This lesson is structured so that three different groups can productively begin the similar triangles application of slope quickly and productively. Group one is designed for students who still struggle to graph linear equations fluently and need extra assistance is getting started. Group one then begins to examine the structure of rate of change on the graph to see and understand why some students say slope is a vertical change over a horizontal change. Group two is a group who graphs linear equations fluently and correctly but need to make the new connection to similar triangles and slope between any two points as viable for finding the unit rate of change. Group 3 is the advanced group who graph fluently and understand the connection to slope between any two points already. This group is really focusing on using skills to answer a more open question that leads to observations about the coordinates of a dilation centered on the origin and the distance a point is from the center when dilated.
I open by explaining that each group is completing a similar activity by structured in a different way so that each group learns something different about the connection between similar triangles and finding the slope of a linear equation. I am explicit that groups will be presenting on day two so they need to focus on making at least two connections between slope and similar triangles so they are ready to present to the class as a group. It is important to be clear about learning intentions and criteria for success. Click on the video link below to watch a clip on how to apply this strategy in the classroom.
The bulk of the class period is spent allowing student groups to work through the activity. I was clear up front that so much time to work means students are to work and not have side conversations about their weekend plans. Long work periods sometimes lead students to eventually gossiping instead of working. I passed out the handout and the method cards for each group up front and this created confusion as the students did not read questions 1, 2 or 3 on the handout but instead assumed the method cards were the questions. I had triangles drawn all over the graphs in all the wrong places because students did not realize the directions were in the handout, the method card was simply “how” to answer these questions. I suggest only passing out the handout first and ask all groups to graph according to their directions. Move about the room checking their graphs and when they are ready to turn the page and begin working on question 1 then give them the method card and make sure to point out that the card is simply “how” to answer the handout question.
As students work through the activity, move about the room assessing formatively and providing feedback that moves student learning forward. As you notice good work and question students about their observations are sure to tell them which ideas you want them to be ready to present to ensure you get a variety of observations throughout the presentation time. You want to know what students will say before you send them to the board on day 2. Throughout the work time, you will utilize a variety of teaching strategies. Click on the links below to watch short videos on how to apply these strategies in the classroom.
Homework: I did not assign homework today. Most groups completed the assignment but for any groups that did not, I allowed the first 5-10 minutes of the following class period to finish and organize their presentation among themselves.
This activity applies similar triangles to finding the slope of a linear equation using any two points along the line. Different groups are meant to make different connections to this main idea but the overall math standard addressed is 8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
The math practice standards used throughout this activity arise from students working in cooperative groups to determine how similar triangles apply to slope of a linear equation and then decide which two connections to present to the rest of the class. Working throughout the activity to make difficult connections addresses practice standard MP1 Make sense of problems and persevere in solving them. Group discussion addresses standard MP3 Construct viable arguments and critique the reasoning of others. Working with similar triangles to see slope and make connections brings in standard MP7 Look for and make use of structure. Using coordinates to graph and find distance to the nearest hundredth bring in accuracy and address standard MP6 Attend to precision. One goal of this activity is that every group will understand why students who graph linear equations fluently use the concept rise/run to graph additional points past the y-intercept. By looking at this structure of similar triangles, the pattern of consistent rise/run should become obvious and this structure is really applying practice standard MP8 Look for and express regularity in repeated reasoning.