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Algebraic Solutions: Systems of Equations in Two Variables
Grade 8Math5 days
In this 8th-grade math project, students explore systems of linear equations through real-world applications and creative activities, such as creating superhero characters whose powers are modeled with linear equations. The project encourages students to use various methods—graphical, algebraic, and inspection—to solve these equations, while understanding their applicability in real-world scenarios. Through engaging entry events like a superhero showdown and an escape room challenge, students develop a deep understanding of how to choose the best strategy for solving equations, culminating in a real-world modeling exercise.Systems of EquationsGraphingAlgebraic MethodsReal-World ProblemsSuperhero CreationInspection
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Inquiry Framework
Question Framework
Driving Question
The overarching question that guides the entire project.How can you use systems of linear equations to solve real-world problems, and what are the best strategies to choose when graphing, using algebra, or solving by inspection?Essential Questions
Supporting questions that break down major concepts.- How can systems of linear equations be used to represent real-world situations?
- What are the different methods to solve a system of linear equations, and what are the advantages and disadvantages of each?
- How does graphing help in estimating the solutions of linear equations?
- In what situations would you choose to solve a system of equations by inspection rather than using algebraic methods?
Standards & Learning Goals
Learning Goals
By the end of this project, students will be able to:- Students will be able to solve systems of two linear equations in two variables algebraically.
- Students will be able to estimate solutions to systems of equations by graphing the equations.
- Students will understand and apply different methods of solving systems of equations, including algebraically and by inspection.
- Students will develop strategies to choose the best method for solving systems of equations based on the context of the problem.
- Students will be able to represent and solve real-world problems using systems of linear equations.
Common Core Standards
CCSS.MATH.CONTENT.8.EE.C.8
Primary
Analyze and solve pairs of simultaneous linear equations.Reason: The project focuses on solving systems of equations, which directly aligns with analyzing and solving pairs of simultaneous linear equations.
CCSS.MATH.CONTENT.8.EE.C.8.A
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: This standard is relevant as students will estimate solutions by graphing, understanding intersections as solutions.
CCSS.MATH.CONTENT.8.EE.C.8.B
Primary
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.Reason: The project is explicitly about solving systems of equations both algebraically and by inspection, as well as using graphing for estimating solutions.
CCSS.MATH.CONTENT.8.EE.C.8.C
Primary
Solve real-world and mathematical problems leading to two linear equations in two variables.Reason: The project aims to apply systems of equations to real-world problems, directly aligning with this standard.
Entry Events
Events that will be used to introduce the project to studentsSuperhero Showdown
Students are tasked with creating a superhero duo where each character's powers represent a set of linear equations. The event begins with a dramatic reveal of these superheroes, and students must determine how the heroes can collaborate to solve challenges through the intersection of their powers, represented by the system of equations.Escape the Equation
An escape room experience is set up in the classroom where each puzzle and step forward requires solving a system of equations. Students are 'locked' in and must use their algebraic skills to decode the solutions and 'escape' in time.📚
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 Superhero Creation
In this activity, students will design a superhero character whose powers represent a linear equation. This creative process will help them understand how equations can model situations.Steps
Here is some basic scaffolding to help students complete the activity.1. Choose two powers for your superhero related to real-world scenarios (e.g., speed and strength).
2. Write a linear equation that represents each power (e.g., "speed = 2x + 3").
3. Design a superhero character with traits and a story that reflect these equations.
Final Product
What students will submit as the final product of the activityA superhero character sheet including linear equations that represent their powers.Alignment
How this activity aligns with the learning objectives & standardsCCSS.MATH.CONTENT.8.EE.C.8.A, as it focuses on representing real-world systems through linear equations.Activity 2
Graphing Adventure Quest
Students will use graphing to find points of intersection for their superhero powers. This will aid them in solving a system of equations by graphing and understanding intersections.Steps
Here is some basic scaffolding to help students complete the activity.1. Plot the two equations representing your superhero's powers on a graph.
2. Find the point of intersection on the graph that shows how the superhero powers combine.
3. Discuss what the intersection point signifies in terms of power collaboration.
Final Product
What students will submit as the final product of the activityA detailed graph displaying the intersection of superhero powers, illustrating collaboration points.Alignment
How this activity aligns with the learning objectives & standardsCCSS.MATH.CONTENT.8.EE.C.8.A, by having students analyze intersection points on a graph to find solutions.Activity 3
Algebraic Strategy Showcase
Students will solve the system of equations representing their superhero powers algebraically. This step involves using substitution or elimination methods.Steps
Here is some basic scaffolding to help students complete the activity.1. Select one algebraic method (substitution or elimination) to solve your system of equations.
2. Perform the chosen algebraic method step by step to find the solution for the variable.
3. Verify your solution by substituting it back into the original equations.
Final Product
What students will submit as the final product of the activityAn algebraically solved system of equations with a clear explanation of steps taken.Alignment
How this activity aligns with the learning objectives & standardsCCSS.MATH.CONTENT.8.EE.C.8.B, focusing on algebraic solutions of linear equations.Activity 4
Escape with Inspection
Students will use inspection to solve simple systems of equations in an escape room-style challenge. They will learn to identify when inspection is the quickest method.Steps
Here is some basic scaffolding to help students complete the activity.1. Examine simple pairs of equations to identify if a solution can be found by inspection.
2. Use logical reasoning to derive a solution without formal algebraic steps.
3. Apply this quick-check method in a time-limited scenario to solve challenges.
Final Product
What students will submit as the final product of the activityA set of solutions derived by inspection in a series of escape room puzzles.Alignment
How this activity aligns with the learning objectives & standardsCCSS.MATH.CONTENT.8.EE.C.8.B, focusing on solving equations by inspection.Activity 5
Real-World Equation Solver
In this activity, students will apply their knowledge of systems of equations to solve a real-world problem by creating a relatable scenario.Steps
Here is some basic scaffolding to help students complete the activity.1. Identify a problem from real life that can be modeled using systems of linear equations (e.g., budgeting time or resources).
2. Formulate a system of equations to represent the problem scenario.
3. Solve the system of equations using an appropriate method chosen by the student.
Final Product
What students will submit as the final product of the activityA written report detailing a real-world problem, the system of equations used to model it, and a solution strategy with justification.Alignment
How this activity aligns with the learning objectives & standardsCCSS.MATH.CONTENT.8.EE.C.8.C, applying systems of equations to real-world problems.🏆
Rubric & Reflection
Portfolio Rubric
Grading criteria for assessing the overall project portfolioSystems of Equations Mastery Rubric
Category 1
Algebraic Solving Skills
Assesses the ability to solve systems of equations algebraically using substitution and elimination methods.Criterion 1
Method Selection and Justification
Evaluating the student's ability to choose and justify the appropriate algebraic method for solving a system of equations.
Exemplary
4 PointsSelects the most efficient algebraic method with insightful justification based on the problem context and demonstrates an innovative understanding of method application.
Proficient
3 PointsSelects an appropriate algebraic method with clear justification, demonstrating a thorough understanding of method use.
Developing
2 PointsSelects an algebraic method with partial justification, showing an emerging understanding of the method application.
Beginning
1 PointsStruggles to select an appropriate method or provide justification, demonstrating initial understanding of algebraic solutions.
Criterion 2
Solution Accuracy and Verification
Measures the accuracy of solving systems of equations and the capacity to verify solutions effectively.
Exemplary
4 PointsAccurately solves complex systems and thoroughly verifies solutions with comprehensive reasoning.
Proficient
3 PointsAccurately solves systems and verifies solutions with clear reasoning.
Developing
2 PointsSolves systems with some accuracy; verification is partial or requires guidance.
Beginning
1 PointsFrequently struggles to accurately solve or verify systems, needing substantial support.
Category 2
Graphical Solution Skills
Evaluates the aptitude in utilizing graphing methods to estimate and interpret solutions for systems of equations.Criterion 1
Graph Construction and Interpretation
Examines the ability to construct accurate graphs representing systems of equations and interpret points of intersection.
Exemplary
4 PointsConstructs precise graphs, accurately interprets intersection points, and demonstrates exceptional insight into graphical solution significance.
Proficient
3 PointsConstructs accurate graphs and interprets intersection points with clarity and understanding.
Developing
2 PointsConstructs graphs with basic accuracy and partial interpretation of intersection points.
Beginning
1 PointsProduces inaccurate graphs and struggles to interpret intersections meaningfully.
Category 3
Inspection Method Application
Focuses on the application of solving systems of equations by inspection and logical reasoning.Criterion 1
Inspection Skill and Speed
Assesses proficiency in identifying solutions by inspection and applying logical reasoning under time constraints.
Exemplary
4 PointsEfficiently and accurately identifies solutions by inspection, demonstrating exceptional speed and logical acumen.
Proficient
3 PointsAccurately identifies solutions by inspection and applies logical reasoning effectively.
Developing
2 PointsIdentifies solutions by inspection with partial accuracy or speed, showing developing reasoning skills.
Beginning
1 PointsStruggles to identify solutions by inspection without significant time or accuracy.
Category 4
Real-World Application
Assesses the ability to apply systems of equations to solve real-world problems and represent scenarios accurately.Criterion 1
Problem Representation and Solution Strategy
Evaluates how accurately students can represent real-world problems using systems of equations and devise effective solution strategies.
Exemplary
4 PointsArticulates complex real-world problems with precision, creating highly accurate systems and devising innovative, effective solutions.
Proficient
3 PointsEffectively represents real-world problems and develops accurate systems and viable solutions.
Developing
2 PointsRepresents problems with partial accuracy, creating systems but offering basic or undeveloped solution strategies.
Beginning
1 PointsStruggles to represent or solve real-world problems accurately, requiring extensive support.
Reflection Prompts
End-of-project reflection questions to get students to think about their learningQuestion 1
Reflect on how creating a superhero character helped you understand how linear equations can model real-world scenarios. What was the most challenging aspect of this process, and how did you overcome it?
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Question 2
On a scale from 1 to 5, how confident do you feel about using graphing to estimate solutions for systems of equations?
Scale
Required
Question 3
Which method (graphing, algebraic, or inspection) did you find most effective when solving the superhero linear equations, and why?
Multiple choice
Required
Options
Graphing
Algebraic (substitution or elimination)
Inspection
Question 4
Reflect on a real-world problem you modeled using systems of equations. How did solving it change your understanding or perspective about the usefulness of mathematics in everyday life?
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Question 5
What did you find most surprising during the escape room challenge involving solving equations by inspection?
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Optional