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.
Two figures are similar if one can fit exactly over the other after rigid transformations and dilations.
In this figure, triangle \(ABC\) is similar to triangle \(DEF\).
If \(ABC\) is rotated around point \(B\) and then dilated with center point \(O\), then it will fit exactly over \(DEF\). This means that they are similar.
The slope of a line is a number we can calculate using any two points on the line. To find the slope, divide the vertical distance between the points by the horizontal distance.
The slope of this line is 2 divided by 3 or \(\frac23\).
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