Business Break-Even: Solving Systems of Equations
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Business Break-Even: Solving Systems of Equations

Grade 9Math21 days
In this project, students apply systems of equations to model business costs, revenue, and profit to determine the break-even point and make informed business decisions. Students use various methods such as graphing, substitution, and elimination to solve systems of equations and utilize technology to analyze business data efficiently. The project culminates in a mini-business plan that incorporates the break-even analysis to inform decisions about pricing, production levels, and marketing strategies. Students justify their business decisions based on the analysis of systems of equations, demonstrating an understanding of how mathematical models can optimize business outcomes.
Systems of EquationsBreak-Even PointBusiness PlanCost AnalysisRevenue ModelingProfit OptimizationAlgebraic Methods
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we, as aspiring business owners, use systems of equations to model costs, revenue, and profit to determine the break-even point and optimize our business decisions?

Essential Questions

Supporting questions that break down major concepts.
  • How can systems of equations model real-world business scenarios?
  • What does the break-even point signify in business, and how can it be determined using systems of equations?
  • How can different methods of solving systems of equations (e.g., graphing, substitution, elimination) be applied to optimize business decisions?
  • In what ways can technology be used to solve systems of equations and analyze business data efficiently?
  • How do business costs, revenue, and profit relate to systems of equations?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Apply systems of equations to model business costs, revenue, and profit.
  • Determine the break-even point using systems of equations.
  • Use different methods (graphing, substitution, elimination) to solve systems of equations for business optimization.
  • Utilize technology to solve systems of equations and analyze business data.
  • Interpret the break-even point in the context of business decision-making.
  • Develop a business plan incorporating systems of equations to model financial aspects.
  • Analyze the relationships between business costs, revenue, and profit within systems of equations.
  • Communicate and justify business decisions based on the analysis of systems of equations.
  • Solve systems of linear equations exactly and approximately with focus on pairs of linear equations in two variables.
  • Explore strategies for solving systems of equations to look at business accounting

Common Core State Standards

CCSS.Math.Content.HSA-REI.C.6
Primary
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Reason: Directly addresses the mathematical skills required for the project.

Entry Events

Events that will be used to introduce the project to students

Lemonade Stand Showdown

The class is presented with two competing lemonade stand business plans with different initial investments and profit margins. Students must use systems of equations to model each business's potential earnings and determine which stand will reach profitability faster, prompting them to consider real-world constraints and make informed recommendations.

Entrepreneur's Dilemma

A local entrepreneur visits the class and shares their business journey, highlighting a moment where understanding break-even analysis was crucial for their survival. Students then brainstorm the mathematical challenges the entrepreneur faced and how systems of equations could have provided solutions.
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Portfolio Activities

Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.
Activity 1

Business Basics: Costs, Revenue, and Profit Equations

Students will begin by understanding the basic components of a business: costs (fixed and variable), revenue, and profit. They will define these terms and create simple linear equations to represent each.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Define fixed costs, variable costs, revenue, and profit in the context of a business. (Step 1)
2. Research typical costs and revenue streams for a business they are interested in. (Step 2)
3. Create linear equations representing total cost (fixed + variable costs) and total revenue based on the number of units sold/services provided. (Step 3)

Final Product

What students will submit as the final product of the activityA definitions list and a set of linear equations modeling cost, revenue, and profit for a hypothetical business.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSA-REI.C.6 (Solve systems of linear equations exactly and approximately) and Learning Goal: Apply systems of equations to model business costs, revenue, and profit.
Activity 2

Graphing the Break-Even Point

Students will graph the cost and revenue equations they created in the previous activity. By finding the intersection point of these two lines, they will visually determine the break-even point.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Graph the cost and revenue equations on the same coordinate plane. (Step 1)
2. Identify the point of intersection (break-even point) on the graph. (Step 2)
3. Explain what the x and y coordinates of the break-even point represent in terms of units sold and total revenue/cost. (Step 3)

Final Product

What students will submit as the final product of the activityA graph showing cost and revenue equations with the break-even point clearly marked.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSA-REI.C.6 (Solve systems of linear equations exactly and approximately) and Learning Goal: Determine the break-even point using systems of equations.
Activity 3

Algebraic Break-Even Analysis: Substitution and Elimination

Students will algebraically solve the system of equations (cost and revenue) using substitution and elimination methods. They will compare the results with their graphical solution from the previous activity.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Solve the system of equations (cost and revenue) using the substitution method. (Step 1)
2. Solve the same system of equations using the elimination method. (Step 2)
3. Compare the algebraic solutions with the graphical solution. Discuss any discrepancies and potential reasons for them. (Step 3)

Final Product

What students will submit as the final product of the activityA written report comparing the break-even point calculated algebraically (using substitution and elimination) with the graphical solution.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSA-REI.C.6 (Solve systems of linear equations exactly and approximately) and Learning Goal: Use different methods (graphing, substitution, elimination) to solve systems of equations for business optimization.
Activity 4

Tech-Enhanced Break-Even Analysis

Students will use graphing calculators or online tools (like Desmos) to solve the systems of equations and analyze how changing parameters (e.g., fixed costs, variable costs, selling price) affects the break-even point.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Use a graphing calculator or online tool to graph the cost and revenue equations. (Step 1)
2. Use the tool to find the intersection point (break-even point). (Step 2)
3. Experiment with changing the parameters of the equations (fixed costs, variable costs, selling price) and observe how the break-even point changes. (Step 3)
4. Prepare a presentation summarizing the process and findings. (Step 4)

Final Product

What students will submit as the final product of the activityA presentation (slides or video) demonstrating how to use technology to solve systems of equations and analyze the impact of changing business parameters on the break-even point.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSA-REI.C.6 (Solve systems of linear equations exactly and approximately) and Learning Goal: Utilize technology to solve systems of equations and analyze business data.
Activity 5

Business Plan: From Break-Even to Business Decisions

Students will create a mini-business plan for their chosen business, incorporating the break-even analysis they performed. They will discuss how understanding the break-even point informs their business decisions (e.g., pricing, production levels).

Steps

Here is some basic scaffolding to help students complete the activity.
1. Write a brief description of their chosen business and its products/services. (Step 1)
2. Include the cost and revenue equations they developed earlier. (Step 2)
3. State the break-even point (units and revenue). (Step 3)
4. Explain how the break-even point informs their decisions about pricing, production levels, and marketing strategies. (Step 4)
5. Justify their business decisions based on the break-even analysis. (Step 5)

Final Product

What students will submit as the final product of the activityA mini-business plan that includes a break-even analysis, a discussion of how the break-even point affects business decisions, and a justification of their pricing and production strategies.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSA-REI.C.6 (Solve systems of linear equations exactly and approximately) and Learning Goal: Interpret the break-even point in the context of business decision-making and Develop a business plan incorporating systems of equations to model financial aspects.
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Break-Even Business Analysis Rubric

Category 1

Understanding of Systems of Equations

Assesses students' comprehension of linear equations related to business costs, revenue, and profit.
Criterion 1

Identification of Cost, Revenue, and Profit Concepts

Evaluates the ability to define and apply these economic concepts within equations.

Exemplary
4 Points

Thoroughly and accurately defines cost, revenue, and profit with precise equations well-aligned to business scenarios.

Proficient
3 Points

Defines cost, revenue, and profit accurately with appropriate equations fitting business contexts.

Developing
2 Points

Partially correct definitions and equations; some confusion evident in business context application.

Beginning
1 Points

Inaccurate or incomplete definitions and equations lacking business context relevance.

Criterion 2

Equation Development

Measures skill in developing linear equations to represent economic factors.

Exemplary
4 Points

Constructs precise and logical linear equations demonstrating advanced connection to business scenarios.

Proficient
3 Points

Develops well-structured equations with clear links to business cases.

Developing
2 Points

Generates equations with partial clarity or relevance to the business context.

Beginning
1 Points

Struggles to create logical equations; limited connection to business context.

Category 2

Method Application for Break-Even Analysis

Focuses on students' use of graphing, substitution, and elimination methods to analyze and solve equations.
Criterion 1

Graphical Representation

Evaluates accuracy in graphing cost and revenue and identifying the break-even point.

Exemplary
4 Points

Graphs are precise; intersection point is clearly identified and explained within the business context.

Proficient
3 Points

Accurate graphing with correctly identified intersection point and reasonable explanation.

Developing
2 Points

Graph contains minor errors; intersection point somewhat identified or explained.

Beginning
1 Points

Incorrect graphs or misidentified intersection introduce significant inaccuracies.

Criterion 2

Algebraic Solution Methods

Assesses the use of substitution and elimination methods for solving systems of equations.

Exemplary
4 Points

Flawless application of both methods with consistent results across algebraic and graphical solutions.

Proficient
3 Points

Correctly applies substitution and elimination with generally consistent outcomes.

Developing
2 Points

Incomplete or partly incorrect application with some inconsistency in results.

Beginning
1 Points

Initial attempts with method application errors leading to conflicting results.

Category 3

Use of Technology in Mathematical Analysis

Assesses student ability to leverage technology for solving and analyzing systems of equations.
Criterion 1

Technological Application

Measures effectiveness in using graphing calculators or software to solve equations and alter parameters.

Exemplary
4 Points

Uses technology adeptly, clearly demonstrating effects of parameter changes on graph outcomes.

Proficient
3 Points

Employs technology effectively with sensible demonstration of parameter impact.

Developing
2 Points

Partial tech use with limited exploration or demonstration of parameter effects.

Beginning
1 Points

Incorrect or minimal use of technology lacking clear outcome demonstrations.

Category 4

Business Decision-Making and Justification

Evaluates the degree to which students explain and support their business decisions based on analysis.
Criterion 1

Business Plan Rationalization

Assesses how students justify business decisions using break-even analysis.

Exemplary
4 Points

Provides insightful, evidence-backed justifications for business decisions, tying analysis to strategic choices.

Proficient
3 Points

Offers sound, logically-supported decisions connected to break-even findings.

Developing
2 Points

Presents partial or sometimes unclear justifications related to break-even point conclusions.

Beginning
1 Points

Lacks coherent rationale or connections between analysis and business decisions.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning
Question 1

How did your understanding of systems of equations evolve as you applied them to real-world business scenarios in this project?

Text
Required
Question 2

What was the most challenging aspect of using systems of equations to determine the break-even point for your business, and how did you overcome it?

Text
Required
Question 3

To what extent do you agree with the statement: 'Systems of equations are a valuable tool for making informed business decisions'?

Scale
Required
Question 4

Which method for solving systems of equations (graphing, substitution, elimination, or technology) did you find most helpful for business optimization, and why?

Multiple choice
Required
Options
Graphing
Substitution
Elimination
Technology