Further Key Knowledge Self Quizzing PDF

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About this document This document is designed to ensure that all of the key information for Further Mathematics is stored in your long-term memory. If you have this info stored in memory, it means that it won’t be using up space in your short-term memory, and this will free up space for more complex problem solving. Another key point to solving problems in maths is to be able to spot solution methods quickly. Your ability to spot a solution method quickly depends, in large part, on what you know and can retrieve automatically from memory. Use the questions (left-hand side) and answers (right-hand side) on the following pages to ensure that you have key info for Further Mathematics in your long-term memory. The way to do this alone is to cover the right-hand side up with a sheet of paper then progressively read the questions, answer them (in your head, out loud, or by writing it out), and then check your answer. Do this once or twice per day over the coming weeks. You may also find it fun to do this with a friend whereby you quiz each other. In this document:

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Know your Data Analysis



Know your Recursion & Financial Modelling

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Know your Matrices

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Know your Networks and Decision Mathematics



Know your Geometry and Measurement



Know your Graphs and Relations

It is advisable to remove the modules that your school is not completing as this document is quite long.

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Prepared by Alex (https://vicmathsnotes.weebly.com/) and Ollie Lovell, 2019

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Know Your Data Analysis........................................................................................ 3 Data Distributions (Univariate Data)...................................................................3 Comparing Data Distributions (Bivariate Data)..................................................6 Linear Association (Least Squares Regression Line and Transformations)..........9 Time Series (Time Series, Smoothing, and Seasonality)...................................11 Know Your Recursion & Financial Modelling.........................................................12 Recurrence and Rules, and Types of Sequences...............................................12 Finance Applications......................................................................................... 15 Amortisation Tables.......................................................................................... 17 TVM Finance Solver.......................................................................................... 18 Know Your Matrices.............................................................................................. 20 Matrix Arithmetic and Types of Matrices...........................................................20 Applications of Matrices.................................................................................... 22 Transition Matrices............................................................................................ 23 Know Your Networks and Decision Mathematics..................................................24 Graphs.............................................................................................................. 24 Walks................................................................................................................ 25 Decision Problems............................................................................................ 26 Activity Networks.............................................................................................. 27 Know Your Geometry and Measurement..............................................................28 Measurement.................................................................................................... 28 Similarity.......................................................................................................... 30 Trigonometry.................................................................................................... 31 Circle Mensuration............................................................................................ 34 Spherical Geometry.......................................................................................... 36 Know Your Graphs and Relations.........................................................................38 Linear and Non-Linear Equations and Graphs...................................................38 Linear Inequalities and Linear Programming....................................................40

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Know Your Data Analysis Data Distributions (Univariate Data) Ordinal data is…

Categorical data that has an order.

Nominal data is…

Categorical data that has no order.

Discrete data is…

Numerical data that is counted.

Continuous data is…

Numerical data that is measured.

Categorical data is displayed using…

Tables, bar charts, segmented bar charts.

The statistics used for analysing categorical data are… Numerical data is displayed using…

Percentages and the mode.

The statistics used for analysing numerical data are...

Shape, centre (mean or median), and spread (range, interquartile range, standard deviation). Symmetric, positively skewed, negatively skewed, bimodal.

The shape of a numerical distribution can be A positively skewed distribution has…

Dot plots, stem plots, histograms, box plots.

Its longer tail pointing to the positive (higher) end.

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A negatively skewed distribution has…

Its longer tail pointing to the negative (lower) end.

The mean is…

An equal re-sharing of the data.

To calculate the mean…

Add up the data and divide by the total frequency. The middle value in an ordered list of data.

The median is…

To calculate the median…

The mode is…

To calculate the mode… The range is…

To calculate the range… The upper and lower quartiles are…

Order the data, then find the data value (average of the middle two values) that splits the list into two equal sized groups. The most common category.

Find the category, or categories, with the highest frequency or percentage. The distance between the highest and lowest data value.

maximum −minimum The median of the lower and upper

groups.

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To calculate the upper and lower quartiles… The interquartile range is…

To calculate the interquartile range… The standard deviation is… How do you convert a logarithm to its original value? The five-number summary is…

After finding the median, find the median of the lower group ( Q 1 ) and the upper group ( Q 3 ). The distance between the upper and lower quartiles.

IQR=Q 3− Q 1 The square root of the average deviation from the mean. Raise 10 to the power of the logarithm ( 10logarithm ). Minimum, lower quartile, median, upper quartile, maximum.

The percentage in each section of a boxplot is equal to… A data point is an outlier if… Upper fence ¿ Lower fence ¿ For approximately normally distributed data, the percentage of data within 1, 2, and 3 standard deviations is…

A

z -score is…

To calculate a

z -score…

It sits above the upper fence, or below the lower fence.

Q 3 +1.5× IQR Q 1−1.5 × IQR 68%, 95%, and 99.7% respectively.

A standardised value used to compare values from different data sets. Subtract the mean from the value then divide the difference by the standard deviation.

z= A population is… A sample is… A simple random sample can be made by… The difference between population parameters and sample statistics is that…

x−´x sx

The entire group of people or items that are in question. A selection of people or items from the whole population. Assign a number to each person in the population, then use a random number generator. Population parameters are the measures of the population. Sample statistics are the measures of the 7

sample that are used to estimate the population parameters.

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Comparing Data Distributions (Bivariate Data) The response variable is…

The variable that acts in response to (changes because of) another variable.

The explanatory variable is…

The variable that explains the changes in another variable.

An association between two variables exists when…

An association exists when the changes in the response variable can be seen by changes in the explanatory variable. There is no association when there are no or insignificant changes in the response variable can be seen by changes in the explanatory variable. Two-way tables or segmented bar charts.

An association between two variables does not exists when…

To visually compare two categorical variables, use…

For a two-way table, the response and explanatory variables are…

EV - columns, RV - rows.

For a segmented bar chart, the response and explanatory variables are…

EV - bars, RV - segments.

To show an association between two categorical variables…

 Show a difference in the modal response of each category in the explanatory variable OR  Show a difference in the percentages of one response category across all categories in the explanatory variable. Parallel dot plots, back-to-back stem plots, or parallel box plots.

To visually compare a categorical and numerical variable, use…

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When comparing a categorical and numerical variable, the response and explanatory variables are… To show an association between a categorical and numerical variable…

To visually compare two numerical variables, us…

EV - categorical, RV - numerical.

Show a difference in the centres (mean or median) or spreads (range, interquartile range, or standard deviation) of the categories. A scatterplot or a time series.

axis, RV - vertical

On a scatterplot, the response and explanatory variables are…

EV - horizontal y axis.

From a worded description of the variables, the response and explanatory variables are…

EV - the variable we want to use to predict another variable with, RV - the variable we wish to predict.

To show an association between two numerical variables… The strength of a scatter plot is …, …., or …

Look at the strength, direction, form of the scatterplot. Strong, moderate, or weak

The direction of a scatterplot is … or …

Negative or positive

The form of a scatterplot is … or …

Linear or non-linear

x

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What statistic is used to measure the linear strength between two numerical variables? The sign of r describes…

The value of r

describes…

When r is between 0.75 and 1 (positive or negative), the strength is…

Pearson's product-moment correlation coefficient, r . The direction of the scatterplot

The strength of the scatterplot:  strong when 0.75 to 1,  moderate when 0.5 to 0.75,  weak when 0.25 to 0.5,  no association when 0 to 0.25. Strong.

When r is between 0.5 and 0.75 (positive or negative), the strength is…

Moderate.

When r is between 0.25 and 0.5 (positive or negative), the strength is…

Weak.

When r is between 0 and 0.25 (positive or negative), the strength is…

None. There’s no association between the variables.

The difference between observation and experimentation when collecting data is that…

Observation looks at existing data to see if there is an association between two variables. Experimentation imposes a treatment in order to observe its effects. Observation does not account for other lurking variables.

Observation not enough to definitively determine cause and effect because… Experimentation needed to definitively determine cause and effect because… The possible non-causal explanations of an association are…

Experimentation allows for careful selection of variables. Common response, confounding factor, and coincidence.

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Linear Association (Least Squares Regression Line and Transformations) The equation of the least squares regression line is To determine the coefficients of the least squares regression line, use… To predict with the equation of a least squares regression line… To sketch the least squares regression line…

RV =a+b × EV ( y=a+b × x ) sy b=r , a= ´y −b x´ sx or a linear regression on the CAS. Substitute the value into the equation, then evaluate or solve it for the other. Use the equation to predict two values. Plot these points and sketch the line.

The slope of the least squares regression line indicates that…

On average, it is predicted that the [response y variable] [in/de]creases by [ b ] [response units] for each 1 [explanatory unit] increase in the [explanatory x variable].

The y -intercept of the least squares regression line indicates that…

On average, it is predicted that, when the [explanatory x variable] is 0 [response unit] the [response y variable] will be [ a ] [response units].

The difference between interpolation and extrapolation is…

Interpolation is predicting within the data set, extrapolation is predicting outside the data set. The pattern may not hold.

Extrapolation less reliable than interpolation as… The coefficient of determination indicates that…

To convert the correlation coefficient to the coefficient of determination…. To convert the coefficient of determination to the correlation coefficient….

[ r 2 × 100 ]% of the variation in the [response variable] can be explained by the variation in the [explanatory variable]. Square the correlation coefficient 2

( correlation coefficient )

Square root the coefficient of determination (use direction to choose positive or negative). 13

+√ coefficient of determination or −√ coefficient of determination

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Residual

¿

The ‘actual’ value in the residual calculation is… The ‘predicted’ value in the residual calculation is…

A linear association exists if… To transform data…

actual − predicted

The response value stated in the question or read from the data set. The estimated response value obtained by predicting the value using the equation of least squares regression line. The residuals are randomly distributed. Use the CAS to find the reciprocal (

1 ), logarithm ( log ( x ) ), or square x (^2) of the explanatory or response variable.

To determine the equation of the least squares regression line with transformed data… To predict with transformed data…

Transform one of the variables, then use the CAS to perform the linear regression. Substitute the value into the equation, then evaluate or solve it for the other. Ensure you find the actual prediction not the transformed prediction.

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Time Series (Time Series, Smoothing, and Seasonality) To describe a time series…

To smooth using an odd numbered moving mean…

Describe the trend (or trends if there is a structural break), any seasonality or irregular changes, and any outliers. For an odd moving mean, find the mean where the value being smoothed is in the middle.

To smooth using an even numbered moving mean…

For an even moving mean, find two means: one where the middle is left of the value being smoothed, and one where the middle is on the right. Then find the mean of the means (apply centring).

To smooth using an odd numbered moving median…

For a group of points, find the median (middle) x and median y value.

A seasonal index…

A number that describes what percent a season is above or below average. Divide the season's value by the yearly average.

To calculate a seasonal index…

seasonal index=

value for season yearly average

If you add up all the seasonal indices, the total is equal to… To de-seasonalise data…

The number of seasons

To correct for seasonality (reseasonalise data)…

Multiply the season's de-seasonalised value by the seasonal index.

Multiply the season's value by the reciprocal of the seasonal index (or divide by it).

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Know Your Recursion & Financial Modelling Recurrence and Rules, and Types of Sequences A recurrence relation is… The two things needed for every recurrence relation are… The difference between V n+1 is that…

V n +1

and

An arithmetic sequence is… A geometric sequence is…

The recurrence relation for an arithmetic sequence is… The recurrence relation for a geometric sequence is… The recurrence relation for a sequence that multiplies then adds? The financial models are arithmetic sequences are… The financial models are geometric sequences are… The financial models are neither arithmetic nor geometric are…

The financial models that are growing are… The financial models that are constant are… The financial models that are decaying are…

An equation that connects one term's value to the next. An initial value and the rule connecting one term's value to the next. V n +1 adds one to the value of V n . V n+1 is one term later than Vn . A sequence that adds the same number (common difference) to each successive term. A sequence that multiplies by the same number (common ratio) to each successive term.

V 0=P , V n+1 =V n+Q V 0=P , V n+1 =R V n V 0=P , V n+1 =R V n+ Q Flat rate depreciation, unit cost depreciation, and simple interest. Reducing balance depreciation and compound interest. Reducing balance loans (and interest only loans), annuities (and perpetuities), and annuity investments. Simple interest, compound interest, and annuity investments. Interest only loans and perpetuities. Flat rate depreciation, unit cost depreciation, reducing balance depreciation.

The graph of arithmetic growth is…

The graph arithmetic decay is…

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The graph of geometric growth is…

The graph of geometric decay is…

This could represent the financial model …

Flat rate depreciation Unit cost depreciation

This could represent the financial model …

Simple interest

This could represent the financial model …

Reducing balance depreciation

This represent the financial model …

Compound interest

This could represent the financial model …

Reducing balance loan Annuity

This could represent the financial model …

Interest only loan Perpetuity 18

This could represent the financial model …

Annuity investment

To "show using recursion" to find the value of a term…

State the starting value, then show the calculation to get the value of each successive term. An equation that connects one term's value with its term number. A recurrence relation goes from one term to the next whereas the rule jumps straight to the desired term.

A rule is… The difference between a recurrence relation and a rule is that… The rule for an arithmetic sequence is… The rule for a geometr...