Arithmetic

Addition and subtraction

  • Subtracting small positive numbers by large negatives
    • Factor out the expression with a negative sign to make calculation easier
    • 0.79 - 79.1 - 58.1
    • -(-0.79 + 79.1 + 58.1) factor out the negative sign
    • -(136.41) calculate the inside of the bracket
    • -136.41

Absolute Value

  • Interpreting absolute value as distance

  • |a|-|b|
    • |a| = absolute value of a is the distance of a from zero
    • |b| = absolute value of b is the distance of b from zero
    • |a-b| = distance between the two values
  • example: What expression is equal to the rectangle’s area?
    • ∣h−d∣⋅∣e−g∣

Multiplication and division

Negative

  • Is an expression negative or positive? tut
    • Count the number of negative signs. An expression with a even number of negative signs is equal, a number with a odd number of negative signs is negative
    • example: $

Fractions

  • fraction = part of a number/thing, used to measure things
  • Fractions are a new way of seeing the world. Sometimes things in the world are not a whole number, they’re less than 1. Instead of viewing the world as just whole numbers, you can view it as a part of something.
  • i.e. a part of cake, a part of the population, a part of your health points
  • When you split a chocolate bar, do you ask for a 4cm chunk or do you ask for half?
  • 1/2 can tell us what proportion of something we’re talking about, without having to talk about the exact numbers
  • example: As Dumbledore, the Headmaster of Hogwarts School of Wizardy, you’re interested in how many students are graduating to become Full Wizards. Which of these answers gives you a better, more intuitive sense, of the world and your school’s success?
    • Answer 1: 980 Students.
    • Answer 2: 8/10 students.
    • Answer one only means something if you know how many students you have overall, is this a big or small number, good or bad? Who knows.
    • Answer 2, however, you know straight away that most of the students will graduate, that’s good news!
  • use: Fractions are used for measuring.
    • For counting we have natural numbers (1,2,3…)
    • Sometimes things are less than 1, they are a just part of one,– a half of one, a third
    • Helps you break something up to make comparisons and gain a better sense of amounts, a more intuitive understanding of how much of this thing you have,
    • useful in recipes, as you don’t have to state exact amounts, 3 out of 4 parts are this pizza is plain cheese, 1 out of 4 has olives.
  • History:
    • 1800 BC: Egyptians, first records of written fracitions, used one hieroglyph image on top of another, was difficult to add together
    • 500 AD: Indians devised brahmi system, 9 symbols and a zero, which is how fractions are wrote today with one number (nume) above the other (domi)

Numerator V.S. Denominator

  • fraction = pieces of 1. 1 being the whole thing.
  • numerator = the number of pieces
  • denominator = the whole thing is divided into denominator pieces
    • \(\frac{3}{8}\) = this Cake is sliced into 8 pieces, and we have three
    • \(\frac{3}{8}\) = 1$ = this cake is sliced into 8 pieces, and we have all 8, so the whole thing
    • \(\frac{0}{8}\) = 0$ = this cake is sliced into 8 pieces, and we have none, so nothing
  • \(\frac{number of pieces remaining}{number of total pieces}\)

Fractions on a number line

  • Choose a part of the number line and split it up into as many pieces as you want
  • Each segment of the number line represents * \(\frac{1}{denominator}\)
  • example: Let’s split the line between 0 and 1 by 4 pieces, writing each piece as a fraction
    • \(4/4 = 1\). Any number divided by its self is one, a whole, complete
    • \(0/4 = 0\). Any number divided by zero is zero, nothing
  • note: You could of chosen any number to split the number line by, and

Dividing Mixed Fractions

Exponents and Powers

Laws of Exponents

  • Exponents with negative bases
    • example 1: \(-2^2\)
    • \(-(2*2)\)
    • \(-4\)
    • note: Order of Operation = exponents are raised before multiplication. Thus \(-2^2\) equals \(-(2^2)\) which will always be negative, no matter if the exponent is odd or equal. If you want a positive result, put paranthesis around base then write exponent as shown below.
    • example 2: \((-2)^2\)
    • \((-2)*(-2)\)
    • \(4\)
    • note: whenever you raise a negative base by an odd exponent, you will get an negative result, because the negatives do not cancel out
      • \((neg)^ odd = neg\)
  • Exponents with negative fractional bases ** example: \((-2/8)^2\)
    • \((-2/8)*(-2/8)\)
    • \(4/16\)

Algebraic Equations tut

  • Useful tool for finding out stuff in real life, even if it isn’t there to count or does not exist
  • Equation is a statement that helps us understand the equality between variables and constants
  • Variables can be any value and are respresented by letters a,b,x, , etc
  • Constants are fixed values. x+3=7 3 and 7 are constants, x variable

  • one side of the equation is equal to the other. To find out the mystery value x, we do the same things to both sides to keep the scales balanced

  • example: You have three pokeballs which cost $9. How much do they cost each?
    • write as equation: 3P = $9
      1. Keep the scales balanced by doing the same to each side
    • Let’s try getting P on it’s own so we’re only left with 1/3rd, aka one Pokeball.
    • Multiply by 1/3 or divide by 3
    • $3P/3 = \(9/3\)
    • $P = \(3\)

Introduction to Algebra

Combining Like terms

To make life easier we combine like-terms to make the expression simpler, like how you might combine pens in one box and pencils in another to make organising your room simpler!

  • Rules
    • Can only add like things together. i.e. you can’t mix water and oil, but you can combine oil with oil and add water to water
  • example 1:
    • 7 + 5 + 2 + 2
    • 7 + 9 . This says the same thing as above, but isn’t it so much more easy on the eyes and noggin?
  • example 2: distributive property
    • \(2(3x+5)\)
    • \(2(3x) + 2(5)\)
    • \(6x + 10\) ### Linear equations and inequalities
  • Linear Equation word problem example: Ash has captured a Squirtle. The Squirtle had an XP of 20 to begin with and grows by 5XP with each battle. His total XP is 60. Write an equation to determine the number of battles(b) you’ve had with Squirtle.
    • \(20+5b=60\)
    • take away the initial 20xp. \(5b=40\)
    • take away the 5xp \(b=8\). You’ve had 8 battles since capturing Squirtle
  • Inequalities explanation
    • Inequalities tell us about the relative size of two values. Maths is not always about “equals”. Sometimes we want to know if something is bigger or smaller.
    • i.e. Harry and Ron have a wand duel, and Harry wins! We don’t know by how many points Harry won, we just know it was more than Ron.
    • this can be written as \(H > R\), where \(H\) is harry’s points and \(R\) is Ron’s points.
  • symbols
    • x < 5 x is to the left of 5 (less than)
    • x > 5 x is to the right of 5 (more than)
  • One step inequalities
    • Red has $1000 to spend, each rare candy cost $4. How many rare candies can he buy?
    • Set up equations, where x = rare candies
    • \(4x < 1000\) = Each rare candy cost $4 and the total must be less than a thousand dollars
    • \(1/4*(4x) < 1/4*(1000)\) Multiply out to get X on it’s own. Multiplying inequality by positive number, so sign remains the same
    • \(x < 250\) = Red must buy less than 250 rare candies.
    • Any amount of rare candies less than 250 is the solution
    • note: You swap inequality sign when multiplying/dividing both sides by a negative number.
    • \(1<2\) = one is less than two
    • \(-1>-2\) = negative two is MORE negative than negative one, hence sign is swapped
  • Two Step Inequalities
  • \(4x+11 < -21\)
    • Step 1: Substract 11 from both sides of inequality
    • \(4x < -32\)
    • Step 2: Isolate x by dividing both sides 4
    • note 4 is not negative, so inequality symbol is not changed
    • \(x < -8\)
  • World Problem Example: Tenar grows 120 apples to eat each month. At the end of the month, Tenar ate 5 apples more than Tehanu, 40 less than her goat and 58 less than Ged. How many apples could Tenar of ate if she did not want to go over the 100 limit?
  • To solve this we need to turn it into a straight-forward inequality
  1. Define variables
    • T = # of apples eaten by Tenar
    • D = # of apples eaten by Tehanu
    • G = # of apples eaten by Goat
    • F = # of apples eaten by Ged
    • \(T+D+G+F <= 120\) want this term to expressed in terms of T (Step 2)
  2. Redefine variables using inequalities
    • Express each variable in terms of apples eaten by Tenar
    • T = 5 + D
    • T = G - 40
    • T = F - 58
    • Solve for D in terms of T, G in terms of T, etc
    • T = 5 + D <-> T - 5 = D
    • T = G - 40 <-> T + 40 = G
    • T = F - 58 <-> T + 58 = F
    • Substitute new terms/variables back into equation
    • \(T + (T - 5) + (T + 40) + (T + 48) <= 120\)
  3. Add variables (T’s) and constants (numbers)
    • \(4T + 83 <= 120\)
  4. Solve the straight-forward inequality
    • \(4T <= 37\)
    • \(T <= 9.25\)
  • Tenar could of eaten 9 or less apples for the family combined to eat less than 100 apples that month

Quadratics

  • Factoring simple expressions = put expression into lowest terms and make it easier to answer
  • Factoring Linear Binomials
    • Binomial = sum of two terms. linear = expression with no exponents or square roots
    • example
    • \(4x+18\)
    • Find greatest common factor and multiply out
    • \(2(2x) + 2(9)\)
    • \(2(2x+9)\)

Algebra Glossary

  • \(2x+3x +5\). x = variable, 2 & 3 = coefficients, 5 = constant, 2x & 3x = like terms.
  • constant = a regular number that is without a variable
  • coefficients = regular/constant number thats multiplied by the variable
  • variables = The thing, normally symbolised by a letter, that you’re trying to solve.
    • i.e. variables = Chuck Norri, money, Pokeballs, slices of cake.
  • like terms = terms that contain the same variable

Proportions

Pikachu is poisoned, so we loose 5 HP for every 2 steps we take. If we have 30 HP, how many steps can we take?

  • Algebra
    • Flip, and destroy
    • \(5/2 = 30/x\)
  • multiplication

Statistics tut1 tut2 Rtut

Experimental Probabillity

  • experimental probabillity = computing the probabillity of a future event based on our obs of past events
  • \(Experimental~Probabillity=\frac{Number~of\,favourable\,outcomes}{Total~ number~of~outcomes}\)
  • Chance of sun tomorrow = number of sunny days this / total number of days

Infer estimates from random sample

Estimate = population size * sample proportion

Example: 20 Pokemoned were surveyed. 8 were grass type. What number of the total pokemon (151) are grass type?

151 * 8/20 = X of total pop that are grass type

Or…use the inverse of the percentage

20% of a sample like drugs. What number is that for the population of 80

estimate = inverse of percentage of sample * total pop

2/10 like drugs

10/2 * 80 = X of total pop who like drugs

Compound events and sample spaces

  • Probabillity of compound event
    • example: You’re at a clothing store. You get to pick from four pieces of clothing (shirt, jean, underwear, hat) and from 3 colours (blue, purple, orange). If you pick by the clothing and colour by random, what are the chances you’ll have an orange hat?
      • probabillity = favourable outcomes / total possible outcomes
      • Draw a compound space

| | shirt | jean | underwear | hat | |——–|————–|————-|——————|————| | blue | blue shirt | blue jean | blue underwear | blue hat | | purple | purple shirt | purple jean | purple underwear | purple hat | | orange | orange shirt | orange jean | orange underwear | orange hat |

    * There is 12 (3*4) possible combinations in total. If we pick randomly all choices are equally likely
    * There is only one favourable outcome of an orange hat 
    * $probabillity of randomly picking a orange hat = 1/12$
  • Counting Principle
    • Counting the total number of possible outcomes/combinations. e.g. ways to get dressed or types of dishes.
    • example: You are designing your next gun. The gun has 3 types of ammo, 13 types of barrels, 7 types of grips, 4 types of sights and 2 types of colour. How many different guns can you create?
      • Total number of guns can be found by multiplying the number of options of each gun component
      • Total number of guns = \(3?13?7?4?2 = 2184\). You can create \(2184\) different guns
  • Log Transformations
    • Data differs by orders of magitude (10, 100, 1000, 10000)
    • To make modelling easier and appearance on graph nicer, log transform
    • Make skewed data normally distributed
    • Makes patterns more visible
  • Confidence Intervals
    • A range of numbers (interval) in which you are 95% sure the true mean will lay
    • There is a 95% chance that average weight of Pokemon will lay between 33 and 53. There is a margin error of 10grams

xmean <- sd(pokemon\(weight) xsd <- sd(pokemon\)weight) n <- length(pokemon$weight)

LCI <- xmean-1.96xsd/sqrt(n) UCI <- xmean+1.96xsd/sqrt(n)

Basic Geometry

  • History
  • Geometery involves the measurements of shapes, patterns, sizes and posistions
  • Building bows, enclosing farms, building shelters, navigating/star-gazing and organising communities all required notions of center, equidistance, length, area, volume, straightness.

Triangles

  • Triangle Inequality Theorem any side of triangle must be less than sum of the two other sides
    • a < (b+c), b < (a+c), c < (c+b)

Area and Perimeter

  • Use
    • How far a wheel will travel in one revolution
    • Calculate the length of fence required to surround house
    • How much space farmland will take up
    • note: Splitting farmland by perimeter is a bad idea, as production is proportional to area. Having a field with a long permeter and small area = small production
    • How long you have to walk around a building
  • Interesting notes
    • The more you cut a shape, the lesser the area but larger perimeter. This happens as the length around the shape increase, as there is more sides, but the space inside the shape decreases.
    • Coral, Kale and other marine organisims exhibit hyperbolic geometry, maximising surface area in a limited volume, providing a greater opportuninity for photosynthesisthis and filter feeding.
  • Same perimeter, different area
    • This is important to know about when designating farmland, for example:
    • You have 24 metres of fencing. What are the different farmlands you can make with that?
    • Farmland 1 = \(1*11m. P = 24. A = 11^2\)
    • Farmland 2 = \(2*10m. P = 24. A = 20^2\)
    • Farmland 3 = \(9*3m. P = 24. A = 27^2\)
    • Although each farm has the same perimeter, the area changes.

Area and Circumfrence of Circles

  • If the circumfrence of a lake is 400m, what is the diameter?
    • \(C = pid\)
    • \(\pi d = C\)
    • $ d = 400 pi m$
  • Impact of Radius Change on the A and C of circles
    • circumfrence increases by the same factor
    • area increases by the factor squared

Scale Drawings

Peter just buried a treasure chest on a remote island and is making a map so he can find it later. One of the key landmarks in the area is a small rectangular hut, 5m by 8m.

The map’s scale is 2 units: 1m. Help him draw…

  • Convert 5m into units
    • \(5 ~m~ * ~2~ units / 1 m\)
    • 5 m * 2 units / 1 m
    • \(= 10 units\)
  • Convert 8m into units
    • \(8m * 2 units / 1 m\)
    • 8 m * 2 units / 1 m
    • \(= 16 units\)

Rates and proportional relationships

Tutorial

  • A proportional relationship is where: \(\frac{Y}{X} = K\) the relationship between y and x is always going to be a constant number
    • constant = a number that stays the same and you multiply by to get an answer
    • K is also know as the constant rate of change or constant rate of proportionality

  • Useful as it tells you what numbers to expect, allowing you to make predictions or fill in missing gaps

  • Proportional relationships can be shown as a equation, table or graph

  • Equation Form
  • \(\frac{Y}{X} = K\)

  • \(Y=KX\)

  • example: You can catch 4 Pokemon every 8 hours. How many can you catch in 48 hours? x = Pokemon Caught. y = Hours.

x 4 8 12 16 20 ? 28
y 8 16 24 32 40 48 56
  • Calculate the constant (rate of change) = 8/4 = 2.
  • 48/2 = 24 pokemon!

Visually identifying proportional relationships through graphs

  • Label a proportional relationship on a coordinate plane

    • The above table would be written as \((4,8), (8,16), (12,24)\)

  • Relationships are proportional if the points can be connected by a (1) straight line through the (2)origin (0,0)

  • Relationships are not proportional if there is no multiplicative constant for us to multiply by to make our coordinates go in a straight line through the origin, i.e. (12,20)
x 4 8 12 16 20 ? 28
y 8 16 20 32 40 48 56

Independent and Dependent Variables

  • Dependent means that one value relies on the other
    • example: If you want to buy three Pokemon for $30, then the cost is dependent on the amount of Pokemon.

Unit Rate

  • Unit rate is the ratio between the values, i.e. every 8 hours I catch four Pokemon.

Improvements

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