of each form of the linear function.