Profit Pursuit PracticeProfit Pursuit Independent Practice: Modeling Business Systems Name: ____________________________________ Date: _____________________________________ Part 1: The Equation Engine Solve each system by setting them equal. Show all algebraic steps. 1. \(y = 6x\) \(y = 2x + 12\) Solution: (____, ____) 2. \(y = 5x\) \(y = x + 24\) Solution: (____, ____) 3. \(y = 12x\) \(y = 4x + 80\) Solution: (____, ____) 4. \(y = 7x\) \(y = 3x + 16\) Solution: (____, ____) 5. \(y = 9x\) \(y = 4x + 35\) Solution: (____, ____) 6. \(y = 4x\) \(y = x + 15\) Solution: (____, ____) Part 2: The Startup Challenge Scenario 1: The Crafty Keychain Co. Maya is starting a 3D-printing business. She needs to find her break-even point. Costs $10 software fee + $1 per print. Revenue Selling for $3 each. 1. Model and Solve Cost: \(y =\) _________________ Revenue: \(y =\) _________________ Quantity Sold (\(x\)): __________ Total Money (\(y\)): __________ Part 3: The Expansion Phase Scenario 2: The Lemonade Stand Maya decides to raise money for charity with a lemonade stand. Costs $12 equipment fee + $0.25 per cup. Revenue Selling for $1.25 per cup. 1. Model and Solve Cost: \(y =\) _________________ Revenue: \(y =\) _________________ Final Summary Maya must sell cups to break even. At this point, she will have spent and earned a total of $.
Startup Success HomeworkStartup Success Homework: Business Analysis & Modeling Student Name: ____________________________________ Due Date: _____________________________________ Part 1: The Warm Up Solve these two systems algebraically by setting them equal. Show your work clearly. \(y = 8x\) \(y = 3x + 25\) Solution: (________, ________) \(y = 15x\) \(y = 5x + 100\) Solution: (________, ________) Part 2: Custom T-Shirt Empire The Scenario You are launching "Empire Threads," a custom T-shirt business. Cost Factors Rent a machine for $50 (fixed). Each blank shirt costs $5. Revenue Factors Sell each finished custom T-shirt for $15. 1. Build the Business Model Cost Equation: \(y =\) ___________________ Revenue Equation: \(y =\) ___________________ 2. Solve for the Break-Even Point Set the equations equal to each other and solve for \(x\). Show all algebraic steps. BREAK-EVEN POINT: (_________, _________) 3. Strategic Analysis A. How many shirts must you sell to cover all your costs? B. If you sell 10 shirts, what will your total profit be?
Profit Pursuit KeyProfit Pursuit Key Teacher Answer Key & Facilitation Guide Practice Key: Scenarios & Drills Part 1 Drills 1. (3, 18) [\(4x = 12\)] 2. (6, 30) [\(4x = 24\)] 3. (10, 120) [\(8x = 80\)] 4. (4, 28) [\(4x = 16\)] 5. (7, 63) [\(5x = 35\)] 6. (5, 20) [\(3x = 15\)] Scenario 1: Keychains \(y = 1x + 10, y = 3x\) Solution: (5, 15) Scenario 2: Lemonade \(y = 0.25x + 12, y = 1.25x\) Solution: (12, 15) Exit Ticket \(y = x + 10, y = 3x\) Solution: (5, 15) Interpretation: Student must sell 5 stickers to break even at $15 total expenses/income. Homework Key Break-Even: (5, 75) A. Cover all costs with 5 shirts. B. Profit at 10 shirts: $50 Facilitation & Coaching Fixed vs Variable Ensure students place the fixed fee (software, rental) as the y-intercept and the unit cost as the slope. X-Axis Meaning In all these models, \(x\) represents the quantity of products sold. Solution Context Remind students that (5, 15) means "5 items for $15." The units alternate (Items, Dollars). Break-Even Logic At the solution, Revenue minus Cost equals zero. This is the first milestone for any startup!
Profit Point Exit TicketProfit Point Exit Ticket: Startup Pulse Check Name: _______________________ Date: _______________________ 1 Solve the System A student is selling stickers. Set the equations equal to find the point where cost equals revenue. Cost: \(y = x + 10\) Revenue: \(y = 3x\) SOLUTION \((x, y)\): (_________, _________) 2 Conceptual Analysis In the sticker business scenario above, explain what your \(x\)-value represents in real-world terms. Business Mathematics Mastery
Market Equilibrium GuideEconomics Reference Guide Market Equilibrium What is Equilibrium? In a market, the Equilibrium Point (\(E\)) is where the downward-sloping Demand curve (\(D\)) intersects the upward-sloping Supply curve (\(S\)). At this exact point, the quantity consumers want to buy equals the quantity producers want to sell. 1 Shifting the Demand Curve Demand shifts when consumer taste, income, or population changes. An Increase in Demand shifts the entire curve to the right. Demand Shifts Right (D → D₂) S D₁ D₂ E₁ E₂ Price (P) Quantity (Q) The Chain of Logic: Demand increases → Consumers want more at every price point. At the original price (\(E₁\)), a shortage occurs because buyers want more than is available. To clear the shortage, sellers raise prices, and producers increase production output. Resulting Shifts Equilibrium Price (\(P\)) ↑ Equilibrium Quantity (\(Q\)) ↑ Market Equilibrium Guide Page 1 of 2 Market Dynamics Supply & Summary Models 2 Shifting the Supply Curve Supply shifts due to changes in production technology, input costs, or the number of sellers. A Decrease in Supply (like more expensive ingredients) shifts the curve to the left. Supply Shifts Left (S₁ → S₂) S₁ S₂ D E₁ E₂ Price (P) Quantity (Q) The Chain of Logic: Supply decreases → Producers make fewer goods at every price point. At the original price (\(E₁\)), a scarcity/shortage occurs because quantity demanded exceeds supply. Buyers bid up prices, and market transactions settle at a higher cost but a lower volume. Resulting Shifts Equilibrium Price (\(P\)) ↑ Equilibrium Quantity (\(Q\)) ↓ 3 The Equilibrium Shift Cheat Sheet Shift TypeImpact on Price (\(P\))Impact on Quantity (\(Q\))Real-world ExampleDemand Increases (→)↑ Increases↑ IncreasesStickers become a massive trend at school.Demand Decreases (←)↓ Decreases↓ DecreasesWinter ends, reducing cold remedy sales.
Equilibrium Shifts WorksheetEquilibrium Shifts Practice Worksheet: Supply & Demand Dynamics Name: ____________________________________ Date: _____________________________________ 1 Scenario A: The Viral Trend Market: Wireless Headphones. Event: A highly popular video creator posts a review praising a specific brand of headphones, causing them to go viral. Draw the shift on the grid below S₁ D₁ E₁ P Q A. Which curve shifts, and in which direction? Curve: ______________ Direction: ______________ B. Explain the logic of this change: Final Equilibrium Impact Equilibrium Price (\(P\)): [ ] Increases [ ] Decreases Equilibrium Quantity (\(Q\)): [ ] Increases [ ] Decreases Market Equilibrium Shift Practice Page 1 of 2 Part 2: Supply Shifts Equilibrium Shifts 2 Scenario B: Rising Cocoa Costs Market: Chocolate Bars. Event: Severe storms damage cocoa crops worldwide, doubling the price of raw cocoa beans for manufacturers. Draw the shift on the grid below S₁ D₁ E₁ P Q A. Which curve shifts, and in which direction? Curve: ______________ Direction: ______________ B. Explain the logic of this change: Final Equilibrium Impact Equilibrium Price (\(P\)): [ ] Increases [ ] Decreases Equilibrium Quantity (\(Q\)): [ ] Increases [ ] Decreases 3 Part 3: Quick Analysis 1. If a technological breakthrough cuts manufacturing costs in half, what occurs? Supply increases, lowering price and increasing quantity. Demand decreases, lowering both price and quantity. 2. What always happens when demand decreases and supply stays the same? Both price and quantity increase. Both price and quantity decrease. Market Equilibrium Shift Practice Page 2 of 2
Profit Pursuit Warm UpDaily Opener DO NOW: Match Point Objective (SWBAT): Model real-world scenarios using two linear equations (e.g., cost vs. revenue). Name: ___________________________ Date: ___________________________ Timing: 10 mins Directions: Solve for the match point X-value. Show all algebraic steps. Problem 1 \(y = 5x\) AND \(y = 2x + 15\) SHOW WORK ABOVE X = ________ Problem 2 \(y = 4x\) AND \(y = x + 12\) SHOW WORK ABOVE X = ________ Problem 3 \(y = 10x\) AND \(y = 2x + 80\) SHOW WORK ABOVE X = ________ Problem 4 \(y = 8x\) AND \(y = 4x + 16\) SHOW WORK ABOVE X = ________ Profit Pursuit Lesson Opener Warm-Up: Systems of Linear Equations
Profit Point Reference NotesConcept Guide & Lesson Notes The Profit & Equilibrium Point Bridging Business Systems (Algebra) and Market Balance (Economics) The Big Idea: Two Curves, One Solution Whether you are finding the break-even point for a business or the market equilibrium for an economy, the math is exactly the same! We are searching for the intersection point \((x, y)\) where two linear forces become perfectly equal. 1 Algebraic Walkthrough: The Match Method Solve the System: \(y = 5x\) and \(y = 2x + 15\) Step A: Set them equal to each other Since both equations equal \(y\), they must equal each other at the intersection point: 5x = 2x + 15 Step B: Solve for \(x\) (Isolate the variable) Subtract \(2x\) from both sides to gather variables on the left side: 5x − 2x = 15 ⇒ 3x = 15 Divide both sides by 3: x = 5 Step C: Find \(y\) by plugging \(x\) back in Plug \(x = 5\) back into either original equation to calculate \(y\): y = 5(5) = 25 Intersection Solution: (5, 25) Profit & Equilibrium Point Notes Page 1 of 2 Concept Guide Visual Concept Map 2 Visualizing Both Models 1. Business Cost vs Revenue Cost (y = mx+b) Revenue (y = mx) Break-Even Point At break-even, the business has covered all expenses. Above this point is Profit, below is Loss. 2. Economic Market Equilibrium Supply (S) Demand (D) Equilibrium Point At equilibrium, the quantity supplied matches the quantity demanded. No shortage or surplus exists. 3 Key Terminology Glossary Business Analytics Fixed Cost: The expenses you pay regardless of sales (e.g., machine rental, rent). Variable Cost: Costs that grow with production (e.g., raw materials per item). Revenue: The total amount of income generated (Price × Quantity sold). Market Dynamics Demand: How much of a product buyers are willing to purchase at different prices. Supply: How much of a product sellers are willing to produce at different prices. Shortage: When demand is higher than supply (causes prices to rise).
Distribution Detectives PracticeStatistical Analysis Distribution Detectives Independent Practice: Identifying Features of Data Distributions Name: ___________________________ Date: ___________________________ Guide: 4 Major Features of a Distribution 1. Shape Symmetric (balanced), Skewed Left (tail points left), or Skewed Right (tail points right). 2. Center The middle or typical value (estimated by Median or the peak/Mode). 3. Spread The range of values from minimum to maximum (Max − Min). 4. Anomalies Outliers (isolated points), gaps (empty intervals), or clusters (grouped data). 1 Part 1: Graphical Feature Profiles Analyze the visual characteristics of each distribution graph below. Distribution A 10 100 A. Overall Shape: B. Spread (Estimated Range): C. Identify the Outlier & Gap: Distribution B 10 90 A. Shape / Peaks (How many?): B. Center (Approx. Median): C. Identify any Clusters or Gaps: Distribution Detectives Guide Page 1 of 2 Part 2 & 3 Scenarios & Data Sets 2 Part 2: Scenario Diagnosis Select the correct distribution shape based on each real-world feature description. Question 1 Most students scored between 80–90 on a test. A few students scored very low, creating a tail on the left side. What is the shape of this distribution? A. Symmetric B. Skewed Left C. Skewed Right Question 2 Most employees at a retail company earn a starting wage of $15/hr. The store owner and two top executives earn over $500,000/year, creating a tail on the right side. What is the shape of this distribution? A. Symmetric B. Skewed Left C. Skewed Right 3 Part 3: Outliers and Center Behavior Case Study: The Custom Mansion Effect In a quiet subdivision, ten identical ranch houses sell for exactly $200,000 each. Suddenly, a massive luxury custom mansion sells for $4,500,000. 1. What specific data feature/anomaly is the $4,500,000 house sale? 2. How does this high sale impact the Mean vs. the Median? Which value is pulled higher? 4 Part 4: Mini Data Detective Analyze this dataset representing homework scores for 7 students: 3, 3, 4, 6, 5, 8, 15 1. Organize the numbers above from least to greatest:
Guided Shapes WorksheetGuided Practice (IEP) Guided Data Shapes Name: ___________________________ Date: ___________________________ Detective Tip: Find the Tail! To find the shape of data, look for the "Tail" (the side with the smallest, shortest bars). Whichever direction the tail points is the direction of the skew! Question 1 Analyze the Spelling Test Distribution Most students studied hard and scored between 80 and 90. A few students scored very low, creating a long tail on the left side. Visual Graph of Test Scores Tail is here! Most scores are here (80-90) Count Score Based on the graph, what is the shape of this distribution? Option A: Symmetric Both sides look exactly equal and balanced like a mirror. Option B: Skewed Left The tail (shortest bars) points to the left side. Option C: Skewed Right The tail (shortest bars) points to the right side. Guided Shapes Practice (Accommodated) Unit 5 Lesson 2
Graph Features Guided SheetGuided Practice Classwork Graph Features Explorer Essential Question: How can I describe the shape and important features of a graph? Name: ___________________________ Date: ___________________________ 1 Part 1: Key Terms & Definitions Cue / QuestionNotes (Fill in the missing key vocabulary words!)What is a distribution?A distribution shows how data is out on a graph.What is a tail?A tail is the part of the graph that away from the main group.Left tail = Skewed ______________ Right tail = Skewed ______________What is symmetric?The graph is on both sides. ⚖️What is a cluster?A of data values that are close together.What is an outlier?A value that is away from the other data.What is a gap?A place where there is or no data.What is a peak?A point where many data values are found. 2 Part 2: Visual Anatomy of a Graph Label the five features on the graph below using the words: Peak, Gap, Outlier, Cluster, Tail. 1. _______ 2. _______ 3. _______ 4. _______ 5. _______ Graph Features Explorer Guided Sheet Page 1 of 2 Part 3 & 4 Shape & Feature Practice 3 Part 3: Match the Clues Draw a line to match each vocabulary word on the left with its definition clue on the right. A. Symmetric (•) B. Peak (•) C. Multiple Peaks (•) D. Outlier (•) (•) A high point on a graph where many values are found. (•) The graph is perfectly balanced on both sides. (•) A value that is far away from all the other data. (•) A graph that contains two or more high points. 4 Part 4: Identify the Shape Read the clue, look at the visual graph, and identify the final distribution shape. 1. The Two-Hill Peak Identify all features seen: Example: Multiple Peaks, Symmetric, etc. 2. The Left-Side Slide Identify all features seen: Example: Single Peak, Skewed Left, etc.
Graph Features KeyTeacher Answer Key Graph Features Key Facilitation Guide & Solutions for the Graph Features Sheet Part 1: Cornell Notes Solutions What is a distribution? ...how data is spread out. What is a tail? ...stretches away from the main group. Left vs. Right Tail: Left tail = Skewed Left | Right tail = Skewed Right What is symmetric? ...is balanced on both sides. What is a cluster? A group of values close together. What is an outlier? A value that is far away from other data. What is a gap? A place with little or no data. What is a peak? A high point on the graph. Part 2: Visual Anatomy Solutions PEAK CLUSTER TAIL GAP OUTLIER Teacher Instructional Script & Verbal Prompts Introduction: "Look at this graph. It represents scores on a spelling test. Let's solve its features together!" 1. Identifying Peak: "Put your finger on the tallest stack of dots on the graph. See how it stands above everything else? That high point is our PEAK." 2. Identifying Cluster: "Now look around that peak. See how those dots are huddled closely together like a group of friends? That is a CLUSTER." 3. Identifying Tail: "Look at the dots to the right. They get shorter and stretch away from our group, like a slide. We call that the TAIL." 4. Identifying Gap & Outlier: "Is there an empty space where no students scored? Yes! That's a GAP. And look at that lonely dot sitting all by itself at the far edge. That's our OUTLIER." Graph Features Key & Guide Page 1 of 2 Answer Solutions Matching & Case Studies Part 3: Match the Clues Answers • A. Symmetric → The graph is perfectly balanced on both sides. • B. Peak → A high point on a graph where many values are found. • C. Multiple Peaks → A graph that contains two or more high points. • D. Outlier → A value that is far away from all the other data. Part 4: Identify the Shape Answers Problem 1: The Two-Hill Peak Key Features Present: Symmetric shape (balanced) Multiple Peaks (Bimodal)
