Derivation of a Line MMAISE
- Author:
- mmaise
This interactive file can be used to see the derivation of the line y=mx+b.
1. Derive the Equation of a Line Through the Origin
a. Download the file Derive Equ of Line Origin.ggb and open it in GeoGebra.
b. Prior to projecting for students, find step 1 of the Construction Protocol located in the right pane (move the arrow keys up or down until step 1 of the Construction Protocol is highlighted or double click on step 1).
c. Arrow down through the Construction Protocol, piece by piece, using the following as a possible discussion guide:
d. Arrow down through Step 3: “Consider a line that passes through the origin and a point B. We will call this line n.”
e. “We will now create a slope triangle with a hypotenuse AB.” Arrow down through steps 4 and 5.
f. Prompt students to observe the coordinates of C (1, 0). From here, it should be easy for the students to see that the x-coordinate of B is also 1 but the y-coordinate may be a little trickier. You can show using different examples that when the x-coordinate is 1, the y¬-coordinate is equivalent to the slope of the line m, therefore leading to the coordinates of B (1, m). Scroll down through steps 6 and 7.
g. “Now we are going to dilate this slope triangle with our center of dilation at the origin and an unknown scale factor so that B slides to a point B’ on line n. Scroll down through step 11.
h. B’ represents any point on the line that satisfies the equation of the line (we can easily change the scale factor in order to move B to be anywhere on the line). Since B’ represents any point on the line we will represent this point with the coordinates (x, y). Scroll down through step 12.
i. Prompt students to think about the coordinates of C’. Scroll down through step 13 to show the coordinates of C’.
j. From our earlier work with dilations, we know that dilations create figures with sides that are proportional. Let’s use the slope triangles we have created to set up a proportion that shows that the slope is constant between any two points on a line. Prompt students to help you set up the proportion and then scroll down through step 14.
k. Prompt students to help you determine the lengths of the segments prior to scrolling down through step 15.
l. By solving for y we have derived the equation of a line passing through the origin. Scroll down to the last step.
2. Derive the equation of a line through a point (0, b).
a. Download the file Derive Equ of Line.ggb and open it in GeoGebra.
b. Prior to projecting for students, find step 1 of the Construction Protocol located in the right pane (move the arrow keys up or down until step 1 of the Construction Protocol is highlighted or double click on step 1).
c. Arrow down through the Construction Protocol, piece by piece, using the following as a possible discussion guide:
d. Arrow down through Step 4: “Consider a line that passes through the point (0, b) and a point F. We will call this line n.”
e. “We will now create a slope triangle with a hypotenuse AB.” Arrow down through step 7.
f. Prompt students to observe the coordinates of C (1, b). Scroll down through step 8.
g. Prompt students to think about the coordinates of B (1, m + b). It will probably help for them to first understand that the length of AB is m, similar to the previous problem. Scroll down through step 9.
h. “Now we are going to dilate this slope triangle with our center of dilation at the origin and an unknown scale factor so that B slides to a point B’ on line n. Scroll down through step 13.
i. B’ represents any point on the line that satisfies the equation of the line (we can easily change the scale factor in order to move B to be anywhere on the line). Since B’ represents any point on the line we will represent this point with the coordinates (x, y). Scroll down through step 14.
j. Prompt students to think about the coordinates of C’. Scroll down through step 15 to show the coordinates of C’.
k. From our earlier work with dilations, we know that dilations create figures with sides that are proportional. Let’s use the slope triangles we have created to set up a proportion that shows that the slope is constant between any two points on a line. Prompt students to help you set up the proportion and then scroll down through step 16.
l. Prompt students to help you determine the lengths of the segments prior to scrolling down through step 17.
m. By solving for y we have derived the equation of a line passing through a point (0, b). Scroll down to the last step.