Model with systems of linear equations
• Model constraints and relationships between quantities with systems of linear equations (A-CED.A.2$^\star$, A-CED.A.3$^\star$).
• Solve systems of linear equations approximately by graphing and exactly by algebraic methods (A-REI.C.6).
Students have worked with systems of linear equations in middle school and solved simple problems with them. In high school they work with more complex modeling problems and become fluent in general methods of solution. This first section on systems focuses on the modeling aspect. The systems are either solved graphically or the algebraic manipulations required to solve them are relatively simple. Furthermore, the modeling emphasis supports conceptual understanding by emphasizing the quantitative meaning of the variables, the equations, and the solutions to the system. This prepares students to take a thinking approach the solution methods in the next section on systems, rather than a purely formal one.
WHAT: In this task, students are given information about the cost of dance tickets (\$5 for individuals and \$8 for a couple), how much money is in the cash box at the door of the dance and how many folks are in attendance. A \$1 bill is found on the ground and students must decide if the \$1 should be in the cash box or not. Students build a system for each possibility. Only one of the systems has a solution that satisfies the constraints imposed by the context (the number of couples tickets must be even).
WHY: The task provides an opportunity for students to engage in an important aspect of mathematical modeling (MP4), namely, continually checking whether the mathematical work makes sense in terms of the context. In the solutions using systems, the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not. However, because the number of tickets sold to couples has to be even, we can determine if the dollar belongs in the cash box or not.
The task lends itself to different solution methods. In addition to using using systems of equations, students can solve the problem by direct quantitative reasoning (MP2). In the context of this section it is recommended that if students solve the problem another way they be shown one of the methods using systems. However the tasks is solved, it necessitates an explanation in order to satisfy the curiosity aroused by the context (MP3).
WHAT: In this problem, students must read and make sense of nutrition labels in order to figure out much quinoa and how much corn there is in a certain type of pasta (MP1). The only information given in this problem are nutrition labels about quinoa and the suggestion to “use the protein content of each ingredient to find out how much quinoa and how much corn is in one serving of the pasta.”
This problem builds off of the Quinoa Pasta 1 problem (seen in middle school) where all relevant information is given. If students struggle with reading nutrition labels, it is suggested that students complete the Quinoa Pasta 1 problem before the Quinoa Pasta 2 problem as this will help to guide them in understanding what to look for on the nutrition labels. The math for both problems is the same but they have to figure out the equations in Quinoa Pasta 2.
WHY: This is a full-blown modeling problem (MP4). Students must come up with the idea on their own that they can set up two equations in two unknowns to solve the problem, and they must then read, understand and extract information from the nutrition labels to set up the equations in the system (MP1). The algebra needed to solve the system is fairly simple substitution; the main difficulty lies in the modeling aspect.
WHAT: In this problem, students compare the cost difference of using different services to watch movies (Netflix, Apple TV and Redbox). Students initially only look at the cost per movie then build on from there adding in the cost for equipment, number of movies, etc. to create and model more realistic cost equations for each movie service (MP4). Students graph the different costs and initially estimate graphically when the cost is the same and follow this up by calculating exactly when the cost is the same. The final stage of this task has students construct an argument based on their equations, graphs and real-world usage of movies about which service is best for them (MP3).
WHY: This is a great problem to reintroduce systems of equations graphically (A-CED.A.3$^\star$). It leverages student’s familiarity with graphing, so the emphasis can be on relating the different situations to each other and what a solution to a system means. Additionally, this problem calls for first approximating and then calculating exactly where the solution is, thus helping students draw connections between graphing and algebraic solutions. The equations are simple equations of the form y = mx + b describing the relationship between quantities (A-CED.A.2$^\star$) and the emphasis is on understanding in terms of the context the meaning of the equations, the system they form, the graphs and the points of intersection.
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WHAT: In this task, students are given information about the time and cost of creating small and large boomerangs, and constraints on the time available. They create a system of equations (A-REI.C.6) that models (MP4) the situation in order to find the quantity of each type of boomerang that maximizes profit. Four student work samples are provided illustrating different solution methods (paragraph, table, graph, equation A-CED.A.3$^\star$).
WHY: This task is very open-ended in nature and allows students an opportunity to make sense of the problem and begin solving it in the way that makes most sense to them initially (table, graph, equation) (MP1). Students demonstrate an understanding of the given data by first constructing a model and then a viable argument about how many of each type of boomerang should be made and why (MP3).