Elementary Mathematics

Vocabulary:Digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Numeral: written symbol for the digitsNumber: an arithmetical value, expressed by a word, symbol, or figureWhole NumbersW= {0,1,2,3,4,5,...}Natural Numbers    (think about counting things in nature how would you start)N= {1,2,3,4,5,...}The place value is the position of each digit in a number.When using base ten blocks, 1000 is a cube, 100 is a large square, a stick is 10, and a tiny cube is 1.Powers:Anything to zero power is 1.                    Exponent (how many times to multiply the base number times itself)BASE          (the number that is multiplied)Base ten system: 0-9 Base two binary system: Uses 0 and 12^3 2^2 2^1 2^0 power8     4     2     1    valueBase 5 Quinary System: uses 0,1,2,3,4

Roman Numerals:any of the letters representing numbers in the Roman numerical system: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000Tally marks:Tally marks are lines used for counting, they are arranged in groups of five. Addition?Combining two or more thingsAddend: the numbers being addedSum: the result of addition3 + 2 = 5Subtraction?The taking away of a part from a whole> greater than​​≥ greater than or equal to< less than ≤ less than or equal toOne is less than three1<3One less than three3-1Rounding:if the digit is less than 5 round down, if it is 5 or greater round upExpanded Form: 243+44= 200+40+340+4 =200+80+7= 287

Properties of addition:Closure Property of Addition: if a and b are whole numbers, then a+b is a whole number. There is only one unique answer.Commutative Property: if a and b are whole number then a+b = b+aAssociative Property: if a,b,and c are whole numbers then (a+b)+c = a+(b+c). It is the moving of parenthesis.Identity Property: a+0=a=0+a. Adding a number by zeroDoubles:1+1=2, 2+2=4, 3+3=6, 4+4=8, 5+5=10 …You can solve 7+8 by taking 7+7= 14+1=15Making tens:_+_=105+5, 7+3, 4+6, 8+2, 1+9What is multiplication?Duplicates of thingsGroups of things-> totalsFactor: numbers being multipliedProduct: result of multiplication1st factor: how many groups                 2x3=62nd factor: what is in each group

Properties for Multiplication of Whole Numbers:Closure Property of Multiplication: a*b= whole number, 2*3=6Commutative Property of Multiplication: a*b=b*a, 3*2=2*3Associative Property of Multiplication: (a*b)*c=a*(b*c), (2*3)*4=2*(3*4)Identity Property of Multiplication: a*1=a=1*aMultiplication property of zero: a*0=0=0*aDivisionDividend: the amount you have that is being shared, divided upDivisor: how what you “have” is being shared, broken upQuotient: result of division6 / 2 = 3Two types of division:Partitive Model: fair sharingHave 6, break up into 2 groupsQuotitive Model: subtraction or measurementHave 6, break up into 2 in each groupProperties:Identity Property: if you divide any number by 1 you will get the number

Multiplication and Division Fact Family:2, 3, 62*3=63*2=66/3=26/2=3Fractions:A rational number is a number that is in the form of p/q, where p and q are integers, and q is not equal to zero.Fractions can be represented using words, pictures, and modeling.A mixed number fraction is a fraction containing a whole number and a proper fraction.

Prime and Composite Numbersfactors: a number that divides the given number evenly or exactly, leaving no remainderPrime number: when a number is only factors are one and itself. Composite number: A number that has more factors than one and itself. For example, 8 has the factors 1, 2, 4, and 8, so it is composite. prime examples:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97composite examples:4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50GCD/GCFGreatest common divisor: the greatest common divisor of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted gcd.Example: The GCD of 20 is 5 because it is the largest number that can divide into itGreatest common factor: The GCF of a set of numbers it the largest factor all of the numbers have in commonExample: The GCF of 12, 20, and 24 is 4. LCM: The least common multiple of two numbers is the smallest number that is a multiple of both of them.Fundamental Theorem of Arithmetic: every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes

numerator: top number of a fractiondenominator: bottom number of a fraction, how many total partsRational Number (fraction) Operations: addition, subtraction, multiplication, and divisioncommon denominators: When fractions have the same denominator, we say they have common denominators. One way to find a common denominator for two (or more) fractions is to list the multiples of each denominator until we find the smallest multiple they have in common.example: common denominator or 7/8 and 3/108 and 10 have 40 and 80 in common so if you multiply the numerator and denominator by one of those numbers you can change the fractions so they have common denominators..

simplest form: A fraction is in simplest form if the top and bottom have no common factors other than 1. In other words, you cannot divide the top and bottom any further and have them still be whole numbers. You might also hear the simplest form called "lowest terms".mixed numbers: whole number and a fractionimproper fractions: fractions whose numerators are bigger than the denominatorfor example, two and a half (2 ½). An improper fraction is a fraction whose top number (numerator) is bigger than its bottom number (denominator). For example, five halves (5/2) is an improper fraction

adding and subtracting with like and unlike denominators:Fractions with the same denominators are called like fractions . To add fractions with like denominators, add the numerators and put it over the denominator. To add fractions with different denominators, such as 1/2 and 1/3, you need to find a common denominator. The least common multiple of 2 and 3 is 6, so you can rewrite the fractions as 3/6 and 2/6, respectively. This allows you to add the numerators together, resulting in a sum of 5/6.Plotting fractions on number line:Suppose we have to mark 5/6 on the number line.Step 1: Draw a horizontal fraction number lineStep 2: Since 5/6 is a proper fraction, it lies between 0 and 1. Mark 0 and 1 on the number line.Step 3: Divide the line segment between 0 and 1 into 6 equal parts since the denominator of the fraction is 6.Step 4: Count 5 parts from the left since the numerator of 5/6 is 5. Name the marking as 5/6.. 

integers: positive and negative whole numbers, including 0origin: beginning or starting point of a number lineabsolute value: distance from the number to the originopposites: number that is the same distance from the originThe further right you go on a number line the larger the numbers get. Addition and subtraction:Chip model: Use + and - chips to add or subtract positive or negative numbersCharged field method: Same thing as chip model but no chip. Only use + and -Rule using absolute values: if the addends have opposite signs, the sign of the sum is the sign of the addend with the greater absolute value. Number line model:smaller to right and larger to left. negative move left and positive move right.

Decimal representations of fractions: Fractions can be represented in decimal form as either exact decimal equivalents, or as rough approximations.2⁄3 = 0.6̅1⁄4 = 0.252⁄4 = 0.53⁄4 = 0.75values of places in decimalsFirst digit after decimal represents tenths place. The next digit after the decimal represents the hundredths place. The remaining digits continue to fill in the place values until there are no digits left.comparing sizes of decimal numbersThe number with a bigger digit in the tenths is the bigger decimal. If they have the same number, you repeat until you get to a place value with a larger number.

explaining the shifting of decimal points:If there IS a decimal point, move it to the right the same number of places that there are zeroes. When dividing by 10, 100, 1000 and so on, move the decimal point to the left as many places as there are 0s. So when dividing by 10, move the decimal point one place, by 100 two places, by 1000 three places and so on.why standard algorithms for adding and subtracting decimal numbers workStandard algorithm works because everything is based off of the base ten system. terminating decimalsTerminating decimal is a decimal that has an end digit. The decimal 1.141 is a terminating decimal since it has a finite number of digits after the decimal point. non-terminating decimalsNon terminating decimals do not have an end digit. They go on for infinity and are able to be show by placing a bar above the repeating number. mental computationWhen you have a decimal where the decimal parts can easily add up to be one whole, you can use mental math to figure out the sum. Think about this. If you had 0.30 + 0.70, you know that 3 + 7 is 10, therefore you know that 0.30 + 0.70 is 1.00how to round decimals:if the last digit is below five round the previous digit down. However, if it's 5 or more than you should round the previous digit up.

Basic operations on positive and negative numbersAddition: Same Signs, Add the Numbers.Addition: Different Signs, Subtract the Numbers.Subtraction: Switch to Addition.Multiplication and Division: Same Sign, Positive Result.Multiplication and Division: Opposite Sign, Negative Resultrepresentations of integers:mental computation:Once you learn all of the rules for integers, it becomes easier to mentally compute. There are tricks such as remembering when to move left (negative) and right (positive) so that your mental computations are correct.

how to round:Positive numbers greater than zero, such as 1, 2, 3, etc., are considered "UP" values and tend to round up. Negative numbers smaller than zero, such as -1, -2, -3, etc., are considered "DOWN" values and tend to round down.algorithm of positive and negative numbers:Input a number If number is less than zero, then it is a negative integer. if the number is greater than zero, then it is a positive integer.If neither, the number is equal to zero.counting positive and negative numbers:

 Irrational Numbers: Numbers that have repeating numbers after the decimal point Real Numbers: Real numbers are numbers that include both rational and irrational numbers. Rational numbers such as integers (-2, 0, 1), fractions(1/2, 2.5) and irrational numbers such as √3, π(22/7), etc., are all real numbers. Order of Operations:meaning of percent:one part in a hundred, the specified amount of something out of one hundredthe three types of percent problems:Find a given percent of a given number. For example, find 25% of 640.Find a percent given two numbers. For example, 15 is what percent of 50?Find a number that is a given percent of another number. For example, 10% of what number is 12? percent increase and decrease:The steps to calculate percentage change are: Find the difference between original and new values. Divide the difference by the original value. Multiply the resulting quotient by 100.adding percentages:Add the given percentages to 100.Convert the percentages to decimals.Multiply to the base value.Multiply the second percentage.