5.1: Points & Lines & Slopes
This activity is intended to supplement Algebra I, Chapter 4, Lesson 3.
ID: 8891
Time required: 45 minutes
Activity Overview
In this activity students will explore the relationship between coordinates of points and locations on the coordinate plane, the relationships of lines with their equations, slopes and \begin{align*}y\end{align*}
Topic: Linear Functions

Graph an equation of the form \begin{align*}y = mx\end{align*}
y=mx and describe how the value of \begin{align*}m\end{align*}m changes as the graph is rotated. 
Graph an equation of the form \begin{align*}y = mx + b\end{align*}
y=mx+b and describe how \begin{align*}b\end{align*}b changes under vertical translations.  Prove that lines with equal slope are parallel.
 Graph lines whose slopes are negative reciprocals.
 Deduce that the slopes of perpendicular lines are negative reciprocals.
Teacher Preparation and Notes
 This investigation offers many possible extensions depending on the level of the student. The activity can be used in the middle grades and even in later elementary grades.

This activity can review coordinates and then go on to introduce or review the relationships of points, lines, slopes, and equations. The graph could also be divided into different zones by graphing \begin{align*}f(x) = x\end{align*}
f(x)=x and \begin{align*}f(x) = x\end{align*}f(x)=−x and having students generalize the slope for each of these zones.  After graphing lines and looking at relationships, another point could be placed on the plane and students could look at the coordinates of the point with respect to the line. This could be used to introduce inequalities.
 To download Cabri Jr., PARALLEL.8xv, and PERPENDI.8xv, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=8891 and select each file. If your calculator already has the Cabri Jr. application, you may still need to download PARALLEL.8xv and PERPENDI.8xv.
Associated Materials
 Student Worksheet: Points & Lines & Slopes, http://www.ck12.org/flexr/chapter/9614
 PARALLEL and PERPENDI (Cabri Jr. files)
Three focus questions define this activity (Some discussion should follow the posing of these questions):
 What are the relationships of the coordinates of points with various locations in the Cartesian plane?

What is the relationship of a line with its slope, equation, and \begin{align*}y\end{align*}
y− intercept?  What is the relationship between the slopes of parallel or perpendicular lines?
Problem 1 – Coordinates of points
Students are to open a new Cabri Jr. file. If the axes are not displayed, they should select F5: Appearance > Hide/Show > Axes.
Step 1 : Students will use the Point on tool (F2: Creation > Point > Point on) to place and label a point \begin{align*}R\end{align*}
Have students now drag point \begin{align*}R\end{align*}
Next, have them drag point \begin{align*}S\end{align*}
Step 2: Instruct students to delete points \begin{align*}R\end{align*}
Step 3: Have students once again drag points \begin{align*}P\end{align*}
Step 4: Students should now be able to make sense of exactly what the coordinates should be in different quadrants as well as on each axis. Give students the coordinates of a point and ask which quadrant it is in and visa versa. This simple activity is good for students to develop their technology skills and establish sign patterns for the four quadrants.
Step 5: Have students use the Perpendicular tool (F3: Construction > Perp.) to construct perpendicular lines through point \begin{align*}P\end{align*}
Note: When using the Segment tool, to select the intersection point of the perpendicular line and the axis, both of these lines must flash and then students should press ENTER.
Step 6: Now have students measure the length of each segment using the D. & Length tool from the F5: Appearance > Measure menu. Drag point \begin{align*}P\end{align*}
Problem 2 – Lines, equations, and slopes
Step 1: Have students delete or hide the coordinates, segments, and measurements. Only points \begin{align*}P\end{align*}
If this is students’ first introduction to slope, provide a brief description of the concept, including “rise over run.” Ask students to visualize a line between points \begin{align*}P\end{align*}
Step 2: Next, have students find the slope (F5: Appearance > Measure > Slope) and equation (F5: Appearance > Coord. & Eq.) of the line. Have them label the slope measurement as shown at right by choosing AlphNum from the F5: Appearance menu and typing SLOPE: in the appropriate place.
Step 3: Now students can investigate the relationship of the line with its slope and equation. First have students drag point \begin{align*}P\end{align*}
Problem 3 – Slopes of parallel and perpendicular lines
Step 1: In the Cabri Jr. file PARALLEL, students are given two parallel lines, their equations, and their slopes. (This construction is done for them. An alternative would be for the students to construct the parallel lines, show the slopes, and explore.)
Students should explore the relationships of parallel lines and slopes while they drag the original line by one of its defining points, \begin{align*}P\end{align*}
Step 2: In the Cabri Jr. file PERPENDI, students are shown two perpendicular lines, their equations, and their slopes. Again, students could be led through this construction, if desired.
Encourage students to drag the lines around and try to identify the relationship between their slopes.
Step 3: Another way to look at the relationship of the slopes is to find the product of the two slopes. Have students select F5: Appearance > Calculate. Next, they should position the cursor over the first slope, and press ENTER. Students should then press \begin{align*}\times\end{align*}
Extensions
As stated in the teacher preparation, there are many extensions to this activity depending on the level of the student. This activity can easily be manipulated to lead students into a deeper study of slopes, parallel lines, and proofs.
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