Project AMP Dr. Antonio R. Quesada Director, Project AMP

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1. Lesson Title: Discovering Slope-Intercept Form

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

3.Key Words: slope, slope-intercept form, y-intercept, linear equations

4.Background knowledge:

Students are presumed to know:

  • how to plot points
  • how to find x and y intercepts
  • how to find slope by counting blocks from one point to another on the line
  • how to graph and use a table on a graphing calculator

5.NCTM Standard(s) Addressed:

Grade 9: Patterns, Functions and Algebra:

#2 Generalize patterns using functions and relationships (linear), and freely translate among tabular, graphical, and symbolic representations.

#6 Write and use equivalent forms of equations and inequalities in problem situations; e.g. changing a linear equation to slope-intercept form.

6. State Strand(s) and Benchmark Addressed:

Grade 8-10: Patterns, Functions, and Algebra:

B. Identify and classify functions as linear or nonlinear, and contrast their properties using

table, graphs, or equations.

E. Analyze and compare functions and their graphs using attributes, such as rate of change,

intercepts, and zeros.

J. Describe and interpret rates of change from graphical and numerical data.

7. Learning Objectives:

1. To find the slope of a line by counting horizontal and vertical distances.

2. To find x and y intercepts algebraically and by interpreting graphs.

3. To write equations in slope-intercept form.

4. To compare slopes based on steepness and direction.

5. To match linear equations to their graphs.

6. To apply slope concepts to real-world application.

8.Materials:

  • Graphing calculator
  • Ruler

9. Suggested procedures:

a.Cite the “attention getter”: “Have you all walked up and down steps before? What concept

is used when constructing a staircase? When did you learn about this concept in your prior

math courses? Over the next couple of days, we are going to revisit the concept of slope

and extend the concept further.”

b.Students may be divided into groups of three/four.

  1. Assessment(s): A quiz will be given to assess that the above objectives were met. The quiz andanswer key are included.

DISCOVERING SLOPE INTERCEPT FORM

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).

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

b) Name the y-intercept as a coordinate.

(0, -1)

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.

6 2

3, which reduces to 1

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

rising

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.

-6 -2

-3, which reduces to -1

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

2 = -2

1 -1 , This is a true statement because a negative divided by a negative is
a positive.

  1. a) Plot the points ( -4, 2) and (5,-1) on the grid.

b)Thinking of what you did in Problems #1, name the slope of the drawn line. In other words, how did you get from one point to the other?

-3

1

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

Falling, because it is going down from left to right.

d)Now go from the opposite point to the other. Name the slope.

3

-1

e)Are the slopes equivalent? Why?

Yes, because both slopes are –1/3. It does not matter if the negative sign is in the numerator or the denominator.

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.

3. Complete the following table.

Original Equation / Rewrite as y= / x-intercept / y-intercept / Slope
a. / 2y-4=x / y= ½x+2 / ( -4 , 0) / ( 0 , 2 ) / ½
b. / 2y+6=10x / y=5x-3 / (3/5, 0) / ( 0 , -3 ) / 5
c. / 2y+8=4x / y=2x-4 / ( 2 , 0) / ( 0 , -4 ) / 2
d. / -y+3=x / y=-x+3 / ( 3, 0) / ( 0 , 3 ) / -1
e. / y-1=-4x / y=-4x+1 / ( 1/4, 0) / ( 0 , 1 ) / -4
f. / 4y-20=-x / y=-1/4x+5 / ( 20 , 0) / ( 0 , 5 ) / -1/4

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

The slopes and the coefficient of x are the same.

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

The y-intercepts are the same as the constant in each equation.

6. Make some conclusions about what you noticed in comparing slope and y-intercept with the equations written in “y=” form.

When an equation is written as “y=”, the coefficient of x is the slope and the constant is the y-intercept.

7. Extension questions:

a. Given the equation y = 2x + 5, where would the graph of this equation cross the y-axis? What is the slope of this line?

The graph would cross the y-axis at 5 and the slope of the line is 2.

b. Graph the equation in part a on your calculator and compare your results to the graph.

I was correct that the line crosses the y-axis at 5 and has a slope of 2.

c. Given the equation y = 5 + 2x, where would the graph of this equation cross the y-axis? What is the slope of this line?

The graph would cross the y-axis at 5 and the slope of the line is 2.

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?

One.

f. What happened? Why did this happen?

The graphs of the two lines lie directly on top of one another because they are the same equation.

g. What arithmetic property of real numbers have you just rediscovered? Give another example using this property. State the y-intercept and the slope. What can you conclude?

The commutative property. Another example is y=4x-2 and y=-2+4x. I can conclude that the coefficient of x is always the slope and the constant term is always the y-intercept, the order of the terms is not important because addition is commutative.

h. Given the equation y = 3x, where would the graph of this equation cross the y-axis? Why did you say what you wrote? Check your graph with the graphing calculator.

The graph of this equation would cross the y-axis at 0 because there is not a constant term. y=3x and y=3x+0 are the same.

Equations in the form y=mx+b are equations in “slope-intercept form”, where m is the slope and b is the y-intercept.

8. Graph equations a, b, and c from the chart in question 3 in your calculator and sketch the graphs below.

9. What do all of the graphs in question 8 have in common? How does this relate to the slope? All of the slopes are rising from left to right.

10. What do you notice about how the slope relates to the steepness of the lines in question 8? The greater the slope, the steeper the line.

11. Graph equations d, e, and f from the chart in question 3 in your calculator and sketch below.

  1. What do all of the graphs in question 11 have in common? How does this relate to the slope? They are all falling from left to right.

13. What do you notice about how the slope relates to the steepness of the lines in question 11?

The greater the absolute value of the slope, the steeper the line. For example, even though -1/4 is a greater number than -4, -4 has a steeper slope.

14. 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.

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

1. x=y2. -2y-10=x3. 3y+3=2x

4. ¼ x=y-45. y+2x=16. y+x=3

CHECK FOR UNDERSTANDING:

  1. In y=mx+b, m represents the slope of the line.
  2. In y=mx+b, b represents the y-intercept of the line.
  3. Rise is the vertical distance.
  4. Run is the horizontal distance.
  5. Create three equations in slope-intercept form.

y=______y=______y=______

Answers will vary.

  1. Circle the slope in each equation.

The coefficient of x should be circled.

  1. Box the y-intercept in each equation.

The constant term should be boxed.

  1. Graph each equation in your calculator and sketch below.

Answers will vary depending on student equations chosen.

  1. For each equation, name two points that are on each line, other than the y-intercept.Answers will vary.

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

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

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

  1. Look at the table feature in your calculator to verify that your points are on the line.
  1. Create two more equations. Circle the slope and box the y-intercept. Graph each equation in the calculator and sketch in the space below. Notice that graph paper has not been provided. 

Answers will vary.

EXTENSION:

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

Maximum Slope
Ramps-wheelchair / 0.125 m
Ramps-walking / 0.3 m
Driveway or street parking / 0.22 m
Stairs / 0.83 m

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?

9 .

4.2 = 2.14286

So, it would not be safe to park your car on one of those streets, because the maximum slope is .22 and 2.14286>0.22

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

.4 .

1.2 = .33333333333

Kelly’s driveway does not meet safety specifications because the maximum slope for a driveway is .22 and .3333333333>.22.

3)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 (72m,4m). The bottom of the ramp has coordinates (22m,1m). Will the ramp meet safety specifications? Graph this situation on the graph paper provided.

3 .

50 = .06

Yes, this ramp will meet safety specification, because the maximum slope for a wheelchair ramp is .125 and .06<0.125

EXTENSION 2:

A stairway is made up of a set of steps. Each steps consists of a step riser and a step tread.

1)A set of stairs is made up of a set of constructed with a step tread of 12 inches and a step riser of 6 inches. What is the slope of the stairway?

6 1

12 = 2 So, the slope of the stairway is ½

2)Measure the step tread and step riser for five stairways in PSH and two handicapped ramps. Then compute each slope. Record on the chart below.Answers will vary, as there are numerous stairways to choose.

STAIRWAY / TREAD / RISER / SLOPE
RAMP / RISE / RUN / SLOPE

3)Graph each slope on one graph with the y-intercept being (0,0) for all seven lines. First graph on calculator, then sketch below using a different color for each line. Compare slopes. Answers will vary.

4)Which set of steps are the most comfortable to walk on?

Students should find that stairways with a smaller slope are more comfortable to walk on.

5)Do our handicapped ramps meet the building specifications from the chart in the first extension? Yes, students will provide the appropriate work to show that the slopes of our handicapped ramps are less than 0.125 m. This may require some conversions if students measured in inches or centimeters.

Summarize all of the concepts that you learned in this lesson.

Slope – rise/runm=positive slope(rises left to right)

negative slope(falls left to right)slope-intercept form – y=mx+b

x-Int – where a line intersects x-axisy-Int – where a line intersects y-axis

Name a real-world situation where slope can be useful (do not include any of the examples in the entire lesson).

ANSWERS WILL VARY

Name ______

LINEAR EQUATIONS QUIZ

  1. Plot the points (4,6) and (-3, -1) and connect them with a line.
  1. What is the slope of the line connecting these points?
  1. Is the slope positive or negative? How do you know?
  1. Write 4x-3y=24 in slope intercept form. Circle the slope and box the y-intercept.
  1. Graph. You may want to check your graph using your calculator.
  1. For each pair of equations, circle the equation that has the steeper slope when graphed. If they have the same steepness, circle both of them.
  1. Rewrite each equation in slope-intercept form, and then match the equation to its graph.

y+2=3x2y-x=4y+4x=-44y-20=-3x

  1. Graph y= 20x-70 on your calculator. Adjust your window so you can view both the x and y intercepts. State three points on the line other than the y-intercept using the table feature in your calculator.

Name ______

LINEAR EQUATIONS QUIZ

  1. Plot the points (-3,-1) and (4, 6) and connect them with a line.
  1. What is the slope of the line connecting these points?

7 = 1

7 1

  1. Is the slope positive or negative? How do you know?

Positive because the line is rising from left to right. To get from (-3, -1) to (4, 6) you go up 7 unit (positive ) and then right 7 units (positive).

  1. Write 12x-6y=24 in slope intercept form. Circle the slope and box the y-intercept.
  1. Graph. You may want to check your graph using your calculator.
  1. For each pair of equations, circle the equation that has the steeper slope when graphed. If they have the same steepness, circle both of them.
  1. Rewrite each equation in slope-intercept form, and then match the equation to its graph.

y+2=3x2y-x=4y+4x=-44y-20=-3x

  1. Graph y= 20x-70 on your calculator. Adjust your window so you can view both the x and y intercepts. State three points on the line other than the y-intercept using the table feature in your calculator.