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The Break-Even Bakery: Linear Equations for Social Impact
Grade 7Math3 days
Seventh-grade students step into the roles of social entrepreneurs to design a bakery business aimed at maximizing donations for a local charity. By researching fixed and variable costs, students construct and graph linear equations to identify their break-even point and visualize profit zones. They apply linear inequalities to calculate specific sales targets required to meet their charitable goals, ultimately presenting their mathematical models and social mission in a professional "Investor Pitch."Linear EquationsBreak-Even PointSocial EntrepreneurshipLinear InequalitiesFinancial LiteracyMathematical ModelingCoordinate Graphing
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Inquiry Framework
Question Framework
Driving Question
The overarching question that guides the entire project.How can we, as social entrepreneurs, use linear equations and inequalities to design a bakery business that maximizes our impact on a local charity?Essential Questions
Supporting questions that break down major concepts.- How do fixed and variable costs influence the total cost of running a business?
- How can we use linear equations to model the relationship between sales, costs, and profits?
- What does the 'break-even point' represent on a graph and in the context of our bakery business?
- How does adjusting the unit price of our goods impact how many items we need to sell to reach our charity goal?
- How can we use inequalities to determine the minimum number of sales needed to generate a specific profit for our cause?
Standards & Learning Goals
Learning Goals
By the end of this project, students will be able to:- Identify and categorize fixed costs (e.g., equipment) and variable costs (e.g., ingredients) to build a comprehensive business budget.
- Construct linear equations in the form y = mx + b to represent the total cost and total revenue of the bakery business.
- Calculate and interpret the break-even point by solving systems of linear equations or by using algebraic substitution.
- Graph cost and revenue functions on a coordinate plane to visually identify the break-even point and the region of profit.
- Apply linear inequalities to determine the minimum number of units sold required to reach a specific donation target for a chosen charity.
- Communicate mathematical reasoning and business decisions through a final "Investor Pitch" or social impact report.
Common Core State Standards (Math)
CCSS.MATH.CONTENT.7.EE.B.4.A
Primary
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.Reason: This is the foundational standard for modeling the bakery's costs (fixed + variable) and revenue to find the break-even point.
CCSS.MATH.CONTENT.7.EE.B.4.B
Primary
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.Reason: This standard is directly used when students determine how many items they must sell to meet or exceed their specific charity profit goal.
CCSS.MATH.CONTENT.7.RP.A.2
Supporting
Recognize and represent proportional relationships between quantities.Reason: Understanding the proportional relationship between the number of items sold and the revenue generated is essential for building the revenue side of the linear equation.
Common Core State Standards (Standards for Mathematical Practice)
CCSS.MATH.PRACTICE.MP4
Secondary
Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.Reason: The project requires students to translate a real-world business scenario into mathematical models (equations and graphs).
Entry Events
Events that will be used to introduce the project to studentsThe Cookie Catastrophe: Forensic Accounting 101
Students enter a classroom staged as a 'failed' bakery with 'For Lease' signs and a messy ledger left on the desk. They must act as 'Forensic Accountants' to analyze the former owner's chaotic receipts and determine the exact 'Point of No Return' where their fixed costs (rent) outweighed their variable sales.📚
Portfolio Activities
Portfolio Activities
These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.Activity 1
The Expense Architect: Building the Cost Model
Students act as the lead purchasers for their bakery. They must research the costs of ingredients (variable costs) and equipment/permits (fixed costs). By the end of this activity, students will distinguish between costs that change based on production and those that stay the same, organizing them into a mathematical model.Steps
Here is some basic scaffolding to help students complete the activity.1. Brainstorm a list of necessary items for a bakery (e.g., flour, sugar, oven, bowls, business license).
2. Categorize each item as a 'Fixed Cost' (one-time/static) or a 'Variable Cost' (per item).
3. Research actual prices using online grocery or equipment stores to assign realistic values to your list.
4. Calculate the 'Total Variable Cost' for a single unit and the 'Total Fixed Cost' for the entire operation.
Final Product
What students will submit as the final product of the activityA 'Baker’s Budget Spreadsheet' that lists all startup costs and the calculated cost per unit produced (e.g., cost per cookie).Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.EE.B.4.A by requiring students to identify fixed costs (q) and variable rates (p) to form the basis of a linear expression. It also supports 'Learning Goal: Identify and categorize fixed costs and variable costs.'Activity 2
The Profit Protagonist: Setting Your Price
Now that students know what it costs to make their goods, they must decide on a selling price. Students will analyze the relationship between the price per unit and total revenue, creating a linear equation to represent their income.Steps
Here is some basic scaffolding to help students complete the activity.1. Research market prices for similar bakery items in your community to ensure competitive pricing.
2. Select a unit price (p) that is higher than your variable cost per unit.
3. Create a revenue equation where 'y' is total revenue and 'x' is the number of items sold.
4. Generate a table of values showing how much revenue is generated if you sell 10, 50, or 100 items.
Final Product
What students will submit as the final product of the activityA 'Revenue Roadmap' document featuring the revenue equation (y = px) and a table showing projected income for various sales volumes.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.RP.A.2 by representing the proportional relationship between the number of items sold and the total revenue. It also addresses 'Learning Goal: Construct linear equations in the form y = mx + b'.Activity 3
Finding the Sweet Spot: The Break-Even Analysis
Students combine their cost and revenue models to find the 'Break-Even Point'—the exact number of items they must sell to cover all expenses before they can start earning profit for charity.Steps
Here is some basic scaffolding to help students complete the activity.1. Write the full cost equation: Total Cost = (Variable Cost * x) + Fixed Costs.
2. Set the Revenue Equation equal to the Total Cost Equation (Revenue = Cost).
3. Solve the equation for 'x' using algebraic properties (e.g., subtracting variables from both sides).
4. Interpret the result: If x is a decimal, explain why you must round up to the next whole number in a real-world business context.
Final Product
What students will submit as the final product of the activityAn 'Equilibrium Report' showing the algebraic steps taken to solve for x, along with a written explanation of what this number means for the business.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.EE.B.4.A by requiring students to solve the equation (Revenue = Cost) to find the specific value of x. It supports 'Learning Goal: Calculate and interpret the break-even point.'Activity 4
The Graphing Gourmet: Visualizing Your Venture
Students will translate their algebraic work into a visual representation. By graphing both the cost and revenue lines on the same coordinate plane, students will visually identify the break-even point as the intersection and color-code the 'Loss' and 'Profit' zones.Steps
Here is some basic scaffolding to help students complete the activity.1. Set up a coordinate plane where the x-axis represents 'Items Sold' and the y-axis represents 'Money ($)'.
2. Graph the Cost Equation starting at the y-intercept (Fixed Costs).
3. Graph the Revenue Equation starting at the origin (0,0).
4. Identify the intersection point and shade the area where Revenue is greater than Cost in green ('The Profit Zone').
Final Product
What students will submit as the final product of the activityA large-scale 'Bakery Growth Graph' (hand-drawn or digital) with labeled axes, both linear functions plotted, and the break-even point clearly marked.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.PRACTICE.MP4 (Modeling with mathematics) and 'Learning Goal: Graph cost and revenue functions on a coordinate plane.'Activity 5
The Charity Challenge: Modeling Social Impact
The bakery isn't just for profit; it's for charity! Students choose a local charity and set a donation goal. They will use inequalities to determine how many items they need to sell to reach that specific goal after all costs are paid.Steps
Here is some basic scaffolding to help students complete the activity.1. Research and select a local charity; decide on a specific dollar amount you want to donate (the Profit Goal).
2. Write an inequality in the form: (Revenue per unit * x) - (Total Cost) ≥ Profit Goal.
3. Solve the inequality for 'x' to find the minimum number of units required.
4. Graph the solution set on a number line and write a sentence explaining the target sales.
Final Product
What students will submit as the final product of the activityA 'Charity Impact Statement' featuring an inequality, its solution, and a number line graph representing the range of sales needed.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.EE.B.4.B by solving and graphing inequalities. It supports 'Learning Goal: Apply linear inequalities to determine the minimum sales for a donation target.'Activity 6
The Big Batch Pitch: Presenting Your Enterprise
Students compile their data, graphs, and impact statements into a professional pitch. They must justify their pricing, explain their break-even point, and demonstrate their potential for social good to a panel of 'investors' (teachers or community members).Steps
Here is some basic scaffolding to help students complete the activity.1. Synthesize the budget, equations, graphs, and charity goals into a cohesive narrative.
2. Prepare visual aids that showcase the Break-Even Graph and the Charity Inequality.
3. Draft a script that explains the 'why' behind your pricing and cost management.
4. Present the pitch to the class, answering questions about how changes in cost or price would affect the charity donation.
Final Product
What students will submit as the final product of the activityA 'Social Enterprise Pitch Deck' (Slide presentation or Video) that integrates the mathematical models with the mission of the bakery.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.PRACTICE.MP4 and 'Learning Goal: Communicate mathematical reasoning and business decisions.' This synthesizes all previous standards into a final communication task.🏆
Rubric & Reflection
Portfolio Rubric
Grading criteria for assessing the overall project portfolioThe Break-Even Bakery: Social Enterprise Rubric
Category 1
Mathematical Application & Communication
This category evaluates the student's ability to translate real-world business constraints into mathematical models and communicate those findings effectively.Criterion 1
Algebraic Modeling: Cost & Revenue Equations
Ability to identify, research, and categorize fixed and variable costs into a functional linear equation (y = mx + b).
Exemplary
4 PointsAccurately identifies all fixed and variable costs with realistic market research. Constructs flawless cost and revenue equations with clearly defined variables. Demonstrates a sophisticated understanding of how the 'y-intercept' represents startup costs and 'slope' represents unit costs.
Proficient
3 PointsCorrectly identifies major fixed and variable costs. Constructs accurate linear equations for cost and revenue. Variables are defined, and the relationship between cost per unit and total cost is clear.
Developing
2 PointsIdentifies most costs but may miscategorize a fixed cost as variable (or vice versa). Equations are present but may contain minor errors in structure or variable placement.
Beginning
1 PointsStruggles to distinguish between fixed and variable costs. Equations are missing, incomplete, or do not reflect the business scenario provided.
Criterion 2
Analytical Solving: Break-Even & Inequalities
Ability to solve for the break-even point (Revenue = Cost) and apply inequalities to determine sales needed for charity goals.
Exemplary
4 PointsSolves the system of equations flawlessly. Provides a sophisticated interpretation of the result, including the necessity of rounding up to the next whole unit. Correctly constructs and solves complex inequalities to meet specific charity profit targets.
Proficient
3 PointsSolves the break-even equation correctly with clear algebraic steps. Interprets the break-even point accurately in the context of the business. Successfully solves inequalities for the charity goal.
Developing
2 PointsAttempts to solve the break-even equation but makes minor computational errors. Can identify the break-even point but struggles to explain its business significance. Inequality setup is partially correct.
Beginning
1 PointsCannot solve the equation or inequality. Does not demonstrate an understanding of what the break-even point represents in a real-world context.
Criterion 3
Visual Representation: Coordinate Graphing
Accuracy and clarity in graphing linear functions on a coordinate plane to visualize business performance.
Exemplary
4 PointsGraphs are perfectly scaled and labeled. The intersection point is clearly marked and matches algebraic findings. Shading for 'Profit' and 'Loss' zones is precise, demonstrating a high level of mathematical modeling.
Proficient
3 PointsGraphs both cost and revenue functions accurately. Axes are labeled, and the break-even point is identified. Visual representation clearly supports the algebraic work.
Developing
2 PointsGraphs are mostly accurate but may have scaling issues or missing labels. The intersection point is visible but may not perfectly align with algebraic solutions.
Beginning
1 PointsGraphs are incorrect, messy, or missing. Does not show the relationship between the two linear functions or the intersection point.
Criterion 4
Communication & Social Impact Pitch
The ability to synthesize mathematical data into a persuasive business case for social impact.
Exemplary
4 PointsDelivers a compelling pitch that seamlessly integrates math and mission. Justifies pricing and cost decisions with data. Answers complex 'what-if' questions regarding changes in variables with ease and accuracy.
Proficient
3 PointsCommunicates mathematical reasoning clearly. Explains the relationship between sales, costs, and charity impact. Presentation is organized and uses the 'Equilibrium Report' effectively.
Developing
2 PointsPresents the business idea but struggles to explain the underlying math. Relying on notes to explain equations. Connection to the charity goal is present but weak.
Beginning
1 PointsPresentation is unorganized or lacks mathematical substance. Cannot explain how the equations relate to the bakery's success or charity impact.
Reflection Prompts
End-of-project reflection questions to get students to think about their learningQuestion 1
Looking back at your 'Break-Even Bakery,' what was the most challenging part of moving from a list of ingredients to a mathematical equation, and how did solving that challenge change how you see 'math in the real world'?
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Question 2
How confident do you feel now in your ability to use linear equations and inequalities to plan a real-world project or solve a complex business problem?
Scale
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Question 3
If your bakery's 'Fixed Costs' (like equipment or permits) suddenly increased, but you kept your selling price the same, what would happen to your Break-Even Point?
Multiple choice
Required
Options
The break-even point would stay exactly the same.
I would need to sell MORE items to reach the break-even point.
I would need to sell FEWER items to reach the break-even point.
The break-even point would decrease because costs are higher.
Question 4
How did using inequalities help you move beyond just 'guessing' and allow you to guarantee a specific level of impact for your chosen charity?
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