Solving Real-World Problems with Systems of Equations
Created byLindsey Sanders
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Solving Real-World Problems with Systems of Equations

Grade 8Math5 days
★★★★★
4.0 (1 rating)
This project invites 8th-grade students to explore real-world problems using systems of equations, guided by the central question of how these systems can effectively model and solve complex situations. Students engage in diverse activities, like the Eco-Engineer Challenge, to design eco-friendly park layouts using mathematical models, and participate in the Equation System Scavenger Hunt to identify scenarios that can be addressed with systems of equations. They learn methods for solving these systems, including graphing, substitution, and elimination, and analyze how changing variables can impact solutions. Through engaging in these activities, students develop skills in applying, interpreting, and communicating mathematical models in practical contexts.
Systems of EquationsReal-World ProblemsMathematical ModelingGraphing MethodsEco-Engineer ChallengeDecision-MakingVariable Analysis
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we use systems of equations to model and solve real-world problems effectively, and what can the solutions tell us about constraints and decision-making in complex situations?

Essential Questions

Supporting questions that break down major concepts.
  • What real-world problems can be modeled using systems of equations?
  • How can multiple equations work together to describe a situation?
  • What methods can we use to solve systems of equations and when is each method most effective?
  • How can variables and constraints affect the solutions to a system of equations in a real-world context?
  • In what ways can the solution to a system of equations inform decision-making in real-life scenarios?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Students will be able to define and identify systems of equations and how they model real-world situations.
  • Students will be able to solve systems of linear equations using graphing, substitution, and elimination methods.
  • Students will be able to interpret the solutions of systems of equations in the context of the original problem.
  • Students will be able to analyze how changes in variables and constraints can affect the outcome of a systems of equations model.
  • Students will be able to apply systems of equations to inform decision-making processes in realistic scenarios.

Common Core Standards

8.EE.C.8
Primary
Analyze and solve pairs of simultaneous linear equations.Reason: This standard directly involves solving systems of equations, which is the core focus of the project.
8.EE.C.8a
Primary
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.Reason: Understanding the intersection of graphs is crucial for visualizing solutions, a key component of solving systems of equations.
8.EE.C.8b
Primary
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.Reason: Students will need to solve systems of equations by various methods including graphing, which aligns directly with this standard.
8.EE.C.8c
Primary
Solve real-world and mathematical problems leading to two linear equations in two variables.Reason: The project centers on using systems of equations to solve real-world problems, which is exactly what this standard addresses.

Entry Events

Events that will be used to introduce the project to students

Eco-Engineer Challenge

Students are tasked with designing an environmentally-friendly park for their community. They will use systems of equations to plan the layout, allocate resources, and ensure the park meets community goals for sustainability and recreation. This project encourages students to incorporate personal insights into eco-friendly solutions and explore innovative uses of math in community planning.
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Portfolio Activities

Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.
Activity 1

Equation System Scavenger Hunt

Students embark on a scavenger hunt to identify real-world scenarios that can be modeled using systems of equations. This fun activity introduces students to the types of problems that can be solved with systems of equations, setting the stage for deeper exploration.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Organize students into small groups and assign each group a different area of the community or school.
2. Task each group with identifying three different situations that could be modeled using systems of equations, such as resource allocation or time management problems.
3. Groups will present their findings to the class, explaining why each situation can be modeled with systems of equations.

Final Product

What students will submit as the final product of the activityA presentation detailing three real-life situations where systems of equations can be applied.

Alignment

How this activity aligns with the learning objectives & standardsAligns with 8.EE.C.8c by emphasizing real-world problems involving systems of equations.
Activity 2

Graph It Out

In this activity, students will explore the graphical interpretation of systems of equations. They will understand how the intersection of graphs represents solutions to the equations.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Provide students with pairs of linear equations and graph paper or graphing software.
2. Have students graph each pair of equations on a single coordinate plane to find the point of intersection.
3. Students label the point of intersection and relate it back to the original equations to show they satisfy both equations.

Final Product

What students will submit as the final product of the activityGraphs with highlighted intersections that demonstrate solutions to systems of equations.

Alignment

How this activity aligns with the learning objectives & standardsSupports 8.EE.C.8a by focusing on identifying solutions as intersections of graphs.
Activity 3

Method Masters

Students will learn and practice three different methods (graphing, substitution, and elimination) to solve systems of equations, understanding the strengths and applications of each method.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Introduce and explain the graphing method, followed by practice problems.
2. Discuss the substitution method and solve corresponding example problems as a class.
3. Present the elimination method and allow students to practice with partners, comparing it against other methods.
4. Wrap up with a discussion about when each method is most effective.

Final Product

What students will submit as the final product of the activityA comparative matrix that outlines the pros and cons of each method for solving systems of equations.

Alignment

How this activity aligns with the learning objectives & standardsAligns with 8.EE.C.8b by teaching multiple methods to solve systems of equations, including estimation by graphing.
Activity 4

Variable Variables

In this activity, students explore how changing variables and constraints impact the solutions of systems of equations, deepening their understanding of the relationships between equations.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Provide students with a base system of equations and ask them to solve it.
2. Have students modify one of the equations (such as changing coefficients or constants) and predict how the solution will change.
3. Students verify predictions by solving the modified systems and discuss findings with peers.

Final Product

What students will submit as the final product of the activityA report that shows original and modified systems of equations, along with their solutions and analysis of changes.

Alignment

How this activity aligns with the learning objectives & standardsAddresses 8.EE.C.8 by emphasizing the impact of variable changes on systems.
Activity 5

Eco-Engineer Presentation

Students apply all they've learned to design environmentally-friendly park layouts using systems of equations, culminating in a presentation to "community leaders."

Steps

Here is some basic scaffolding to help students complete the activity.
1. Revisit the Eco-Engineer Challenge guidelines and tasks within groups.
2. Using knowledge of systems of equations, students design a park layout, considering constraints like area, budget, and resources using mathematical models.
3. Prepare a presentation that includes the park design, the systems of equations used, and justification for their choices.
4. Present designs to classmates acting as community leaders who will provide feedback.

Final Product

What students will submit as the final product of the activityA detailed presentation of an eco-friendly park layout supported by systems of equations modeling.

Alignment

How this activity aligns with the learning objectives & standardsDirectly supports 8.EE.C.8c by applying systems to solve and justify real-world problems.
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Systems of Equations Portfolio Rubric

Category 1

Understanding and Identification

Assessment of the student's ability to identify and define systems of equations and their applications in real-world scenarios.
Criterion 1

Identifying Real-World Applications

Evaluating the student's ability to recognize situations that can be modeled using systems of equations.

Exemplary
4 Points

Consistently identifies several complex real-world problems suitable for systems of equations, with insightful rationale.

Proficient
3 Points

Accurately identifies appropriate real-world problems for systems of equations with clear explanations.

Developing
2 Points

Identifies basic real-world problems but with partial understanding and explanation.

Beginning
1 Points

Struggles to identify suitable problems for systems of equations with limited reasoning.

Criterion 2

Defining Systems of Equations

Evaluating the student's comprehension of the components and structure of systems of equations.

Exemplary
4 Points

Demonstrates an in-depth understanding of systems of equations, accurately defining components and relationships.

Proficient
3 Points

Shows a clear understanding of systems of equations, defining components with minor inaccuracies.

Developing
2 Points

Shows partial understanding with frequent errors in defining systems and their components.

Beginning
1 Points

Demonstrates limited understanding with minimal ability to define components of systems.

Category 2

Method Application

Assessment of the student's ability to apply various methods to solve systems of equations and demonstrate the process.
Criterion 1

Graphing Solutions

Evaluating the student's ability to use graphing to solve and understand systems of equations.

Exemplary
4 Points

Consistently graphs with precision, including detailed explanations of intersections and solution accuracy.

Proficient
3 Points

Accurately graphs solutions with clear identification of intersections and solutions.

Developing
2 Points

Graphs with some accuracy but lacks clear identification and explanation of solutions.

Beginning
1 Points

Struggles with graphing and detailing intersections, showing basic misunderstandings.

Criterion 2

Using Substitution and Elimination Methods

Evaluating the student's ability to apply substitution and elimination methods correctly and efficiently.

Exemplary
4 Points

Applies substitution and elimination perfectly with sophisticated understanding of circumstances for use.

Proficient
3 Points

Uses substitution and elimination correctly with minor misunderstandings.

Developing
2 Points

Shows partial skill in applying solving methods with occasional errors.

Beginning
1 Points

Struggles to apply substitution and elimination with significant errors.

Category 3

Interpretation and Analysis

Assessment of the student's ability to interpret solutions of systems of equations in contextual scenarios and analyze implications.
Criterion 1

Interpreting Solutions

Evaluating the student's ability to analyze and contextualize solutions derived from systems of equations.

Exemplary
4 Points

Provides comprehensive and insightful interpretation of solutions with deep contextual understanding.

Proficient
3 Points

Offers clear interpretation of solutions with appropriate contextual relevance.

Developing
2 Points

Provides basic interpretation with limited context application.

Beginning
1 Points

Struggles to interpret solutions contextually, offering minimal insights.

Criterion 2

Analysis of Variable Changes

Evaluating the student's capacity to analyze how changes in variables affect solutions and model behavior.

Exemplary
4 Points

Thoroughly analyzes variable changes, predicting effects with well-reasoned explanations and correct solutions.

Proficient
3 Points

Analyzes variable changes accurately with clear predictions and relates to solutions effectively.

Developing
2 Points

Shows basic understanding of variable effects with some predictive inaccuracies.

Beginning
1 Points

Struggles with predicting effects of variable changes with major inaccuracies.

Category 4

Communication and Justification

Assessment of the student's ability to communicate reasoning and justify decisions regarding systems of equations.
Criterion 1

Communicating Mathematical Reasoning

Evaluating how effectively the student communicates the mathematical processes and reasoning.

Exemplary
4 Points

Presents logical, coherent explanations of processes with persuasive justification.

Proficient
3 Points

Provides clear explanations with appropriate justification of methods.

Developing
2 Points

Offers explanations with some clarity but lacks depth or detail.

Beginning
1 Points

Struggles to explain reasoning and justify methods succinctly.

Criterion 2

Presentation and Collaboration

Evaluating the student's effectiveness in presenting work and collaborating during group activities.

Exemplary
4 Points

Delivers compelling presentations and exhibits leadership within collaborative settings.

Proficient
3 Points

Participates effectively and delivers clear presentations in group settings.

Developing
2 Points

Contributes to presentations but with limited engagement or clarity.

Beginning
1 Points

Shows minimal involvement in presentations and group work.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning
Question 1

Reflect on how your understanding of using systems of equations to model and solve real-world problems has evolved throughout this project. What are some key insights you gained?

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Question 2

How confident are you in applying different methods (graphing, substitution, elimination) to solve systems of equations?

Scale
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Question 3

Which method of solving systems of equations do you find most effective in real-world situations, and why?

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Question 4

In what ways did the Eco-Engineer Challenge help you understand the application of systems of equations in community planning and decision-making?

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Question 5

Thinking about the Variable Variables activity, how do changes in variables and constraints impact the solutions to a system of equations? Provide an example.

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Question 6

How useful did you find collaboratively working with your peers during the project?

Scale
Optional