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MathPresentation MathPresentation Presentation Transcript

  • Lights. Cameras. Action!
  • Hi, my name is Nica and I’m a performance arts teacher. This is my first year teaching at this school and I am a little bit nervous for what this year may have in store for me. Hopefully with your help, my first year here will be a success. Smoking Sanity Inn by flickr user: °<~ Gabrielle Sinatra ~>°
  • Feel Music by flickr user: XŤЯΣΛМ i Every year the school holds a spring concert for the students to perform. Since this is my first year teaching, I’d like to do something both the students and I will enjoy. I asked the students in my class if they would like to do a musical or a play and out of the 50 students, 39 of them would prefer to do a musical. However, how would the rest of the high school students react to a musical? Would they enjoy watching a musical? If I surveyed another 750 students, how certain can I be that the results will be approximately the same?
  • Finding the mean and the standard deviation To answer this question, we first need to know if : np > 5 and nq > 5 Where: n = number of trails p = probability of success q = probability of failure If np or nq are not equal or greater than five, we will not be able to find the mean and the standard deviation that's needed to answer the rest of the problem.
    • n = 750 students
    • p = 39/50 = 0.78
    • Probability of success is 78% because out of the 50 students, 39 of them said yes to a musical
    • q = 11/50 = 0.22
    • Probability of failure is 22% because out of the 50 students, 11 of them said no to a musical.
    np > 5 ? 750(0.78) > 5 = 585 > 5 nq > 5 ? 750(0.22) > 5 = 165 > 5 Calculations p and q always equal to one in the same event. Finding the mean and the standard deviation
  • The formula to find the mean is : The formula to find the mean, is the same formula we used to see if np was greater or equal to 5 . Now that we know both np and nq are both greater than five, we can now find the mean and the standard deviation . Calculations :
  • The formula to find the standard deviation is : Calculations : Now that we know both np and nq are both greater than five, we can now find the mean and the standard deviation .
  • Finding the margin of error The formula to find the margin of error is:
    • We use the number 1.96 because between the two z-scores, -1.96 and +1.96, you’ll find that they will give you exactly 0.95 of all data in a standard normal distribution.
    Calculations :
  • Now that we have the margin of error, we can find the 95% confidence interval using the formula : Using the 95% confidence interval method, we’ll help Nica be 95% certain of the number of students who will enjoy watching a musical because they will be included in the interval. Finding the 95% confidence interval
  • Calculations : 95% confidence interval
    • Notice that we dropped the decimal places because since we’re working with people, we cannot have half a person. Therefore, we use whole numbers as our final answer.
  • Final Answer Day 47/366.....Disney's Aladdin: A Musical Spectacular by flickr user: ldcross Nica can be 95% confident that from 576 to 594 students, will enjoy watching a high school musical with a margin of error of 4.211.
  • Now that I am confident that the school will enjoy a musical, the musical I have chosen for the students to perform is called Face-2-Face by flickr user: °<~ Gabrielle Sinatra ~>° A Deal With the Devil
  • The classroom that I am working with is broken up into 4 groups, each group consists of : 15 choir members 13 band members 12 dancers There will also be 10 students from the drama class who also be involved in the musical. And I never want to say goodbye by flickr user: °<~ Gabrielle Sinatra ~>°
  • That means that I’ll need to order in 50 different costumes for the students to wear, assuming that each group member will be wearing the same costume. I looked through two different catalogues and noticed that each store has the same costumes that I’m looking for but at different prices. Harajuku fashion - gothic & lolita by flickr user: Suviko
  • Looking at the prices above, which store would be cheaper if I ordered all the costumes from one store? What is the total cost of all the costumes in both the stores? Harajuku fashion - gothic & lolita by flickr user: Suviko Store A Store B Choir’s Costume $ 15.00 $ 13.49 $ 11.50 $ 12.50 Band’s Costume Dancer’s Costume Drama’s Costume $ 19.25 $ 24.99 $ 17.45 $ 20.95
  • Looking at the prices above, which store would be cheaper if I ordered all the costumes from one store? What is the total cost of all the costumes in both the stores? To solve this problem, we will be using a method known as matrices . Harajuku fashion - gothic & lolita by flickr user: Suviko Store A Store B Choir’s Costume $ 15.00 $ 13.49 $ 11.50 $ 12.50 Band’s Costume Dancer’s Costume Drama’s Costume $ 19.25 $ 24.99 $ 17.45 $ 20.95
  • Matrices is a way of organizing numbers in a table so that calculations can be made. Spiekermann House Numbers by flickr user: Stewf
  • Our first matrix we’ll set up, will be the number of students in the four different groups, which are: 15 choir members 13 band members 12 dancers 10 drama students Before we begin, we first need to set up two different matrices.
  • Our first matrix we’ll set up, will be the number of students in the four different groups, which are: 15 choir members 13 band members 12 dancers 10 drama students Choir Band Dancers Drama 15 13 12 10 # of students S =
  • Our first matrix we’ll set up, will be the number of students in the four different groups, which are: 15 choir members 13 band members 12 dancers 10 drama students
    • Features
    • The dimensions is 1 x 4.
    • This means that it has 1 row and 4 columns
    • (number of rows always goes first when stating the dimensions).
    • Square brackets are always used to represent a matrix
    • This matrix is labeled “S” which stands for the students
    Choir Band Dancers Drama 15 13 12 10 # of students S =
  • Our second matrix will be the cost of the costumes , using the table below Store A Store B Choir’s Costume $ 15.00 $ 13.49 $ 11.50 $ 12.50 Band’s Costume Dancer’s Costume Drama’s Costume $ 19.25 $ 24.99 $ 17.45 $ 20.95
  • Our second matrix will be the cost of the costumes , using the table below Store A Store B Choir’s Costume $ 15.00 $ 13.49 $ 11.50 $ 12.50 Band’s Costume Dancer’s Costume Drama’s Costume $ 19.25 $ 24.99 $ 17.45 $ 20.95 Store A Store B Choir 15.00 13.49 11.50 12.50 Band Dancers Drama 19.25 24.99 17.45 20.95 P=
  • Our second matrix will be the cost of the costumes , using the table below
    • Features:
    • This matrix is labeled “P” for the prices of the costumes.
    • The dimensions are 4 x 2.
    Store A Store B Choir’s Costume $ 15.00 $ 13.49 $ 11.50 $ 12.50 Band’s Costume Dancer’s Costume Drama’s Costume $ 19.25 $ 24.99 $ 17.45 $ 20.95 Store A Store B Choir 15.00 13.49 11.50 12.50 Band Dancers Drama 19.25 24.99 17.45 20.95 P=
  • Now that we have both matrices set up, we can use the method matrix multiplication , to determine the total cost of the costumes in both the stores. However, when dealing with matrix multiplication, there are a few rules to keep in mind. Choir Band Dancers Drama 15 13 12 10 # of students S = Store A Store B Choir 15.00 13.49 11.50 12.50 Band Dancers Drama 19.25 24.99 17.45 20.95 P=
  • Rule #1 - The number of columns in matrix S, must be equal to the number of rows in matrix P. 1 x 4 4 x 2 the same [S] [P]
  • Rule #1 Rule #2
    • When setting up your matrices, make sure you label both the matrices in the correct order .
    • The order that we’ll be using is :
    1) choir 2) band 3) dancers 4) drama - The number of columns in matrix S, must be equal to the number of rows in matrix P. 1 x 4 4 x 2 the same [S] [P]
  • Rule #1 Rule #2
    • When setting up your matrices, make sure you label both the matrices in the correct order .
    • The order that we’ll be using is :
    1) choir 2) band 3) dancers 4) drama Rule #3 - Therefore the product matrix will have the dimensions of :
    • The dimensions of the product matrix , which is the final answer, will be determined by the number of rows in matrix S and the number of columns in matrix P .
    1 x 2 [S] [P] - The number of columns in matrix S, must be equal to the number of rows in matrix P. 1 x 4 4 x 2 the same [S] [P] 1 x 4 4 x 2
  • Now that we have the basics down, we can use matrix multiplication to determine the total cost of the costumes in both the stores. Matrix multiplication involves both addition and multiplication .
  • Store A Store B Choir 15.00 13.49 11.50 12.50 Band Dancers Drama 19.25 24.99 17.45 20.95 P= Step 1: Take the entire row in matrix S and entire column of Store A, in matrix P . Choir Band Dancers Drama 15 13 12 10 # of students S =
  • Calculations : = ( 15 x 12.50 ) + ( 13 x 15.00 ) + ( 12 x 19.25 ) + ( 10 x 24.99 ) 187.50 195.00 231.00 249.90 = + + + Total cost of costumes in Store A = $ 863.40 Store A Store B Choir 15.00 13.49 11.50 12.50 Band Dancers Drama 19.25 24.99 17.45 20.95 P= Choir Band Dancers Drama 15 13 12 10 # of students S =
  • Choir Band Dancers Drama 15 13 12 10 # of students S = = ( 15 x 12.50 ) + ( 13 x 15.00 ) + ( 12 x 19.25 ) + ( 10 x 24.99 ) 187.50 195.00 231.00 249.90 = + + + As you can see, I’m taking the 1 st number in matrix S and multiplying it by the 1 st number in matrix P. Calculations : Total cost of costumes in Store A = $ 863.40 Store A Choir 15.00 12.50 Band Dancers Drama 19.25 24.99 P=
  • Choir Band Dancers Drama 15 13 12 10 # of students S = = ( 15 x 12.50 ) + ( 13 x 15.00 ) + ( 12 x 19.25 ) + ( 10 x 24.99 ) 187.50 195.00 231.00 249.90 = + + + Total cost of costumes in Store A = $ 863.40 Then I take the 2 nd number in matrix S and multiply it by the 2 nd number in matrix P. Calculations : Store A Choir 15.00 12.50 Band Dancers Drama 19.25 24.99 P=
  • Choir Band Dancers Drama 15 13 12 10 # of students S = = ( 15 x 12.50 ) + ( 13 x 15.00 ) + ( 12 x 19.25 ) + ( 10 x 24.99 ) 187.50 195.00 231.00 249.90 = + + + Total cost of costumes in Store A = $ 863.40 Calculations : And so on and so on, all the way to the 4 th number in both matrices Store A Choir 15.00 12.50 Band Dancers Drama 19.25 24.99 P=
  • Choir Band Dancers Drama 15 13 12 10 # of students S = 187.50 195.00 231.00 249.90 = + + + Total cost of costumes in Store A = $ 863.40 Calculations : Once all the multiplying is done, I add up all the answers I got, to get the total cost of the costumes in Store A. Store A Choir 15.00 12.50 Band Dancers Drama 19.25 24.99 P=
  • Store A Store B Choir 15.00 13.49 11.50 12.50 Band Dancers Drama 19.25 24.99 17.45 20.95 P= Now we need to find the total cost of the costumes in Store B. Which is exactly the same thing as you did before, the only difference now is that you’re going to take the entire column of Store B, in matrix P . Choir Band Dancers Drama 15 13 12 10 # of students S =
  • Calculations : = ( 15 x 11.50 ) + ( 13 x 13.49 ) + ( 12 x 17.45 ) + ( 10 x 20.95 ) 172.50 175.37 209.40 209.50 = + + + Total cost of costumes in Store B = $ 766.77 Store A Store B Choir 15.00 13.49 11.50 12.50 Band Dancers Drama 19.25 24.99 17.45 20.95 P= Choir Band Dancers Drama 15 13 12 10 # of students S =
  • The Product Matrix 863.40 766.77 Store A Store B Total Cost
  • Final Answer Nica should buy all the costumes from Store B because she’ll only need to spend $766.77, saving her $96.63. Una's dress by flickr user: °<~ Gabrielle Sinatra ~>°
  • The musical, A Deal With the Devil, has three main characters. This means that 3 lucky students from the drama class will be given the leading roles. Stella & Cynthia by flickr user: °<~ Gabrielle Sinatra ~>°
  • If the drama class consists of 10 students, how many combinations can Nica choose 3 students to be given the leading roles? Friday night II by flickr user: °<~ Gabrielle Sinatra ~>°
  • To answer this question, we’ll be using a formula known as the “ choose formula ” which looks like: n! ( n - r )! r!
  • Where r = the # of objects that are being arranged n = the # of objects that are available to be arranged ! = symbol used for factorial notation n! ( n - r )! r!
    • Factorial Notation is when we want to multiply all the natural numbers from a particular number down to 1.
    • ex) 5! Is the same as 5 x 4 x 3 x 2 x 1
    n! ( n - r )! r!
  • Now that we have our formula, we plug in what we know. r = 3 n= 10 Calculations : = = = 120 different ways n! ( n - r )! r! 10! ( 10 - 3 )! 3! 10! 7! x 3!
  • There are 120 different ways for Nica to choose 3 students for the leading roles from the 10 drama students. Final Answer SLartist Idol @ mylife.com by flickr user: °<~ Gabrielle Sinatra ~>°
  • Although the spring concert is months away, I need to start thinking about other expenses such as the cost of costumes, hiring make-up artists, hairdressers, electricians, lighting, props and advertisements. Hands/Make up by flickr user: spaceodissey
  • To help me pay these expenses, the bank loans me $3000.00 at an interest rate of 8.4% compounded monthly. If I want to pay back the loan before the year ends, how much should my monthly payments be? Money! By flickr user: Tracy O
  • To answer this question, we will be using a function on a T1-83 Plus graphing calculator , known as the TVM Solver . To get this function, you press the 1) [ APPS ] button 2) [ enter ] 3) [ #1: TVM_Solver ]
  • Once you’re done that, you should get a screen that looks something like: N: I%: PV: PMT: F/V: P/Y: C/Y: PMT: END BEGIN
  • where : N = total number of payments I%: interest rate as a percent PV: present value PMT: payments made FV: future value P/Y: # of payments/yr. C/Y: # of compounding periods/yr. N: I%: PV: PMT: F/V: P/Y: C/Y: PMT: END BEGIN
  • Now that we know what each valuable stands for, we can begin to fill out what we know by taking apart the question. To help me pay these expenses, the bank loans me $3000.00 at an interest rate of 8.4% compounded monthly . If I want to pay back the loan before in one year , how much should my monthly payments be?
    • We left PMT blank because that’s what we’re trying to solve for.
    • FV is zero because Nica doesn’t want to owe anything at the end of the one year.
    • N: 12
    • I%: 8.4%
    • PV: 3000
    • PMT:
    • F/V: 0
    • P/Y: 12
    • C/Y: 12
    • PMT: END BEGIN
    • First step, make sure the cursor is set on PMT.
    • N: 12
    • I%: 8.4%
    • PV: 3000
    • PMT:
    • F/V: 0
    • P/Y: 12
    • C/Y: 12
    • PMT: END BEGIN
    • First step to solve this problem on your calculator, is make sure the cursor is set on PMT.
    Second step, press the buttons [ ALPHA ] then [ ENTER ]
    • N: 12
    • I%: 8.4%
    • PV: 3000
    • PMT:
    • F/V: 0
    • P/Y: 12
    • C/Y: 12
    • PMT: END BEGIN
    • First step to solve this problem on your calculator, is make sure the cursor is set on PMT.
    • Second step, you press the buttons on your calculator
    [ ALPHA ] then [ ENTER ] . . . And the answer you should get is
    • You get a negative answer because that’s money that you’ll be losing every month.
    - 261.52
    • N: 12
    • I%: 8.4%
    • PV: 3000
    • PMT:
    • F/V: 0
    • P/Y: 12
    • C/Y: 12
    • PMT: END BEGIN
  • Final Answer Nica will be making monthly payments of $261.52 for the next 12 months to pay back her loan of $3000.00. (creepy) objectivist institute in SL by flickr user: melmcbride
  • The clock has struck 3:30 and the bell has rang, dismissing the students of the school. It’s been a long day but I’ve made it through =) Thank you for your help, if it wasn’t for you, I’d still be struggling with my first problem ;) Be sure to come back to watch our musical, Deal With the Devil, this coming spring 2008. Farewell. Tired... b y flickr user: °<~ Gabrielle Sinatra ~>°
  • All the pictures were taken off of www.flickr.com toy camera tower by: .m for mattijs Gerber RECON Flashlight by: JoeBGood by: Andyrob
  • the tools by flickr user: andreas_krautwald
  • Pro(i)mo(ptic) stuff II by flickr user: °<~ Gabrielle Sinatra ~>° Pro(i)mo(ptic) stuff by flickr user: °<~ Gabrielle Sinatra ~>°