WPP#10: Unit L Concepts 9-14: Probability

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WPP#9: Unit L Concepts 4-8: Counting Principles

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SP#6: Unit K Concept 10: Converting Repeating Decimals into Rational Fractions

 

Step 1: Start with the repeating decimal 5.212121... and find the a sub 1 and a sub 2 terms by taking the repeating terms out as chunks, using zeroes as placeholders where necessary.
Step 2: Divide the a sub 2 value by the a sub 1 value to get the ratio.
Step 3: Plug the terms into the summation notation for infinite geometric sums.
Step 4: Plug the terms into our sum formula and simplify.

For this problem, you needed to pay attention to the formulas used. Make sure you use the right ones! If you don't, you will get the wrong answers.

Fibonacci Beauty Ratio Activity

Name: Jesus
Foot to Navel: 109 cm  Navel to top of head: 59 cm  Ratio: 1.847
Navel to chin: 43 cm   Chin to top of head: 22 cm    Ratio: 1.955
Knee to navel: 57 cm   Foot to knee: 52 cm               Ratio: 1.09
Average: 1.631

Name: Leslie
Foot to Navel: 96 cm    Navel to top of head: 64 cm  Ratio: 1.5
Navel to chin: 45 cm   Chin to top of head: 19 cm      Ratio: 2.368
Knee to navel: 51 cm   Foot to knee: 46 cm               Ratio: 1.109
Average: 1.659

Name: Sammy
Foot to Navel: 100 cm  Navel to top of head: 68 cm  Ratio: 1.471
Navel to chin: 49 cm    Chin to top of head: 24 cm    Ratio: 2.042
Knee to navel: 50 cm   Foot to knee: 48 cm               Ratio: 1.142
Average: 1.518

Name: Vivian
Foot to Navel: 104 cm  Navel to top of head: 68 cm  Ratio: 1.529
Navel to chin: 47 cm    Chin to top of head: 21 cm    Ratio: 2.238
Knee to navel: 58 cm   Foot to knee: 50 cm               Ratio: 1.16
Average: 1.642

Name: Christine
Foot to Navel: 98 cm    Navel to top of head: 59 cm  Ratio: 1.661
Navel to chin: 38 cm    Chin to top of head: 21 cm    Ratio: 1.809
Knee to navel: 50 cm   Foot to knee: 48 cm               Ratio: 1.042
Average: 1.504

The Golden Ratio is a number that is found using the numbers in the Fibonacci series. It is used to determine mathematical beauty by analysing proportions. The person closest to the Golden Ratio of 1.618 was Jesus, who had an average of 1.631.
In my opinion, the Beauty Ratio is potentially useful for mathematical purposes. It can also be good for artistic purposes. However, it shouldn't be taken as a real life standard.

Fibonacci Haiku: Originality

Try.
Write.
It's hard.
New ideas?
Forget about it.
Keep an eye on those syllables!
(Unless you're cheating and using word limit instead.)
Just kidding, it's not technically cheating; it's just taking the easier way out.

How to Draw the Perfect Pinecone: Fibonacci Numbers in Nature

I found this video while looking up Fibonacci and I think it's really neat! This person posts a lot of other math related things too (check out her video on snowflakes if you're looking for something quick to make for the upcoming winter holidays. :D)

SP#5: Unit J Concept 6: Partial Fraction Decomposition (Repeated Factors ver.)

*FYI: The step numbers do not necessarily correlate with the numbers in the image.*

Step 1: Seperate your denominator factors into fractions- in this case, one of the factors is cubed, so when writing the fractions out remember to count up the factors. [1]
Step 2: Find a common denominator. In this case, you simply have to multiply the numerator by the factors that aren't already in the denominator. Then once you've multiplied and FOIL'd all the numbers and letters, set it equal to your original equation.
Step 3: Seperate the like terms into 4 different equations. At this point, you can take out the 'x cubed', 'x squared', and 'x' values and leave just the coefficients. [2]
Step 4: Two of your equations should have 2 terms and the other two should have 4 terms. Take the latter two equations and use elimination to get rid of a term. In this case, I multiplied the purple equation by two, then added it to the other 4 term equation to get rid of the D term. [3a]
Step 5: Now you have a 3 term equation, but no other one lying around ready to use. Add the 2 term equations together to make another 3 term equation. [3b] Use elimination again to get rid of another value- hopefully you should see where I'm going with this. In this case, I multiplied the 3 term equation by 4 to get rid of the C term.
Step 6: Now you have a 2 term equation. Take any one of your first 2 term equations and use elimination again to get rid of a term. In this case, I multiplied the pink equation by 4 to get rid of the B term, leaving me with the A term.
Step 7: Solve for that final term, then back substitute until you have solved for all the terms. That's it, you're done~!

SP#4: Unit J Concept 5: Partial Fraction Decomposition (Distinct Factors ver.)

Step 1: Find a common denominator. In this case, you have to multiply the numerator by the factors that aren't already in the denominator. Then once you've multiplied and FOIL'd all the numbers, add the common terms together.



Step 2: Set the numerators to letter values (in this case, A, B, and C) and do the same factoring and/or FOILing method as before. However, instead of adding the common terms together, set the fractions equal to your answer from Step 1. Then separate the common terms into three equations.



Step 3: Take the coefficients and put then into matrix form. Input the values into your calculator and put it into reduced row echelon form. Finally, take the answer column and put each one over a factor in fraction form. Look familiar? It should, that's the equation we started with.

SV#5: Unit J Concepts 3-4: Solving Matrix Problems

~>To watch my video, click the link here<~

This video will show you how to solve Problem 5 of the DP problems. In order to watch this video, you should probably pay attention to all the negative and positive signs. Check your math often because I tend to trip over my words and might say something incorrect before going back over it.

WPP#6: Unit I Concepts 3-5: Investment and Interest

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SV#4: Unit I Concept 2: Solving and Graphing Log Equations

~To watch my video, please click here~

For this video, you should make sure you understand how to find the h and k values. Knowing whether a log equation has a vertical or horizontal asymptote helps as well. Don't forget to try solving it yourself before watching the video!

SP#3: Unit I Concept 1: Solving an Exponential Equation

To solve this problem, you need to understand how to find the a, b, h, and k values. This is because they are key to finding the asymptote, domain, and range. You also need to pay attention to the coloured parts of the image, because chances they are important.

 

Step 1: Start with the equation (shown in the box titled 'equation') and determine if the graph lies below or above the asymptote. In this case, since the value of a is negative, the graph lies below the aymptote.
Step 2: Determine if the asymptote is vertical or horizontal by checking if it a log or exponential. In this case, the equation is an exponential one, and it's asymptote is equal to the k value. Therefore, the asymptote is 'y = -3'
Step 3: Solve for the x-intercept. Set y equal to 0 and add 3 to both sides. Then divide both sides by -2. At this point, we would normally take the log of both sides, but since you can't take the log or a negative (-3/2), there is no x-intercept.
Step 4: Solve for the y-intercept. Set x equal to 0 and simplify the exponent to 3. 1/2 to the power of 3 equals 1/8, and when you multiply that by -2 you get -2/8, or -1/4. Subtract 3 from -1/4 (I changed 3 to 12/4 to make it simpler) and you end up with -13/4. This means (0, -13/4) is your y-intercept- you can convert this to decimal form on your calculator to make it easier to graph.
Step 5: Determine the domain and range of the graph. Since this is an exponential equation, the domain is unrestricted. The range, however, depends on the asymptote. In this case, the range is -∞, -3.
Step 6: Graph the graph, haha that's an amusing phrase. Plug the equation into your graphing calculator and press the buttons labelled '2ND' and 'TABLE'. This will give you a table of all the points on the graph. Select a few, put them in your table, and plot them. Draw the aymptote, connect the dots, and voilà, you have solved this exponential equation.

SV#3: Unit H Concept 7: Treasure Hunt Logs

~*To watch my video, click here*~

This video shows how to solve a log using given clues. It also indirectly shows how to utilise the power, quotient, and product properties.

For this video, you should make sure you know and understand the properties of logs. Otherwise, some of the things I do in this video will not make sense. Also, it may be helpful for you to write the problem and steps down on a seperate piece of paper, as I have exceedingly bad handwriting and am entirely unapologetic for it.

SV#2: Unit G Concepts 1-7: Solving and Graphing a Rational Function

*To watch my video, please click the link here*

This video is about how to solve a rational function. It also shows how to graph the function.

For this video, you should make sure to write down all the pieces of the problem as I solve them. This is because I often clear the screen to give myself more room to work. If you don't write the needed parts down, you will probably find yourself replaying parts of the video more times than you'd like.

SV#1: Unit F Concept 10: Finding Real and Imaginary Zeroes

*~To view my video, please click on the link here~*

This problem is about how to find the real zeroes of a polynomial. It also shows how to find the imaginary zeroes of a polynomial.
For this video, you should make sure to pay close attention to the speaker. Do not be tempted to turn away from the screen, no matter how mind-numbingly boring the speaker is. You'll also want to follow along on a piece of paper.

SP#2: Unit E Concept 7: Zeroes and Multiplicity

This problem is about using randomly selected zeroes to create a polynomial. The zeroes should also be used to find the x-intercepts and the y-intercept. Finally, it should be possible to glean the end behavior and graph the equation as well.
For this problem, you should make sure you know how the multiplicity of a zero affects the graph of the equation. A multiplicity of one, for example, means that the graph will go through that point. Knowing the meanings of the multiplicity will make the graph more accurate.

WPP#4: Unit E Concept 3: Maximum Area

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SP#1: Unit E Concept 1: Quadratic

This problem is about putting a function, f(x)=x^2-6x+5, into the parent function equation and using the parent form to find the vertex, axis of reflection, y intercept, and x intercepts. To solve the function, we must first use the 'completing the square' method (as shown on the left side of the image) to factor the equation out. This will make it easier to put it in the parent function equation (as shown on the right side of the image) and subsequently find the needed values.
One thing you, dear viewer, might want to keep an eye on is the x intercept values. Make sure you do the math carefully and don't end up with a strange square root or an imaginary number (neither of which are in the answers.) Don't forget that there are two x intercepts as well, unlike the singular y intercept we all know and love.

WPP#3: Unit E Concept 2: Path of Bullet

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

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WPP#1: Unit A Concept 6: Writing and Solving Linear Models

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