What s up? Further Mathematics Past-student perspective. Hayley Short. Motivation. Studying Process. Study Techniques. But don t stop there!

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1 1 What s up? Reflect on today s VCE Seminar Further Mathematics Past-student perspective Hayley Short Recent VCE Graduate Think about what you want to achieve from today / Further / VCE / life in general Take a moment to say hi! Hayley Short Graduated in 2013 Studying at Monash University Bachelor of Business (Accounting) Seminars Director with Engage Motivation You re almost there! Know what motivates you: Competition? A specific end goal? Constructive fear? Harry Potter Nerd Engage Education Foundation Studying Process Read Text Highlight Key Books Points Chapter Summary Study Techniques How can I study effectively to score well on exams? Practice Exams Read all notes multiple times Questions from text book But don t stop there! H O W Learn from your mistakes: Correct your work Work out where you lost marks Re-write those answers until they are the mark you want to achieve Re-read that response to remind yourself Find similar questions and make sure you can do them Mark question types you tend to lose marks on I L E A R N T T H E N

2 2 H O W Studying - Procrastination I L E A R N N O W DISTRACTIONS!!! Know What Works for YOU! Quiet or Music? At home or at school? Cold or hot? Hungry or full? Rewards or punishment? Facebook or a run? Cold shower or no snacks? Time Management It s super important. But actually. Engage Education Foundation What to do now? Exam 1 is on October 30 Two Weeks to Exam That is just 19 days away! How can you maximise the time you have left?

3 3 One Week to Exam Exam Week Study timetable Be honest Be specific Time Management Other Tips Have an overall plan Write to do lists every day Make time for yourself Commit to it Make ALL subjects a priority Exams Unavoidable Not that scary really Engage Education Foundation Know the exam What questions come up every year? - VCAA Exam : Variable data includes distance, sex, number of children, type of car, postcode How many are categorical? 0, 1, 2, 3 or 4? Where can you find exams? Before the exam Pack the night before (make sure you have a fully charged calculator!) Check the details (where is it, what time?) Read over your bound reference Don t speak to people who will stress you out: Did you see the exam that said? I heard the examiners are including a section about In the exam Don t waste your reading time- test how to best use these precious 15 minutes. Read carefully: Questions with instructions (number of decimal places) How many marks is that? Highlight or underline key information

4 The Engage Wiki Make sure you check it out! 12 subjects 150,000 words of notes 25+ hours of video Top tips from top scores (video series) Cheat sheets for every SAC and the EXAM Topic specific VCAA questions (like checkpoints) 75 FREE practice exams ATAR calculator VCE Explained video series 30 short videos answering all your questions e.g. How is my study score calculated Free practice exams go to wiki.ee.org.au More subjects Question Time Questions? us at me at Post on the Engage Facebook page The Engage Team will answer your questions Favourite Harry Potter book? Some final words DO NOT BE MR BEAN! READ the question carefully ANSWER what the question asks LEARN from your mistakes Ask for help: Your teacher Your tutor Your friends Me! Further Maths FURTHER MATHEMATICS CORE Data Analysis Emily Condos Sunday, October 11 th

5 5 CLASSIFYING DATA Data can be classified as Categorical or Numerical. Categorical data involves names or labels. Numerical data involves numbers. Data which involves counting is called Discrete. Data which involves measuring is called Continuous. Categorical eg AFL team you support CLASSIFYING DATA Categorical Data Some categorical data can appear numerical: eg the TV channels use numbers for their names Channel 7 and Channel 9 also, you all probably have student numbers at your school eg However, these numbers are not used for calculations, so they are Categorical Data. DATA Numerical Discrete Continuous eg number of brothers eg height Numerical Data Discrete data can only take certain values within a range remember: COUNTING whilst Continuous data can possibly take all values within a range. remember: MEASURING CLASSIFYING DATA UNIVARIATE DATA Data which describes one variable. What you have to be able to do: 1. Display the data. 2. Interpret a data display. 3. Calculate Univariate statistics. VCAA Exam 1, DISPLAYING UNIVARIATE DATA DISPLAYING UNIVARIATE DATA Questions asking you to display data in a certain way can only be asked on Exam 2. Bar Graph Dot Plot Histogram Box Plot Segmented Bar Graph Stem Plot Frequency Table Number of children Number of families VCAA Exam 2, DISPLAYING UNIVARIATE DATA Questions asking you to display data in a certain way can only be asked on Exam 2. DISPLAYING UNIVARIATE DATA Questions asking you to display data in a certain way can only be asked on Exam 2. VCAA Exam 2, The value for Canada in this question was in a table on the page before. It was 16. VCAA Exam 2, 2010.

6 DISPLAYING UNIVARIATE DATA Questions asking you to display data in a certain way can only be asked on Exam 2. You are not asked to draw big, complex diagrams in this section so do not spend your revision time doing this type of activity. UNIVARIATE DATA Data which describes one variable. What you have to be able to do: 1. Display the data 2. Interpret a data display 3. Calculate Univariate statistics INTERPRETING UNIVARIATE DATA DISPLAYS 1. Read the heading to the question carefully. 2. Read the labels on the axes or the headings of a table carefully. 3. Read the units on the axes carefully. 4. Look at the Key if there is one. VCAA Exam 1, VCAA Exam 1, Segmented Bar Graphs VCAA Exam 1, Segmented Bar Graphs VCAA Exam 2, Segmented Bar Graphs VCAA Exam 1, INTERPRETING UNIVARIATE DATA DISPLAYS Shape of the distribution NEGATIVELY SKEWED SYMMETRICAL POSITIVELY SKEWED Stem Leaf Stem Leaf Stem Leaf INTERPRETING UNIVARIATE DATA DISPLAYS Stemplots Stem Leaf Stem Leaf Key: 2 4 = 24 Key: 2 4 = 2.4 This stemplot displays the values 7, 12, 22, 24, 30, 32, 33, 44, 47, 48, 49, 49, 52, 57, 58, 58, 58, 59, 61, 63, 65, 67. This stemplot displays the values 0.7, 1.2, 2.2, 2.4, 3.0, 3.2, 3.3, 4.4, 4.7, 4.8, 4.9, 4.9, 5.2, 5.7, 5.8, 5.8, 5.8, 5.9, 6.1, 6.3, 6.5,

7 7 CALCULATING UNIVARIATE STATISTICS MODE the value which occurs most often the value which has the highest column the value which has the greatest frequency Stem Leaf Key: 2 4 = 240 What is the mode? CALCULATING UNIVARIATE STATISTICS MEDIAN the middle value of the ordered distribution 50% of the distribution is less than the median 50% of the distribution is greater than the median LOWER QUARTILE (Q 1 or Q L ) middle of the lower half of the distribution 25% of the distribution is less than the median UPPER QUARTILE (Q 3 or Q L ) middle of the upper half of the distribution 25% of the distribution is greater than the median INTER QUARTILE RANGE (IQR) = UPPER QUARTILE LOWER QUARTILE RANGE = MAXIMUM VALUE MINIMUM VALUE VCAA Exam 1, MIN MEDIAN MAX MIN MEDIAN MAX LOWER QUARTILE UPPER QUARTILE LOWER QUARTILE UPPER QUARTILE Median = 26 IQR = = 14 Range = = 22 7 values. Middle is at the position. 8 th Median = ( ) 2 = IQR = = 161 Range = = values. Middle is at the position th VCAA Exam 1, VCAA Exam 1, values. Middle is at the position. That is a 2. th VCAA Exam 1, VCAA Exam 1, 2011.

8 8 INTERPRETING UNIVARIATE DATA DISPLAYS Box Plots One very important fact about box plots is that 25% of the values lie in each section. This is easy to consider for a symmetric box plot, but is also true for a skewed box plot. VCAA Exam 1, VCAA Exam 1, VCAA Exam 1, INTERPRETING UNIVARIATE DATA DISPLAYS Box Plots Another important aspect of box plots are OUTLIERS. Outliers are values which are a considerable distance below or above the other scores. They are often the result of data input error. Mathematically, we test MIN MEDIAN MAX IQR = = 14 IQR 1.5 = 21 LOWER QUARTILE UPPER QUARTILE So, to be an outlier, a value would have to be less than = 2 or greater than = 54. Clearly, none of these values are outliers. CALCULATING UNIVARIATE STATISTICS MEAN also called the AVERAGE add up all the values and divide the total by the number of values STANDARD DEVIATION calculate the SD using your CAS calculator measures the spread of values about the mean especially significant for a bell shaped distribution The % rule for a bell shaped curve states that approximately: (a) 68% of data lie within 1 standard deviation either side of the mean (b) 95% of the data lie within 2 standard deviations either side of the mean (c) 99.7% of data lie within 3 standard deviations either side of the mean. VCAA Exam 1, 2008.

9 9 VCAA Exam 1, Learn to use the Stats List Editor on your CAS calculator to find: Mean *** Standard Deviation *** and Median, Q1, Q3, min, max etc. VCAA Exam 1, When a distribution is symmetrical, like a bell shaped distribution, both the mean and the median are good measures of the central value. If the distribution is skewed, the median becomes the better measure of a central value. The mean loses its reliability. Z scores Another statistic which comes from the bell shaped curve is a Z score. Z scores (also known as standardised scores) are used to compare values from different distributions. A z score of 0 indicates the value is equal to the mean. A z score of 1 indicates a value equal to 1 standard deviation above the mean. A z score of 1 indicates a value equal to 1 standard deviation below the mean and so on... A z score is calculated using the formula ( ) VCAA Exam 2, VCAA Exam 2, BIVARIATE DATA Data which describes two variables. Bivariate data can be displayed using Back to Back Stem Plots Parallel Box Plots BIVARIATE DATA In Exam 2, students are often asked to compare the two or more data statistics in bivariate and multivariate data. When you do so it is important to: 1. Make a comparison statement eg is greater than and 2. Support your statement with specific values. a numerical variable and a categorical variable with 2 categories. a numerical variable and a categorical variable with at least 2 categories. VCAA Exam 2, VCAA Exam 2, 2006.

10 10 Scatterplots Used to display the relationship between two numerical variables Visitors Temperature ( C) Number of swimmers at Prahran Pool and the corresponding daily temperature. Before graphing, identify the independent variable and the dependent variable. Reasoning: It is reasonable to expect that the number of visitors at the swimming pool on any day will depend on the temperature on that day (and not the other way around). Answer: Daily temperature is the independent variable; Number of visitors is the dependent variable. Number of visitors at Prahran pool Each dot on a scatterplot represents 2 values. eg the circled dot represents 30 months old and 92 cm tall. Daily temperature Positive scatterplot is rising left to right. Negative scatterplot is falling left to right. Strong dots are lined up. The relationship between two variables is called the correlation. When describing the correlation, you need to comment on Strength strong, moderate or weak Direction positive or negative Form linear or non linear Outliers. From this scatterplot, age and height could be described as having a Moderate to strong, positive, linear correlation. One outlier is present. Moderate dots are tightly clustered around an imaginary line. Weak dots are widely clustered around an imaginary line. A positive correlation between two variables, as above, can be described by a sentence that reads: As the age increases, the height increases. Remember, the height is the dependent variable, so it depends on the changes in the independent age variable, if there is a good correlation. Here is a scatterplot from which it appears that there is a moderate, positive, non linear correlation between income and average age. Of woman at first marriage. No outliers are completely obvious. So, again because the scatterplot shows a positive relationship we could write: As the income increases, the average age of women at first marriage increases. Correlation, even when it is strong, does not mean that one variable CAUSES another. It simply tells us that there is a relationship between the two variables when they occur together. Be careful of Multiple choice options which imply one variable causes the other! Here is a scatterplot from which it appears that there is a moderate, negative, non linear correlation between television hours per week and homework hours per week. This time, because the scatterplot shows a negative relationship we could write: As the television hours per week increases the homework hours per week decrease. Again note that the dependent variable depends on the changes in the independent variable and not the other way around.

11 11 r value r value Called the Correlation Co efficient. Gives a numerical value to the correlation between two variables. Perfect positive linear correlation r = 1 No relationship r = 0 Perfect negative linear correlation r = 1 Learn to use your CAS calculator to find: r value The r value is used to make the description of the correlation between two variables more objective. However, the r value is only valid if the correlation is linear in nature. If it is non linear, we have to transform it before we can use statistics like the r value. r 2 value Called the Co efficient of Determination. Is calculated by squaring the r value, and so it is always a statistic between 0 and 1 (0% to 100%). Scatterplots An important statistical device we calculate for a scatterplot is a line of best fit. We call it a Regression Line. The line summarises the points in the scatterplot. It is only relevant to fit a line of best fit if the correlation is Linear. Again, is only valid in assessing bivariate data with a linear relationship. Is used in 2 ways: It tells us 1. The proportion of variation in one variable which can be explained by the variation in the other variable. OR 2. How well the linear rule linking the two variables can predict the value of the dependent variable when we are given the value of the independent variable. VCAA Exam 2, Scatterplots The line has the equation y = mx+ c OR y = c + mx. Scatterplots The line has the equation y = mx+ c OR y = c + mx. m is the slope of the line. It indicates the rate at which the dependent variable is increasing or decreasing in relation to the independent variable. c is the y intercept. It indicates the approximate value of the dependent variable when the independent variable equals 0. Scatterplots The equation can be used to calculate values for which data does not exist (predictions). If the predicted value is inside the range of the existing data, it is called an INTERPOLATION. If the predicted value is outside the range of the existing data, it is called an EXTRAPOLATION. Interpolations are more reliable predictions than Extrapolations.

12 12 Scatterplots FINDING THE EQUATION There are 3 ways to find the equation of the line which you have to learn MEDIAN METHOD a) from the Graph b) from the original data 2. LEAST SQUARES REGRESSION a) from the original data 3. From Summary Statistics VCAA Exam 2, FINDING THE EQUATION 3 MEDIAN METHOD (from the graph) The 3 Median Method is one way of finding a line of best fit for data which is linear in nature. It requires at least 6 data points to get a meaningful line. It is not adversely affected by a small number of outliers. Divide the points into 3 groups using vertical lines. (a) If the number of points is divisible by 3, divide them into 3 equal groups, eg 3, 3, 3 or 7, 7, 7. (b) If there is 1 extra point, put the extra point in the middle group, eg 3, 4, 3 or 7, 8, 7. (c) If there are 2 extra points, put 1 extra point in each of the outer groups, eg 4, 3, 4 or 8. Calculate the median x value of the points in each of the 3 groups (when moving left to right). Calculate the median y value of the points in each of the 3 groups (when moving bottom to top). FINDING THE EQUATION 3 MEDIAN METHOD Drawing the Line Place a ruler so that it passes through the lower and upper medians. Maintaining the same slope, slide the ruler one third of the way towards the middle median. Finding the Equation The equation of the line for the 3 Median method is in the form y = m x + c where and FINDING THE EQUATION 3 MEDIAN METHOD (from the original data) LEAST SQUARES REGRESSION Equation FINDING THE EQUATION From Summary Statistics To use the formulae, you would be given: Each of these equations for the Line of Best Fit can be found by entering the data into lists on your CAS calculator and using the appropriate options. The equations will be slightly different for each method for the same set of data because they use different approaches. The 3 Median Method is based on medians; the Least Squares Regression Equation is based on means. Since medians are less affected by outliers than means, the 3 Median Line is a better choice if there are outliers in the data. Both equations are in the form y = mx+ c OR y = c + m x. Learn to use your CAS calculator to find: 3 Median equation & Least Squares Regression equation x y the mean of the independent variable (x variable) the mean of the dependent variable (y variable) S x the standard deviation of the independent variable S y the standard deviation of the dependent variable r Pearson s product moment correlation coefficient. Finding the Equation The equation of the line for the 3 Median method is in the form y = m x + c where and VCAA Exam 1, EVALUATING THE LINEARITY OF THE DATA Because our statistics are not relevant if the distribution of data is not linear, it is important to a) Check how linear our data is b) Try and improve the linearity of our data so that we can have greater confidence in the predictions and conclusions we make from the data. The closer the r value is to 1 or to 1, the more linear the data is. The closer the r 2 value is to 1, the more linear the data is.

13 13 EVALUATING THE LINEARITY OF THE DATA Residual Analysis EVALUATING THE LINEARITY OF THE DATA Residual Analysis Residual Analysis is used to further check whether the nature of the data is linear or not. A residual is the vertical difference between each data point and the regression line. each Residual = Actual Predicted y value y value (from the original data) (calculated from the equation) This gives a set of points, positive and negative, which we can graph and analyse. VCAA Exam 2, EVALUATING THE LINEARITY OF THE DATA Residual Analysis One of 3 patterns usually emerges: TRANSFORMING THE LINEARITY OF THE DATA If data appears to be non linear, a transformation may improve the linearity. The residuals are randomly above and below the x axis. Conclusion: Data is probably LINEAR. The residuals show a curved pattern ( ), with a series of negative, then positive and back to negative along the x axis. Conclusion: Data is probably NON LINEAR. The residuals show a curved pattern ( ), with a series of positive, then negative and back to positive along the x axis. Conclusion: Data is probably NON LINEAR. For this shape of Residual Analysis, y v log 10 x OR 1 y v x transformations may improve linearity. For this shape of Residual Analysis, y v x 2 transformation may improve linearity. Improvements in the r value or r 2 value would measure whether the linearity had been improved or not. Once a transformation has been done, you include the transformed variable in the equation. TRANSFORMING THE LINEARITY OF THE DATA If you have the original scatterplot of the data, a decision about which transformation is suitable could be made from the following table. r 2 = = % of the variation in height can be explained by variations in age. TIME SERIES A Time Series is an example of bivariate data where the independent variable is always some unit of time eg days, months or years.

14 TIME SERIES TIME SERIES Trends During your preparation for Exams DON T spend a lot of time drawing graphs. DO take time to understand the processes. TIME SERIES Fitting a Trend Line A Trend Line is like a line of best fit for a scatterplot. The same methods, 3 Median Method OR Least Squares Regression, are used to: Find and draw the line and/or To calculate its equation. X TIME SERIES Fitting a Trend Line A Trend Line has an equation which is also in the form y = mx+ c OR y = c + mx. A Time Code is usually applied to the time variable to make calculations with the equation possible. X X VCAA Exam 1, VCAA Exam 1, TIME SERIES Smoothing Time period Rainfall (mm) 3 point median 5 point median Median Smoothing The medians are found from successive groups of medians and lined up against the middle of the group VCAA Exam 1, VCAA Exam 1, VCAA Exam 1,

15 15 TIME SERIES Deseasonalising TIME SERIES Deseasonalising = Deseasonalised Actual Value Value Seasonal Index The reason for collecting data over a period of time is to attempt to forecast the future using current trends. This new data has the seasonal influences removed and underlying trends can be revealed eg increasing or decreasing over time. Seasonalised Deseasonalised Seasonal Value Value Index (Forecast prediction) = x VCAA Exam 1, VCAA Exam 1, TERMINOLOGY FURTHER MATHEMATICS Matrices TERMINOLOGY

16 16 BASIC CALCULATIONS MATRIX MULTIPLICATION this is 1 3 X X 3 4 because the two numbers in the centre do not match MATRIX MULTIPLICATION MATRIX MULTIPLICATION SOLVING SYSTEMS OF EQUATIONS SOLVING SYSTEMS OF EQUATIONS

17 17 SOLVING SYSTEMS OF EQUATIONS TRANSITION and STATE MATRICES TRANSITION and STATE MATRICES

18 18 FURTHER MATHEMATICS Linear Graphs & Relations VCAA EXAM 2, 2008 THE LINEAR EQUATION y = mx + c, The general equation of a straight line is where m is the gradient (slope) and c is the y intercept. In real life situations, the gradient represents the change (increase or decrease) in y, as x increases by 1 unit. The steeper the line, the greater the rate of change. VCAA Exam 2, 2008.

19 19 THE LINEAR EQUATION The y intercept is the value of y where the graph cuts the y axis. When applied to real life situations, the y intercept often represents the initial (or original) value of something. Parallel lines have the same gradient, but different y intercepts. VCAA Exam 1, THE LINEAR EQUATION VCAA Exam 2, VCAA Exam 1, THE LINEAR EQUATION SKETCHING LINEAR GRAPHS VCAA Exam 1, Gradient Intercept method This method is used if the equation is in y = mx + c form. 1. The first point plotted is the y intercept, given by the value of c. Plot it on a set of axes. 2. The gradient is given by m. Write the gradient as a fraction and identify the values of the rise and the run. 3. To obtain the second point, start from the y intercept and move up (or down) and across, as suggested by the gradient. 4. Join the two points together with a straight line and label the graph. VCAA Exam 1, SKETCHING LINEAR GRAPHS x and y intercept method This method is used if the equation is in ax + by = c form, or if you are required to show both the x and y intercepts. VCAA Exam 1, If a point is on the y axis, its x coordinate is 0. To find the y intercept, substitute 0 for x and solve the resultant equation. 2. If a point is on the x axis, its y coordinate is 0. To find the x intercept, substitute 0 for y and solve the resultant equation. 3. Plot the x intercept and the y intercept on a set of axes. 4. Join the two points together with a straight line and label the graph.

20 20 SKETCHING LINEAR GRAPHS Sketching a line over a required interval If a graph needs to be sketched between two given x values, its end points must be shown. Since only two points are needed to sketch a line, we can obtain the coordinates of the end points and join them together. To construct a graph of a straight line between a and b, follow these steps: 1. Rearrange the equation to make y the subject. 2. Substitute each of the two given x values (that is, a and b) into the equation and find corresponding values of y. 3. Plot the end points on a set of axes. 4. Join the two points together with a straight line and label the graph. VCAA Exam 1, A PAIR OF LINEAR GRAPHS A PAIR OF LINEAR EQUATIONS VCAA Exam 1, VCAA Exam 1, When would 2 equations have no simultaneous solution? A: If they are parallel. A PAIR OF LINEAR GRAPHS Break Even Analysis is an application of Simultaneous Equations. 1. Making a profit depends on the costs associated with the business (labour, raw materials and plant) and its revenue (the money it earns through sales). Profit = Revenue Costs 2. The break even point occurs when Costs = Revenue. 3. Graphically, the break even point is the point of intersection of the straight lines representing costs and revenue. 4. Neither a profit nor a loss is made at the break even point. VCAA Exam 2, Break Even Analysis LINE SEGMENT GRAPHS VCAA Exam 1, 2010.

21 21 STEP GRAPHS VCAA Exam 1, NON LINEAR GRAPHS VCAA Exam 1, POWER GRAPHS to LINEAR GRAPHS When dealing with the following group of questions, 1. Check the axes carefully 2. Use values which have been given to you as point s co ordinates. VCAA Exam 1, VCAA Exam 1, 2008.

22 22 greater than or equal to less than or equal to LINEAR INEQUATIONS > greater than < less than 1. Linear inequations are in the form ax + by < c, where the < sign can also be >, or. 2. The solution to a linear inequation is a region (or half plane) either above or below the graph of the corresponding linear equation. 3. To determine the solution region follow these steps. (a) Plot the corresponding linear equation (boundary). (b) Pick a test point above this equation line. (c) If the point satisfies the linear inequation, then the solution region is above the line, otherwise it is below the line. Shade the region not required. (d) Inequations with the signs < or > do not have the equation line as part of the solution region and are indicated with a dashed line. (e) Inequations with the signs or do have the equation line as part of the solution region and are indicated with a solid line. VCAA Exam 1, Always include a key indicating the region required. VCAA Exam 2, LINEAR PROGRAMMING Linear Programming is an application of linear inequations where an objective function, often associated with profit or costs, is maximised or minimised given a set of constraints that limit the possible solutions. 1. Linear programming problems are made up of three components: (a) a set of decision variables (x and y) (b) a set of constraints on these variables, expressed as a set of simultaneous linear inequations (c) an objective function of these variables, which is either minimised or maximised. LINEAR PROGRAMMING 2. To solve the general linear programming problem: (a) define the decision variables (b) define the constraint inequations (c) graph the constraints as a set of linear inequations (d) determine the intersections (vertices) of the solution region (e) define the objective function and decide whether it is to be minimised or maximised (f) the best (or optimal) value of the objective function is at one of the vertices of the solution region, so substitute each vertex in turn and calculate the objective function. (g) select the values of x and y which optimise (minimise or maximise) the objective function. LINEAR PROGRAMMING 2. To solve the general linear programming problem: (b) define the constraint inequations (c) graph the constraints as a set of linear inequations. LINEAR PROGRAMMING You can be asked to write the inequations, which act as constraints (ie the conditions which limit the possible answers to the overall question). VCAA Exam 2, 2006.

23 LINEAR PROGRAMMING You can be asked to indicate the feasible region by shading. LINEAR PROGRAMMING (0, 8) (2.9, 5.8) Solve(20x+25y=200 and y=2x,{x,y}) Key: Feasible region (0, 8) (2.9, 5.8) Key: Feasible region LINEAR PROGRAMMING LINEAR PROGRAMMING You can be asked to write the objective equation ie the focus (object) of the overall question. This is often about maximising profit or minimising costs. The solution to the objective equation can only be found in the feasible region, most commonly at the vertices. However, be careful about whether only whole number answers make sense or whether decimal answers can be included. If the vertices only involve whole numbers, this concern doesn t occur. (0, 8) (2.9, 5.8) Key: Feasible region LINEAR PROGRAMMING (0, 8) (2.9, 5.8) Key: Feasible region LINEAR PROGRAMMING Notice you are being asked what is the maximum profit, not at what values do you find the maximum profit. READ THE QUESTION VCAA Exam 1, LINEAR PROGRAMMING The optimal solution is regularly found at the vertices of the feasible region. Test the objective function at each of the vertices. FURTHER MATHEMATICS Business related Maths 23

24 24 ACCURACY OF BUSINESS CALCULATIONS PERCENTAGES in BUSINESS CALCULATIONS To calculate X% of an amount we use X amount 100 To calculate amount A as a percentage of Amount B we use amount amount A B 100 % Discount A discount is an amount of money by which the price of an item is reduced. If expressed as a percentage of the original price, it is called a percentage discount. Discount ($) = Original price Sale price Discount ($) 100 Original Price ($) Percentage Discount = Capital Gain A capital gain (or loss) is the difference between the cost price and the selling price of an item. Capital gain ($) = Sale price Cost price Commission A commission is an amount of money which a salesman earns as a percentage of the sales he makes. CommissionRate Commission($) = TotalSales($) 100 VCAA Exam 1, 2006 VCAA Exam 1, 2007 PERCENTAGE INCREASES and DECREASES For example: A 3% increase multiply by 1.03 A 3% decrease multiply by 0.97 A 15% increase multiply by 1.15 A 15% decrease multiply by 0.85 A 60% increase multiply by 1.60 A 60% decrease multiply by 0.40 Inflation Inflation is a measure of the average increase in the price of goods and services from one year to the next. It is an example of percentage increase calculated year after year. VCAA Exam 1, 2011 $ = $162 Mon Tues Wed Thurs Fri & Sat PERCENTAGE INCREASES and DECREASES Goods & Services Tax (GST) Final Price of an item or service = the pre GST amount + the GST (what the customer (10% of the pays for the item) base price) The retailer collects the sale price, keeps the purchase price and passes the GST on to the Government Tax Office. VCAA Exam 1, 2011

25 25 Goods & Services Tax (GST) To calculate the GST from a Final Price of a item or Service, we divide the amount by 11. Not 10! Stamp Duty Stamp Duty is a tax levied on documents involved in the transfer of real estate, businesses, insurance policies, mortgages and motor vehicles. VCAA Exam 1, 2006 VCAA Exam 1, 2006 SIMPLE INTEREST Simple interest can be calculated in the case of : * an investment the investor is repaid the amount he invested plus interest * a loan the borrower must repay the amount borrowed plus interest The formula for calculating simple interest is I P r T 100 I = Simple interest earned on an investment or charged for a loan ($) P = Principal (amount of money invested or loaned) ($) r = Rate of interest per time period (% per period) T = Time, the number of periods (years, months, days) over which the agreement operates. NB The interest rate, r, and time period, T, must be stated and calculated in the same time terms. Total amount of loan or investment = Principal + Interest (charged or earned) A = P + I VCAA Exam 1, 2006 VCAA Exam 1, 2007 The SIMPLE INTEREST formula is used to calculate interest on BANK ACCOUNTS. The SIMPLE INTEREST formula is used to calculate interest on PERPETUITIES. A perpetuity is an investment which never decreases because as interest is earned it is given to the investor. Examples are pensions and scholarships. VCAA Exam 1, 2006 Use the minimum balance amount for the month as the Principal. VCAA Exam 1, 2006 The SIMPLE INTEREST formula is used to calculate interest on PERPETUITIES. The SIMPLE INTEREST formula is used to calculate interest on HIRE PURCHASE purchases. People buy on hire purchase when they cannot afford to buy the goods for cash. A deposit is usually paid and the balance is paid over a fixed period of time. The retailer arranges a contract with a financial institution and the purchaser pays regular installments including interest at a flat rate to the financial institution. A flat rate is the same as simple interest rate. VCAA Exam 1, 2006 The main stages of interest and total price calculations are: 1. Loan amount = price of goods deposit paid 2. Flat rate interest on the loan is calculated using the simple interest formula. 3. Instalment amount = 4. Total cost of goods = price of goods + interest

26 26 The main stages of interest and total price calculations are: 1. Loan amount = price of goods deposit paid 2. Flat rate interest on the loan is calculated using the simple interest formula. 3. Installment amount = 4. Total cost of goods = price of goods + interest VCAA Exam 1, 2008 VCAA Exam 1, 2006 VCAA Exam 1, 2009 COMPOUND INTEREST Compound interest differs from simple interest in that it is calculated on the current amount of an investment or loan, rather than on the original amount. COMPOUND INTEREST The formula for calculating compound interest is r A P ( 1 ) T 100 A = Total amount of an investment or a loan ($) P = Principal (amount of money invested or loaned) ($) r = Rate of interest per time period (% per period) T = Time, the number of periods (years, months, days) over which the agreement operates. NB The interest rate, r, and time period, T, must be stated and calculated in the same time terms. Amount of Interest ($) = Total amount of loan or investment Principal I = A P VCAA Exam 1, 2011 VCAA Exam 1, 2007 COMPOUND INTEREST EFFECTIVE INTEREST RATE In a simple interest loan, the amount borrowed reduces over the term of the loan, but the customer is still paying interest on the total initial loan amount. The effective interest rate is the equivalent reducing balance interest rate taken over the contract period. The formula for calculating the effective interest rate is VCAA Exam 1, 2008 where n is the number of payments. This makes the Effective interest rate a little less than 2 Simple interest rate. EFFECTIVE INTEREST RATE REDUCING BALANCE LOANS A Reducing Balance Loan, also known as an Annuity, involves the borrowing of a Principal, which is repaid with regular repayments which contain an amount of the principal + an amount of interest, which is calculated on the current balance of the loan. You will use the FINANCE($) app on the CAS calculator to do Reducing Balance Loan calculations. VCAA Exam 1, 2008 The FINANCE app screen shows the following headings: N = N is the number of time periods when interest is added I% = I is the interest rate per annum PV = PV is the Principal Value PMT = PMT is the payment contribution per time period FV = FV is the Final Value of investment at the end of the time P/Y = P/Y is the number of payment contributions per year C/Y = C/Y is the number of interest compounding per year PMT: END BEGIN END means the calculations are done at the end of each time period PV + (money coming TO you) PMT (money going AWAY from you) FV (money going AWAY from you)

27 27 ANNUITY INVESTMENTS An Annuity investment involves the regular (eg monthly or quarterly) payment of money into an account over an extended period of time. This investment earns interest continuously over the life of the investment, based on the current balance. The interest is added to the Principal as it is earned. Superannuation is an example of an Annuity investment. You will use the FINANCE($) app on the CAS calculator to do Reducing Balance Loan calculations. The FINANCE app screen shows the following headings: N = N is the number of time periods when interest is added I% = I is the interest rate per annum PV = PV is the Principal Value PMT = PMT is the payment contribution per time period FV = FV is the Final Value of investment at the end of the time P/Y = P/Y is the number of payment contributions per year C/Y = C/Y is the number of interest compounding per year PMT: END BEGIN END means the calculations are done at the end of each time period PV (money going AWAY from you) PMT (money going AWAY from you) FV + (money coming TO you) VCAA Exam 1, 2010 DEPRECIATION The estimated loss in value of assets is called depreciation. The original cost of an item is sometimes called its prime cost. The estimated value of an item at any point in time is called its book value. When the book value becomes zero, the item is said to be written off. At the end of an item s useful or effective life its book value is then called its scrap value. This may be zero but often is not. 3 Methods of calculating Depreciation 1. Flat Rate depreciation (also referred to as Straight Line depreciation and Prime Cost depreciation) 2. Reducing balance depreciation ( also called diminishing value depreciation.) 3. Unit Cost Depreciation Total depreciation = prime cost current value FLAT RATE DEPRECIATION Flat Rate Depreciation is a method which involves subtracting a fixed amount each time period (normally each year). The fixed amount can be a number of dollars OR a percentage of the cost price. BV T = P d T BV T = book value ($) after time, T P = cost price ($) or prime cost ($) T = time since purchase (years) d = fixed amount per year OR percentage of P per year Rate of depreciation = total depreciation number of years % rate of depreciation = annual amount of depreciation prime cost 100% VCAA Exam 1, 2010 REDUCING BALANCE DEPRECIATION If an item depreciates by the reducing balance method then its value decreases by a fixed percentage rate each time interval, generally each year. This rate is a percentage of the previous book value of the item. BV T = book value ($) after time, T P = cost price ($) or prime cost ($) T = time since purchase (years) r = rate of depreciation VCAA Exam 1, 2006

28 28 UNIT COST DEPRECIATION Unit Cost Depreciation is based on depreciating the value of an item based on its production eg depreciating a photocopier based on the number of copies it makes or depreciating a machine on the number of toys it produces or depreciating a van on the number of kilometres it travels. FURTHER MATHEMATICS Geometry & Trigonometry VCAA Exam 1, 2009 GEOMETRY Angles VCAA Exam 1, 2010 GEOMETRY Triangles VCAA Exam 1, 2011 GEOMETRY Quadrilaterals GEOMETRY Polygons

29 29 RIGHT ANGLE TRIANGLES PYTHAGORAS THEOREM In the Examiner s comments for this question, it was stated that many students assumed ACB was right angled. Also, many assumed XY = AY. a c (hypotenuse) c 2 = a 2 + b 2 Neither assumption was true. VCAA Exam 2, 2006 b a 2 = c 2 b 2 PYTHAGORAS THEOREM PYTHAGORAS THEOREM VCAA Exam 2, 2006 VCAA Exam 2, 2007 SOH CAH TOA VCAA Exam 1, 2010 MEASUREMENT FORMULAE VCAA Exam 1, 2010

30 30 MEASUREMENT FORMULAE VOLUME MEASUREMENT FORMULAE VOLUME MEASUREMENT FORMULAE VCAA Exam 1, 2010 SIMILAR TRIANGLES SCALE FACTOR Length In part (a) we were asked to show AW = 34 cm. SCALE FACTOR Area & Volume VCAA Exam 2, 2007 VCAA Exam 1, 2011 VCAA Exam 1, 2010

31 31 NON RIGHT ANGLED TRIANGLES NON RIGHT ANGLED TRIANGLES The SINE RULE Problems in which the Sine Rule should be used will involve 2 sides and 2 angles. OR NON RIGHT ANGLED TRIANGLES The SINE RULE NON RIGHT ANGLED TRIANGLES The COSINE RULE Problems in which the Cosine Rule should be used will involve 3 sides and 1 angle. AND To find a side length To find an angle VCAA Exam 1, 2011 NON RIGHT ANGLED TRIANGLES The COSINE RULE AREA FORMULAE for TRIANGLES VCAA Exam 2, 2008 BEARINGS VCAA Exam 1, 2010

32 32 ADVICE: VCAA Exam 1, 2006 VCAA Exam 2, 2008 ANGLES of ELEVATION and DEPRESSION ANGLES of ELEVATION and DEPRESSION The angle of elevation is the angle above the horizon. The angle of depression is the angle below the horizon. VCAA Exam 2, 2008 TRIANGULATION CONTOUR MAPS c c 20m 15 m = 15 m B Rise Slope Run VCAA Exam 2, 2011

33 33 UNDIRECTED GRAPHS and NETWORKS FURTHER MATHEMATICS Networks & Decision Mathematics The graph may be represented in an adjacency matrix form as: A B C D A B C D UNDIRECTED GRAPHS and NETWORKS A loop is considered as a single edge but it contributes 2 to the degree of the endpoint. UNDIRECTED GRAPHS and NETWORKS EULER S FORMULA

34 34 UNDIRECTED GRAPHS and NETWORKS Paths & Circuits UNDIRECTED GRAPHS and NETWORKS Paths & Circuits SHORTEST PATH UNDIRECTED GRAPHS and NETWORKS Trees DIRECTED GRAPHS Reachability DIRECTED GRAPHS Routes

35 35 DIRECTED GRAPHS Connectivity DIRECTED GRAPHS Dominance C Connectivity Matrix 0 C + C 2 = C 2 DIRECTED GRAPHS Network Flow AIM: find the maximum flow possible through a network Network Flow In each cut, only count the flows that are heading towards the sink side of the cut. eg In the diagram, for Cut 1 we should ignore flow BD; in Cut 2 we should ignore flow EB.

36 36 CRITICAL PATH ANALYSIS AIM: find the shortest time to complete a series of tasks. CRITICAL PATH ANALYSIS BIPARTITE GRAPHS AIM: to optimise the allocation of tasks A Bipartite Graph is a directed graph which has two distinct groups. Connections exist between the two groups, not within a group. BIPARTITE GRAPHS The Hungarian Algorithm

37 37 MATHS QUEST 12 Novak, Bakogianis, Boucher, Nolan, Phillips ARITHMETIC SEQUENCES 1. An arithmetic sequence is a sequence of numbers for which the difference between successive terms is the same. FURTHER MATHEMATICS Number Patterns 2. The first term of an arithmetic sequence is referred to as a. 3. The common difference between successive terms is referred to as d. 4. t n is the n th term; for example, t 6 refers to the 6th term in the sequence. To test whether a sequence of numbers is an arithmetic calculate t 2 t 1 and t 3 t 2 and so on through the sequence. If the answers are the same then it is an arithmetic sequence. eg 8, 15, 22, 29, To find the n th term of an arithmetic sequence we can use the formula t n = a + (n 1)d

38 38 FINDING THE RULE OF AN ARITHMETIC SEQUENCES VCAA Exam 1, 2006 Substitute your values into t n = a + (n 1)d and then simplify the equation. VCAA Exam 1, 2011 VCAA Exam 1, 2007 VCAA Exam 1, 2008 FINDING a and d if you know two terms Find a and d in an arithmetic sequence if the 4 th term is 132 and the 11 th term is 188. ADDING THE TERMS OF ARITHMETIC SEQUENCES The sum of an arithmetic sequence is called an arithmetic series. So 2, 5, 8, 11, 14 is an example of an arithmetic sequence is an example of an arithmetic series. To find the sum of a given number of terms of an arithmetic sequence we can use one of two formulae. If we know the first and last term and how many terms there are, we can use where S n = the sum of n terms n = the given number of terms a = the first term l = the last term ie the nth term. If we know the values of a and d and how many terms there are, we can use VCAA Exam 1, 2007 GEOMETRIC SEQUENCES 1. A geometric sequence is a sequence of numbers for which the first term is multiplied by a number, known as the common ratio, to create the second term which is multiplied by the common ratio to create the third term, and so on. 2. The first term of a geometric sequence is referred to as a. VCAA Exam 1, The common ratio between successive terms is referred to as r. 4. t n is the n th term; for example, t 6 refers to the 6th term in the sequence. VCAA Exam 1, 2011 To test whether a sequence of numbers is an geometric calculate t 2 t 1 and t 3 t 2 and so on through the sequence. If the answers are the same then it is a geometric sequence. eg 4, 12, 36, 108, To find the n th term of an geometric sequence we can use the formula t n = a r (n 1) ADDING THE TERMS OF GEOMETRIC SEQUENCES The sum of an geometric sequence is called an geometric series. To find the sum of n terms of a geometric sequence, we can use the formula VCAA Exam 1, 2008 VCAA Exam 1, 2010 where S n = the sum of n terms n = the given number of terms a = the first term l = the last term ie the nth term. VCAA Exam 1, 2006 VCAA Exam 1, 2010

39 39 INFINITE GEOMETRIC SEQUENCES GRAPHS OF ARITHMETIC SEQUENCES If the common ratio, r, is a value between 1 and 1, then it is possible to find the sum of its infinite number of terms. This is called the sum to infinity. The formula for it is t n t n if d is positive n if d is negative n VCAA Exam 1, 2010 GRAPHS OF GEOMETRIC SEQUENCES t n a > 1, r > 1 t n a > 1, r > 1 t n a < 1, r > 1 n n n t n r < -1 t n -1 < r < 1 n n VCAA Exam 1, 2010 DIFFERENCE EQUATIONS Any sequence of numbers can be represented like this..., t n 2, t n 1, t n, t n + 1, t n + 2, eg t n+1 = t n + 20 t 1 = 5 would give 5, then =25, then = 45, and so on A first order difference equation consists of an equation linking 2 consecutive terms (eg t n and t n+1 OR t n 1 and t n ) ie eg 5, 25, 45, 65, 85, t n = 3t n t 0 = 2 a starting term (the starting term might be denoted by t 0 or t 1 ). or t n+1 = t n + 20 t 1 = 5 would give 2, then =11, then = 38, and so on ie 2, 11, 38, 109, 332, eg t n = 3t n t 0 = 2 ARITHMETIC SEQUENCES as FIRST ORDER DIFFERENCE EQUATIONS An arithmetic sequence with a common difference of d t n = a + (n 1)d f(n+1) f(n) = 5 f(n+1) = 5 + f(n) VCAA Exam 1, 2006 n fn f(n+1) = 5 + f(n) 1 f1= 1 f(2) = = 4 2 f2= 4 gives the same values as a first order difference equation of the form: t n+ 1 = t n + b where b is the common difference and for b > 0 it is an increasing sequence b < 0 it is a decreasing sequence. So d and b are representing the same thing, namely the common difference in an arithmetic sequence.

40 40 GEOMETRIC SEQUENCES as FIRST ORDER DIFFERENCE EQUATIONS PERCENTAGE INCREASES and DECREASES A geometric sequence with a common ratio of r t n = a r (n 1) gives the same values as a first order difference equation of the form: t n+ 1 = at n where a is the common ratio and for a > 1 it is an increasing sequence 0 < a < 1 it is a decreasing sequence a < 0 it is a sequence alternating between positive and negative values. For example: A 3% increase multiply by 1.03 A 3% decrease multiply by 0.97 A 15% increase multiply by 1.15 A 15% decrease multiply by 0.85 A 60% increase multiply by 1.60 A 60% decrease multiply by 0.40 So r and a are representing the same thing, namely the common ratio in an geometric sequence. OTHER SEQUENCES as FIRST ORDER DIFFERENCE EQUATIONS VCAA Exam 1, 2007 FINDING THE FIRST ORDER DIFFERENCE EQUATION FROM A GRAPH OF VALUES OR A SEQUENCE OF NUMBERS To find the first order difference equation follow these steps: 1. Write down the values of the terms in order as a sequence. 2a. If the graph is a straight line or has a common difference (b) by t 2 t 1, t 3 t 2 etc. Then write your equation t n+1 = t n + b t 1 = OR t n = t n 1 + b t 0 = 2b. If the graph is a curved line, find the common ratio (a) by t 2 t 1, t 3 t 2 etc. Then write your equation t n+1 = at n t 1 = OR t n = at n 1 t 0 = ALWAYS CHECK THE QUESTION TO SEE IF IT STIPULATES WHICH FORM IS REQUIRED BETWEEN t n+1 and t n OR t n and t n 1. VCAA Exam 1, 2006 ALWAYS REMEMBER TO INCLUDE A FIRST TERM VALUE VCAA Exam 1, 2007 VCAA Exam 1, 2006

41 41 SECOND ORDER DIFFERENCE EQUATIONS A Fibonacci Sequence is one in which each new term is created by adding the previous two terms. A second order difference equation for a Fibonacci sequence is set out using the following notation: f n + 2 = f n + f n + 1 given f 1 and f 2 OR f n = f n 1 + f n 2 given f 0 and f 1 VCAA Exam 1, 2006 For example, the classic Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, OR The second order difference equation for this would be f n + 2 = f n + f n + 1 f 1 = 1, f 2 = 1 f n = f n 1 + f n 2 f 0 = 1, f 1 = 1 VCAA Exam 2, 2008