# Lesson 13

Intersection Points

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

• Use graphical methods to solve a system of equations consisting of one linear and one quadratic equation.

### Student Facing

• Let’s look at how circles and parabolas interact with lines.

### Student Facing

• I can use a graph to find the intersection points of a line and a circle.

Building On

Building Towards

### Glossary Entries

• opposite

Two numbers are opposites of each other if they are the same distance from 0 on the number line, but on opposite sides.

The opposite of 3 is -3 and the opposite of -5 is 5.

• point-slope form

The form of an equation for a line with slope $$m$$ through the point $$(h,k)$$. Point-slope form is usually written as $$y-k = m(x-h)$$. It can also be written as $$y = k + m(x-h)$$.

• reciprocal

If $$p$$ is a rational number that is not zero, then the reciprocal of $$p$$ is the number $$\frac{1}{p}$$.

### Print Formatted Materials

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