Students should begin by working to understand what the writing_linear_equations_warmup is asking them to find and developing a plan for solving the problem (MP1). Allow students to work by themselves for approximately the first 3-5 minutes. Then have them do a think-pair-share with their partner where they can discuss and critique each others strategies (MP3). While students are working both individually and in pairs, observe how students are attempting to solve the problem. Some will think about finding the first term in the sequence additively, others will think multiplicatively. When you call on students to share out their answers, begin with the additive ideas and move on to the multiplicative ideas. As always, have students comment on the preceding group's work first before sharing their own ideas. This will help to create a culture where students listen to one another during discussions.
The thinking required for this warm up is similar to what is required in this lesson. Students will need to find the equation of a line in y=mx+b form when given a point on the line and the rate of change or slope. This warm up requires the same thinking with the exception that students need to investigate the pattern in the sequence to find the common difference (MP7).
Without giving students any instruction, have them try to solve the problem on the first slide with their partner (MP3). Remind them that in order to write the equation of a line in slope-intercept form you need to know two things: (1) The slope of the line (2) The y-intercept. Their task is to find both of those values.
Just as in the warm-up, students will approach the problem in different ways. Some students will plot the point and use the slope to work backwards. Others will determine the y-intercept by subtracting 0.5 from the y-value for each change of -1 in the x-value. Others may even determine that since they have to go back 4 units on the x-axis they must go down 2 units on the y-axis.
When students share out their answers, encourage the generation of different ideas. Try to scaffold answers in a strategic way. For example, have a group that knew a change of 4 in the x-direction would result in a change of 2 in the y-direction could build on a group that subtracted 0.5 four times. This strategic scaffolding of answers will help all members of the class understand the concept at a deeper level.
Once students have arrived at a solution of y=0.5x+5 ask them how they could check their work. Refer back to the idea that a line is really a collection of points that all make the equation of the line true. Give students a minute to think and then guide them towards the idea of plugging in the point (4, 7) to the equation to show that it makes a true statement.
In the second slide, we are going to work backwards from this understanding. Ask students to pretend that they did not know the answer to the question and ask them how they could use y=mx+b to solve the problem. Start by making a connection back to the previous question showing that the slope and the x-value will result in a certain y-value (so mx would be 0.5(4) which is equal to 2) The next concept is to discuss the idea of making a true equation. If we move 4 in the x-direction we move 2 in the y-direction but we want the y-value to be 7. How far up the y-axis do we have to start to make this a true statement (aka 7 = 2 + b). Students should be able to see that the y-value is 5. They should be starting to make a connection based on the work they did in the first part of the problem with no previous instruction. Students can now picture in a more concrete way what this abstract approach represents (MP2).
This formula is almost certainly new for students. In order to keep ideas more clear in students mind we are using y-k=m(x-h). In this form (h,k) is the point on the line and m is the slope. This notation will be consistent with that of a parabola written in vertex form (y=a(x-h)^2 +k). Let students experiment with this formula by substituting in the value (4,7) and m=0.5 and showing that the solution is consistent (MP8).
Slide 4 and 5
The next two slides offer students the opportunity to practice these two methods independently or with a partner. In both cases, after students obtain two answers that match they should check their results graphically. This means that students should plot the point and use the slope to work towards the y-axis in order to find the y-intercept. This step is crucial because it is helping students to develop an understanding of the concept through multiple representations (algebraically and graphically).
This ticket out will require students to take what they have learned and apply it to a contextualized situation. It is up to the teacher's discretion whether or not to require students to use one of the methods discussed in class (slope-intercept form or point-slope form). I choose not to give any instruction because I want to see which students are choosing to use one of these methods and which students are still approaching the problem either additively or multiplicatively. In either case, students will be reasoning about the problem correctly (MP1) and I can provide further opportunities for practice for students who are not comfortable applying the formulas at this point.
Due to the fact that this is a step up from the classwork, I will let students work in pairs and hand in one solution. This way they can develop a plan together and bounce ideas off one another during the solution process (MP3).