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Created byYolanda Williams
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The Rate Race: Graphing and Comparing Proportional Relationships
Grade 7Math5 days
★★★★★
5.0 (1 rating)Proportional RelationshipsUnit RateConstant Of ProportionalityMathematical ModelingFinancial LiteracyLinear GraphingBusiness Consulting
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Inquiry Framework
Question Framework
Driving Question
The overarching question that guides the entire project.How can we, as business consultants, use proportional relationships to design a competitive service and convince potential clients that our pricing model offers the best value?Essential Questions
Supporting questions that break down major concepts.- How does finding a unit rate help us compare products or services that aren't packaged the same?
- In what ways can we identify a proportional relationship when looking at a table, a graph, or an equation?
- What does the 'constant of proportionality' represent in a real-world context, and why is it important for making predictions?
- How do the steepness and features of a graphed line help us compare the rates of two different businesses or scenarios?
- How can we use proportional equations (y = kx) to solve multi-step problems involving taxes, tips, or discounts?
Standards & Learning Goals
Learning Goals
By the end of this project, students will be able to:- Calculate and compare unit rates for different service offerings to determine the most cost-effective business model.
- Identify proportional relationships across multiple representations, including tables, equations (y = kx), and coordinate graphs.
- Determine and interpret the constant of proportionality (unit rate) within the context of a business pricing structure.
- Construct and analyze graphs of proportional relationships, using the steepness of the line to compare competing business rates.
- Apply proportional reasoning to solve multi-step real-world financial problems involving taxes, tips, and discounts for client proposals.
Common Core State Standards for Mathematics
CCSS.MATH.CONTENT.7.RP.A.1
Primary
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.Reason: Students must calculate the unit price for their services to compare their business against competitors and ensure they are offering a fair rate.
CCSS.MATH.CONTENT.7.RP.A.2
Primary
Recognize and represent proportional relationships between quantities.Reason: This is the foundational standard for the project, requiring students to prove their pricing models are proportional through tables, graphs, and equations.
CCSS.MATH.CONTENT.7.RP.A.2.B
Primary
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Reason: As consultants, students need to identify the 'k' value (price per unit) to explain their pricing strategy to potential clients.
CCSS.MATH.CONTENT.7.RP.A.2.D
Primary
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.Reason: Students will use graphs to visualize their pricing and must explain to clients what specific points (such as the unit rate at x=1) represent in their service contract.
CCSS.MATH.CONTENT.7.RP.A.3
Primary
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error.Reason: The project requires students to calculate final costs for clients, which include applying discounts for bulk services or adding taxes and service fees.
Common Core State Standards for Mathematical Practice
CCSS.MATH.PRACTICE.MP4
Supporting
Model with mathematics.Reason: Students are applying mathematical structures to a real-world business simulation, using equations and graphs to model financial decisions.
Entry Events
Events that will be used to introduce the project to studentsThe Sneaker Bot Sabotage
A famous sneaker brand releases a 'limited drop' where the price increases proportionally based on demand levels in different cities. Students are given 'glitched' data tables and must identify the constant of proportionality to predict the final resell price, comparing different city growth lines to see where the best profit margin lies.The Influencer Audit: Truth in Trends
Students are presented with two competing social media influencers claiming to have the 'fastest-growing' fan base. Using 'leaked' data sets with different scales and mismatched timeframes, students must find the constant of proportionality to debunk the influencer who is visually manipulating their growth graphs to look more successful.📚
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 Unit Rate Scout: Finding the 'K' Factor
In this introductory activity, students act as lead researchers for their consulting firm. They are tasked with analyzing 'raw data' from various suppliers or service providers to find the most efficient unit rates. They will encounter ratios with fractions and different units (e.g., $15.50 for 2.5 hours of consulting) and must convert these into a standard unit rate (the price for 1 unit) to establish their baseline constant of proportionality (k).Steps
Here is some basic scaffolding to help students complete the activity.1. Select a business niche (e.g., social media management, sneaker reselling, or tech support) and review the provided 'Supplier Data Cards' containing fractional rates and mixed units.
2. Use division to calculate the unit rate (k) for each supplier. For example, if a supplier charges $120 for 3/4 of a day of work, calculate the rate for 1 full day.
3. Compare the unit rates and select the one that offers the best value for your consulting firm's business model.
4. Define your 'k' value and write a brief explanation of what this number represents in the context of your specific business.
Final Product
What students will submit as the final product of the activityA 'Business Cost Analysis' sheet that lists at least three different supplier rates, the calculated unit rate for each, and a justified selection of which rate will serve as their business's 'Constant of Proportionality.'Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.RP.A.1 (Compute unit rates associated with ratios of fractions) and 7.RP.A.2.B (Identify the constant of proportionality).Activity 2
The Proportionality Blueprint: Building the Matrix
Now that students have their constant of proportionality (k), they must prove their business model is truly proportional. They will create a 'Pricing Matrix' that shows the relationship between the number of units sold (x) and the total cost (y). Students will practice moving between verbal descriptions, data tables, and the algebraic equation y = kx to demonstrate consistency in their pricing.Steps
Here is some basic scaffolding to help students complete the activity.1. Create a data table with an 'Input' column (number of units/hours) and an 'Output' column (total cost). Ensure the table includes the 'zero' point (0 units = $0).
2. Apply your 'k' value from Activity 1 to at least five different quantities to fill in your table.
3. Write the equation in the form y = kx that represents your table, where 'y' is the total cost and 'x' is the quantity.
4. Test your equation by plugging in a new 'x' value to see if it predicts the correct 'y' value.
Final Product
What students will submit as the final product of the activityA 'Pricing Strategy Matrix' which includes a data table (with at least 5 entries), the formal equation (y = kx) for their business, and a 'Proportionality Proof' explaining why their model starts at (0,0).Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.RP.A.2 (Recognize and represent proportional relationships) and 7.RP.A.2.C (Represent proportional relationships by equations).Activity 3
The Visual Showdown: Mapping the Market
Students will now visualize their business model. Using their data from the Pricing Matrix, they will graph their proportional relationship on a coordinate plane. To make it a 'showdown,' they will also graph a competitor’s line (provided by the teacher) on the same grid. Students must analyze the 'steepness' of the lines and identify the specific coordinates that represent the unit rate.Steps
Here is some basic scaffolding to help students complete the activity.1. Plot the points from your Pricing Matrix onto a coordinate plane, ensuring your 'x' (quantity) and 'y' (cost) axes are correctly labeled.
2. Draw a straight line through the points and the origin to represent your 'Business Growth Line.'
3. Plot the competitor’s data on the same graph using a different color.
4. Identify and circle the point (1, r) on your line. Write a 'Point Profile' explaining exactly what this coordinate means (e.g., 'At 1 hour of service, the cost is $45').
5. Compare the steepness (slope) of your line to the competitor's. Explain which business is more expensive and how the graph shows this visually.
Final Product
What students will submit as the final product of the activityA 'Competitive Landscape Graph' featuring two distinct lines, labeled axes, and a written analysis identifying the points (0,0) and (1, r) for their business.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.RP.A.2.B (Identify k in graphs) and 7.RP.A.2.D (Explain what a point (x, y) means on a graph).Activity 4
The Consultant's Closing: The Final Invoice
In the final phase, students apply their model to a real-world client scenario. A client wants to hire their firm, but the transaction involves 'real-world' math: a bulk discount (markdown), a service tax (markup), and a final comparison. Students must use their unit rate and equation to calculate the subtotal and then apply percentages to find the final price.Steps
Here is some basic scaffolding to help students complete the activity.1. Calculate the subtotal for a large client order (e.g., 50 units or 100 hours) using your equation y = kx.
2. Apply a 'Bulk Discount' or 'New Client Discount' by calculating the percentage and subtracting it from the subtotal.
3. Calculate the required sales tax or service fee (markup) based on the discounted price and add it to the total.
4. Write a final 'Value Pitch' to the client, explaining why your total price—even with tax—is a better deal than the competitor's based on your earlier graph analysis.
Final Product
What students will submit as the final product of the activityA 'Professional Client Proposal & Invoice' that breaks down the subtotal, applies a 10% 'New Client Discount,' adds a 5% 'Service Tax,' and presents the final 'All-In' price to the client.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.7.RP.A.3 (Use proportional relationships to solve multistep ratio and percent problems).🏆
Rubric & Reflection
Portfolio Rubric
Grading criteria for assessing the overall project portfolioRatio & Proportion: Business Consultant Portfolio Rubric
Category 1
Foundational Proportional Reasoning
Focuses on the foundational mathematical skill of determining unit rates and establishing the constant of proportionality from various data sources.Criterion 1
Unit Rate Computation and Selection (k)
Measures the ability to calculate unit rates from complex ratios (including fractions) and identify the constant of proportionality (k) within a business context.
Exemplary
4 PointsFlawlessly calculates unit rates from complex fractional data; provides a sophisticated justification for the selected 'k' value that demonstrates a deep understanding of business efficiency.
Proficient
3 PointsAccurately calculates unit rates from fractional data; identifies the constant of proportionality and provides a clear reason for the business selection.
Developing
2 PointsCalculates unit rates with minor errors in fractional division; identifies a 'k' value but the business justification is thin or inconsistent.
Beginning
1 PointsStruggles to calculate unit rates from raw data; cannot clearly identify a constant of proportionality for the business model.
Category 2
Structural Modeling
Assesses the student's ability to translate a verbal business model into mathematical structures like tables and equations.Criterion 1
Tabular and Algebraic Modeling
Evaluates the accuracy and completeness of representing proportional relationships through data tables and the algebraic equation y = kx.
Exemplary
4 PointsCreates a comprehensive table and a perfect y=kx equation; provides an innovative 'Proportionality Proof' that connects (0,0) to real-world business startup costs.
Proficient
3 PointsConstructs an accurate data table with 5+ entries and the correct y=kx equation; explains the proportional nature of the model starting at the origin.
Developing
2 PointsTable contains some calculation errors or the equation is improperly formatted (e.g., missing variables); shows basic understanding of proportionality.
Beginning
1 PointsTable and equation are incomplete or do not represent a proportional relationship (e.g., non-constant rate).
Category 3
Visual Market Analysis
Focuses on the visual demonstration of mathematical relationships and the ability to extract meaning from coordinate planes.Criterion 1
Graphical Representation and Analysis
Measures the skill of graphing proportional relationships and interpreting the meaning of specific coordinate points (0,0) and (1,r) in context.
Exemplary
4 PointsGraphs are perfectly scaled and labeled; provides a sophisticated 'Point Profile' that explains the unit rate's impact on business scalability compared to a competitor.
Proficient
3 PointsGraphs the business and competitor lines accurately; correctly identifies and explains the meaning of (0,0) and (1,r) in the context of the service.
Developing
2 PointsGraphing is largely accurate but lacks precision in slope or labeling; identifies points but struggles to explain their real-world significance.
Beginning
1 PointsGraph is missing key components (axes labels, origin) or lines do not correctly represent the identified unit rates.
Category 4
Applied Financial Literacy
Evaluates the application of proportional reasoning to real-world financial transactions involving markups and markdowns.Criterion 1
Financial Application and Percentages
Assesses the ability to solve multi-step problems involving percentages (discounts and taxes) to create a final financial document.
Exemplary
4 PointsCalculates final invoice with 100% accuracy; demonstrates advanced integration of skills by explaining the sequential impact of discounts and taxes on the bottom line.
Proficient
3 PointsCorrectly applies a percentage discount and a service tax to the subtotal; produces an accurate and professional final invoice.
Developing
2 PointsCalculates subtotal correctly but makes errors in applying percentages (e.g., adding discount instead of subtracting) or misses one of the multi-step components.
Beginning
1 PointsUnable to apply percentage calculations to the subtotal; final invoice is incomplete or mathematically unsound.
Category 5
Mathematical Modeling & Professionalism
Assesses the overarching ability to model with mathematics and communicate findings effectively in a professional simulation.Criterion 1
Consultative Communication and Argumentation
Measures the student's ability to act as a 'Business Consultant' by using mathematical evidence to argue for the value of their service.
Exemplary
4 PointsPresents a compelling, evidence-based 'Value Pitch' that masterfully uses graphs and unit rates to prove market dominance; exhibits high-level professional communication.
Proficient
3 PointsProvides a clear 'Value Pitch' using mathematical evidence from previous activities to justify the pricing model to the client.
Developing
2 PointsPitch is primarily descriptive rather than evidence-based; makes limited reference to the mathematical data gathered during the project.
Beginning
1 PointsFails to provide a justification for the pricing model; communication is unclear or lacks mathematical support.
Reflection Prompts
End-of-project reflection questions to get students to think about their learningQuestion 1
How has your understanding of the 'Constant of Proportionality' (k) changed after acting as a business consultant? What does it represent beyond just a math answer?
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Question 2
How confident do you feel in your ability to take a business scenario and represent it as both an equation (y = kx) and a coordinate graph?
Scale
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Question 3
In your final 'Value Pitch,' which mathematical representation do you think was most convincing to your client, and why?
Multiple choice
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Options
The Pricing Matrix (Table): It shows exact numbers for every scenario.
The Proportional Equation (y = kx): It allows for the fastest calculations.
The Business Growth Line (Graph): It shows the 'steepness' and value compared to competitors visually.
Question 4
In 'The Consultant's Closing,' you had to apply discounts and taxes. How did having a solid unit rate and equation make it easier to handle these final price changes?
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Question 5
Looking at your 'Visual Showdown' graph, what did the 'steepness' of your line versus the competitor's line tell you about who was charging more? How did the point (1, r) help you prove your point?
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