StANDARD Maths BASIC revision notes



Algebra-Variable/pronumeral- a symbol for a number that is currently not known Algebraic expressions- a mathematical sentence involving numbers and symbols, but does not include an equal sign.Solution- a value that can replace the variable/pronumeral to make a number sentence true.Linear equations- a mathematical sentence involving numbers, symbols and the equal sign. It is linear because the variable does not have a power.Term- each part of an expression or equation separated by an operation symbol.Substitution- replacing a variable with a number.The distributive law- a(b+c) = a x (b+c) = ab + ac Frieds rule for infants up to 24 months (2yrs) – child dosage= age in months (divided by) 150, x adult dosage. Young’s rule for infants up to 2yrs – child dosage= age in years (divided by) age in years + 12, x adult dosage Clarks rule for children 2yrs and over – child dosage- weight in kgs (divided by) 70, x adult dosage.Making equationsAngles in a triangle add up to 180 degrees Angles in quadrilaterals add up to 360 degrees Alternate angles and corresponding angles are equal Co-interior angles add up to 180 degreesChanging the subjectThe subject of an equation is the variable that is on its own on one side of the equal sign – v= u + at Linear functionsLinear functions are any function where (x) and (y) do not have a power greater than 1. All linear functions typically look like this – y = m(gradient)x + b(y-intercept)The gradient is a rate or slope, which compares the change in (y) with the change in (x)m= rise over run, (Y2-Y1 divided by X2-X1) Linear modellingdirect linear variation means that one variable is directly proportional to another variable.Independent variable- x Dependent variable- y Data-Categorical data is presented in words or symbolsNominal data cannot be ordered Ordinal data can be ordered Numerical data is presented in numbersDiscrete data (counted: separate, values, gaps)Continuous data (measured: smooth scale, no gaps)Sector graph: frequency divide by total x 360 Divided bar graph: frequency divide by total x length of graphPareto chartA Pareto chart combines a frequency histogram and cumulative frequency line graphThe Pareto principle states that the issues that need the most attention are the ones that account for the first 80% of the cumulative frequency.EXAMPLE-Tennis netballfootballSwimming golfCricket 5102045158Sports frequencyc.fc.%Swimming 454545/103 x100= 43.7%football206565/103 x100= 63.1%golf158080/103 x100= 77.7%netball109090/103 x100=87.4%cricket89898/103 x100=95.1%tennis8103103/103 x100=100%Misleading graphs can give a wrong impression by:Not having a scaleHaving an uneven scale or showing only part of the scaleNot showing the correct position of zero on the scaleA census is a survey of a population in which every item is includedSampling-Surveys a representative group of items from a populationProvides an estimate or approximate information about the population.Simple and not expensive Taking a census-Surveys all items in a populationProvides exact information about the population Complex and expensive If the whole population is too large or too difficult to survey, a sample of items is selected from the populationFor a sample to be representative of the population, then each item in the population must have an equal chance of being chosenThe sample size impacts the accuracy of results. The larger the sample, the more likely you have accurate resultsIn a random sample, each item is chosen completely at random from the populationIn a systematic sample, the first item is chosen at random and all other items are chosen at regular intervalsIn a stratified sample, the population is divided into strata (layers) according to some characteristics (for example, gender or age group) and an random sample is taken from each layer using representative proportions or percentages.In a self-selected sample, people choose to participate in a surveyDot plots and stem-leaf plots are used forNumerical (quantitative) dataSmall sets of dataShowing gaps and clustersShowing outliers (scores that are much different from the rest)the mean is the average of all scores in a dataset the mode is the most frequent or repeated valuethe median is the middle scorequartiles of a dataset are values separated into 4 groups or quarters. Represented as: Q1, Q2, Q3deciles of a dataset are values separated into 10 equal groups or tenths. Represented as: D1, D2……D9. D2= lowest 20% of scores (or top 80%)percentiles of a dataset are values separated into hundredths. Represented as: P1, P2……P99. P20= lowest 20% of scores (or top 80%)range= highest score – lowest score interquartile range= Q3-Q1 Outliersan outlier is a score that is eitherless than Q1 – 1.5 x IQR orgreater than Q3 + 1.5 x IQRoutliers can affect the measures of a central tendency of a dataset.The mean is most affected by outliers (because its value depends on every score)The median can be affected, but not by muchThe mode is not affected at allBox plotsA box plot gives a 5-number summary of a dataset:The lower extreme (lowest score)The lower quartile (Q1)The median (Q2)The upper quartile (Q3)The upper extreme (highest score)Standard deviationThe sample mean, and the sample standard deviation are called statisticsThe population mean and the population standard deviation are called par meters.SkewUnimodal: one peak, one modeBimodal: two modesMultimodal: more than one peakSymmetrical: data is evenly spread and balanced about the centre of a distribution where the mean, mode, and median are equal.Positively skewed: data is clustered together to one side and the tail of the graph points towards the positive side of the number line and the mean is greater than the median, which is greater than the mode.Negatively skewed: data is clustered together to one side and the tail of the graph points towards the negative side of the number line and the mean in smaller than the median, which is smaller than the mode.Measurement-Metric units- lengthMillimetre (mm)Centimetre (cm) – 1cm = 10mmMetre (m) – 1m= 100cm= 1000mmKilometre – 1km= 1000mMetric units- massMilligram (mg)Gram (g) – 1g= 1000mgKilogram (kg) – 1kg= 1000gTonne (t) – 1t= 1000kg Megatonne (Mt) – 1Mt= 1000,000t ................
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