Fluency
The ability to solve problems quickly and accurately by knowing which methods to use and how to use them
Accuracy
Getting the correct answer
Automaticity
Selecting a problem-solving method and performing computations without requiring much time to think and process through
Rate
How quickly computations are made
Flexibility
Being able to solve problems in more than one way and selecting the most appropriate method
Reasoning
The ability to hypothesize test theories, and draw conclusions
Inductive Reasoning
Conclusions made based on observation
Deductive Reasoning
Conclusions based on logical synthesis of prior knowledge of facts and truths
Adaptive Reasoning
The ability to think logically about relationships between concepts and to adapt when problems and situations change
Concrete Model
Use objects to demonstrate operations (Cubes, Base-Ten Blocks)
Semi-Concrete Model
Use pictures, instead of actual objects, to demonstrate operations
Semi-Abstract Model
Use single symbols (tally marks) to represent numbers of objects while performing operations
Abstract Model
Using numbers only to perform operations
Pictorial
Number lines, bar graphs, pie charts; representations of numerical values
Put-Together Problem
There are 2 oranges and 3 bananas in a fruit bowl. How many pieces of fruit are there all together?
Take Apart Problem
8 students were asked whether they like dogs or cats better. 5 students picked dogs. How many picked cats?
Array
One way to model a multiplication problem visually
Tiling
A rectangle is divided into unit squares which can be counted to find the area
Measurement Division
Used when students know how many objects are in each group but do not know how many groups there are
Partitive Division
When students know how many groups there need to be, but not how many objects are in each group.
Algorithms
Way to solve problems without visual models
Iteration
Computational process in which the same steps are repeated until a final answer is found
Subsitizing
The ability to instantly see the number of objects in a small set without having to count them
Natural Numbers
The numbers we use to count (1,2,3...)
Whole Numbers
Natural numbers and zero
Integers
Whole numbers and their corresponding negatives
Fractions
Portions of integers expressed with a numerator and a denominator
Decimals
Portions of integers expressed as numbers following a decimal point
Rational Numbers
All integers and fractions
Irrational Numbers
Numbers that cannot be expressed as fractions, such as infinite, non repeating numbers
Exponents
Number that raise another number to a power making it multiply itself a certain number of times
Addition Property of Zero
Adding 0 to a number does not change the numbers value
Multiplication Property of Zero
Any number multiplied by 0 equals 0
Additive Inverse
The sum of any number and its additive inverse equals 0. x + -x = 0
Powers of Zero
0 raised to any power equals 0
Division by Zero
Dividing any number by 0 results in a quotient that is undefined
Zero as a Dividend
Dividing 0 by any number results in a quotient of 0
Multiplication Property of One
Multiplication by 1 does not change the value of the number
Multiplication Inverse
The product of any number and its multiplicative inverse is 1
Powers of One
1 raised to any power equals 1
Quotient of One
Any number (other than 0) divided by itself, equals 1
Absolute Value
A numbers distance from 0
Addition
Regroup by Carrying
Subtraction
Regroup by Borrowing
Multiplication
Repeated Addition
Commutative Property
The order of the numbers being added or multiplied does not affect the final results
Distributive Property
a (b+c) = ab + ac
Associative Property
If the operations are the same, the terms can be regrouped by moving the parentheses. (a+b) + c = a + (b+c)
Order of Operations
PEMDAS
Factors
Whole numbers that are multiplied together to get a product.
Prime Factorization
Factoring a number and then factoring its factors until all factors are prime
Greatest Common Factor (GCF)
When comparing 2 numbers, the largest factor they have in common
Multiples
Multiplying a number by whole numbers
Least Common Multiple (LCM)
When comparing the multiples of two numbers, the smallest multiple that they share
Divisible by 2
All even numbers
Divisible by 3
If all the digits add up to something divisible by 3, then the number is divisible by 3.111: 1 + 1 + 1 = 3
Divisible by 4
If the last two digits of the number are evenly divisible by 4, then the number is divisible by 43480: 80 is divisible by 4
Divisible by 5
If the number ends in 0 or 5
Divisible by 6
If the number is divisible by both 2 and 3, then it is divisible by 6
Divisible by 8
If the last three digits are divisible by 8, then the number is divisible by 8
Divisible by 9
If all of the digits add up to a number divisible by 9234: 2 + 3 + 4 = 9
Variables
A letter used to refer to an unknown quantity
Expressions
Mathematical phrases that can contain numbers, variables, and operations
Equations
Two mathematical expressions that are set equal to one another using the equal sign
Reflexive Property
Every number is equal to itself
Symmetric Property
If a number is equal to another number, then the converse is also true
Transitive Property
If number "a" is equal to number "b," and number "b" is equal to number "c," then number "a" is also equal to number "c
Substitution Property
If two numbers are equal to one another, they are interchangeable
Difference of Two Perfect Squares
a^2 - b^2 = (a + b) (a - b)
Quadratic Formula
To solve quadratic equations (ax^2 + bx + c = 0)
Pythagorean Theorem
To find the missing length of a right triangle a^2 + b^2 = c^2
Product Laws of Exponents
(a^n)(a^m) = a^n+m(a^n)(b^n) = (ab)^n
Quotient Laws of Exponents
a^n / a^m = a^n-ma^n / b^n= (a/b)^n
Negative Exponents
b^ -n = 1/b^n
Zero Exponent
b^0 = 10^n = 0
One Rules Exponents
b^1 = b1^n = 1
Midpoint Formula
(X2 - X1/2), (Y2-Y1/2)
Slope Formula
m = (Y2-Y1)/(X2-X1)
Slope Intercept Formula
y = mx + bm=slopeb= y-intercept
Inequality
Comparison of two expressions Whatever you do to one side, you must do to the other
Multiplying or Dividing Inequalities by a Negative
The sign will become reversed
Scientific Notation
One place value before a decimal point and then the number is written in terms of multiplication by 10 to a power
Expanded Form
Breaking a number apart into an addition expression based on the place value of each digit748 = 700 + 40 + 8
Arithmetic Sequence
The same value is added each time
Geometric Sequence
The same value is multiplied each time
Squares Sequence
Square each number
Cubes Sequence
Cube each number
Fibonacci Sequence
Each number is the sum of the two numbers before it
Functions
Algebraic equations that have an input and an output (f(x))
Vertical Line Test
If a vertical line would only pass through the graph once, then it is a function
Add or Subtract Fractions
If there are like denominators, just add the numerator If there are unlike denominators, find the LCM of the denominator and multiply
Multiply Fractions
Multiply across to create a new fraction
Divide Fractions
Keep, Change, Change
Measurement Prefixes
Kilo, Hecto, Deka, Base Unit, Deci, Centi, Milli
Mean
Average
Median
Middle of the set when they are arranged in numerical order
Mode
The number that appears most frequently
Range
Describes the span between the numbers; subtracts the lowest number from the highest number
Precision
How close a measured value is to an actual value
Accuracy
How close measured values are to each other
Complimentary Angle
Angles that add up to 90'
Supplementary Angles
Angles that add up to 180'
Polygon
2D shape which all sides are straight, non over lapping lines
Acute Triangle
All three angles are less than 90'
Right Triangle
One angle is 90'
Obtuse Triangle
One angle is greater than 180'
Equilateral Triangle
All three sides are equal in length
Isosceles Triangle
2 sides are equal in length
Scalene Triangle
No sides are equal in length
Geometric Nets
2D figures that represent the faces of a 3D shape
Congruent Shapes
Exactly the same shape and size
Similar Shapes
The same shape but different sizes
Area of a Triangle
A = 1/2bh
Area of a Rectangle
A = lw
Area of a Square
A = S^2
Area of a Circle
A = 3.14r^2
Area of a Parallelogram
A = bh