Objectives:



MAT 117 WEEK 1 LESSON PLAN

Total estimated time: 148 minutes

Objectives:

Cartesian Plane

• Plot points in the Cartesian plane

• Find distance between two points in the Cartesian Plane - Distance Formula and then use the Distance formula to solve Geometry and real-life application problems

• Find the Midpoint of the segment joining two points in the Cartesian Plane and use the Midpoint formula to solve application problems

Graphs of equations

• Determine whether a point lies on the graph of an equation. Sketch graphs using a table of values and a graphing utility. Find the x and y-intercepts of the graph of an equation (algebraically and graphically)

• Determine symmetries in the graph of an equation (algebraically and graphically)

• Write the General Form Equation of a circle its Standard Form and determine the center and radius

Motivation: [2 minutes] With the Cartesian Plane we can develop a branch of mathematics that brings together algebra and geometry in a unified way – a way that visualizes numbers as points on a graph, and equations as geometric figures. In this course as well as in later courses we will be working extensively with the Cartesian coordinate system locating points, finding the distance between points and most importantly graphing equations. So, for example, in this course we will graph lines, circles, then extend our graphs to more elaborate functions, such as: polynomials, exponential, logarithmic functions and then use these graphs to solve systems of equations.

Warm up discussion: [5 minutes] Draw two perpendicular lines. Have the students identify the axes, the quadrants and the origin. Then plot a couple of points and have the students identify the coordinates. Have the students close their eyes, give them the coordinates of the points and have them identify the quadrant. Remind students to open their eyes.

Then, since the students have already seen this preliminary material before, have them arrive at the definition of x and y coordinates.

Distance formula

Warm up example or activity:

[5 minutes] Draw two points P and rm the students of the goal: finding the distance between them. Ask the students where to plot the third point so that a right triangle is formed. Ask the students: “What theorem, or even what word, seems to be blatantly staring at you on the chalkboard?” Hopefully the answer will be the Pythagorean Theorem.

Derive the distance using the theorem.

Formal concept:

[3 minutes] Draw two generic points [pic] and have the students derive the general distance formula.

Example:

1. [10 minutes] Two ships leave port at the same time. Ship A travels east at 12 miles per hour. Ship B travels north at 8 miles per hour. How far apart are they after 4.5 hours?

2. [10 minutes] Determine whether the 3 points are vertices of a right triangle: A(0, 0), B(1, 1), and C(2, -2). You may decide to give a hint to the students about the Pythagorean Theorem, you may not.

Midpoint formula

Warm up example or activity:

[5 minutes] Draw two points on a vertical line. Have the students identify the coordinates of the midpoint between them. Have them discover that we are just averaging coordinates. Do the same for points on a horizontal line. Draw two random points P and R. Have them tell you the coordinates of the midpoint.

Formal concept

[3 minutes] Draw two generic points [pic] and have the students derive the general distance formula

Examples:

1. [5 minutes] The total revenue of a law firm in 1994 was $520,000. In 1996 the revenue was $980,000. Assuming linear growth, use the midpoint formula to estimate the revenue in 1995.

Determine whether a point lies on the graph and sketch the graph using a table of values or a calculator; determine x and y intercepts algebraically and graphically

Warm up example or activity:

[5 minutes] Write on the board a linear equation. To get the pulse of the class, ask the students what is the shape of the graph? what is it called?

If they answer anything with the word line in it, proceed on to asking them how to check if a specific point belongs to the line. So, give them one point on the line, and one not on the line.

Now tell the class you are going to give them a set of points that are on the line and ask them if we can graph the line using those points. They should answer yes. Proceed to draw the line and emphasize we are graphing the line by plotting points and connecting them. Introduce the concepts of x and y intercepts.

Formal concept:

Determine if a point lies on a graph: [2 minutes] Substitute the coordinates of the point into the equation to test if you get a true statement.

Determining x and y intercepts: [5 minutes] Ask the students how to find the x and y intercepts and make sure they verbalize the following:

algebraically: - x intercept: let y=0 and find x.

- y intercept: let x=0 and find y.

graphically: - x intercept: trace or root finder (depending on student’s level)

- y intercept: trace or table feature (depending on student’s level)

Examples:

Graphing by plotting points and with the graphing calculator, and determine x and y intercepts.

1. [10 minutes] Consider the equation: [pic]. Construct a table of values by assigning values to x and finding the corresponding y. Make sure the table includes the x and y intercepts. Graph by connecting the points with a smooth curve. Make sure the students are aware that the smallest x value they can use is –2 and use this to foreshadow the concept of domain, which will be covered later in the course.

2. [10 minutes] Consider the equation: [pic]. Construct a table of values by assigning values to y and finding the corresponding x. Make sure the table includes the x and y intercepts. Graph by connecting the points with a smooth curve.

3. [10 minutes] Have the students graph both with their graphing calculator. Be certain to go slowly, for most students it is the first time they use it.

4. [10 minutes] Consider the equation [pic][pic]. Graphically find the x and y intercepts.

Determine the symmetry of the graph of an equation algebraically and graphically

Warm up example or activity:

[10 minutes] Draw three sets of perpendicular x-y axes. Then sketch the following:

[pic] [pic] [pic]

Have the students visually observe the inherent symmetry.

Discuss x-axis, y-axis and origin symmetry with them as a reflections about the y-axis, x-axis and about the x and y axes respectively.

Consider the point (2, 4), (2,4) and (2, 6) respectively for the three graphs.

Ask the students if the graph has y axis symmetry, what corresponding point must be on the graph?

Ask the students if the graph has x axis symmetry, what corresponding point must be on the graph?

Ask the students if the graph has origin symmetry, what corresponding point must be on the graph?

Formal concept

[5 minutes] Have the students discover the pattern of corresponding pairs of points for each symmetry; specifically they should recognize that for y axis symmetry, if (x, y) is on the graph, then (-x, y) should be on the graph too, and so on.

Write down the tests for symmetry.

Examples:

1. [20 minutes] For the following equations, determine if the graphs of the equations display symmetry. Justify algebraically and/or graphically.

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic]

f) [pic]

g) [pic] - Please note: a good transition problem into the next topic (if circles is the next topic).

Write the General Form Equation of a circle in Standard Form and determine the center and radius of the circle

Warm up example or activity:

[5 minutes] Start with a circle drawn on the board, with center off the origin, but clearly labeled. Then label a point on the circle P(x, y). Then, remind the students what a circle is: a set of points whose distance from the center is constant and that distance we loosely refer to as the radius. Using the distance formula, have them help you derive the standard equation of a circle.

Formal concept:

[3 minutes] Ask the students, “If the center is (h, k), and the radius is r, what is the equation of a circle? Write down the standard equation of a circle on the board. Circle it. Mention the general form of the equation of the circle and then contrast the two forms.

Examples:

1. [15 minutes] First, briefly review the concept of completing the square. Write the following equations in standard form, and then determine the Center and Radius from standard form.

a) [pic]

b) [pic]

Follow up assessment: Because the material is so basic, the assigned homework problems should be sufficient as a follow up assessment.

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