Using Equations for Lines
In the previous two lessons, students saw that all slope triangles for a line give the same slope value, and this value is called the slope of the line. They also began writing relationships satisfied by all points \((x,y)\) on a line. In this lesson, they continue to write equations but with less scaffolding, that is no similar triangles are selected so students need to figure out what to do given a line and a few points on the line.
The properties of slope triangles that make the slope of a line meaningful have to do with dilations. In particular, dilations do not change the quotient of the vertical side length and horizontal side length of a slope triangle. Students return to dilations in this lesson, applied to a single slope triangle with varying scale factor. This gives a different way of seeing how the coordinates of points on a line vary.
Both techniques, using equations and studying all of the dilations of a single slope triangle, give expressions representing points on a line.
- Create an equation of a line with positive slope on a coordinate grid using knowledge of similar triangles.
- Generalize (orally) a process for dilating a slope triangle $ABC$ on a coordinate plane with center of dilation $A$ and scale factor $s$.
- Justify (orally) that a point$ (x,y)$ is on a line by verifying that the values of $x$ and $y$ satisfy the equation of the line.
Let’s write equations for lines.
- I can find an equation for a line and use that to decide which points are on that line.
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