In this lesson, students combine concepts of geometry and algebra to find solutions to a system of equations consisting of a linear and a quadratic equation. Students begin by considering the number of ways in which a circle and a line can intersect. Then, they solve 2 simple systems graphically, using algebraic methods to verify their estimates of the solutions. Finally, they write their own equations that meet certain constraints.
Students have an opportunity to reason abstractly (MP2) when they decide how they can verify if a point truly represents the intersection of a circle and a line, and when they draw connections between geometric figures and algebra in the lesson synthesis.
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 13.1 Warmup: Which One Doesn’t Belong: Lines and Curves (5 minutes)
 13.2 Activity: Circles and Lines (15 minutes)

13.3 Activity: Creating Lines (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 13.4 Cooldown: Find and Verify (5 minutes)
Learning goals:
 Use graphical methods to solve a system of equations consisting of one linear and one quadratic equation.
Learning goals (student facing):
 Let’s look at how circles and parabolas interact with lines.
Learning targets (student facing):
 I can use a graph to find the intersection points of a line and a circle.
Standards:
 This lesson builds on the standards: CCSS.HSGGPE.A.1MS.GGPE.1MO.G.GPE.A.1
 This lesson builds towards the standards: CCSS.HSAREI.C.7MS.AREI.7MO.A1.REI.B.4
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.