5.1: Points & Lines & Slopes

Difficulty Level: At Grade Created by: CK-12

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*}intercepts, and lastly, the slopes of parallel and perpendicular lines.

Topic: Linear Functions

• Graph an equation of the form \begin{align*}y = mx\end{align*} and describe how the value of \begin{align*}m\end{align*} changes as the graph is rotated.
• Graph an equation of the form \begin{align*}y = mx + b\end{align*} and describe how \begin{align*}b\end{align*} 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*} and \begin{align*}f(x) = -x\end{align*} 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.

Associated Materials

Three focus questions define this activity (Some discussion should follow the posing of these questions):

1. What are the relationships of the coordinates of points with various locations in the Cartesian plane?
2. What is the relationship of a line with its slope, equation, and \begin{align*}y-\end{align*}intercept?
3. 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*} on the \begin{align*}x-\end{align*}axis and a point \begin{align*}S\end{align*} on the \begin{align*}y-\end{align*}axis. (The labels \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} may be typed by choosing the Alpha-Num tool from the F5: Appearance menu directly after placing each point, pressing ENTER, and typing the letter.) Then have students label the coordinates of each point, by selecting F5: Appearance > Coord. & Eq.

Have students now drag point \begin{align*}R\end{align*} and answer questions from the worksheet about \begin{align*}x-\end{align*}intercepts. (This foreshadows two important concepts later: \begin{align*}x-\end{align*}intercepts of graphs and equations of vertical lines.) To drag a point, move the cursor over it and press ALPHA. Then use the arrow keys to move the point. Press ALPHA to let the point go.

Next, have them drag point \begin{align*}S\end{align*} and answer questions about \begin{align*}y-\end{align*}intercepts. (This foreshadows the concept of the \begin{align*}y-\end{align*}intercept of a graph.)

Step 2: Instruct students to delete points \begin{align*}R\end{align*} and \begin{align*}S\end{align*} (move the cursor over each point and press DEL) and place and label points \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} anywhere in the coordinate system using the Point tool from the F2: Creation menu. Then have students label the coordinates of each point, by selecting F5: Appearance > Coord. & Eq.

Step 3: Have students once again drag points \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} around the plane and answer the questions on their worksheet about the sign of the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}coordinates in each quadrant.

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*} to each axis. After the perpendicular lines are in place, direct students to construct segments from point \begin{align*}P\end{align*} to each axis (F2: Creation > Segment) and then hide the “excess” perpendicular lines using the Hide/Show tool, available from F5: Appearance > Hide/Show.

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*} and have students conjecture about the distances and the coordinates. (There is a one-to-one correspondence between the distances from \begin{align*}P\end{align*} to each axis and the absolute value of the coordinates of \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*} and \begin{align*}Q\end{align*} with their labels should appear.

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*} and \begin{align*}Q\end{align*} and conjecture as to whether the slope is positive or negative, small or large, etc. Then have them draw a line through points \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} using the Line tool from the F2: Creation menu.

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 Alph-Num 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*} or \begin{align*}Q\end{align*} and notice what changes and what does not. Next drag by the line itself and observe what changes. Students should also drag point \begin{align*}Q\end{align*} to the \begin{align*}y-\end{align*}axis to investigate the relationship between a line’s \begin{align*}y-\end{align*}intercept and its equation.

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*} or \begin{align*}Q\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*} to specify the operation and move the cursor to the second slope and press ENTER again. This will yield the product of the two slopes—have students drag it to a convenient location on the screen. Finally, have students drag the line through points \begin{align*}P\end{align*} or \begin{align*}Q\end{align*} by grabbing and dragging either point. Encourage them to notice what happens to the product as the lines change!

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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Feb 22, 2012
Oct 31, 2014
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TI.MAT.ENG.TE.1.Algebra-I.5.1