WPP 10: Concepts 9-14: Calculating Probability

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WPP 9: Unit L Concepts 4-8: Calculating Possibilities

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SP 6: Unit K Concept 10: Writing a Repeating Decimal as a Rational Number Using Geometric Series

The parts of this problem that a student needs to pay attention to is plugging in the correct numbers into the formula. The student must first correctly identify the rate by taking any term in the series and dividing it by the one that precedes. In addition, it is easier to leave the numbers as fractions, since students are not allowed to use their calculators, and solve the problem in this way.

Fibonacci Haiku: The Cosmic Order

Fate

is

not one

for pity.

Our souls are like stars,

binary stars, woefully caught

in the most beautifully tragic cosmic collapse.

But nobody in the world had ever said that this love was going to be easy.

http://img.dictionary.com/binary_star-308870-400-234.jpg

(Syllables)

SP 5: Unit J Concept 6: Decomposing Fractions with Repeated Factors

The trickiest part of this problem is remembering to count up the powers when you write a new expression using new variables in place of the numerator. In addition, solving the system of equations is also a difficult aspect of the problem. Instead of using matrices, you have to solve it by eliminating one variable at a time, then finding an answer.

SP 4: Unit J Concept 4: Partial Fraction Decomposition with Distinct Factors

The trickiest part of this problem is decomposing the fraction back again. The student needs to pay attention to how to set up the problem using new variables, such as A, B, and C. The factoring and the distribution must all be correct, because one simply mistake in multiplication could cause the whole problem to be wrong. Finally, the student needs to set up the system of equations correctly using terms that correspond with one another, and then plug that system into the calculator to solve.

SV 5: Unit J Concept 3: Solving 3-Variable Systems with Matrices

The part of this problem that a student needs to pay attention to is how to find the equations to get certain numbers in their spots. The student needs to make sure that when solving for the row, he or she is careful with their math and doesn't make a mistake, because one error or miscalculation could make the whole matrix wrong, and then he or she would have to start over. In addition, the student needs to pay attention to the equations in the system before putting them in the matrix, seeing which ones would be more convenient to place in the rows. For example, if an equation has "0x", it is smart to put it in the third row, because the first step is to find a 0 in the bottom left hand corner. Since there is already a 0, the student will have less work to do.

WPP 6: Concepts 3-5: Compound Interest, Continuously Compounding Interest, and Investment Application Problems

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SV 4: Unit I Concept 2: Graphing Logarithmic Equations

The parts of the problem a student needs to pay attention to is how to plug in the equation into a calculator, since it is a little bit trickier than exponential equations. A student needs to know how to apply the change of base formula to input the equation into a calculator and determine other important points on the graph. Furthermore, a student needs to pay attention to the domain and range. There are no restrictions on the y-values, or the range, but there will be a restriction on the x-values, or the domain. The vertical asymptote permits any x-values that are less than or equal to negative 1.

SP 3: Unit I Concept 1: Graphing Exponential Functions

The viewer needs to pay special attention to how and why the graph does NOT have an x-intercept. Since taking the natural log of a negative number would be undefined, the graph therefore never crosses the x axis. Logically speaking, the viewer knows that the graph will be above the horizontal asymptote because "a" is positive. And since the graph will never cross over y=2, they can see that there will not be an x-intercept.

SV 3: Unit H Concept 7: Finding logs given approximations

The trickiest concept to learn in this problem would be remembering to add in the extra clues to your list. Remember, logb(b)=1, so we can always use that when solving for the log. Another thing you have to watch out for is expanding the logs correctly. If the log has an exponent, you have to bring it to the front of the term and turn it into a coefficient, therefore getting rid of the exponent and being able to substitute the variables in correctly later on. This will ensure that your final answer with the variables is correct.

SV 2: Unit G Concepts 1-7: Graphing Rational Functions

This problem is about graphing a rational function with a slant asymptote and hole. There are many aspects to this problem that a student needs to find before graphing the function itself. For example, there are slant and vertical asymptotes that restrict where the graph goes, which can be mentioned hand-in-hand with the holes of the graph. Also, there is the domain, y-intercept, and x-intercept to find in the rational function so you the specific points of the graph.

The things a student needs to look for in the problem is the possibility of a hole depending on the vertical asymptote. If you cancel something out when simplifying, you will have a hole and need to pay attention to that as well. The last thing that is notable in the problem are the windows: in this case, we will have to adjust them to see the graph accordingly. In addition, we will have to set our 10 by 10 graph to a different scale and probably count by twos if we want to be able to draw it there.

SV #1: Unit F Concept 10: Given Polynomial of 4th or 5th Degree, Find all Zeroes (Real and Complex)

This problem is about finding the zeroes and factorization of complex polynomials with the highest degree of 4. In concept 10, we will have either irrational or imaginary numbers as one or more of our zeroes. We will be applying the rational roots and factor theorem, along with Descartes Rule of Signs, to solve this polynomial. Dealing with imaginary numbers is also a useful skill to have. As a result, it is imperative that a student has studied the previous concepts well, as this concept will combine all of them into solving a polynomial to the 4th degree.

A student needs to pay special attention to the synthetic division of the polynomial by one of the possible zeroes. It is crucial to use the next synthetic division bar once he or she has found of the zero heroes. Then, after dividing the 4th degree polynomial twice, the student now has a solvable quadratic polynomial and can easily find the rest of the zeroes in this way.

SP#2: Unit E Concept 7: Graphing Polynomials

This problem is about solving a polynomial with a fourth degree and graphing it. A student needs to factor out the polynomial and find the x-intercepts, in addition to their multiplicities. That way, he or she will know how the x-intercepts act on the graph (through, bounce, or curve).
The viewer needs to pay special attention to the end behavior of the graph to understand the general formation of the graph. Since this end behavior is even positive, both ends of the graph will be pointing upwards. In addition, a viewer needs to pay special to the x-intercepts and their multiplicities to see what the graph will look like in between the two ends: whether the graph will go through, bounce off, or curve out of the x-intercepts. These apply to the multiplicities of one, two, and three, respectively.

WPP 4: Unit E Concept 3: Maximizing Area

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WPP 3: Unit E Concept 2: Path of a "Football"

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SP #1: Unit E Concept 1: Identifying x-intercepts, y-intercepts, Vertex (Max/Min), Axis of Quadratics and Graphing Them

This problem is about algebraically changing a equation in standard form into parent form by completing the square. This way, the equation easier to graph when necessary. Important key points a student needs to find are: the vertex, y-intercept, axis of symmetry, and x-intercepts. There are a number of steps involved in finding the solutions to these problems, but they will make graphing the parabola less difficult and complicated in the end.
A student needs to pay attention to the key point of the graphed parabola, which is the vertex. The vertex can either be the maximum or the minimum of the graph, depending on whether a is positive or negative. It is also essential to remember the axis of symmetry on a parabola, since this will help a student graph more points by reflecting them across the parabola. In doing so, a student will have a better understanding of the parent graph function and the graph itself and be able to have an accurate drawing of his or her equation.

WPP#2: Unit A Concept 7 - Profit, Revenue, Cost

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WPP#1: Unit A Concept 6 - Linear Models

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