Project Guide
Real World Inequality Challenges Project Guide
Introduction
Welcome, problem solvers! In this project, you will apply your knowledge of systems of inequalities to solve real-world problems. For each scenario below, you will need to:
- Define your variables.
- Write a system of linear inequalities that represents the situation.
- Graph the system of inequalities.
- Identify and explain at least two possible solutions that satisfy all conditions.
- Reflect on the limitations or assumptions of your model.
Let's get started!
Scenario 1: The Ultimate Arcade Challenge
You and your friend are at an arcade. You have $20 to spend and 30 minutes of time. The basketball game costs $2 and takes 3 minutes to play. The racing game costs $3 and takes 5 minutes to play.
Your Task: How many of each game can you play while staying within your budget and time limits? Show your work by defining variables, writing the system, graphing, and identifying possible solutions.
Scenario 2: Pet Shop Ponderings
A local pet shop is designing a new habitat. It can be a maximum of 30 square feet and hold a maximum of 50 pounds. Hamsters require 2 square feet and weigh 0.5 pounds each. Guinea pigs require 4 square feet and weigh 2 pounds each.
Your Task: How many of each animal (hamsters and guinea pigs) can they comfortably house in the new habitat? Show your work by defining variables, writing the system, graphing, and identifying possible solutions.
Scenario 3: Baking Bonanza!
You're helping to bake cookies and brownies for a school bake sale. You have 10 cups of flour and 6 cups of sugar available. Each batch of cookies requires 1 cup of flour and 0.5 cups of sugar. Each batch of brownies requires 1.5 cups of flour and 1 cup of sugar. You want to make at least 15 total batches of baked goods.
Your Task: How many batches of cookies and brownies can you make to meet all the requirements? Show your work by defining variables, writing the system, graphing, and identifying possible solutions.
Scenario 4: School Fundraiser Frenzy
A school club is selling t-shirts and hoodies to raise money. Each t-shirt sells for $15 and each hoodie sells for $25. They need to sell a total of at least 30 items and want to raise at least $500 in total. How many t-shirts and hoodies can they sell to meet these goals?
Scenario 5: Healthy Snack Stand
You are preparing fruit skewers and veggie sticks for a school event. Each fruit skewer requires 2 pieces of fruit and costs $0.75 to make. Each veggie stick portion requires 3 veggie sticks and costs $0.50 to make. You have a budget of $20 and need to prepare enough for at least 40 total pieces of fruit/veggies. How many of each snack can you prepare?
Scenario 6: Game Night Galore!
For a family game night, you want to play board games and card games. Each board game takes 45 minutes to play and requires 3 players. Each card game takes 20 minutes to play and requires 2 players. You have a total of 3 hours (180 minutes) available and a maximum of 5 people playing. You want to play at least 4 games in total. How many of each type of game can you play?
Reflection
After completing all scenarios, consider the following:
- What challenges did you face when creating your inequalities or graphing them?
- How do these real-world examples help you understand the importance of systems of inequalities?
use Lenny to create lessons.
No credit card needed
Rubric
Inequality Challenges Rubric
This rubric will be used to assess your understanding and application of systems of linear inequalities in real-world scenarios.
| Criteria | 4 - Exceeds Expectations | 3 - Meets Expectations | 2 - Partially Meets Expectations | 1 - Does Not Meet Expectations |
|---|---|---|---|---|
| Defining Variables | Clearly and accurately defines all variables for each scenario. | Accurately defines most variables for each scenario. | Defines some variables, but with minor inaccuracies or omissions. | Fails to define variables or definitions are largely incorrect. |
| Writing System of Inequalities | Writes correct and complete systems of linear inequalities for all scenarios, demonstrating a deep understanding. | Writes correct and complete systems for most scenarios with minor errors in one or two. | Writes incomplete or partially correct systems for some scenarios, with significant errors. | Fails to write systems of inequalities or they are largely incorrect and do not represent the scenario. |
| Graphing Systems | Creates accurate and clearly labeled graphs for all systems of inequalities, including shaded regions and boundaries. | Creates mostly accurate and labeled graphs for most systems, with minor graphing errors. | Graphs are present but contain several inaccuracies, poor labeling, or unclear shaded regions. | Fails to graph systems or graphs are entirely inaccurate and uninterpretable. |
| Identifying Possible Solutions | Identifies at least two distinct and valid solutions for each scenario, explaining their real-world meaning thoroughly. | Identifies at least two valid solutions for most scenarios, with reasonable explanations. | Identifies one valid solution for some scenarios, or explanations are unclear or incorrect. | Fails to identify any valid solutions or explanations are completely absent or incorrect. |
| Reflection | Provides insightful and detailed reflections on challenges and the importance of inequalities. | Provides clear and thoughtful reflections on challenges and the importance of inequalities. | Provides basic reflections, but lacks depth or clarity on challenges and importance. | Fails to provide reflections, or reflections are minimal and lack understanding. |
Scoring:
- Total Points Possible: 20 points
- Grade Conversion:
- 18-20 points: A
- 15-17 points: B
- 12-14 points: C
- 9-11 points: D
- 0-8 points: F
Answer Key
Real World Inequality Challenges Answer Key
This answer key provides detailed solutions and explanations for each scenario in the Real World Inequality Challenges Project Guide.
Scenario 1: The Ultimate Arcade Challenge
- Variables:
x= basketball games,y= racing games - System:
2x + 3y <= 20(Cost),3x + 5y <= 30(Time),x >= 0,y >= 0 - Graphing: Graph boundary lines
2x + 3y = 20and3x + 5y = 30. Shade below both in Quadrant I. The feasible region is the overlap. - Possible Solutions (examples): (5, 3) - Cost $19, Time 30 min. (2, 4) - Cost $16, Time 26 min.
Scenario 2: Pet Shop Ponderings
- Variables:
h= hamsters,g= guinea pigs - System:
2h + 4g <= 30(Space),0.5h + 2g <= 50(Weight),h >= 0,g >= 0 - Graphing: Graph boundary lines
2h + 4g = 30and0.5h + 2g = 50. Shade below both in Quadrant I. The feasible region is the overlap. - Possible Solutions (examples): (10, 2) - Space 28 sq ft, Weight 9 lbs. (5, 5) - Space 30 sq ft, Weight 12.5 lbs.
Scenario 3: Baking Bonanza!
- Variables:
c= cookies,b= brownies - System:
1c + 1.5b <= 10(Flour),0.5c + 1b <= 6(Sugar),c + b >= 15(Total Batches),c >= 0,b >= 0 - Graphing: Graph boundary lines for flour, sugar, and total batches. Shade below flour/sugar lines and above total batches line.
- Possible Solutions (examples): For these numbers, the feasible region is non-existent. The constraints conflict, meaning it's impossible to make at least 15 batches with the given flour and sugar limits.
Scenario 4: School Fundraiser Frenzy
- Variables:
t= t-shirts,h= hoodies - System:
t + h >= 30(Total Items),15t + 25h >= 500(Total Revenue),t >= 0,h >= 0 - Graphing: Graph boundary lines
t + h = 30and15t + 25h = 500. Shade above both lines in Quadrant I. The feasible region is the overlap. - Possible Solutions (examples): (20, 10) - 30 items, $550 revenue. (10, 20) - 30 items, $650 revenue.
Scenario 5: Healthy Snack Stand
- Variables:
f= fruit skewers,v= veggie sticks - System:
0.75f + 0.50v <= 20(Budget),2f + 3v >= 40(Total Pieces),f >= 0,v >= 0 - Graphing: Graph boundary lines
0.75f + 0.50v = 20and2f + 3v = 40. Shade below budget line and above total pieces line in Quadrant I. The feasible region is the overlap. - Possible Solutions (examples): (10, 10) - Cost $12.50, 50 pieces. (5, 15) - Cost $11.25, 55 pieces.
Scenario 6: Game Night Galore!
- Variables:
b= board games,c= card games - System:
45b + 20c <= 180(Time),3b + 2c <= 5(Players),b + c >= 4(Total Games),b >= 0,c >= 0 - Graphing: Graph boundary lines for time, players, and total games. Shade below time/player lines and above total games line.
- Possible Solutions (examples): The player constraint (
3b + 2c <= 5) makes it highly unlikely to play at least 4 games. The feasible region is likely non-existent, illustrating that some goals cannot be met simultaneously given tight restrictions. Students should show how the regions do not overlap.
Reflection
- Challenges in creating inequalities or graphing them: Students may struggle with accurately translating word problems into mathematical inequalities, especially determining the correct inequality symbol (
<,>,<=,>=). Graphing challenges can include correctly identifying intercepts, drawing lines, and shading the appropriate region, particularly when dealing with multiple inequalities and determining the overlapping feasible region. Identifying points in the feasible region, especially non-integer points, can also be challenging. - How these real-world examples help understand the importance of systems of inequalities: These examples demonstrate that systems of inequalities are powerful tools for modeling situations with multiple constraints and making decisions. They show how to find all possible outcomes (feasible solutions) that satisfy all conditions, or to understand why certain goals may not be achievable. This helps in resource allocation, planning, and problem-solving in practical situations where perfect solutions are rare.