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.
- Generalize (using words and other representations) that a line can be represented by an equation in point-slope form.
- Let’s investigate equations of lines.
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.
- I can use the definition of slope to write the equation for a line in 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 \(y = k + m(x-h)\).