Once questions on yesterday's Homework Assignment are cleared up, I begin Day 2 of this lesson by asking the class to discuss the answers to the following questions with their elbow partner in a Think-Pair-Share fashion:
Once I've allowed time for the pairs to discuss their thoughts, I will call on volunteer students to share responses. The first question is usually answered quickly. Students may or may not stumble on Question 2. If some students are confused I may say, "Well, the solutions to the systems we've solved graphically have been located where?" Usually several students will point out "at the intersections." Then, I will ask, "Given that, can anyone imagine a linear system with no solutions? What would that look like?" I'll be patient with this question. Once students begin to get the idea, I will ask what that means for the equations in the system:
Is there anything that we can say about the equations in a system if it has no solutions?
Before ending the launch section I will ask if a system of linear equations can have an infinite number of solutions? Sometimes students have an idea that this is possible right away. If so, I will ask for an example so that everyone can try to understand what is going on using a idea from a peer. If it is not easy to come up with an example, I will suggest that students might go to the board (or use the document camera) to show how what a system with infinite solutions might look like.
We'll be engaging in two different activities for the New Info section today. For Part 1, I will ask students to pair up with the same partners they worked with in Day 1 of the lesson. As they organize themselves, I will project the following system on the board.
y = 12x
y = 10x + 6
I address the class, ask that notebooks, graph paper and pencils are taken out, and tell each of the groups that they must create a word problem for this system of equations. The problem can be similar to those we've done in class, yet I encourage them to try and think of another kind of real world circustance that the two equations can represent. I walk around checking out what students have come up with, aiding those groups who may have difficulty thinking up a problem. For these groups, I may suggest circumstances like: parking lot fees, utility bills, personal savings, or gas-mileage problems.
Once they've created a problem, the next step is to create a table of values for each equation and graph these on a coordinate plane. Before going calling anyone up to share their problem and graph on the document camera, I ask that each group answer the following quesions:
1. What does the solution to the system represent in your problem.
2. What did you observe when creating the tables of values?
3. What difficulties did you run into as you completed the task?
See the Handout Application Sharing Equations 2 as an alternative to using the whiteboard for this section.
Once students have completed their work, I call on a couple of groups to volunteer and share their problem and the answers to the questions on the document camera. I like to choose a group that may have been struggling throughout the process, just to help their confidence and of course, their comprehension of the task. Students may have questions or comments with respect to the presentations which I try and handle intelligently due to the time limit, I then take advantage of the responses for questions 2 and 3 and ask that they all take out their graphing calculators. At this point I indicate that many times, graphing on paper is a weak method because it's difficult to draw accurate graphs. I add that in the past, graphing was used to determine the number of solutions, not to get the actual solutions. Fortunately, we now have graphing utilities like your calculator that give us exact solutions.
I ask that each group graph both equations on y1 and y2 and then observe the graph. I ask if their graph looks like the one in the calculator.
I then tell students to go to "TABLE" in the graphing calculator and they should see the tables of values of both equations. I ask to find the solution to our problem. They usually have no problem finding it. When students have difficulty, I ask other students to help out. Many "window" settings" problems may come up, but there are always other students willing to point things out and help their classmates. If too many students are having trouble, I use the document camera and my calculator and go through each step.
To close the lesson, I will hand each student an E x i t P a s s P r o b l e m s with two problems targeting the lesson's "big idea" which is finding the solution to a system by making a table of values on paper and using the "Table" tool on their graphing calculator. Students should be able to answer both questions within 10 minutes. I walk around observing student work and checking how they are using their calculators. Students will usually voice that they've found the matching y values on the calculator table with excitement.
I purposely made the solution values large so that the students have to scroll down the table to find it, and not simply spot the solution on the graph. In past experiences with this situation, the "tech smart" students use their calculators to fill out the table in Question 1. This is no big deal, but I do announce as I hand out the Exit pass, that I'd like everyone to not use their calculators for Question 1. I walk around monitoring this because and make sure the students are working independently.
Every student should be given this extension to the lesson at some point. I remind students that solving systems by graphing on a coordinate plane is not the most effective method….why?
I add that this is quite the case when the solution does not involve integers. We can only obtain a close estimate of the true solution, if we’ve graphed with care and accuracy. In subsequent lessons, we will solve systems with effective algebraic methods, but in this extension we will use our graphing calculators to find the intersection of our lines.
The Solve System using Graphing Calculator video will briefly explain the procedure to the class using this system of equations as an example:
y = 3x – 5
y = x – 3
This homework assignment consist of 3 questions: Homework sharing equations2
The first question actually occurred in the recent NY Common Core State exam. The other two are practice questions similar to the problems students worked with during class.