Algebra 2

Writing Linear Equations & Solving Systems of Linear Equations

Course: Algebra 2

Lesson for Unit #1: Writing & Solving Systems of Linear Equations

Overarching Question:
What mathematical methods can be used to find solutions to systems of linear equations?
Previous Unit:
Linear Functions / This Unit:
Solving Systems of Linear Equations / Next Unit:
Quadratic Functions
Questions to Focus Assessment and Instruction:
  1. What are the possible types of solutions for a system of two linear equations? Give sample equations that demonstrate these types.
  2. Explore rewriting linear expressions in equivalent forms for solving linear equations.
/ Intellectual Processes(Standards for Mathematical Practice)
Model with mathematics: solve systems of equations problems that arise in mathematics and make connections to real-world situations
Construct viable arguments andcritique the argumentsof others: organize and consolidate mathematical thinking about systems of equations through communication
Attend to precision: use clear definitions in discussion with others and in their own reasoning; specify units of measure and label axes to clarify the correspondence with quantities in a problem.
Key Concepts:
equivalent linear expressions
connect solutions of equations to real-world situations / Write linear equations in slope-intercept form
Write linear equations in point-slope form
Solve systems of linear equations

Lesson Abstract

This investigation uses real-world examples to write equations and systems of equations using the ideas of supply and demand.

Common Core State Standards

HSA-CED.A.2 Create equations that describe numbers or relationships. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HSA-REI.C.5 Solve systems of equations. Solve systems of linear equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables.

HSA-REI.D.11 Represent and solve equations and inequalities graphically. Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions appropriately.

Lesson Title: Supply and Demand: Applications (Part 1 only)

Key:

Selecting and Setting up a Mathematical Task:
  • By the end of this lesson what do you want your students to understand, know, and be able to do?
  • In what ways does the task build on student’s previous knowledge?
  • What questions will you ask to help students access their prior knowledge?
  • What misconceptions might students have? What errors might students make?
/
  • Students will make appropriate mathematical choices in writing and solving for a system of linear equations.
  • Students will be able to explain what the solution to a system of linear equations means, in particular in the case where the linear equation is related to a real-world context.
  • Students will check the solution to a system of equations and verify that it makes the equations true.
  • Students will use a CAS system, if available, to solve a system of linear equations, and explain when this might be a more efficient method for solving an equation.
  • Students will be able to evaluate the reasoning of others in solving a system of linear equations, decide if it makes sense and ask useful questions to clarify the arguments used.
  • Students have had experience in identifying and combining like terms.
  • Students have studied and used the distributive property.
  • Students have studied the algebraic properties of equality and now use those properties to algebraically find the solutions to equations.
  • What are the 3 forms of linear equations?
  • How can you write an equation from a table of values?
  • How can you write an equation from a graph?
  • Which form is easiest to use given a graph? Given a table?
  • Students may not pay attention to the sign of a term in distributing.
  • Students may not distribute to all terms in parentheses.

Launch:
  • How will you introduce students to the activity so as to provide access to all students while maintaining the cognitive demands of the task?
  • What will be seen or heard that indicates that the students understood what the task is asking them to do?
/
  • Provide students with a linear graph. Ask them to write an equation for the line and compare their equation with a partner.
  • Provide students with a data table for points on the line given above. Ask them to write an equation for the data in the table and compare it with a partner. (Think, pair, share)\
  • Watch a short video clip explaining supply and demand (stop at 2:21)
  • Students will develop equivalent equations for the given line and table.
  • Students will be able to observe the solutions of other students and ask the other students for clarifications or confirm that algebraic properties were correctly used in writing the equation.

Supporting Student’s Exploration of the Task:
  • What questions will be asked to focus students’ thinking on the key mathematics ideas?
  • What questions will be asked to assess student’s understanding of key mathematics ideas?
  • What questions will be asked to encourage all students to share their thinking with others or to assess their understanding of their peer’s ideas?
  • How will you extend the task to provide additional challenge?
/
  • Why is the intersection of the demand line and supply line of interest?
  • What is the goal in solving a linearsystem of equations algebraically?
  • Does it make a difference which operation is done first in finding the solution to an equation? Give an example to support your response.
  • When you have the solution to an equation, what do you know?
  • How can you confirm that you have the correct solution to the equation?
  • What does the solution to Part 1,#4c mean in this real-world situation?
  • How will this information help Ms. King make a decision about the dolls?
  • How could you describe your method of solving a linear system of equations to someone who does not know how to correctly solve one?
  • If you and your classmate get different answers for the same system, how could you reconcile the difference?
  • Could you solve this problem without a graph? Why or why not?
  • Can you describe your solution as an ordered pair? Explain.
  • What might cause the equilibrium to change?
  • Part 3 – non-linear system

Sharing and Discussing the Task:
  • What specific questions will be asked so that all students will:
  • Make sense of the mathematical ideas that you wanted them to learn?
  • Expand on, debate, and question the solutions being shared?
  • Make connections between the different strategies that are presented?
  • Look for patterns?
  • Begin to form generalizations?
  • What will be seen or heard that indicates all students understand the mathematical ideas you intended them to learn?
/
  • What types and numbers of solutions are possible when solving a system of linear equations? Give examples to support your answer.
  • When you find a value for the variable in a system of linear equations, how can you ensure your solution is accurate?
  • Under what circumstances do you think solving a system of equations algebraically is more efficient or accurate than using a table or graph?
  • How is it possible for two people to solve a system of equations in a different sequence and still get the same solution?
  • Ask a group to share their supply equation. Ask other groups if they agree with the first group. If there are differences (which are expected) ask the group to explain the difference(s).
  • When solving a system of equations that was written to represent a real-world situation, why should you check both whether the solution is correct and whether it makes sense in the real-world context?
  • Students will solve a system of equations using rules of algebra and verify their solution is correct.
  • Students will compare their algebraic solution to the solution found on a graph or in a table and correctly show the connection between these other forms.

Assessment
Possible assessment question with solution:

Page 1of 4

10/4/18