What is a Solution? (Lesson 6.1)

Unit 6 - Day 1

​Learning Objectives​
  • Define a solution as an ordered pair that satisfies an equation and is thus on the graph of that equation

  • Use algebraic and graphical methods to find solutions to systems of equations

  • Determine when a system of equations will be inconsistent

Quick Lesson Plan
Activity: Can You Solve the Puzzle?

     

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Lesson Handout

Answer Key

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Experience First

In this lesson we will solidify important ideas about what it means for an ordered pair to be a solution to an equation and to a system of equations. We conceptualize solutions as x and y values that solve the puzzle, i.e. satisfy the equation. We purposefully phrase the puzzle in a way that encourages students to pick an x-value and then find the corresponding output, noticing that there are many solutions to the puzzle. Students use dot stickers to plot their puzzle solutions and discuss that the graph represents all possible solutions to the puzzle.

 

This process is repeated twice with two different puzzles. We use red and blue dot stickers so students can easily see the solution points, as well as which points are solutions to both puzzles.

 

To prep for this activity, sketch a coordinate grid on chart paper using the scale shown in the activity. This class graph can then be hung up and referred back to in future lessons.

Formalize Later

Although students are familiar with the idea of a solution and can easily identify solutions to a system of equations as the intersections of their graphs, push students to articulate this reasoning out loud. We want students to use the language of points that make both equations true or that satisfy both equations. We love asking multiple students: “What does this dot represent?” when pointing to various dots on the class graph.

 

Students should understand that a solution to a system occurs at the x-value where two functions have the same output. This has many important interpretations in contextual problems like question 3 in the Check Your Understanding. Additionally, we often ask students to write their own system that has a given solution.

 

In question 5 of the activity, many students used the graph to find the intersection instead of solving algebraically. This works well for this problem, but take time in the debrief to show how to solve algebraically. Much more practice with this will take place tomorrow.

 

The idea of an inconsistent system is also an important learning goal of today’s lesson. Students should be able to reason through this graphically and analytically. In question 4, students must choose the value of a parameter in an equation so that two graphs do not intersect. Many students confused this with finding the solution of the equation. Questions like these make great assessment questions because students need a deep conceptual understanding of what an inconsistent system is!