MATH REVIEWER
MATH REVIEWER
Data Collection
Data collection methods are techniques and procedures for gathering information for research purposes. They can range from
simple self-reported surveys to more complex quantitative or qualitative experiments.
Two Types:
Quantitative data collection gathers numeric data that puts consumer insights into a quantifiable context.
Answer the questions, “How much/many?”.
Example:
1. AGE
2. HEIGHT
3. DISTANCE
4. WEIGHT
5. ETC.
Qualitative data is information that describes and explains something. It can be seen, observed, and written down.
Answer the questions, “WHY”.
Example:
1. GENDER
2. COLOR
3. CATEGORY
4. DESCRIPTION
5. ETC
Probability Sampling:
Simple Random Sampling - Simple random sampling gives all members of the population an equal chance of being selected.
Stratified Random Sampling - Stratified sampling draws a sample from each group (or stratum) separately to ensure every subgroup is represented.
Systematic Random Sampling - Systematic sampling, each member of the population is assigned a number and then selected at regular intervals to form a sample.
Cluster Random Sampling - Like stratified sampling, cluster sampling separates the population into subgroups or clusters. But that’s where the two probability sampling methods diverge.
Non-Probability Sampling:
Convenience Sampling - In this approach, researchers pull together a sample from individuals who are available and willing to participate.
Snowball Sampling - This type of sampling relies on people in your population to identify others to sample.
Purposive Sampling - Purposive sampling refers to intentionally selecting participants based on their characteristics, knowledge, experiences, or some other criteria.
Quota Sampling - Like stratified sampling, quota sampling divides the population into subgroups based on known characteristics, traits, or interests.
Ungrouped Data - Ungrouped frequency distribution: It shows the frequency of an item in each separate data value rather than groups of data values.
Group Data - The data is arranged and separated into groups called class intervals. The frequency of data belonging to each class interval is noted in a frequency distribution table. The grouped frequency table shows the distribution of frequencies in class intervals.
Absolute Value - The absolute value of a number n, written as |n|, is the “distance on the number line from 0 to n”.
Like-signed - Add the absolute value of the integers and affix the common sign.
Examples: 4 + 5 = 9 // -4 + (-5) = -9 // -7 + (-9) = -16 // 4 + (-5) = -1 // -3 + 3 = 0
Subtraction of Integers - Change the sign of the subtrahend and continue as if in addition. Keep the minuend, change the operation to addition, and change the sign of the subtrahend.
Example: 6 - 10 = ? 6 - 10 = ? 6 - (-10) = ? 6 + (-10) = -4
Multiplication of Integers
If we multiply two positive integers, the product is positive.
If we multiply two negative integers, the product is positive.
If we multiply two integers with unlike signs, the product is negative.
Division of Integers
If we divide two positive integers, the quotient is positive.
If we divide two negative integers, the quotient is positive.
If we divide two integers with unlike signs, the quotient is negative.
Algebra is the use of letters and symbols to represent values and their relations.
Musa al-Khowarizmi, Father of Algebra
hisab al-jabr w'al-muqabala (The Compendious Book on Calculation by Completion and Balancing)
Constant - Any symbol representing one fixed value.
Example: 5x2; 5 and 2 are constants. -3yx; -3 is the constant.
Variable - Any symbol representing possible values of a quantity.
Example: 4y + 2; y is the variable. -9ab; a and b are variables.
Algebraic Expression - An algebraic expression is a Mathematical phrase that contains constants, variables, and/or operators.
Example: 2x + 5
Terms - Either a single number or a variable, or numbers and variables multiplied together. Separated by their sign.
Examples: 2x; – 3y; 7/5 // 2x; – 3y // 2x
Constant Terms - A term that the value of which is constant and will never change.
Example: “3x + 2,” 2 is the constant term.
Coefficient - Coefficient is another term for cofactor.
Types:
- Literal Coefficient
- Numerical Coefficient
An expression is any combination of one or more constants and variables along with at least one mathematical operation.
An equation is a statement that two numbers or two expressions are equal.
In the spoken language, an algebraic expression is to a phrase as an equation is to a sentence.
Expression - “the sum of 3 and x”
Equation - “The sum of 3 and x is equal to 15.”
expression : phrase :: equation : sentence
Equivalence Relations
Reflexive Properties - A number equals itself.
Symmetric Property - Order of equality does not matter.
Transitive Property - Two numbers equal to the same number are equal to each other.
Addition Property - If 2 + 3 = 5, then 2 + 3 + 1 = 5 + 1
Subtraction Property - If 2 + 3 = 5, then 2 + 3 − 1 = 5 − 1
Multiplication Property - If 2 + 3 = 5, then (2 + 3)(4) = (5)(4)
Division Property - If 2 + 3 = 5, then 2 + 3 /10 =5/10
Substitution Property - If x = 5, then the equation 2 + 3 = x is the same as 2 + 3 = 5
Identity Property - 4 + __ = 4 5(__) = 5
Inverse Property - 4 + ___ = 0 5(___) = 1
Distribution Property - 2 3 + 4 = 2 3 + 2 4
Examples:
m increased by 5 = m + 5
t decreased by 7 = t − 7
11 less than a number = n − 11
1 less than 4 times a number = 4n − 11
two more than a number = n + 2
Examples:
a + 5 = 11 a = 6
15 = b + 8 b = 7
x − 7 = 10 x = 17
y − (−9) = 11 y = 2
z + 1/2 = 2 z = 3/2
Example:
Twenty less than a number is 35. Find the number.
Representation:
Let x = the number
Equation:
x – 20 = 35
Solve:
x – 20 = 35
x = 35 + 20
x = 55
Prove:
x – 20 = 35
55 – 20 = 35 35 = 35
Thus, the number is 55.