Gesturing is a powerful way for students to learn. I have students gesture with their hands often, particularly when we examine behavior of graphs. One of our teachers was working to help students remember the slope-intercept form of a line. She found this video on Youtube:
The teacher had the students do the gestures to help remember the parts of the slope-intercept form. I wanted to help the students understand the graphical effects of changing the values of m and b on the line y = mx + b. I started with a blank Geogebra sheet and constructed the graph of y = mx + b actively in front of the students.
Students had no trouble with the effect altering the value of m had on the graph. However, something interesting happened when I asked the class, "What is your prediction for the effect changing b will have on the graph?" One of the students quickly replied, "The black point will move along the x-axis." I think my face betrayed my attempt to remain neutral. I said with my mouth, "You're right, the black point will move along the x-axis. What else will happen?" The students eventually pointed out the relationship between the b-value on the slider and the y-coordinate of the pink point.
But that student's comment has stuck with me since yesterday. I hadn't really considered before how we would go about defining the movement of the black point along the x-axis.
Let's take a second look at the graph.
We can define the x-coordinate at point A using x = (y/m) - (b/m). I defined the text box using the LaTEX editor in Geogebra.
where object i is the y-coordinate of point A. Since point A is the x-intercept, its y-coordinate is always zero. Then the root of y = mx+b is given by x = -1*(b/m).
This result also empowers my Math Bowl participants because they can now rattle off the x-intercept of a line written in slope-intercept form very quickly.
Here is a copy of the Geogebra sheet if you would like to play.
This setting can be used to generate many bowl-type questions where the student is asked to compute the area of the triangle bounded by the y-axis, the x-axis, and a line usually expressed in slope-intercept form or standard form. Below is a screenshot from my additional work on this problem.
Being able to compute the x-intercept immediately gives the student a good chance to solve the area problem very quickly. Since the x-axis and y-axis are perpendicular, the student can quickly find the area of the triangle by taking (1/2)*(b)*(b/m).
Here is a link to the Geogebra sheet displaying the area of the triangle bounded by the x-axis, the y-axis, and the line y = mx+b.