Student Page 2.4.5 Slopes and Tree Trunks
Near the Calder Plaza in downtown Grand Rapids, Michigan a stand of trees is planted in a 5 by 5 grid as shown in Figure 2.4.5.1. Each green dot represents the location of a tree. The trunks of the trees are uniform in size. All of the trunks are straight and perpendicular to the ground.
1.
The line of sight from the origin to the trees with coordinates (1, 1), (2, 2), (3, 3), (4, 4), and (5, 5) is shown. Which of these tree trunks can you see if you are standing at the origin, (0, 0)? Why do you think so?
2.
Other than the tree trunks in Row 1 and Column 1, which tree trunks can you see if you are standing at the origin? Why do you think these tree trunks are visible?
3.
What is the slope of the line of sight from the origin (0, 0) to the tree trunk at point (4, 2)? What other tree trunks are along this line of sight? How do you know?
4.
At what point on one of the axes would you have to be standing in order for the tree trunks at points (2, 5) and (1, 3) to be on the same line of sight? What is the slope of this line of sight? What do you think the equation of this line of site is? Why do you think so?
5.
Draw a line of sight through point (2, 0) parallel to the line you found in Student Page Exercise 2.4.5.4. What trees are along this line of sight?
6.
Find the slope of the equation of the line in Student Page Exercise 2.4.5.5. What do you think is the equation of this line of site? Why do you think so?
7.
How are the slopes of the lines in Student Page Exercise 2.4.5.4 and Student Page Exercise 2.4.5.6 related?