Common Core Standards: Math
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional cost constraints on combinations of different foods.
Students have already translated words into algebraic equations (sometimes with more than one variable!) and have actually taken the time to solve the problem. We hate to make you the villain here, but you have to tell them their work isn't over. Students now have to interpret the results. This standard is about one thing: analysis. Well… actually it's about three things:
- Creating equations/inequalities or systems of equations/inequalities
- Solving these equations/inequalities or system of equations
- Interpreting the answer properly
To analyze problems in which multiple relationships affect multiple variables, students must be able to create systems of equations, solve them, and interpret the results appropriately.
Creating Systems of Equations
In order to create systems of equations from word problems or other contexts, students need to be able to differentiate the relations and create equations for each. To support this, should already be able to create equations from word problems.
Creating equations from a word problem or similar context is a three-step translation process:
- Translate the equality or inequality (=, <, >, ≤, or ≥)
- Translate the operations (+, –, ×, ÷, xn, nx)
- Translate the numbers and variables
Systems of equations are identified during step one of this process. Students need to be able to read a problem and identify how many equality and inequality relations are described. Then, they should write each down separately.
Once these are written down, students perform steps two and three (translating the operations, numbers, and variables) independently for each equation. They should also simplify each equation individually before working to solve the system of equations.
Solving Systems of Equations
Solving systems of equations can be done through substitution or adding the two equations together to cancel out one of the variables. The goal is to eliminate one variable so that we can find the solution for the other and then substitute that answer back in to find the value of the second variable.
Hopefully, students already know how to solve systems of equations. After all, it's necessary in order to interpret results from a set of equations.
Interpreting Results from Systems of Equations
In many situations, students struggle to understand what an algebraic result means in the context of a word problem. This is especially true when systems of equations are involved and when they arrive at solutions that are correct algebraically, but incorrect in context.
For instance, a question about how many tops hats a giraffe can wear might produce the number 6.25 as the answer. This might make sense algebraically, but in the context of giraffes wearing top hats—how can a giraffe wear one fourth of a hat? The logical answer would then have to be 6 (although logic might not be our biggest concern if we're talking about giraffes in top hats).
Such algebraic solutions also present a challenge when multiple roots are encountered or when cancelling rational expressions. Describing what values of a variable are allowed when recording the variable information is one strategy for dealing with this problem. (For instance, we could write that giraffes only wear top hats in whole numbers.)
A common error is to report an answer based on a different variable in the problem. Recording the variable information helps to prevent such errors. If this is a common issue, students should try highlighting the quantity of interest in the problem, the matching variable name, and the eventual result. They can then check all highlighted items to ensure that they match.
Multiple variables present even more of a challenge, as students often need to find information for more than one variable in a problem. Highlighting the different information requested in different colors is one strategy. Another possibility is to have students solve for all variables before interpreting a solution; however, this becomes troublesome in more complex problems.
Interpreting results should be performed throughout the algebra curriculum. With enough practice, students will perform whatever strategies work best for them naturally.