Constant Velocity Mathematical Model, Day 1
Lesson 7 of 10
Objective: Students will be able to write a mathematical expression to describe a position vs. time graph and use those expressions as models to solve calculation problems.
My goals for this lesson are for students to see the development of equations from the mathematical model created from the graphs in the Dune Buggy Lab, to solve and calculate constant velocity problems, and to ask quality questions that they need answered before the test in three days. The main focus of this lesson is to solve and calculate constant velocity equations, concepts relating to the Common Core standards of rearranging and solving equations.
To start off class, I pass out the Unit 1 Checkpoint #2 from the previous day and spend time talking about the sections that many students struggled on. This usually consists of writing the mathematical model and calculating the slope. I tell them that when writing the mathematical model they need to make sure to replace the variables of x and y with the variable names and replace m and b with their corresponding numbers and appropriate units. For slope, I point them to the x and y axis and ask them how much each box on the x-axis represents and how much each box on the y-axis represents. When they see that they are not 1 for each box, they recognize their error in calculating rise over run. A student sample is shown below.
Once we are done discussing yesterday’s checkpoint, we move to discussing and focusing on the mathematical model. I use the checkpoint as something to base our next discussion on. I write the notes on the board and ask the students write these ideas in their notebooks throughout our discussion. I have students participate in the discussion as I reveal the mathematical models and like to have students write these concepts down as notes for them to use as a reference.
Students start by stating what are 3 ways we could define velocity from what we already know or the reading from the textbook the night before. After multiple students volunteer, the list contains the three following definitions: slope of a position vs. time graph, change in position during a time interval, and speed in a given direction (or speed + direction). I ask them if they think that velocity and speed are the same thing and I get mixed answers, but I usually have a student that says that direction is the main difference.
The goal is to have students focus on the mathematical model and where it comes from. I have students think back to the checkpoint and to the Dune Buggy Lab and what the position vs. time graph looked like. I draw a line with a positive slope as a reminder. From there, we review the mathematical model that we came up with which is: final position = velocity x time + initial position. After we select letters to represent position, velocity and time, I write the equation in symbols and make sure to emphasize that the f and i are to distinguish between the final and initial positions. Then I go through the how to rearrange the equations as in the Constant Velocity Mathematical Model Notes. I show them a circle of how to easily rearrange the equation similar to the idea below:
Finally I ask them if we use the same equation for speed. When they say no, I ask them if they saw one in the reading and they typically come up with the equation: average speed = total distance/time. I tell them that it is similar to the velocity equation, the only difference being that velocity has a direction so it could be positive or negative and speed does not have a direction so it has no sign. By the end of this section of a lesson, students notes should look like the Constant Velocity Mathematical Model Notes, which they can use to help them in the next part of the lesson.
Guided Problem Solving
I show students two guided practice problems after we discuss the mathematical model they will use so that students have a reference on how to set up their work. I tell students that the minimum amount of work I need to see is the equation in symbols, the equation with numbers substituted and the answer with units circled. I choose each of these problems because they help students to see the difference between speed and velocity and serve to help student check that the units are matching. My approach is demonstrated in the Constant Velocity Mathematical Model Notes picture below and the video, Guided Problem Solving with Constant Velocity Mathematical Model.
I use these two problems for different reasons. The first problem helps students to see that there is a difference between speed and velocity even when calculating. The second problem helps students see how important having matching units are and remind them how to convert. I use the guided problem solving to help them see my thought process in how I would solve the problems to help them as they start their own set of practice problems.
After the students have completed the guided practice problems with me, I tell them that they will work with their table groups to complete Worksheet #3 Constant Velocity Calculations. I make sure to emphasize that they will be graded on showing their work by putting the equation that they chose to use in symbols, calculating the correct answer, and applying the correct units to their answer. If students do not have any questions, I tell them they may begin work in their groups. I have students work with their groups so they can get some time to practice with the models we developed.
As students are working, I walk around to see if there are any questions that their group members cannot answer with Unit 1 Worksheet #3 KEY. When students call me over to ask a question, I always ask if they asked their group members for help first before asking me. I do this because I think it is important for students to use their peers as resources in the classroom since that is a skill they will use long after high school.
Questioning Exit Slip
At the end of class, I have students take out their learning targets. When they are looking at their learning targets I ask them to write 1 question on a Post-it note that they must have answered before they take the test. I have students do this for a couple of reasons. First, I want to know what areas I need to focus on in any review the next day. Second, I need a list of student generated questions for other students to answer in another activity in the next lesson. Finally, I want to create a resource that answers all of the Student Questions from all of my classes so that when they complete the activity in the next lesson, they have the answers to all of the questions.