Slopes of Lines
Pacing:This lesson should take one to two class periods
Goal:Students should feel comfortable with slopes and lines. Use this lesson as a review of key concepts needed to
determine parallel and perpendicular lines in the coordinate plane.
Fun Fact!The word slope comes from the Middle English wordsloop, meaning at an angle.
Most students have experienced slope in Algebra. However, students rarely have seen the delta symbol when
determining slope. Use the following to stress the connection to pre-calculus:
Slope=rate of change
Rise
run=
4 y
4 x=
y 2 −y 1
x 2 −x 1Vocabulary Connection! Ask students to brainstorm the many different interpretations of the word slope. Apply
these to real world situations such as the slope of a mountain, or the part of a continent draining into a particular
ocean (Alaska’s North Slope), the slope of a wheelchair ramp, etc.
Include the synonym for slope – grade. Students should come up with more examples using this word.
Real Life Connection! Eldred Street in Los Angeles, California has a grade of 33%, Baldwin Street In Dunedin,
New Zealand boasts a 35% incline, and Banton Avenue in Pittsburg, Pennsylvania officially measurers 37%! Have
students reconstruct the incline of these streets using the rise over run notion of slope.
When discussing the rise over run triangles, begin making the right triangle connection to students, demonstrating
that every rise/run triangle will form a 90 degree angle. When students are asked to find the distance between two
oblique points, the distance formula is a derivation of Pythagorean’s Theorem.
Fun Tip!To illustrate why vertical lines have an undefined slope, ask for volunteers for the following demonstration.
To illustrate a horizontal line, run a length of masking tape on your floor. Ask a student to walk over the line.
Onlookers should see the student is walking at a zero incline (or slope).
To illustrate an oblique line, lay a 2” by 4” piece of wood on top of a chair, or something sturdy, creating a 2%−3%
incline. Ask a student to walk up the hill. Relate the percentage to a fraction, relating rise over run.
To illustrate a vertical, ask students to place their feet on a wall, lying parallel to the floor. Instruct the students
to walk up the wall in this position, similar to what Spiderman can do. Students will tell you this is impossible!
Dividing by zero is also impossible, thus illustrating why vertical lines have undefined (impossible) slopes.
To further demonstrate perpendicular slopes, use the formula to your advantage slope=((xy^22 −−yx^11 ))so the slope of the
line perpendicular must be−((yx^22 −−xy^11 ))
Students may find that making anxyT-chart is an easy way to construct a line. Whichever your preference, make
sure students can see a variety of ways to begin to solve a problem.
Equations of Lines
Pacing:This lesson should take one to two class periods
Goal:This lesson reinforces key concepts learned during Algebra to prepare students for geometric connections.
Students will review slope intercept form, standard form for a linear equation, and introduce equations for parallel
and perpendicular lines.
1.3. Parallel and Perpendicular Lines