Objectives of the Lesson
By the end of our lesson, we would like for students to be able to:
- write the equation of their line from the checkerboard data in slope-intercept form using information from their
1) graph
2) slope and a y-intercept
3) slope and any given point on the line
- recognize what parallel lines have in common
- recognize the relationship between perpendicular lines
The first objective above seems like a simple task to expect, but this was the students first introduction to "slope-intercept form." This lesson was also the first of three that explored the ideas of slope and this form of linear equations. So we designed the task to allow for student development of content and connections to the mathematical materials listed in the first objective of our lesson. The task given to students was presented in a manor that allowed the students to explore the mathematical concepts in the objectives without being explicitly directed by the teacher(s). Because the tasks were designed in this manor we felt that the student discovery and exploration out weighed an attempt to fit more objectives into one day.
In the video you can see that there were times when the students were able to do the project without any teacher assistants, and that the groups were able to come to a consensus on why and how they did the math that they chose to do. Seeing the students solve the task using the ideas our objectives listed above was exciting to watch. The group in the front of the room struggled through the idea that six squares were required to enclose zero checkers. But they eventually were able to back up the information from their graph with an intuitive understanding of the way that their data was changing.
We did not explicitly give any real world connections or build them with the tasks. But the next time these tasks are used the teacher could simply lead a discussion on linear programming/modeling which is what we had the students do, or alter the tasks to have more of an anchor and relevance to the real world.
The previous unit dealt with slopes of lines and their relationship to other lines. Therefore the final two objectives of the lesson were review form the previous unit, and we attempted to cover in the closer of the lesson. Through this experience our group came to the conclusion that we needed to be more mindful of the clock in order to have a more meaningful closer to the lesson, which you will see in the video.
Mathematical Tasks
The first task that we had the students try was a checkerboard activity. Students were given checkerboards, checkers, rulers, graph paper, and a chart. The students were asked to find out how many squares it would take to enclose 1,2,3,4,5,and 6 checkers. The only rule about the placement of the checkers was that they were in a horizontal or vertical row. The students were then asked to graph their findings. After they graphed their data students were asked to project how many squares would enclose 10, 15,20, or 0 checkers using their data. And finally the students were asked to find out a way to calculate the number of squares it would take to enclose x squares.
The math for the first few questions seems rather elementary, but because the students were working in groups they had to clarify to the other members of the group why they organized their data, and how they would use their data to find the answers to the numbers that they couldn’t check. Some students argued using the patterns that they saw in their data charts, some went straight to the algebraic equation of a line, and others such as the group closest to the camera in the video argued it geometrically by extrapolating their graph and reading the values off of it.
Then for the majority of the groups came the big jump of trying to find the generic equation. This is a task that I believe should be asked of our students more often than it is. You could see the struggle on the faces of the group members as they made their first attempts at finding a general solution. They had to work through their own understanding of the situation and what would happen to their solution if they multiplied or added or any operation. They then were forced to piece it all together and explain it to their classmates. This explanation and discussion about math is important to building the connected framework in the students’ minds so that they can solve future problems using similar reasoning.
Discourse
When we were planning the lesson we wanted most of the discussion to occur in the small groups, so we chose two groups to look at that we thought had the most intriguing discussions. These two groups really got into talking about how many chips would enclose 0, 15, and 20 circles as well as talking about how to find the slope intercept form. The two groups that are being focused on in the video is the group that sits in the front of the classroom, which we will call Group 1, and the group that sits closest to the camera, which we will call Group 2.
The first observation that we came across while looking at this protion of the video was that the students often had some trouble once they could no longer place the chips on the checkerboard. In the worksheet that was given to them, the students were asked to find how many squares would enclose 15 and 20 circles. The students used their graphs to find the answers to these two questions and most of the groups were able to find the answer right away. However, Group 2 needed to fix their graph a little bit before they got the right answer.
Then came the issue of zero circles, and this really baffled some of the students. Group 1 assumed right away that the answer was zero, but then someone noticed that that answer did not coincide with the graph. The graph had 6 as the amount of squares that it would take to enclose zero circles, so now the group was confused. Then they asked for some help. Peter was able to help them by having them go back to placing one cirlce on the checkerboard and have them explain why they got that amount of squares. Peter then took away the cirlce and asked the group a couple of questions. Group 1 got to talking about what was going on and then one student came up with the correct answer. He was then able to explain it to the other memebers of his group. This was great because Peter was able to help them without giving them the answer. The students were able to talk it through and then explain it to the other members who did not understand.
In Group 2, they understood how the circles worked, and they realized that they needed to come up with an equation to find "something". We were able to lead them to what that "something" was without telling them. They knew that they were finding the amount of circles and that the points meant something. Peter was able to lead them through a series of questions to help them get to the answer. The students began to put it together, but they were doing it in different ways. This was a little confusing to them, but through some discussion and finding out what each part of the equation meant they realized that the they were doing the same thing. Once they figued out the equation, they were able to tell us what the slope and y-intercept of the line was in similar equations.
We really liked that the students struggled with this a little bit, and that we did not push them in a certain direction to get to the answer. The students were able to discover for themselves the equation for slope-intercept form and what each part of the slope-intercept form represented. We thought that through this activity the students would be able physically see what was going on with a line and how to find its slope. We wanted the students to see that there is a way to use everything that they had previously learned about lines in a combined equation.
Environment
In the classroom, the students were already in groups of four. We kept the desks the way they were and the same groups because the groups worked well in the classroom. The desks were arranged in tables throughout the classroom. We kept them in small groups to encouage discussion and risk taking and to keep the amount of supplies that would be needed for the classroom minimal.
In the small groups, the students were able to really talk about what they did not understand and they also got help from the other members of their group. This helps when there is only one teacher in the classroom because their peers can often communicate better with each other than the teacher can with the students. The small groups also seemed to allow the students to make mistakes without being afraid. They also seemed more open to asking questions.
We were able to encourage student learning by questioning the students as they were going through the work. We were albe to keep them on task because they were in small groups and there were a few teachers that were albe to walk around the classroom.
This was very evident in the two segments that we picked out for the two groups where they were discussing the zero cirlces and the slope-intercept form. The students were all engaged in the task and we were able to get them back on task if they were talking about other things. We were also abel to pull in the students that were not as talkative through questions.
Analysis of Student Learning
Having several instructors on site, we were able to analyze student learning in a variety of ways. During the checkerboard activity, students were given a brief introduction to the days activity, then they had the remainder of the period to work on the assignment. Student learning was analyzed when the instructors walked around and observed the students at work.
If the students were stuck or confused on any of the steps or questions, the instructors were there to help guide the students to the right path hopefully without giving away the answer. If other students finished the assignment early, then the instructors would ask the students questions to see if they really did comprehend what they were supposed to.
On the videotape, it shows that the instructors gathered information about what the students were learning by observing how the students were doing on the activity. We also gathered information by how well they were answering the questions on the worksheet, and by the graph they were making of their data. All of the students had the same project and worksheet, so all of the students' graphs should have been the same (for Day 1).
We chose the particular method of observation in order to see if the students understood what they were doing, and also to ensure that by doing this in class, they would be able to do the homework that was given to them for that evening. By observing individual groups at a time, we were able to come up with questions to ask other groups to help lead them in the right direction. We, as instructors, learned that students learn just as well, if not better, from working with other students. Students can often explain to one another in simpler, more understandable terms how to do a problem. We were just there to ensure that they stayed on the right track.
In the checkerboard activity, we wanted students to see the positive pattern between the increasing number of squares that enclose an increasing number of circles. When the students created the tables of their data and then plotted their points on the graph, they should have noticed that that the number of circles and the number of squares had a positive linear relationship (with a slope of +2).
We expected the students to pick up on many of the relationships between slope and lines. From these relationships, we were hoping to have the students build the slope intercept equation, and then conceptualize the equation in their understanding of these relationships.
In our objectives stated in our lesson plan, we wanted the students to be able to write equations in slope-intercept form using a graph, a slope and y-intercept, two points, and a slope at any given point. Being that this was only the first day of a three-day lesson for accomplishing these objectives, we believe that this was a very good activity. Students could already use their graph, two points, and the slope and y-intercept to write an equation in slope-intercept form.
We discovered the following day that most of the students did not completely finish the homework that was assigned, and we would have liked to have seen how well each of the students understood the problems when they were not in class. The students also needed help building graphs. Students needed to label their graphs better, so they could could answer questions later on their worksheet instead of just trying to guess the answers.
Analysis of Teaching
For Day 1 and Day 2, we pretty much followed our lesson plans during the class, making very few, if any, changes to the lesson as we were teaching. The only day we had to modify the lesson from the original plan was on Day 3.
For Day 3, we had originally planned that the entire class would work as one big group. However, after the first period attempted this, we realized that changes needed to be made for this activity to me more effective. Therefore, the remainder of the periods doing this activity worked in smaller groups instead of one big group. Each group had to come up with an equation, then each student had to come up with the other group's equation (in slope intercept form given a series of points). We changed the lesson so everyone or every group was doing something (either creating their own line or solving someone elses)in order to keep everyone on task.
Based on student performances, we learned that we needed better wrap-ups at the end of each lesson. A lot of the students did not think they had to finish the questions on the worksheet for homework if they did not get all of them answered in class.
Next time we teach this unit, we would spread the lesson out to four to five days instead of three. Each of the lessons could have used better explanations and wrap-ups, and we feel the studetns should have been given the opportunity to present their Day 1 and 2 projects to the class. We would have liked the students to explain to us and to the class how they did each of the activities and how they answered the questions on their worksheet.