# Lesson 9

Equations of Lines

### Lesson Narrative

In grade 8, students used similar triangles to explain why the slope $$m$$ is the same between any two distinct points on a non-vertical line in the coordinate plane, and they derived the equation $$y = mx + b$$ for a line intercepting the vertical axis at $$b$$. In this lesson, students develop the point-slope form of a linear equation: $$y-k=m(x-h)$$. Students will be writing equations of lines in the next several lessons, and intercepts will not always be readily available. Point-slope form will require the least algebraic manipulation and allow students to focus on geometric properties.

Slope calculations are an important part of this lesson, so students begin with a warm-up that helps them recall this concept. Then they use the definition of slope to build the point-slope equation, mirroring the way they developed the equation of a circle. Finally, they practice writing and interpreting equations of lines in point-slope form. Students have the opportunity to construct a viable argument (MP3) when they explain their methods of calculating slope.

### Learning Goals

Teacher Facing

• Generalize (using words and other representations) that a line can be represented by an equation in point-slope form.

### Student Facing

• Let’s investigate equations of lines.

### Required Preparation

Set up a display for graphing technology so you can dynamically graph several student-generated equations, both in one of the activities and in the lesson synthesis.

### Student Facing

• I can use the definition of slope to write the equation for a line in point-slope form.

Building On

Building Towards

### Glossary Entries

• 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 $$y = k + m(x-h)$$.