Discovery Activity: Slope

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1 Page 1 of 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: ~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 Internet Access: desmos.com Ruler Markers, colored pencils, etc

2 Page 2 of 14 DISCOVERING SLOPE INTERCEPT FORM This lesson is a review of slope. It will guide you through discovering slopeintercept 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. 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 named the slope for this line. Is the slope rising (positive) or falling (negative)? e) Now count the units moving from (2,3) to (-1, -3), in other words, count the units 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. This is the slope for the line. f) Now write your results in proportion form (setting the two fractions equal to one another). Is this a true statement? How so?

3 Page 3 of a) Plot the points ( -4, 2) and (5,-1) on the grid. Connect them. b) Thinking of the slope of the line in Problems #1, what is the slope of this line? In other words, how did you get from one point to the other? c) Is the slope positive (rising) or negative (falling)? Why? d) Now go from the opposite point to the other. Name the slope. 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 ( y) and the denominator tells you the horizontal distance ( x). We describe slope as rise run y x (y y ) (x x ) to help us remember this.

4 Page 4 of Complete the following table. Original Equation Rewrite as y= x- intercept y-intercept Slope y 2 y 1 x 2 x 1 a. 2y-4=x (, 0) ( 0, ) b. 2y+6=10x (, 0) ( 0, ) c. 2y+8=4x (, 0) ( 0, ) d. -y+3=x (, 0) ( 0, ) e. y-1=-4x (, 0) ( 0, ) f. 4y-20=-x (, 0) ( 0, ) 4. In each row, compare the slope and the numbers in the equation of the form y=. What do you notice? 5. In each row, compare the y-intercept and numbers in the equation of the form y=. What do you notice? 6. Make some conclusions about what you noticed in comparing slope and y-intercept with the equations written in y= form. 7. Classwork: a. Given the equation y = 2x + 5, where (as in an ordered pair, or a point) would the graph of this equation cross the y-axis? What is the slope of this line?

5 Page 5 of 14 b. Graph the equation in part a on your graphing calculator, or desmos and compare your results to the graph. Same? Different? 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? d. Graph the equation in part c in your calculator, or desmos on the same grid as part a. e. How many lines do you have on your calculator? f. What happened? Why did this happen? 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? 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, or desmos.

6 Page 6 of 14 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 the 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? 10. What do you notice about how the slope relates to the steepness of the lines in question 8?

7 Page 7 of Graph equations d, e, and f from the chart in question 3 in your calculator, or desmos, and sketch below. 12. What do all of the graphs in question 11 have in common? How does this relate to the slope? 13. What do you notice about how the slope relates to the steepness of the lines in question 11? 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. y 3x 1 or y 2x 3 y 1 x 3 or y 4 x 8 2 y 5x 1 or y 5x 3 3y 9 6x or 2y 12 8x

Page 8 of 14 15. Rewrite each of the following equations in slope-intercept form; then match each equation to its graph. 1. x=y 2. -2y+10=x 3. 3y+3=2x 4. ¼ x=y-4 5. y+2x=1 6.

8 Page 8 of Rewrite each of the following equations in slope-intercept form; then match each equation to its graph. 1. x=y 2. -2y+10=x 3. 3y+3=2x 4. ¼ x=y-4 5. y+2x=1 6. y+x=1 CHECKING FOR UNDERSTANDING: a. In y=mx+b, m represents the of the line. b. In y=mx+b, b represents the of the line. c. Rise is the distance or change. d. Run is the distance or change.

9 Page 9 of 14 e. Create three equations in slope-intercept form. y= y= y= f. Circle the slope in each equation above. g. Box the y-intercept in each equation. h. Graph each equation in your calculator or desmos, and sketch below. i. For each equation, name two points that are on each line, other than the y- intercept. Equation #1:(, ), (, ) Equation #2:(, ), (, ) Equation #3:(, ), (, )

10 Page 10 of 14 j. ~If you are using a graphing calculator, look at the table feature in your calculator to verify that your points are on the line. ~If you are using desmos, look at the point you choose. Are the points on the line of the equation?

Page 11 of 14 Homework #1: The building codes and safety standards for slope are listed below: Maximum Slope Ramps-wheelchair 0.125 Ramps-walking 0.3 Driveway or street parking 0.22 Stairs 0.

11 Page 11 of 14 Homework #1: The building codes and safety standards for slope are listed below: Maximum Slope Ramps-wheelchair Ramps-walking 0.3 Driveway or street parking 0.22 Stairs ) Some streets in San Francisco are on hills with a run of 9 m and a rise of 4.2m. Would it be safe to park your car on one of those streets? 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? 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.

12 Page 12 of 14 Homework #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? 2) Measure the step tread and step riser for five stairways and two handicapped ramps. Then compute each slope. Record on the chart below. STAIRWAY TREAD RISER SLOPE RAMP RISE RUN SLOPE

13 Page 13 of 14 3) Graph each slope on one graph with the y-intercept being (0,0) for all seven lines. First graph on calculator, or desmos, then sketch below using a different color for each line. Compare slopes (same, different?) 4) Which set of steps are the most comfortable to walk on? 5) Do our handicapped ramps meet the building specifications from the chart in the homework #1?

14 Page 14 of 14 Summarize all of the concepts that you learned in this lesson. Name a real-world situation where slope can be useful (do not include any of the examples from this lesson).

Chapter 9 Linear equations/graphing. 1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane

Chapter 9 Linear equations/graphing 1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane Rectangular Coordinate System Quadrant II (-,+) y-axis Quadrant