PRINT VERSION - Understanding Forms of Linear Equations

Lesson

Based on what students already know, students may only need an intervention on one or two of these forms, and these sessions do not need to be consecutive to be beneficial.

Slope-Intercept Form (60 minutes)

1. Start by providing students with the Graphing Lines Phet simulation for Slope-Intercept. Give students 5 minutes to experiment with the simulator. Ask them, “What do you notice? What do you wonder?”
2. Give students a copy of the handout - Slope-Intercept Form. Answer key provided for reference.
3. Students should feel free to use calculators and rulers.
4. Students (ideally in groups) should work at their own pace. Draw their attention to the locations where teacher initials are needed. Emphasize that they should not move on from these places until you have reviewed their work on that section and that groups should all be ready for initials before you are called over.
5. In the first scenario, students have all four forms of the information provided for them (verbal/situational, algebraic, graphical, tabular). Their task here is to make connections between the four elements - identifying rate of change and y-intercept in the different forms.
6. The second scenario requires students to complete their own table and graph, although labels and x-values are provided. Students should still be making connections between the situation and the different representations. The graph on this problem will be challenging for students. Encourage them to plot a point every 12 months instead of every 6 months, thus the increase is 1 instead of 0.5.
7. Students then work through the third and fourth scenarios in their groups. If questions arise about something that was given in a previous problem, refer the students to those problems. Use the previous information as a road map for moving forward.
8. As class time wanes, take a few minutes to consolidate the learning about Slope-Intercept form: What can be seen in the equation and where you find that information.

Point-Slope Form (50 minutes)

1. Start by providing students with the Graphing Lines Phet simulation for Point-Slope. Give students 5 minutes to experiment with the simulator. Ask them, “What do you notice? What do you wonder?” (There may be some differences between the Phet simulation and what the student is used to)
2. Give students a copy of the handout Point-Slope Form. Answer Key provided for reference.
3. Students should feel free to use rulers and calculators and ideally are working in groups.
4. Keep the introduction short. Tell students they are going to look at a different form of linear equations. The data is still linear but the equation will not be y=mx+b. Their job is to explore the new equation form - find similarities and differences. (Note: this will be a significant challenge for some students as they have spent a significant amount of time with slope-intercept form in prior courses.)
5. Students have all of the information provided for them to move through the first scenario. The main goal is to notice the new structure, specifically the placement of the slope and a known coordinate pair. The last question of the first scenario is intended to help students internalize that although equations do not appear identical they can still be equivalent. This is important not just for point-slope form but also to note that the different forms (slope-intercept, point-slope, standard) can all identify the same line.
6. Initial papers as groups finish (this reduces the load on you if students are collaborating on their answers). Encourage students to consult with their group members as questions come up.
7. Students have slightly less information in the second scenario. (Note: the rate of change is not given in the prompt although it is provided in the equation.) This again is designed to help students see the structure of point-slope form. Students may calculate the rate without using the equation - this is great! The connection will be even stronger when they notice the rate value later in the equation. There is no need (at any time) for students to algebraically change point-slope form to slope-intercept form (although some students may choose to do so). When graphing, have students notice that they can find slope between two points, determine additional points, and graph an accurate line even without the y-intercept being explicitly given. 
8. As class wraps up, give a quick review of point-slope form - possibly put up a couple of equations and ask for the rate and “a point” that the line passes through. Also check in on the understanding that although equations look different they can still be equivalent.

Standard Form (50 minutes)

1. Give students a copy of the handout Standard Form. Answer Key provided for reference.
2. Students should feel free to use rulers and calculators and ideally are working in groups.
3. Keep the introduction short. Tell students they are going to look at yet a different form of linear equations - the data is still linear but the equation will not be slope-intercept nor point-slope form. Their job is to explore the new equation form in order to find similarities and differences. You will probably get some moans and groans here and complaints like, “Why do you have to show us so many ways? Why can’t you just show us one?”  Remind students that the purpose of different forms is that each one provides different information and is most useful for certain data. That is exactly what they are needing to process while exploring these different forms.
4. Students have all of the information provided for the first scenario. As necessary, remind them to reference all forms of the data. There may be some difficulty with the algebraic form: “Why is that part marked out??” Have students go back to the previous questions when Keana was only buying one type of paint - we did not need to consider the other type.
5. The second scenario removes some of the scaffolds, but still has students identifying the x- and y-intercepts and using that information to move forward to slope and other points. (Intercepts are the most easily accessible information provided by standard form.)
6. Students may struggle with the questions where they are asked to provide two different equations in slope-intercept and point-slope forms. Ideally, they still have those handouts and can review the parts of those forms. Then students should look for the necessary information from their work thus far in the problem (slope, y-intercept, and ‘another point’).
7. For the third scenario, students may panic when they are looking at just an equation. Refer them back to the previous problems. Ask, “What kind of information can we easily find from this equation?” Have them evaluate when one of the variables is equal to zero (e.g., x=0 or y=0) and solve. Ask where to place that value on the graph or in the table. Then encourage them to repeat with the other term.
8. As class draws to a close, to consolidate the lesson, ask what information is easily found from standard form. Ask how this information can be used to find the rate of change.

Linear I SPY (50 minutes)

1. Prepare cards for I SPY game. Students match each of the question cards to one of the functions. The functions are used more than once. Answer key is provided for reference.
2. Students can be overwhelmed by all the cards at first. Begin the session by having them look at all the cards and then choose a place to start - it does not have to be question one and the cards do not have to be matched in order. 
3. If students have a favorite form (table, equation, graph) - have them pull those cards and only focus on those. 
4. Encourage students to use the process of elimination to reduce options - if you already matched one graph then that function is not an option for the remaining graphs.
5. Students will likely struggle with standard form the most. Ask them what can easily be found from this form (x- and y-intercepts). If the graphs have already been matched - use the graph to find the correct intercepts and the graph then leads back to the equation.

• A natural extension of this work is analysis of scatter plots that trend as linear. Equations can be analyzed with this new knowledge.