This mathematical problem would be a great real-world application for 8th and/or 9th grade Algebra 1 students. I would adapt this application by allowing students to work in groups. It would promote mathematical discourse, which “includes the purposeful exchange of ideas through classroom discussion, as well as through other forms of verbal, visual, and written communication” (National Council of Teachers of Mathematics [NCTM], 2014, pp. 29). This application builds and honors student thinking while making sure the mathematical ideas are at the center of the lesson during discussions.

   The use of technology is very important to this lesson. “Technology is an inescapable fact of life in the world in which we live and should be embraced as a powerful tool for doing mathematics” (NCTM, pp. 82). Teachers need to recognize how taking full advantage of the power of technology can effectively enhance students’ mathematical learning. During this lesson, I used the Numbers App and WolframAlpha. Integrating technology is also a Standard of Mathematical Practice (SMP). The standard states an excellent mathematical program promotes the use of mathematical tools and technology as essential resources to aid students’ learning and sense making of mathematical ideas (NCTM, pp. 5). Another SMP this mathematical application reflects is make sense of problems and persevere in solving them. The quadratic portion of the problem could be extremely frustrating for students who have never seen a real-world example of it. Teachers need to encourage their students during this productive struggle because this practice aids in students’ mathematical understanding. One last SMP I will mention is this problem helps students construct viable arguments and critique the reasoning of others. In a classroom, teachers need to build an atmosphere where students aren’t afraid to be critiqued. We live in an imperfect world therefore students should experience getting problems incorrect. By analyzing what went wrong and how to fix it, it aids students in understanding topics on a relational level.

For more information on the Standards of Mathematical Practices, click the link to investigate.

Alabama Course of Study Standards this mathematical application applies to:

12. Create equations and inequalities in one variable, and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [A-CED1]

• I constructed linear, exponential, and quadratic equations for the spreadsheet to solve this problem.

16. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1]

 27. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. [F-IF3]

• I was able to see from the information given, there were 3 functions and was able to define all 3 recursively.

34. Write a function that describes a relationship between two quantities.* [F-BF1]

a. Determine an explicit expression, a recursive process, or steps for calculation from a context. [F-BF1a]

b. Combine standard function types using arithmetic operations. [F-BF1b]

• Explicit formula is the same thing as a closed formula. For each of the functions, I was able to find the closed formulas. As stated above, I was also able to find recursive formulas for all of the plans as well.

35. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between two forms.* [F-BF2]

• This mathematical problem is a great way to model linear, exponential and quadratic functions in a real world application.

37. Distinguish between situations that can be modeled with linear functions and exponential functions. [F-LE1]

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. [F-LE1a]

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. [F-LE1b]

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [F-LE1c]

• I showed how I was able to distinguish between the three functions and when graphed, you could visually see the difference between each graph.

39. Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. [F-LE3]

 

Refer back to the Mathematical Solution to see how Choose Your Allowance reflects these standards.

Click here to continue to the Conclusion.