MAT 117 WEEK 2 LESSON PLAN
Time expected to complete the lesson: Approx. 125 min.
Subject: Lines in the plane
Objectives:
1. Determine the slope of a line passing through two points.
2. Find the point-slope form equation of a line.
3. Find the slope-intercept form equation of a line.
4. Sketch a line from an equation.
5. Determine if two lines are parallel, perpendicular or neither.
6. Write the equation of a line that passes through a point and is parallel or perpendicular to a given line.
Motivation:(2 min.) The concept of slope enters into a person's life in many ways. Whether viewing the change in the value of an investment, assessing the difficulty of climbing a mountain path or estimating the time required to drive from one city to another, slope comes into play. For students going on to calculus, viewing the slope of a secant line and extrapolating it to the slope of a tangent line is important to understanding the concept of a derivative. Parallel and perpendicular lines show up on street maps and forms the basis of the Cartesian Coordinate System.
Slope and Equations of Lines
Warm up discussion:(5 min.)
Ask the students to assume they each have invested in a mutual fund in which the net asset value per share is $10. Six months later, the NAV/S has dropped to $8.
1. Ask what this would look like on a graph.
2. Determine the decrease per month in the value of a share.
3. Discuss how this is, in effect, the slope of the line.
Formal concept:(8 min.)
Draw two generic points (x1,y1), (x2,y2) connected by a line. Ask the students to determine the change in the values of x and y as they move from one point to the other.
Point out that the ratio of change of y values divided by change of x values is the definition of the slope of a line, which is represented by m.
It is important for the students to recognize that the slope is identical regardless of whether one travels from point A to point B, or from point B to point A.
Discuss the meaning of a line that slopes upward to the right (positive slope) as opposed to sloping downward to the right (negative slope). You might use an example such as finding the slope of the line that passes through (3, 5) and (5, 8) and contrasting it with the slope of the line that passes through (3, 5) and (5, -8). Talk about the difference between a zero slope (horizontal line) and an undefined slope (vertical line). Examples would be helpful.
Examples:(10 min.)
1. Show the students how to find the slopes of the two segments given in the formal concept, specifying the difference between positive and negative slopes.
a) (3,5) and (5,8)
b) (3,5) and (5,-8)
What type of slope does each segment demonstrate?
2. Find the slope of a line passing through the following pairs of points:
a) (1, 0) and (3,1)
b) (-1, 2) and (2,-5)
c) (-2, 4) and (2, 4)
d) (2, 4) and (2, 7)
Ask the students to think about what they expect their slopes to be (positive, negative, zero, or undefined). Does their calculated slope agree with their expected answer?
Transition to Point-Slope Form and Slope-Intercept Form:(1 min.)
If you know the slope of a line you also know the coordinates of a single point on the line, you can find the equation of the line. This is very useful in finding other points on the line or for drawing an accurate picture of the graph.
Formal concept: (8 min.)
Have the students use the slope formula to derive the point-slope form of an equation by isolating the (y2 - y1). Therefore, (y2-y1) = m(x2-x1). This is called the point-slope equation and allows an equation to be derived when a point on a line and the slope of the line are known.
Once we know the point-slope equation, we can easily derive the slope-intercept form by using the values of a point and the slope to put the equation in the form y = mx + b. In this form m is still the slope and b is the y-intercept which is the value of y when x = 0.
Examples:(10-15 min.)
1. Use the point-slope form with (x1,x2) = (1,-2) and m=3.
Demonstrate the first one to the class.
2. Use the point-slope form with (x1,x2) = (-2, 8) and m=-1.
3. During the first quarter of the year a company had sales of $3.4 million. During the second quarter, the same company had sales of $3.7 million. Use the point-slope form to write and equation of a line. (Hint: How do we find slope when we aren't given it directly?)
Sketching Graphs of Lines
Warm up example or activity: (5 min.)
To sketch a graph of a line, we need to ensure that the graph is in slope-intercept form, y=mx+b, where m=slope and b=y-intercept. What is a y-intercept? How do we move on a graph (the Cartesian Plane) when we know the slope?
Formal Concept: (10 min.)
Have the students discover the "rise over run" or the "change in y over the change in x" concept by trying a few examples. Then demonstrate that by identifying the y-intercept it givesus a starting point (our first coordinate point) and the slope provides the second coordinate point to form two points on the line for graphing. Special cases will need to be discussed such as the form of vertical and horizontal lines.
Examples:(10 min.)
1. Sketch the graph of the following linear equations:
a) y = 2x + 1
b) x + y = 2
c) -3y = 4x + 6
d) y = 2
Parallel and Perpendicular Lines
Warm up example or activity:(5 min.)
Demonstrate that the streets in the local area are like our mathematical mapping system. For example, ask what the streets Rural Rd and University look like on the map. (They are perpendicular.) Then ask about University and McClintock, which are parallel. Streets such as Grand and Glendale intersect, but are not parallel or perpendicular. Finally, Rural turns into Scottsdale so they are the same road. Mathematical lines interact in the same fashion.
Formal Concept (10 min.)
Parallel Lines - Same slope, but different y-intercepts
Perpendicular Lines - The slopes are negative reciprocals of each other (the product of the slopes is -1)
Intersecting Lines - Lines in the same plane that are neither parallel nor perpendicular, therefore their slopes are different and are not negative reciprocals of each other
(Two lines that have the same slope and the same y-intercept are the same line.)
Examples:(15 min.)
1. Show that 3x - y = 4 and 6x - 2y = 7 are parallel.
2. Show that 3x - y = 4 and x + 3y = 6 are perpendicular.
3. Show that 3x - y = 4 and 3x + y = 4 are neither parallel nor perpendicular.
4. Show that 3x - y = 4 and 6x - 2y = 8 are the same line.
(or you could ask to the students to identify whether the above problems are parallel, perpendicular, intersecting but not perpendicular, or the same line)
Transition to Writing the Equations of Lines Through a Point and Either Parallel or Perpendicular to a Given Line(1 min.)
Now it is time to tie it all together!
Formal Concepts(5-8 min.)
To write a line that is parallel or perpendicular to another and passes through a given point, one must identify the slope of the given line and then treat it according to what the problem is asking for:
Parallel – the slope remains the same
Perpendicular – use the negative reciprocal of the slope of the original line
Use the point and the appropriate slope to find the y-intercept and then write the equation of the line in slope-intercept form.
Examples:(8-10 min.)
Write the equation of a line through the point (3, 7) and
1) parallel to the line 3x - y = 4
2) perpendicular to the line 3x - y = 4
Follow Up Assessment: Because the material is so basic, the assigned homework problems should be sufficient as a follow up assessment. A short quiz or a warm-up problem the following class period would be a good indication of whether or not the students understood the material.