I had a summative test on the second week of school!

• 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.

• 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:
  1. Represent the first number using either positive or negative tiles.
  2. 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:
  1. Represent the multiplication of the two in zero pairs.
  2. 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:
  1. Represent the first number using the appropriate number tiles.
  2. For each group of tiles representing the second number, circle it.
  3. The answer will be the number of groups.

• To multiply or divide two fractions, they must both be an improper fraction.
• You can multiply two improper fractions in the lowest terms:
  1. 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)
  2. Do the same for the numerator of the second fraction and the denominator of the first.
  3. The answer's numerator is the first numerator multiplied by the second numerator.
  4. 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:
  1. Make the second fraction its reciprocal (reverse/flip/inverse) of the second fraction. (1/2 -> 2/1)
  2. 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.

• 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)

• 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. 1
    2. 4
    3. 9
    4. 16
    5. 25
    6. 36
    7. 49
    8. 64
    9. 81
    10. 100
    11. 121
    12. 144
    13. 169
    14. 196
    15. 225
    16. 256
    17. 289
    18. 324
    19. 361
    20. 400
  • Prime factorization can be used to determine if a number is a perfect square:
    1. Factorize the number into two numbers. (484 = 2(242))
    2. Attempt to factorize each of those numbers. (242 = 2(121))
    3. Keep repeating and attempting with each new number and stop when all the numbers are primes. (2, 2, 11, 11)
    4. 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)
    5. 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)

• 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 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.