2.3 Discovering Slope

2.3 DISCOVERING SLOPE

This lesson is a review of slope. It will guide you through discovering slope-intercept form using paper/pencil and a graphing calculator. It includes looking at positive/negative slope, comparing the steepness of slopes, and relating slope to real-world applications (handicapped ramps, stairs).

Section 1

a)Plot the points ( -1, -3) and (2,3) on the grid and then connect them using a ruler.

b)Write the y-intercept as an ordered pair. ______

c)Count blocks up and then right to move from ( -1, -3) to (2,3). Up is a positive direction; right is a positive direction. Write these results in rise/run (fraction) form.

d)You have just named the slope for this line. Is the slope rising or falling?

e)Now count blocks moving from (2,3) to (-1, -3), in other words, count blocks down and then left. Down is a negative direction (what sign should you then write before your number?); left is a negative direction (what sign should you then write before your number?). Write these results in rise/run (fraction) form. You have just named the slope for this line.

f)Now write your results in proportion form (setting the two fractions equal to one another). Is this a true statement? How so?

Section 2

a)Plot the points ( -4, 2) and (5,-1) on the gridand then connect them using a ruler.

b)Thinking of what you did in Section 1, name the slope of the drawn line. In other words, how did you get from one point to the other? Up is a positive direction; right is a positive direction.

c)Is the slope positive (rising) or negative (falling)? Why?

d)Now go from the opposite point to the other. Name the slope. Down is a negative direction; left is a negative direction.

e)Are the slopes equivalent? Why?

Slope is a fraction that tells you how steep a line is. The numerator tells you the vertical distance and the denominator tells you the horizontal distance.

We describe slope as to help us remember this.

Slope Formula:

Section 3—Complete the following table.

a)In each row, compare the slope and the numbers in the equation of the form “y=”. What do you notice?

b)In each row, compare the y-intercept and numbers in the equation of the form “y=”. What do you notice?

c)Make some conclusions about what you noticed in comparing slope and y-intercept with the equations written in “y=” column above.

Section 4—Extension questions:

a)Given the equation, where would the graph of this equation cross the y-axis? What is the slope of this line?

b)Graph the equation in part (a) on your calculator and compare your results to the graph. Were you correct in (a)?

c)Given the equation, where would the graph of this equation cross the y-axis? What is the slope of this line?

d)Graph the equation in part (c) in your calculator on the same grid as part (a).

e)How many lines do you have on your calculator? ______

f)Explain what you think happened? Why did this happen?

g)What arithmetic property of real numbers have you just rediscovered? ______

h)Give another example using this property. State the y-intercept and the slope. What can you conclude?

i)Given the equation, where would the graph of this equation cross the y-axis? Why did you say what you wrote?

j)Check your graph with the graphing calculator. Were you correct?

Equations in the form are equations in “slope-intercept form”, where is the slope and is the y-intercept.

Section 5

a)Graph the equations (1), (2), and (3) from the chart’s “y=” column in Section 3 in your calculator and sketch the graphs below. Write the equations in the blanks below.

(1)______(2)______(3)______

b)Look for a similarity between the graphs in question (a). How does this relate to the slope?

c)What do you notice about how the slope relates to the steepness of the lines in question (a)?

d)Graph equations (4), (5), and (6) from the chart’s “y=” column inSection 3 in your calculator and sketch below.Write the equations in the blanks below.

(4)______(5) ______(6) ______

e)Look for a similarity between the graphs in question (d). How does this relate to the slope?

f)What do you notice about how the slope relates to the steepness of the lines in question (d)?

Section 6

a)For each pair of equations, circle the equation of the line that would be steeper when graphed. If they have the same steepness, circle both of them.

b)Use your calculator to graph each of the pairs of the equations above and see if you were correct.

……………………………………………………………………………………………………………………………………………………………...

c)Rewrite each of the following equations in slope-intercept form; then match each equation to its graph.

1. 2. 3.

4. 5. 6.

Section 7—Check For Understand:

a)In , represents the ______of the line.

b)In , represents the ______of the line.

c)Rise is the ______distance.

d)Run is the ______distance.

e)Create three equations in slope-intercept form.

#1: ______#2: ______#3: ______

f)Circle the slope in each equation.

g)Box the y-intercept in each equation.

h)Graph each of your equations in your calculator and sketch below. (use a ruler)

#1 #2#3

i)For each equation, name two points that are on each line, other than the y-intercept.

Equation #1: (_____,_____), (_____,_____)

Equation #2: (_____,_____), (_____,_____)

Equation #3: (_____,_____), (_____,_____)

j)Look at the table feature in your calculator to verify that your points are on the line.

Extension

Real Life Example

The building codes and safety standards for slope are listed below:

1. Some streets in San Francisco are on hills with a run of 9 m and a rise of 4.2 m. Would it be safe to park your car on one of those streets?

1. The local park is adding stairs on one side of the river so visitors are able to park closer to the river. The stairs have a run of 12 inches and a rise of 6 inches. Do the new stairs meet the safety specifications?

1. The Smith’s driveway has a run of 1.2 m and a rise of 0.4 m. Does it meet the safety specifications?

1. A ramp is to be built at the library for wheelchair accessibility. When a grid is placed over the architect’s plans, the top of the ramp has coordinates of (). The bottom of the ramp has coordinates (). Graph this situation on the graph paper provided.

1. Will the ramp meet safety specifications?