Introduction
Subject: High School Algebra I (Grades 8-10)
Topics: Linear equations, graphing, systems of equations
Time: 4 sessions of 45-50 minutes
Grouping: Teams of 4
Would you waste ₱6,500 without noticing?
Your school’s Student Council is ordering class T-shirts for a fundraiser. Two printing shops sent quotes:
| Setup fee | Price per shirt | |
|---|---|---|
| Shop A | ₱1,500 | ₱120 |
| Shop B | ₱0 | ₱200 |
Shop B looks like the winner because it has no setup fee. For 10 shirts, it really is cheaper (₱2,000 vs. ₱2,700). But for 100 shirts, Shop B costs ₱6,500 more than Shop A. That’s the cost of a whole school event.
Somewhere between 10 and 100 shirts, the cheaper shop switches. Where? The Council doesn’t know, and it’s counting on someone to find out.
That someone is you.
Your role
You and your team of four are a consulting firm hired by the Student Council. You’ll use linear equations, graphs, and systems of equations to answer one question:
Which shop should the Council choose, and how many shirts must it sell to reach its ₱5,000 profit goal?
What you’ll learn
- How to turn a real price quote into a linear equation (y = mx + b)
- How to graph two equations and read their intersection
- How to solve a system of equations to find an exact break-even point
- How to back up a recommendation with math
Where you’re headed
Your team will present a Consultant Report to the “Student Council” (your classmates). Go to the Task page to see what your firm must deliver.
Task
Each Group will produce a Consultant Report and a 5-minute presentation. Each report must contain all six parts below.
| # | Part | What you must show |
|---|---|---|
| 1 | Cost equations | One equation for each shop, y = mx + b, where x is the number of shirts and y is the total cost. Explain what the slope and y-intercept mean in this situation. |
| 2 | Graph | Both equations on one coordinate plane with labeled axes, labeled lines, and the intersection point marked. |
| 3 | Break-even solution | The number of shirts where both shops cost the same, solved algebraically (substitution or elimination) with every step shown. Check it against your graph. |
| 4 | Profit analysis | The Council will sell shirts for ₱350 each. Write a revenue equation, then a profit equation for each shop (profit = revenue − cost). Find how many shirts the Council must sell to earn a ₱5,000 profit with each shop. |
| 5 | Recommendation | One paragraph telling the Council which shop to choose if it expects to sell 20, 50, or 100 shirts. Use calculated costs and profits as evidence. |
| 6 | Challenge (optional) | Shop C charges ₱800 setup and ₱150 per shirt. Add it to your graph and analysis. Is there any order size where Shop C is the best choice? |
Final product: a report and presentation that a Student Council member with no algebra background could follow and trust.
Process
Follow these steps in order. Complete each checkpoint before moving on, and show it to your teacher or a peer.
Session 1: Get organized and get ready
Step 1. Form your firm. Choose a role. Everyone must be able to explain all the math, whatever their role.
| Role | Job |
|---|---|
| Equation Analyst | Writes the equations and checks that they match the quotes |
| Graph Designer | Builds the graphs in Desmos or GeoGebra |
| Systems Solver | Leads the algebraic solution |
| Lead Presenter | Organizes the report and leads the presentation |
Step 2. Review the skills you need. Each member does one practice set from the Review and Practice resources (slope, y-intercept, or systems of equations). Share one thing you learned with your team.
Checkpoint: Each member explains slope and y-intercept in their own words.
Session 2: Model and graph
Step 3. Write the equations. For each shop, decide which number is the slope (the cost that changes with each shirt) and which is the y-intercept (the cost you pay before any shirt is made). Write one equation per shop. Swap with a teammate and check each other’s work.
Checkpoint: Two equations, each checked by a second person.
Step 4. Graph and predict. Enter both equations in Desmos. Adjust the window until you can see where the lines cross. Record your predicted break-even point and answer: Which shop is cheaper to the left of the crossing? To the right?
Checkpoint: A screenshot or sketch of the graph with the intersection labeled.
Session 3: Solve and analyze
Step 5. Solve algebraically. Set the two equations equal (or use substitution) and solve for x. Compare with your prediction. If they don’t match, find the mistake. Note that you can’t sell part of a shirt, so decide how to interpret a decimal answer.
Checkpoint: Complete step-by-step work, with the solution verified against the graph.
Step 6. Add revenue and profit. The Council sells each shirt for ₱350.
- Write the revenue equation: R = 350x.
- Write a profit equation for each shop: P = R − cost.
- Set each profit equal to 5,000 and solve for x. Round to a whole number of shirts that makes sense.
Checkpoint: Two profit equations and two solutions, each with a one-sentence interpretation.
Session 4: Recommend and present
Step 7. Test and recommend. Calculate the total cost and the profit at 20, 50, and 100 shirts for each shop. Organize results in a table. Write your recommendation paragraph. If you do the challenge, include Shop C.
Checkpoint: A results table and a recommendation that cites at least three numbers from it.
Step 8. Present. Present for 5 minutes. Every member speaks. Classmates act as the Student Council and ask each team one question.
Evaluation
Team rubric (24 points)
| Criteria | 4 Excellent | 3 Proficient | 2 Developing | 1 Beginning |
|---|---|---|---|---|
| Equations | Both equations correct; slope and y-intercept clearly explained in context | Both correct; explanation has minor gaps | One equation has an error | Missing or mostly incorrect |
| Graph | Accurate, with labeled axes and lines, intersection clearly marked | Mostly accurate; one small labeling issue | Several errors or missing labels | Missing or inaccurate |
| Break-even solution | Correct answer, all steps shown, verified against graph, decimal interpreted sensibly | Correct answer, most steps shown | Correct method with arithmetic errors | Incorrect method or no work |
| Profit analysis | Revenue and profit equations correct; ₱5,000 goal solved correctly for both shops | Correct for one shop; minor errors in the other | Partially complete | Missing or incorrect |
| Recommendation | Clear choice for 20, 50, and 100 shirts, supported by at least three calculated numbers | Clear choice; evidence incomplete | Vague or weakly supported | No clear recommendation |
| Presentation and teamwork | Organized and clear; every member speaks and can explain the math | Mostly organized; fairly even contribution | Disorganized or uneven participation | Unclear; one member did most of the work |
Individual score (6 points)
| Component | 3 | 2 | 1 |
|---|---|---|---|
| On-the-spot question (e.g., “What does the slope mean here?”) | Accurate and clear | Partly correct | Incorrect or unable to answer |
| Written reflection (see Conclusion) | Thoughtful and specific, answers all questions | Answers most questions | Brief or incomplete |
Optional bonus: +2 points to the team for a correct Shop C analysis.
Grading scale (team + individual = 30 points)
| Score | Rating |
|---|---|
| 27-30 | Outstanding |
| 23-26 | Very Satisfactory |
| 18-22 | Satisfactory |
| 13-17 | Fairly Satisfactory |
| Below 13 | Needs Improvement |
Conclusion
What you accomplished
You started with two price quotes and ended with a recommendation that could guide real spending. Along the way, you translated a situation into linear equations, graphed them, solved a system to find the break-even point, and used profit equations to set a target. The big lesson: the “cheapest” option depends on how much you plan to buy or make. A small up-front cost can be a bargain at large quantities, and no up-front cost can be expensive in the long run.
Reflect (answer individually, one paragraph each)
In this problem, what did the slope mean? What did the y-intercept mean?
Your graph and your algebra should agree. Why? What would it mean if they didn’t?
Where else could a break-even analysis help you (a phone plan, a small business, a gym membership, a canteen stall)?
What was the hardest part, and how did your team work through it?
Go further
Find a real pair of offers in your community, such as two data plans, two ride fares, or two printing quotes. Build your own break-even analysis and share what you find.
Credits
Review and practice
Khan Academy, Algebra 1 (slope-intercept form, systems of equations): https://www.khanacademy.org/math/algebra
Math Is Fun, Algebra: https://www.mathsisfun.com/algebra/
Purplemath, free algebra lessons: https://www.purplemath.com
OpenStax, Algebra and Trigonometry (free textbook; linear functions and systems of equations): https://openstax.org/details/books/algebra-and-trigonometry
Graphing tools
Desmos Graphing Calculator: https://www.desmos.com/calculator
GeoGebra: https://www.geogebra.org
PhET “Graphing Lines” simulation (to build intuition for slope and intercept): https://phet.colorado.edu/en/simulations/graphing-lines
Presentation tools
Teacher Page
Standards: CCSS HSA.CED.A.2, HSA.REI.C.6, HSF.LE.A.2, HSF.IF.C.7a. Align with your local DepEd competencies.
Answer key
- Shop A: y = 120x + 1500. Shop B: y = 200x.
- Break-even: 120x + 1500 = 200x → x = 18.75, so Shop A is cheaper from 19 shirts on.
- Revenue: R = 350x. Profit A: P = 230x − 1500. Profit B: P = 150x.
- ₱5,000 goal: Shop A needs x ≈ 28.26, so 29 shirts. Shop B needs x ≈ 33.33, so 34 shirts.
| Shirts | Cost A | Cost B | Profit A | Profit B | Better |
|---|---|---|---|---|---|
| 20 | ₱3,900 | ₱4,000 | ₱3,100 | ₱3,000 | A (slightly) |
| 50 | ₱7,500 | ₱10,000 | ₱10,000 | ₱7,500 | A |
| 100 | ₱13,500 | ₱20,000 | ₱21,500 | ₱15,000 | A |
- Challenge (Shop C: y = 800 + 150x): C ties B at 16 shirts and ties A at about 23.3 shirts. So C is the cheapest for 17-23 shirts (at 20 shirts: C = ₱3,800, beating A = ₱3,900), and A is cheapest from 24 shirts on. Note that this changes the 20-shirt recommendation if Shop C is included.
Note: Give struggling learners equation templates and a partially completed table. Advanced learners can do the Shop C challenge or explore what happens if the selling price changes.