Session 9: First topic after exam 1 is complex numbers. The Khan academy videos can be found here up through adding/subtracting, for multiplication, and division. However like us on the philosophy a bit. Here is a video that goes into a bit of the history as well. It points out that one point, negative numbers were not accepted as real, so the fact that sqrt(-1) is “imaginary” should make us skeptical. That they are really important for solving algebraic equations should not be in doubt.
Can you connect how we plotted complex numbers in the complex plane in class with what is being illustrated here? [Solutions to the equation may not exist in the original world where the graph is plotted, but if we add another dimension perpendicular to the plane, the solutions exist there.] Can you find examples of other situations where at first glance, you are stuck but if you take a broader view, all of sudden escape routes appear?
Session 10: Here are the Khan Academy videos for solving quadratic equations by factoring, using the sq rt property and completing the sq. Here is an alternative for the square root property:
I found the range of examples nice and the pace of presentation about right. Describe in words the situation when you can use the square root property.
To set up this situation, that is what completing the square is all about. Here is a gentle presentation to some difficult problems:
Please make a list of steps that would work for solving these problems.
Session 11: The following is a long 15 minute video from NJIT. The first 9 minutes will be a review for solving using completing the square and the square root formula. For those of you already of comfortable with method, skip to the 9th minute and watch the presentation of the quadratic formula:
The video only applies to one situation, which is where the discriminant, the part under the square root, is a perfect square. This situation coincides with when the problem can be solved using factoring. Use factoring to solve the same problem. What are other 2 situations? Create and solve an example for each case on your own by playing around with a, b and c. Here are the Khan academy videos with examples for all situations.
Session 12: There are many approaches to graphing quadratic equations. Watch the Khan academy videos that are selected here. Now explain in your own words
- What it means for an equation to be in vertex form?
- What is the axis of symmetry? How will it help when you are graphing?
- What is the y intercept of a graph? How do find the y-intercept?
- One major decision that needs be made when graphing a parabola is whether it like a valley (U) or a hill (∩). How do you know which it is?
- Perhaps the trickiest part is the shape (stretched or flattened vertically). How do you know whether it is stretched or flattened? I have chosen a short video below that does a good job explaining the concept::
Session 13 (snow day)
Session 14: There are 3 topics. One is to do more examples on putting quadratic functions into vertex form. Two is to provide an alternative way to find the vertex which consists of finding its x-value by using a formula and then plugging in to get the y-value. Three is to do some applications/modeling.
Topic 1: more examples on putting quadratic functions into vertex form.
This exposition is to show you how to complete the square for a quadratic function that is non-monic, meaning that the leading coefficient is not 1. I suggest that you watch this khan academy video as a warmup. The function treated is particularly nice in that the roots can be found by factoring and the leading coefficient can be factored out, so no fractions pop up.
By the way, to see all of the quadratic material at Khan academy, go to:
https://www.khanacademy.org/math/algebra/quadratics/
Now to the real example. The next video shows how to put the function in vertex form, which is something that you are asked to do in webwork. A caveat (warning): the Khan academy treatment is slightly different from the webwork approach. Below, I step-by-step show you how to proceed to be able to do this same problem in webwork:
y=-3x^2+24x-27
Step 1: Separate non-x-terms from x-terms
y+27=-3x^2+24x
Step 2: Divide off the coefficient of x^2
-y/3-9=x^2-8x
Step 3: Identify the value that completes the square
half of -8 squared is 16
Step 4: Complete the Square
-y/3-9+16=x^2-8x+16
Step 5: Factor and start solving for y
-y/3+7=(x-4)^2
-y/3=(x-4)^2-7
Step 6: Finish solving for y
y=-3(x-4)^2+21
Step 7: Identify the vertex
Vertex is (4, 21)
Here is another problem:
y=4x^2-32x+69
Step 1: Separate non-x-terms from x-terms
y-69=4x^2-32x
Step 2: Divide off the coefficient of x^2
y/4-69/4=x^2-8x
Step 3: Identify the value that completes the square
half of -8 squared is 16
Step 4: Complete the Square
y/4-69/4+16=x^2-8x+16
Step 5: Factor and start solving for y
y/4-5/4=(x-4)^2
y/4=(x-4)^2+5/4
Step 6: Finish solving for y
y=4(x-4)^2+5
Step 7: Identify the vertex
Vertex is (4, 5)
Topic 2: provide an alternative way to find the vertex which consists of finding its x-value by using a formula and then plugging in to get the y-value. NOTE: there is a webwork problem set for this topic that opens today 3/14 at 3:45. There is a videos that I have picked out. The camera on this is a bit jittery so if you get seasick or dislike her exposition, you can try an alternative.
One reason I chose Dr. Sutcliffe’s video is that she doesn’t just use the formula, but also shows where it comes from. This derivation uses the same technique, namely completing the square, that we used to come up with the quadratic formula. For your writing assignment, I would like you to divide your paper into 2 columns. In the left column, derive the quadratic formula as we did in class. In the write column, start with a quadratic function in general form and in parallel fashion to column 1, complete the square. Finally, line up the formula in vertex form so that you can derive the formula for the x-coordinate. Note that it is not recommended that you bother with the y-coordinate formula. Instead, just plug your x-value into the original general equation.
Topic 3: quadratic applications/modeling. Start by going through the Khan academy unit on the topic. How realistic are these models? For your writing assignment, consider whether the mosquito population will indeed start falling if the rain amounts continue to increase. Given your answer to this together with the fact that populations can not be negative and rainfall can not be negative (although you can have a deficit of rainfall) to decide on a realistic range of numbers for which this model might apply. Ditto for the ball being tossed off a roof. For what range of (time) values is this model realistic. Even within this range, what factors could make the model not so realistic. Hint: consider friction and its affect on how high the ball will reach as well as the time it takes to reach the ground. By the way, there is no webwork for this topic. Hence make sure to work through at least some of the problems in the course schedule: p. 398: 61, 65, 67, 69, 71 and p. 595: 41, 43, 47. In particular, problem 47 is another ball thrown up in the air problem.
Session 15: The math department’s video page for geometry is here. There are 4 topics, distance and midpoint formulas, perpendicular bisector and circles.
Topic 1: distance formula. As the videos point out, the formula is really an application of the Pythagorean formula for the hypotenuse of a right triangle. There is webwork for this topic.
Topic 2: midpoint formula. Watch the videos and do the exercise. There is no webwork for this topic. I think of the midpoint formula as basically finding the average of the x and y values of the endpoints, respectively.
Topic 3: perpendicular bisector (an application of the midpoint formula plus some review of equations of lines, in particular fact that perpendicular lines have slopes that are negative reciprocals of each other). Watch this video and download the department’s special sheet. It has the procedure written out step by step and some exercises that you should do. There is no webwork for this topic.
Topic 4: graphing circles. Watch the video on how to put a general equation into center/radius form. Here are some non-Khan videos. The first does an example which has its center on one of the axes:
Which axis does this circle have its center on? Given an equation in general form, how do you whether the circle’s center is on an axis and which axis it will be? Here’s another example, this time the center will not be on an axis. Why?
Notice how the 4 points on the circle are marked and labeled. Write down a step-by-step procedure for doing so. Here is another video of an example whose center is not on an axis. The 4 points are marked but not labeled with coordinates. Use the procedure you have written down to find those coordinates.
Session 16: Systems of 3 linear equations in 3 variables. Make sure that you review 2 equations in 2 variables first, especially if you haven’t worked with systems in awhile. There are 2 methods, namely elimination and substitution. We will emphasize elimination. I suggest you check out the Khan academy videos selected here. As an alternative, I have selected:
This instructor has elected to eliminate y. For your writing assignment, I would like you to step by step explain what has to be done to solve a system using the elimination method. You may assume that one of the variables somewhere has either one or minus one as a coefficient.
Final topic is nonlinear systems. The Khan academy videos treat the case of a line and either a circle or a parabola. They emphasize the substitution method. I will instead focus on examples that treat 2 nonlinear equations better solved using elimination:
Once you watch the video and learn how to do it using elimination, go back and try to solve it with substitution. What variable in what equation could you easily isolated? What happens when you substitute into the other equation? In particular what is the degree of that equation? Will it be easily solved? Here is another example that uses elimination. Again watch to learn that approach and then describe what happens if you use substitution instead.
