Math 8
I had a summative test on the second week of school!
Ratios and Rates
- A ratio is a colon-separated group of terms (quantities/numbers) of the same unit that can be of two types:
- A part to part ratio compares each part of a group to the other parts of the same group. (three red marbles and four blue marbles in a bag is 3:4 or 4:3)
- A part to whole ratio compares a part or two to the whole group. (three red marbles and four blue marbles in a bag is 4:7 or 3:7)
- Ratios can have two or three terms. (9:8, 1:2:3)
- A unit ratio is a ratio where one of the terms is the value of one. (3:1, 6:1)
- Equivalent ratios are essentially equivalent fractions that can be easily written as another form. (1:2 = 2:4, 5:3:11 = 15:9:33)
- The lowest terms of a ratio is the smallest possible equivalent ratio of it.
- A rate compares two quantities measured in different units. (200 meters/2.3 minutes)
- A unit rate is a rate in which the second term is one. ($3.50/1 pounds = $7.00/2 pounds)
- You can use unit rates to find out the better deal when buying things. (store A offers four bananas for $5.00, and store B offers five bananas for $6.00 -> $5.00/4 bananas = $1.25/1 banana, $6.00/5 bananas = $1.20/1 banana -> B is cheaper than A)
- The uses of a unit rate isn’t limited to buying things. (Jesse can type 187 words in 5 minutes, and Brent can type 444 words in 10 minutes -> 187(2) = 374 words/10 minutes -> 444 words/10 minutes > 374 words/10 minutes)
- The use of unit rates is similar to using fractions and division.
Integers
- If both numbers are of the same sign in multiplication or division, the answer is positive, otherwise it’s negative. (3(3) = 9, 4(-2) = -8, -5 / -1 = 5, -8/8 = -1)
- Multiplying a positive or negative number by a negative number can be done using number tiles:
- Represent the first number using either positive or negative tiles.
- Copy that group of tiles by the number of times specified by the second number.
- To multiply a positive or negative number by a negative number with number tiles, you will essentially be “removing” the number from zero:
- Represent the multiplication of the two in zero pairs.
- Remove groups of either the positive or negative tiles such that the end result will be of the correct sign.
- A positive or negative number can be divided by a positive number using number tiles by splitting things up into groups:
- Represent the first number using the appropriate number tiles.
- For each group of tiles representing the second number, circle it.
- The answer will be the number of groups.
Fractions
- To multiply or divide two fractions, they must both be an improper fraction.
- You can multiply two improper fractions in the lowest terms:
- Reduce the numerator of the first fraction and the denominator of the second as if it was an equivalent fraction. (2 and 4 = 1 and 2)
- Do the same for the numerator of the second fraction and the denominator of the first.
- The answer’s numerator is the first numerator multiplied by the second numerator.
- The answer’s denominator is the first denominator multiplied by the second denominator.
- The process of dividing two fractions is the same as multiplying but with an extra step:
- Make the second fraction its reciprocal (reverse/flip/inverse) of the second fraction. (1/2 -> 2/1)
- Multiply the resulting two fractions together.
- You can also multiply fractions by filling in a grid and divide fractions by grouping parts of a grid.
Equations
- There are several types of equations:
- A multiplication or division one-step equation, represented by ax = b or x/a = b, is the simplest type of equation and only involves one step to solve. (2n = 6 -> n = 3)
- A multiplication or division two-step equation, represented by ax + b = c or x/a + b = c, is a type of equation that requires one step to solve. (3n + 4 = 13 -> 3n = 9 -> n = 3)
- There are also distributive property two-step equations, represented by a(x + b) = c, which are equations that can optionally be solved differently. (2(n + 3) = 14 -> 2n + 6 = 14 -> 2n = 8 -> n = 4)
Powers and Pythagoras
- A power or exponent represents repeated multiplication. (three to the power of four = 3^4 = 3(3)(3)(3) = 81, four to the power of three = 4^3 = 4(4)(4) = 64)
- An exponent of two is a square. (five squared = 5^2 = 5(5) = 25, ten squared = 10^2 = 10(10) = 100)
- Squares can be represented by a square grid.
- A perfect square is a number that has two equal factors, and those factors can be squared to equal the perfect square. (121, 256, 36, 81, 400)
- Here’s a list of all the perfect squares from one to twenty:
- 1
- 4
- 9
- 16
- 25
- 36
- 49
- 64
- 81
- 100
- 121
- 144
- 169
- 196
- 225
- 256
- 289
- 324
- 361
- 400
- Prime factorization can be used to determine if a number is a perfect square:
- Factorize the number into two numbers. (484 = 2(242))
- Attempt to factorize each of those numbers. (242 = 2(121))
- Keep repeating and attempting with each new number and stop when all the numbers are primes. (2, 2, 11, 11)
- The number is a perfect square if it has an even number of each prime number. (two (even) twos, two (even) elevens -> 484 is a perfect square)
- You’ll be left with what appears to be a tree with branches and leaves.
- The square of the square root of a number equals the number; this only results in a whole number when done on a perfect square. (sqrt(16) = 4, sqrt(49) = 7, sqrt(225) = 15)
- Number lines can be used to estimate the square roots of non-perfect numbers.
- In a right-angled triangle (ignoring trigonometry side names), the two sides touching the right angle are referred to as a and b, while the remaining longest side referred to as c and is the hypotenuse.
- The Pythagorean theorem (named after a big-brain Greek guy that you will soon rival in the future after reading these study notes) states that in any right-angled triangle, the square of a plus the square of b is equal to the square of c (a^2 + b^2 = c^2). (a = 3, b = 4, c = 5 -> 3^2 + 4^2 = 5^2 = 9 + 16 = 25)
- This can be used to find the length of one of the sides using the other sides by getting the square root of a number. (a = 3, b = 4 -> c = sqrt(a^2 + b^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 25)
- A Pythagorean triple is a group of three whole numbers that satisfy the Pythagorean Theorem. (3, 4, 5)
Surface Area and Volume
- There are some terms used to define different dimensions:
- Zero-dimensional (0D) objects are points with no directional measurement.
- One-dimensional (1D) objects are lines with one direction of measurement.
- Two-dimensional (2D) objects are flat planes with two perpendicular directions of measurement.
- Three-dimensional (3D) objects have an addition direction of measurement.
- For 3D objects, there are six different views: the top, bottom, left, right, front, and back.
- Prisms are 3D objects that have two parallel bases and a number of perpendicular faces.
- Rectangular prisms are composed entirely of rectangular faces and have six sides.
- Cubes are rectangular prisms but are composed entirely of squares.
- Triangular prisms have a triangle as their base.
- Cylinders have circular bases.
- The surface area of a 3D object is the 2D measure (square units) of all the combined area of the object’s faces, and multiple shapes in contact with each other will also “merge” some faces too.
- For rectangular prisms, you can use SA = 2(wh + wd + hd), where w, h, and d each correspond to width, height, and depth. (2x4x3 meter rectangular prism -> SA = 2(2(4) + 2(3) + 4(3)) = 2(8 + 6 + 12) = 2(16) = 32 square meters)
- To find the surface area of a cube, you can use SA = 6s, where s is the side/edge length. (cube with side length of four meters -> SA = 6(4) = 24 square meters)
- You can calculate the surface area of a triangular prism with SA = ab + ad + bd + cd. (triangular prism with base of 1x2x3 meters and depth of four meters -> SA = 1(2) + 1(4) + 2(4) + 3(4) = 2 + 4 + 8 + 12 = 26 square meters)
- The surface areas of cylinders require the circumference and area, so SA = 2pi(r^2) + pi(d)(h) where pi is… pi (3.14…), r is radius, d is diameter, and h is height. (cylinder with diameter of two meters (radius of one meter) and height of three meters -> SA = 2(3.14)(1^2) + 3.14(2)(3) = 6.28(1^2) + 18.84 = 12.56 + 18.84 = 31.4 square meters)
- The volume of a 3D object is the 3D measure (cubic units) of the space occupied by the object. It can be calculated for prisms by multiplying the base shape’s area by the depth/length of the object.
- To find the volume of a rectangular prism, you can use V = whd. (2x4x3 meter rectangular prism -> V = 2(4)(3) = 24 cubic meters)
- For cubes, you can use V = s^3. (cube with side length of seven meters -> V = 7^3 = 343 cubic meters)
- The surface areas of triangular prisms require V = d(wh/2). (9x8x7 meter triangular prism -> V = 7(9(8) / 2) = 7(72/2) = 7(36) = 252 cubic meters)
- You can calculate the volume of a cylinder using V = d(pi)(r^2). (cylinder with radius of six and depth of ten -> V = 10(3.14)(6^2) = 31.4(6^2) = 31.4(36) = 1130.4 cubic meters)
Probability and Data
- Probability events can be represented in fractions, decimals, or percentages. (rolling a die and drawing two cards from a deck without replacement will equal one specific arrangement = 1/6(1/52)(1/51) = 1/15912 = 0.000063 = 0.0063%)
- Different graphs better serve different purposes:
- Bar graphs are good for comparing quantities in categories. (price of the same product from different competitors)
- Line graphs are good for showing how something changes over time. (price of something over time)
- Circle/pie graphs good for showing things out of a whole. (time spent doing things)
- Pictographs are good for looking “appealing”. (star ratings of places)
- Data in charts can look misleading through the use of tricks:
- Increments can be made tiny or huge, and in some bad cases increments are broken up.
- The percentages in a circle/pie graph may not add up to 100% sometimes, and its slices may be disproportionately sized.
- Icons in pictographs may have different sizes to put emphasis on specific categories.
- A bar graph might make the horrifying decision to use pictures instead of bars.