study-notes

Math 7

This subject probably involves the least amount of studying because you’ll be thinking more than the other subjects instead of just remembering. Practicing word problems is also cool for practice. Note that in this document, a number after a variable (normal size) means exponent. MathsIsFun is an excellent resource to complement this, and it isn’t “for babies”.

Number Concepts

There are two main types of numbers:
- A prime number is a whole number above 1 that can’t be made by multiplying other whole numbers together.
- Composite numbers are whole numbers that can be made by multiplying other whole numbers together.
- 1 isn’t a prime nor a composite number.
The following are rules to tell if a number is divisible by some certain number.
- 1 when it’s an integer. (123, 543, 2,490)
- 2 when the last digit is even. (6, 92, 412,342)
- 3 when the sum of all of its digits are a multiple of 3. (135, 5,706, 16,587)
- 4 when the number made by the last two digits are divisible by 4. (416, 56, 2,924)
- 5 when the last digit is either a 5, or a 0. (125, 85, 1,050)
- 6 when it’s divisible by both 2 and 3. (312, 5,706, 625,872)
- 8 when the number made by the last 3 digits are divisible by 8. (3,032, 304, 62,937)
- 9 when the sum of all digits is a multiple of 9. (496, 18, 51,840)
- 10 when the last digit is a 0. (40, 500, 48,620)
You can use a Venn Diagram to sort numbers by their divisibility (the two circles chart thingy with a separate spot in the middle).

Integers

Positive numbers are above 0, and negative numbers are below 0 and are prefixed with a -.
It may sometimes be clearer to surround numbers with parentheses.
If you’re adding/subtracting two numbers that are positive and negative, you can reverse the sign (negative, or not) of the second number, and reverse the operation (addition to subtraction and vice versa).
Number tiles are a way to represent numbers by having specific tiles for either -1, or +1.
- For example, red/black tiles can be positive, and yellow/white tiles can be negative.
- One tile from each together make a zero pair (which are usually circled together into a “bubble”).
- A zero pair is equal to… Well, 0.
- So for example, 5 red tiles and 2 yellow tiles make 3
- To add two numbers represented by number tiles:
  1. Model both numbers with number tiles.
  2. Circle all zero pairs.
  3. Count the remaining number tiles; that’s the answer.
- Adding example: -6 + 2: draw 6 yellow tiles, 2 red tiles are added, remove the two zero pairs, there are now 4 yellow tiles, the answer is -4.
- To subtract two numbers represented by number tiles:
  1. Model the first number with number tiles.
  2. Can the amount from the second number be taken away from the first number? If so, do so.
  3. If not, make enough zero pairs to subtract the second number (by adding negative tiles if more positive tiles are needed to be subtracted, or by adding positive tiles if more negative tiles are needed to be subtracted).
- Subtracting example: 5 - -2: draw 5 red tiles, there are no yellow (negative) tiles to take away (you need two), draw 2 zero pairs for 2 yellow tiles, remove the 2 yellow tiles, you now have 7 red tiles, the answer is 7.

Decimals

You should know basic math operations by now (like long addition, subtraction, division, and multiplication).
Some types of estimation:
- Front-end estimations is only using the whole number of a decimal number. (1.4 -> 1, 5.6 -> 5, 66.9 -> 66)
- Rounding estimation is rounding to the nearest nth place (tenth, hundredth, thousandth) place. (1.4 -> 1, 5.6 -> 6, 66.9 -> 67)
PEMDAS (how dare people say BEDMAS!) is the order in which operations are applied in an equation.
- Parentheses, Exponents, Division/Multiplication, Addition/Subtraction.
- Note that when division/multiplication or addition/subtraction is reached, operations are applied left to right.

Fractions, Decimals, and Percents

Percent means out of 100.
There are two types of decimal numbers:
- A terminating decimal number has a finite number of decimal places. (4.6, 16.72, 134.154)
- A repeating decimal number has a specific pattern of numbers that repeat forever. (pi, 10/3, 2/9)
  - Repeating decimal numbers have a straight line on top of the repeating decimals to show that they’re repeated.
Common factors are factors (numbers multiplied together to get a product) that are present in two, or more numbers.
Some ways to convert between fractions, decimals, and percentages:
- To convert a fraction to a decimal, convert the denominator to the nearest power of 10 by multiplying or dividing, then do the specific operation on the numerator, then make the decimal the numerator in the appropriate place value. (13/200 = 65/1000 = 0.065)
  - If the denominator can’t become a power of 10, divide the numerator by the denominator. (1/7 = 1/7 = 0.142,857)
- To convert a decimal to fraction, the place value of the last digit becomes the denominator and the original number becomes the numerator. (0.561 = 561/1000)
- The following are rules for each denominator to be applied to the numerator to make the decimal.
  - /2 -> half the number.
  - /3 -> 0.3 repeated, 0.6 repeated.
  - /4 -> multiply by 0.25.
  - /5 -> multiply by 2.
  - /8 -> numerator multiplied by 0.125.
  - /9 -> numerator repeats after decimal.
  - /10 -> prefixed by 0.
  - /11 -> multiplied by 9 and then repeats.
- To convert a fraction to a percent, convert it to decimal, then move the decimal 2 places to the right, then add the % sign. (4/5 = 0.8 = 80%)
- To convert from percent to fraction, the number becomes the numerator and the denominator becomes 100. (55% = 55/100)
- To convert from a decimal to a percent, either multiply the decimal by 100 or move the decimal 2 places to the right and then always add the % sign. (0.18 = 18%)
- To convert from percent to decimal, drop the % sign, then either move the decimal 2 places to the left, or divide the number by 100. (75% = 0.75)
When comparing fractions, decimals, and percents, it’s best to convert them all to one of those formats, then reorder them, then turn them back to their original form.

Fraction Operations

Equivalent fractions are fractions that all represent the same amount, despite being different (imagine groups of small pizza slices being in the shape of one big pizza slice). (3/9, 1/3, 2/6)
- An equivalent fraction can be made from a fraction by multiplying/dividing both the numerator and the denominator by the same number. (2/5 = 4/10)
- A simplified fraction is the smallest possible equivalent fraction for a fraction.
- To find one for a fraction, divide the numerator and denominator by the largest common factor. (65/100’s largest common factor is 5 -> 65/5 = 13, 100/5 = 20 -> simplified fraction: 13/20)
There are 2 ways to represent a number larger than 1 with a fraction.
- An improper fraction has a numerator larger than, or equal to the denominator. (3/2, 5/5, 6/2)
- A mixed number is a whole number along with a proper fraction beside it. (5 3/4, 2 4/6, 1 6/10)
- To convert from improper to mixed, divide the numerator by the denominator, that’s the whole number, then the remainder is the numerator. (5/4 = 1 1/4)
- To convert from mixed to improper, multiply the whole number by the denominator, add that to the numerator, then write the result on top of the denominator. (2 3/5 = 13/5)
A common denominator is a denominator that’s the same between two numbers. (3/5, 1/5)
- To make a fraction’s denominator common with another, multiply/divide both the numerator and denominator till it is. (3/6, 1/12 = 6/12, 1/12)
To add fractions, make sure there is a common denominator, then just add the numerators, but keep the denominators the same. (2/7 + 3/7 = 5/7)
To subtract fractions, make sure there is a common denominator, then just subtract the numerators, but keep the denominators the same (4/6 - 3/6 = 1/6)

Percent Problems

You can get a percentage of a number by multiplying the number by the percent as a decimal. (60% of 50 = 50(0.6) = 30)
GST is a percentage tax imposed on buying products applied on their prices. Right now it’s 5%.
- To get the GST of a product, multiply the price by 0.05. To get the total end price, either add the price of the GST with the item price, or multiply the item price by 1.05. ($40 with GST = 40(1.05) = $42)
To get the amount saved by a discount or coupon, multiply the price by the discount or coupon as a decimal. To get the total end price, either subtract original item price by the amount saved, or multiply the original item price by the “opposite of the percentage” as a decimal. ($60 and a 20% coupon = 20(0.8) = $48)

Statistics and Probability

There are different measures of central tendency:
- The mean of a group of numbers can be considered the “true average”.
- It can be calculated by summing up all the numbers in a group together and dividing it by the number of numbers. (2, 5, 1, 7 -> 16/4 = 4)
- The median is the number in the middle in a group of ordered (or order them yourself) numbers. (4, 5, 9, 1, 0 -> 9)
- The mode of a group of numbers is the number that occurs most time. (3, 9, 3, 1, 0 -> 3)
The range (not a Measure of Central Tendency), is the difference between the biggest and smallest numbers in a group of numbers. (3, 7, 1, 9, 5 -> 8)
An outlier is an abnormally large or small number in a group of numbers, compared to the other numbers. (3, 9, 1, 435, 4 -> 435)
A circle graph shows how categories of data compare to each other as a whole.
- It’s a circle divided up into sections.
- They need a title, labels/a legend, the sum of 360 for the total degrees, and data is reported as percentages.
- To find the angle in degrees for a sector, turn the percent into a decimal, multiply each decimal by 360, and then round the number. Use a protractor to measure each angle. (56% of circle = 0.56(360) = 201.6 degrees)
- You can draw the circle by tracing a circular object, or using a compass.
With chances and probabilities:
- The favorable outcome is the event you’re looking to occur in a probability experiment.
- Possible outcomes are the possible outcomes in a probability experiment.
- Probability(event) = favorable outcomes/possible outcomes.
- An independent event is an event in a group of events that doesn’t affect the probability of the other events, while a dependent event affects the probability of the other events.
- Either a tree diagram, or a table of outcomes can be used to represent possible outcomes of events.
  - A tree diagram starts and branches out for each event. The amount of branches from each branch is the possible number of outcomes. The amount of branches at the end is the total overall amount of possible outcomes.
  - A table of outcomes only works with 2 events (unless you want to traverse the nth dimension). It’s a table where the rows represent the possible outcomes of the first event, and the columns represent the second event.

Patterns and Relations

An expression is a mathematical phrase made up of numbers and/or variables connected by operations. The variable can represent any number we choose. (3 - 5, 8, 2 - x)
An equation is a mathematical statement, or sentence, containing two expressions separated by an equals sign. The variable only has one value. (3 - 5 = -2, 3x = 6, x = 8)
There are different parts to an expression or equation:
- A constant term is a number that doesn’t change. It increases or decreases the value of an expression. (3, 62, 92)
- A variable is a letter or symbol representing an unknown number. (x, y, n)
- A numerical coefficient is a number that multiplies the variable. (3x, 5y, 9i)
- A term is a constant or variable prefixed by a +, or - sign. (-6, -z, 3)
When multiplying, you shouldn’t use an X because it can be confused with the common variable name, x. Instead, use a numerical coefficient with or without parentheses. (3(6), 7x, 2(9))
To evaluate an expression, substitute all occurrences of variable with their given value. (3n where n is 12 = 36)
A table of values is a table showing two sets of related numbers.
- Patterns can be described by expressions which can be displayed in a table of values.
- The left column has the inputs (x), the right column has the outputs (y).
- So x is usually used as an input, and y as an output.
- They can be solved with an expression applied with every input.
A relation is a rule that allows you to use a number to get another number. It has 2 variables on opposite sides of an = (y = 4x + 8)
- A relation can be written by analyzing a pattern forming from a table of values.
- A relation can be graphed by having inputs on the x-axis, and on the y-axis the answers.
Continuous data is data that can be measured and has a line going through the coordinates on the graph. (people’s heights, widths, temperature)
Discrete data is data that can be counted and only has dot coordinates on the graph. (students in class, tickets, month)
Graphs need a title, a manipulated (independent) variable, a label on the x-axis, a responding (dependent) variable, a label on the y-axis, proper scales for data, bars are evenly spaced, and everything is neat and organized with a ruler.

Equations

Equations can be represented using number tiles by using a long tile as an x tile. (one long x tile and 3 negative white tiles = 4 positive black tiles, means x - 3 = 4)
- This makes two expressions on either side of the equation.
To solve an equation using number tiles, all tiles on the side with the x tile must be removed by making zero tiles, so that the x can be isolated.
- Add the opposite of the tiles of the side, then add the same to the other side. (to solve: a long x tile with 2 positive black tiles = 6 negative white tiles, add 2 negative white tiles to make all the tiles on the left zero pairs, while increasing the right side to -8, making x = -8)
A balance scale can be used to represent equations by having a balanced balance scale with each side having weights with a number on them, with one side having an x weight.
- To solve an equation using a balance scale, we must isolate the x on one side by removing all other numbers on that side.
- We remove the required numbers on the x side, then the same ones on the other side; this keeps it balanced.
To solve an equation normally, we also must isolate the x variable, by performing the reverse of PEMDAS, and the reverse operators. (3x + 3 = 9 -> 3x = 6 -> x = 2)

Geometry

A Cartesian plane is a 2-dimensional coordinate plane, with 4 quadrants.
- The first is top right, second is top left, third is bottom left, and fourth is bottom right.
- The y-axis increases the higher it is and decreases the lower it is.
- The x-axis increases the more to the right it is and decreases the more to the left it is.
- When representing coordinates, we put x in the left, and y in the right, of a pair of parentheses. ((3, 5), (2, -5), (-2, -6))
- The point of origin is the center. (0, 0)
Shapes can be plotted on a plane by putting points on it and connecting them. ((4, 5), (3, 5), (-2, -5))
A shape’s points are usually represented by capital letters. (A, B, C, D)
A transformation is a change to a shape’s points to change the shape.
- The points of the second shape made after a transformation must be the same letters but suffixed by a ‘ and are called {letter} + “prime”. (A’, B’, C’, D’)
- Two transformations make a double prime. (A”, B”, C”, D”)
- A translation slides a shape along a straight line without turning. (move 3 up and 8 down = (3, -8))
A reflection flips a shape across a line of reflection, mirroring the shape. (reflect over x-axis, reflect over y-axis, line of reflection y = 2)
A rotation rotates a shape around a point. The point is sometimes the origin point. It must rotate at an angle and in a direction. (rotate around (3, -6) 90 degrees counter-clockwise)
Bisect means to cut into 2, since bi means 2, and sect means cut.
A line is infinite, and its representation has 2 arrows.
A ray is infinite from a point, and its representation has 1 arrow.
A line segment is finite, and its representation has 0 arrows.
Drawing a line (bisector) to divide a line segment into 2 equal parts is bisecting the segment.
- When it’s drawn at a 90-degree right angle, it’s called a perpendicular bisector.
- So perpendicular lines are lines that intersect to form a 90-degree right angle.
- After drawing a segment, it can have a perpendicular bisector made using a compass, by opening it up to greater than half the segment length, then drawing an arch from one point, then from the other. Where they intersect are the points for the bisector.
- Making one with a protractor should be straightforward.
- Notation to represent the whole thing is the 2 segment point letters, followed by a small horizontal line with a small vertical line coming out its top, followed by the 2 points of the bisector. (AB+CD (replace the plus sign))

Parallel lines are lines that never intersect onto each other and are always the same distance away.

They can be made with a protractor by making a perpendicular line through a line, and then making another through the perpendicular line just made.
With a compass, make a perpendicular line through a line, and then making another through the perpendicular line just made.

Parallel lines can be labeled with the 2 letters for the points of the first line, then

, then the 2 letters for the points of the second line. (AB

EF)

An acute angle is an angle that’s less than a 90-degree right angle. (35, 56, 61)
An obtuse angle is an angle that’s more than a 90-degree right angle. (105, 352, 126)
Equal angles are represented by the same symbol.
An angle bisector is a line that divides an angle into 2 equal parts.
- To make one with a protractor, simply measure half of the angle, then draw the bisector.
- To make one with a compass, draw an arc from the point of the angle, then for each of the 2 intersections made, draw another arc at the point of intersection. Where the final 2 arcs intersect, that’s the point of the bisector.
- An angle bisector is represented by an < then first the points of the two angle lines with the angle point in between and = + the original amount of degrees, then the first angle line followed by the angle point, the bisector = the second angle line followed by the angle point, and the bisector, followed by = + the bisected angle in degrees. (ABC = 90, ABD = CBD = 45)

Area and Perimeter

Length is the length of the sides, and width is of the top and bottom.
Base is the width of the bottom, and height is how tall the whole shape is (not a side).
Perimeter is the length around a shape, area is the total space inside.
- Formula for square perimeter: P = 4s. (4(3) = 12)
- Formula for rectangle/square perimeter: P = 2b + 2h or P = 2w + 2h. (2(3) + 2(7) = 26)
- Formula for square area: A = s2. (5 to the power of 2 = 25)
- Formula for rectangle/square area: A = bh or A = wh. (3(5) = 15)
By removing the rearranging the 2 triangles on the sides of a parallelogram, you make a rectangle.
- Formula for parallelogram perimeter: P = 2b + 2h or P = 2w + 2h. (2(6) + 2(9) = 30)
- Formula for parallelogram area: A = bh. (3(8) = 24)
Two triangles will always make a parallelogram, or a rectangle.
- Formula for triangle perimeter: a + b + c. (3 + 8 + 2)
- Formula for triangle area: bh/2. (8(6) / 2 = 24)
The radius of a circle is the length from the center to the edge.
The diameter is across the circle and is twice the radius.
The circumference is the length around a circle.
- The circumference divided by the diameter is always equal to the irrational (repeating without a pattern) number known as pi (represented by the Greek character of the same name).
  - The first 10 digits of pi are 3.141592653, but usually 3.14 is just used.
- The formula for circle circumference: pi(d), or 2pi(r). (2(3.14(6)) = 37.68)
The formula for circle area: pi(r2). (3.14(3 to the power of 2) = 28.26)
Some shapes are made up of other shapes, so the area/perimeter of those shapes can be added, or subtracted together.

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