Elementary Mathematics
Spring 2025

Algorithm is a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer.Addition Algorithms:American StandardPartial SumsPartial Sums with Place ValueSolve Left to RightExpanded NotationLattice MethodsSubtraction Algorithms:American StandardReverse IndianSolve Left to RightExpanded NotationInteger SubtractionMultiplication Algorithms:American StandardPlace ValueExpanded NotationLattice Method

Fractions: A fraction is a part of something. A penny is a fraction of a dollar. The two numbers in a fraction are called the numerator and denominator.Numerator is over the denominator:Numerator - which tells how many parts you haveDenominator - which tells how many parts in the wholeForms of Fractions:Proper fraction The top number is less than the bottom number.Improper fraction The top number is equal to or larger than the bottom number.Mixed number A whole number is written next to a proper fractionReducing a fraction means writing it an easier way - with smaller numbers. When you reduce a fraction the value does not change. A reduced fraction is equal to the original fraction. When you have reduced a fraction as much as possible, the fraction is then in lowest terms.Reduce: 20 30 In multiplication, when both the top and bottom numbers end with 0's, cross out the 0's. Then check to see if you can continue to reduce.

EXAMPLE of Addition Expanded Notation:478 + 394 =Solve using expanded notation:400 + 70 + 8 =478300 + 90 + 4 = 394800 + 70 + 2 = 872EXAMPLE of Subtraction Expanded Notation:645 - 279 = 366Solve using Expanded Notation of Subtraction:_ 600 + 40 + 5 = 200 + 70 + 9 = 300 + 60 + 6 = 366

PROPERTIES:Properties of Multiplication:Identity property: when I multiply a number by one, the number does not change.Commutative property (aka Order Property): the order property is the order in which I multiply the number does not matter. Changing the order of factors doesn't change the product.Associative property: when grouping, what changes is the way I group. The outcome is always the same in the problem.Zero property: when I multipy a number by zero, the answer is always zero.Division does not have properties. Division, in a sense is the opposite of multiplication.Properties of Addition:Identity property: Adding zero to any number results in the original number.Commutative property: the order does not change the value. The order of the numbers being added does not change the sum.Associative property: means changing the grouping does not change the value. The order in which numbers are grouped when added does not change the sum.Distributive property: The sum of two numbers multiplied by a third number is equal to the sum of each number multiplied by the third number individually. Even if you distribute the numbers differently, they will solve to the same.Subtraction does not have properties. Subtraction in a sense is the opposite of addition.

Algorithm is a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer.Addition Algorithms:American StandardPartial SumsPartial Sums with Place ValueSolve Left to RightExpanded NotationLattice MethodsSubtraction Algorithms:American StandardReverse IndianSolve Left to RightExpanded NotationInteger SubtractionMultiplication Algorithms:American StandardPlace ValueExpanded NotationLattice Method

Raising fractions to higher terms is the opposite of reducing a fraction to lowest terms. Steps follow:Step 1: Divide the old bottom number into the new one (the number you want to raise to). Keep the bottom number as the number you want to raise to ( the new one). Step 2: Multiply the answer by the old top number. Place it on the top.Check: Reduce the new fraction to see if you get the original fraction.Changing improper fraction to whole or mixed numbers.An improper fraction is a fraction with the top number that is big or bigger than the bottom number. An improper fraction is equal to or larger than one whole.Change an improper fraction by dividing the bottom number into the top number and writing the remainder. Steps follow: Step 1: Divide the bottom into the topStep 2: Write the remainder as a fraction over the original bottom.Step 3: Reduce the remaining fraction.Changing mixed numbers to improper fractions. Steps follow: Step 1: Multipy the bottom number by the whole number.Step 2: Add the result to the top number.Step 3: Place the total over the bottom number.

A process standard, an essential component in every classroom.Problem Solving Steps:1.      Understand/Communicate2.      Devise a Plan/Strategize3.      Implement Plan4.      Look Back/Check your answer.Is this reasonable?Invented by George Polya (Einstein's Son's advisor)

A numeration system is a system for representing quantity of a number in a consistent mannerThere are 10 digits (numerals): 0,1,2,3,4,5,6,7,8,9Using the ten digits every number can be written.Base 10 means using the ten different digits to write numbers. Digits (numerals): 0,1,2,3,4,5,6,7,8,9. Base 10 = Decimal SystemDecimal System is positional which means that the placement of digits (or numerals) in the number specifies the value of the number.Base-5 (Quinary /ˈkwaɪnəri/ or pental) is a numeral system with five as the base.5^0 = Ones = five to the 05^1 = Fives = five to the one5^2 = 25's = five to the two5^3 = 125's = five to the 3Digits used in numeration system:In Base 10 - 0,1,2,3,4,5,6,7,8,9In Base 2 - 0,1In Base 3 - 0,1,2In Base 4 - 0,1,3In Base 5 - 0,1,2,3,4Expanded Notation:Example:723 = 700+20+3723 = (7x100) + (2x10) + (3x1)723 = (7x10^2) + (2x10^1) + (3x10^0)IMPORTANT: Exponent operations do not have a commutative property. AND, the ^ symbol is used to represent an exponent.