In today's lesson, students explore systems of equations that do not have unique solutions. I especially like the approach taken in the task that we will use because it combines the concept students have been using (solving systems using elimination) with students developing a context from which they can understand why the system does not have a unique solution.
I begin class by reading through Taken Out of Context together. In my experience, students do not have a lot of experience writing their own word problems, so we might work on writing a scenario for the first problem together. Students can then use elimination to solve the first system (which does have a unique solution).
I explain to students that I want them to do this for the remaining three problems. I might warn them that they may find something interesting happens with some of the systems and they will want to use the problem situations they develop to understand what is happening with them.
Next, I let students get to work on Questions #2 - #4 of Taken Out of Context in pairs or small groups. As they work, I look for students who are struggling to write contexts for the problems. If students are struggling with this piece, I might have them take a look at their past work with Shopping for Cats and Dogs to see that they need to write problems like the ones given to them in that assignment. I also encourage them to use items that are familiar to them, rather than staying with the pet food analogy.
If students get stuck on the solving piece, I might ask them some guiding questions. For example, in Question #2, there are an infinite number of solutions. In order to get students thinking along those lines, I might ask them to think about the problem without thinking about algebra for a minute. I might say something like, "What do you notice about the amount of items x and y that you purchased the second time as they compare to the first time?" Here, I am trying to get them to see that if everything in both equations is doubled (or the same multiple of the first equation), then there are an infinite number of prices that would work in both equations. If students are having trouble seeing this, I might ask them to play around with real prices. For example, if item x cost $1, item y would have to cost $4. But, at the same time, if item x cost $4, item y would have to cost $2. This is the first time we have seen two different possible pairs of prices.
Question #3 is sometimes easier for students to understand because they realize the set up is impossible. I might try to guide them through this reasoning if they are struggling but having them notice that again the coefficients of x and y are fixed multiples of each other but the total prices are not. I might ask them, "How could this be possible?" and try to use some numbers to demonstrate that there is no way the prices can work out to the correct totals.
Since students have already worked on solving systems of equations by graphing, and much of the unit has focused on getting visual representations of algebra, some students ay be eager to look at the graphs of these equations. I like to have students who finish early graph the systems to see what is happening. This will also be a part of the discussion section of the lesson.
Additionally, students who are ready for a challenge can continue on to Question #5 which is a system with three equations and three unknowns.
In the Discussion section of today's lesson, I ask students to share out their findings and how they understand systems that do not have a unique solution. We spend most of our time discussing Questions #2 and #3. We have three ways to look at these problems:
1. Thinking through the problem situations we created and using reasoning to come to an answer.
2. Looking at what happens when we try to use the elimination method with the algebraic representations of the situation.
3. Looking at the graphs of the equations.
I like to explore all three methods so that students with different ways of understanding can find the explanation that makes the most sense to them.
I generally guide the discussion using the Inconsistent and Dependent Systems.
If there is time, I'll have students who worked on Question #5 share out their way of arriving at a solution. There are different ways to solve this problem and it's fun to see if students approached it differently.
I often find my students need some practice with this work outside of class. I like to assign the homework assignment Determine Number of Solutions as a way for students to think about how they can figure out the number of solutions a system has by looking at slope-intercept form.
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