Summary: Define slope as the ratio of the vertical change to the horizontal
change. Recognize slope from tables, ordered pairs, or graphs. Show that slope is constant
using similarity of right triangles
Main Curriculum Tie: Mathematics  PreAlgebra Standard 2 Objective 3 Recognize similar polygons and use properties of similar triangles to solve problems and define the slope of a line. Materials:
Attachments
Background For Teachers: Understanding (Big
Ideas):
slope of a line
Essential Questions:
 How can a ratio be used to describe the steepness of a line?
 Why does the slope of a line remain constant?
 How can right triangles be used to show that slope is constant?
Skill Focus:
Write a ratio for slope of a
line. Use similar triangles
to show slope as constant.
Vocabulary Focus:
slope, change in y, change in x, rise, run, similar triangles,
constant
Ways to Gain/Maintain Attention (Primacy):
treasure map, connection to skiing,
predicting, note taking, cooperative activity, music and movement, game
Instructional Procedures:
Starter:
Have your materials manager pick up the Little Graph Treasure Map paper, a
ruler and your graphing calculator. Follow the directions on the map to arrive
at the point where the treasure is. If you end up at the right point, you will
get the treasure.
Lesson Segment 1: How would you describe the graph of the ordered pairs for
a relationship where the ratio of change is constant?
Have a student come to the overhead to place a point where each of the points on in
the starter should be. If a student correctly places the points, the team gets a little
treat.
Q. ThinkTeamshare Q. How would you describe a line that contains each of these
points?
Have the students sketch a line using their rulers through the points on their treasure
map and plot several other points that lie on that line. Have them write the directions
for finding the next point as a ratio comparing the number of spaces they moved up to
the number they moved across. Explain that the change or distance from one point to
another moving up and moving across can be written as a ratio, ½.
Q: ThinkTeamShare. What make the line straight rather than being curvy?
Remind students that moving vertically is moving parallel to the yaxis, and moving
horizontally is moving parallel to the xaxis. So, rather than simply using the words,
“change” or “move”, we could use the words, “change in y over change in x”. Refer to
the vocabulary on the board.
Ask students to sketch a prediction on the back of the Little Treasure Map for what the
line would look like if change Y and the change in X were to be different from point to
point. Then, have them start at the original point and use the slope 1/1 to plot another
point, then 2/5 to plot a third point, then 3/2 to plot a fourth point.
Lesson Segment 2: How can a ratio be used to describe the “steepness” of a
line? Why does slope remain constant?
Explain that the change in y compared to the change is x is called the slope. The
slope refers to the steepness of the line. When the slope is a small number, the line is
less steep. When the slope is a larger number, the line is more steep. Show students
the Skiing graphic (attached). Have them predict whether the slope of each line will be
a smaller or a larger number, or a number close to 0. Overlay each line of the skiing
transparency on a graph paper transparency and demonstrate identifying two points
then counting the slope from one point to another emphasizing how Y changes and how
X changes from one point to the other.
Slopes from graphs:
We write the steepness of a line as a ratio telling how the rise of the line compares to
the run of the line. Hand out the “Slopes From Graphs” worksheet. Help students
identify four points where the line intersects two gridlines and have them count the
change in Y and the change in x and write the change as a ratio for each. Have
students compare the ratio for change in y/change in x with each point asking whether
or not the ratios are equivalent. Remind students that they decided if the ratios were
not equivalent, the line would not be a straight line.
Discuss lines with a positive, negative, 0, or not slope at all. A mnemonic for this is the
skiing scenario
When you are moving upward from right to left, this is a POSITIVE slope.
When you are moving downward from right to left, this is a NEGATIVE slope.
When you are skiing horizontally, this is cross country skiing with 0 slope.
When you are falling off a cliff (vertical line) this is NO SLOPE AT ALL
Dance: “Slope Dance” Students stand and face the front of the room. You stand
behind them. Put on music and have them use their arms to show a positive, negative,
0, or no slope line as you call each of these out.
Journal: Fold and cut the Finding the three Ways To Find Slope foldable. Fill in the
example for finding slope from a graph of a line.
Lesson Segment 3: How can ordered pairs for a linear relation show that
slope of a line is constant?
Slope from a table
Using The Table Ask Feature On the Graphing Calculator to Investigate Slope
To find patterns leading to the concept of slope, type an equation in the Y=. In
Table Set, set the Independent and Dependent to “ask”. Press the Table key and type
values in the X list. Press the Enter key to put a few intermittent values in Y list.
Leave some of the values out of the Y list. Have students determine what the missing
values are. Ask, “How did you decide that?” Some students will remember this activity
from the last lesson for writing an equation. Others will reply that they saw the pattern
in the change in the Y values rather than in relating Y to X. Tell them for slope they
will be using the change from Y2 to Y1 and from X2 to X1 Have them copy the tables
on an assignment paper and write the change in Y to the change in X as a ratio. Do
several equations in this manner. Some possible equations you may wish to try are:
y = x, y = x , y = 2x, y = 3x, y = ½ x, y = 1/3x, y = 5, y = x, y = x
Connect the “change in Y over change in X” from the table to counting that change
when they were looking at the graph. Discuss that just as the ratio of rise to run on
the graph was always the same ratio, the ratios of change in Y to change in X in the
table must be equivalent. On their assignment paper ask them to begin with X being 0
and y being any number they choose and construct tables of values that having the
following slope:
 ¾
 1/3
 – ½
 2/3
 0
Game: Truth or Dare
Give each team one of the tables from the Tables and Slope Transparencies. The team
works together to determine if the tables show a linear relationship by checking for a
constant ratio for change in y to change in x. A team member is then selected to bring
the transparency to the overhead and ask the class members to determine if the table
shows a linear relation or not and how they determined their answer. Class members
are given 30 seconds to check with their team to reduce risk. The student at the
overhead then selects a person from the class to either tell the truth or take a dare. If
the selected student can tell the truth and explain their reasoning, they needn’t take
the dare. If not, the challenging team gives the dare such as, “Jump up and down
while barking like a dog.” All dares must be respectful and the teacher can veto a dare
if is inappropriate. Students should copy all tables on an assignment paper and write
the slope of the line IF the table indicates a linear relation.
Journal: Fill in the example for finding slope from a table on the foldable.
Slope from ordered pairs.
Have the students look again at their “Slopes From Graphs” paper. On the back of that
paper, have them write the ordered pair for the points they identified in each graph.
Help them identify how y has changed and how x has changed by looking at the
difference in two of those ordered pairs. Help them write a math expression to
compute the change (use slope formula and placing ordered pairs vertically as if in a
table and subtracting). Then have them compare their computed slope value to the
counted slope value from the front of the page.
FourCorners practice: Use the ordered pairs on the 10 cards (attached). Ask person
# 1 from a team to come draw out any two ordered pairs. Then, have person #2 from
another team come do the same, and person # 3 from a team, and person # 4 from
another team. Do Four Corners where all the #1’s go to a corner, the # 2’s to another
corner, the # 3’s to a third corner and the # 4’s to the fourth corner of the room. In
the corner they look at the 2 ordered pairs that were drawn. Together, they find the
slope of the line that would contain those two points. Have the students return to their
desks and teach their teams how to find the slopes. These four problems should also
be recorded on the back of the Slopes from Graphs worksheet.
Journal:
Fill in the example for finding slope using two ordered pairs.
Lesson Segment 4: How can right triangles be used to show that slope is
constant?
Manipulative activity
Give each pair of students a few centimeter cubes. Each student should be given the
Slope and Similar Triangles worksheet to record the work. Students will be building
slope with Centimeter Cubes, Sketching a line along the lower vertices of the cubes and
tracing the right triangles formed by the cubes. Students will then compare sides from
several right triangles for a line from their sketches to determine if the triangles are
similar or not. Remind students that similar triangles must have corresponding sides in
proportion. If the ratios of rise to run are equivalent ratios, slope must be constant for
the line.
Assessment Plan: performance, observation, questioning
Bibliography: This lesson plan was created by Linda Bolin. Author: Utah LessonPlans
Created Date : May 15 2009 13:01 PM
