Elementary Mathematics
By: Ashley Walker

3.1 Numeration: Numerals are what we see when we look at numbers. It is the name for the symbol used when we see numbers such as 2 or 9. Face value is the value of each number. Different numeration systems include the Tally numeration system, Egyptian numeration system, Babylonian numeration system, Mayan numeration system, and Hindu- Arabic numeration system. Place Value: based on powers of 10Ten thousand, thousands, hundreds, tens, ones. Number base systems: We usually count in base 10: 0, 1,2,3,4,5,6,7,8,9,10 and so on In other bases it would be as shown:Base 4: the only numbers available are 0,1,2,3.Base 5: the available numbers are 0,1,2,3,4. Base 2: 0,1

Regroup and Trade!Adding with whole numbers: Properties of addition: Closure: a+b=cCommutative: a+b= b+aAssociative: (a+b) +c = (b+c) +aIdentity: a+0= aDifferent models of addition: Set Model Number line Examples of both: https://docs.google.com/document/d/1OZia6YB7HBZXaj7zL8rkkweGoJveo0enMn7IV9fjL1g/edit?usp=sharingSubtracting with whole numbers: None of the addition properties work when it comes to subtraction Different types of models: ComparisonMissing addendTake awayNumber lineExamples of all: https://docs.google.com/document/d/13wKuCEAdsgKZr38KN5zzych7hEfR_sbY3qyUqe13_B0/edit?usp=sharingAlternate methods of subtracting: European method: add to both sides: example: 73 - 27-------------------------- becomes 74 - 37-----------------------------`Troubles coming method : example shown here: https://docs.google.com/document/d/1ra2dPSSBjS9J5LOQ_Sik34F8J3dzEKpBd89H3kpAwKg/edit?usp=sharing

Multiplication: 6 Properties of multiplication: associative:(a x b)x c= a x (b x c)communative: a x b= b x a distributive: a(b+c)= (a x b) + (a x b)identity: a x 1 = aassociativezero: a x 0 = 0Repeated addition: 5+5+5 = 5 x 3Division: Two models of division: Sharing/set (partition model): answers how many in each group Example: Noah has 18 fruit snacks and wants to share them with his 3 friends. How many will each friend get? : 18/3= 6 fruit snacks each Repeated subtraction: answers how many groups Example: Noah has 18 fruit snacks, and he wants to give each person 6. How many of his friends can he share with? : 18/6= 3 friends Anything divided by 0 is undefined

Divisibility: Divisibility refers to whether or not a number can be divided. Example: 2 divides 12 = 2/12 Divisibility tests:A number is divisible by 3 if the sum of the digits is divisible by 3. A number is divisible by 4 if the two-digit number in the tens and one's place is divisible by 4. A number is divisible by 5 if the one's digit is a 0 or a 5. A number is divisible by 6 if it is divisible by both 2 and 3. A number is divisible by 8 if the three-digit number made by the last three digits of the number is divisible by 3. A number is divisible by 9 if the sum of the digits is divisible by 9.Prime and Composite Numbers: Prime: A number that only has two factors being 1 and itself. Example: 2 is a prime number Composite: A number that has another factor other than 1 and itself. Example: 9 has factors of 1,3,9. Factors: Whole numbers that we multiply to get another whole number Factorization of a number: composite numbers expressed as products of 2 or more whole numbers greater than 1. Factor Tree example: https://docs.google.com/document/d/149m8kPx9EwRe4fRcPVvWCqTHSIf8pviv7pc1KQBU_P0/edit?usp=sharingUpsidedown division example: https://docs.google.com/document/d/149m8kPx9EwRe4fRcPVvWCqTHSIf8pviv7pc1KQBU_P0/edit?usp=sharing

GCF & LCM: GCF= Greatest common factor. The greatest whole number that divides two numbers LCM= Least common multiple.The least whole number that is a multiple of both the two given numbers. Ways to find GCF and LCM: Bar model method: Build a model of 2 or more whole numbers with different colored rods.Slide Method: find a common factor that divides each number, divide them all then write them underneath, repeat if you have any more in common. Prime factorization method: Find the prime factorization of each number. GCF= what prime factors they have in common LCM= which one has the most, then multiply them. Examples of all here: https://docs.google.com/document/d/10JjU1xWLiRQtPZVybHgRF-TdKw1pHpbJ1mDfTHt2z8E/edit?usp=sharing

Integer addition and subtraction: Integer: a whole number Negative integers= the opposite of positive ones. Example: -5 = the opposite of 5 Absolute value: The distance between the point of an integer and 0. Example: absolute value of 4 = 4 shown as |4|= 4 Integer addition models: Chip model: positive shown with yellow chips, Negative shown with red chips. One red neutralizes one yellow and makes a zero pair. Numberline model: Always start at zero, if the number is positive walk forward. If the number is negative walk backward. Examples of both are here: https://docs.google.com/document/d/1jQzP90452hE1c26rNnTtekByYSqiPujhf2R9F1AN3SY/edit?usp=sharingProperties of integer addition: Closure property: a+b= a unique numberCommunicative property: a+b=b+a Associative property: (a+b)+c= a+(b+c) Identity property: 0+a=a=a+0 Integer subtraction models: Chip modelNumberline model: same rules as addition, but subtraction is modeled by turning around and facing the left/ negative direction. Subtraction using the opposite approach: a-b=a+-bExamples of all here: https://docs.google.com/document/d/1jQzP90452hE1c26rNnTtekByYSqiPujhf2R9F1AN3SY/edit?usp=sharing

Integer multiplication and division: Models to show multiplication: Chip modelNumberline Both shown here :https://docs.google.com/document/d/17HyfFVKD5R-vndJte5EtOApONlsBmKYHZ9gSbbTRklE/edit?usp=sharingProperties of integer multiplication: Closure: ab=a unique numberCommunicative: ab=baAssociative: (ab)c=a(bc)Identity: a*1=aDistribuative: a(b+c)=ab+acZero: a*0=0=0*aModels to show division: Sharing/ Partition: answers how many in each group Repeated subtraction: answers how many groups Examples shown here: https://docs.google.com/document/d/1l9fPkzCeuW8UO8sR2l6M9Gsxi85T2y5VHI3NLctIvNQ/edit?usp=sharing

Fractions: Rational numbers: introduced as fractionsProper: Rational number that is less than 1 Improper: Rational number greater than 1Numerator: how many of these parts are under consideration Denominator: how many equal parts has the unit been subdivided into Numerator------------------------ DenominatorWays to represent fractions/rational numbers: Bar/ area modelNumberline modelSet modelExamples of all here: https://docs.google.com/document/d/13t0x8pMpwiXBR5m1e4zQlBYJnPFGjufe7AJmtMJBhCg/edit?usp=sharingSimplifying fractions: When a and b or the numerator and denominator have common factors, divide both numbers by them till the fraction is in its simplest form. Simpliest form: Lowest terms/ is a and b have no more common factors left. Comparing Fractions: ways to find if a fraction is less than or greater than another More same sized parts--If the numbers have the same denominator, which has the greater numerator.Samer number but different sized parts - If the numbers have the same numerator, which has the smaller denominator. More and less than one half or one whole-find which fraction is over 1 or under 1/2Closeness to one half or one whole -find which fraction is closer to either 1 or 1/2

Adding and Subtracting Fractions: Adding Fractions: Adding with like denominators: a/b + c/b = a+c/b Example: 2/7+ 3/7 = 5/7 Adding with unlike denominators: we have to replace the fraction it gives us with an equivalent fraction that has like denominators Example: 3/5+ 1/2= 3(2)/5(2) + 1(5)/2(5) = 6/10 + 5/10= 11/10 Ways to show the addition of fractions: Fraction Circle ModelNumberline ModelArea ModelAll shown here: https://docs.google.com/document/d/1L-zwjykVERma78gA-5ycvBaGU4zGw4mn-ZIc-uXfSMI/edit?usp=sharingMixed numbers: the sum of an integer and a proper fraction Adding mixed numbers: additive inverse property of rational numbers= a/b + (-a/b) = 0 Example: 2/3+ (-2/3) = 0 Subtracting Fractions: Subtracting with like denominators is the same as adding so, a/b-c/b= a-c/b Example: 5/7 - 2/7 = 3/7Subtracting with unlike denominators is also the same since we need to make both fractions have the same denominator by using an equivalent fraction.Example: 3/5-1/2= 3(2)/5(2) - 1(5)/2(5) = 6/10 -5/10= 1/10 Example of number line subtraction: https://docs.google.com/document/d/15qjVPKjpjWqYEILwHIXCxa3bT4f1bgn9wxCVIaOoIgU/edit?usp=sharing

Multiplying and Dividing Fractions: Multiplying Fractions: Properties of multiplication of fractions: Multiplication identity property: 1*a/b=a/bInverse property: a/b*b/a=1Distributive property: a/b(c/d+e/f)= a/b *c/d +a/b * e/fExamples of multiplying fractions all shown here: https://docs.google.com/document/d/1oUQkY-Nhlr4nb5p0FaM1_bUrtbe-aaacQAsWmDVAJNE/edit?usp=sharingDividing Fractions: 1 3/4 divided by 1/2 : think of it as how many 1/2 foot lengths are there in something that is 1 3/4 feet long? algorithms for division of rational numbers: Invert and multiply: 2/3 divided by 3/7= x if 2/3= 3/7xEqual denominators a/b divided by c/d= ad/bd divided by bc/bd* bd/bc= ad/bcMore notes on dividing fractions are shown here: https://docs.google.com/document/d/1h3oGKRS9Bw0JX1PH7cuTDWqK5Lo1CBaMeOpEpMiscIE/edit?usp=sharing

Ratio and Proportional Reasoning: Ratio: a comparison of two quantities often shown as a:b Part- to Whole ratio example: the ratio of girls to boys in the class is 1:3; and girls to children is 1:4. All children are the whole and girls are the part. Part-to-Part Ratio example: the ratio of the number of girls to the number of boys Part- to- Whole model is shown here: The comparison model is shown here: https://docs.google.com/document/d/1CSvgAzWL-kB_9kaEA8E2TAVpt88uLbIVhNdEw7ddNIY/edit?usp=sharingProportion: a statement where two given ratios are equal. more notes on proportions here: https://docs.google.com/document/d/1ngtySyitzUqdJ_hak7jbvwwkgE9W7WIehU9nJZRDDQ4/edit?usp=sharing

Adding and Subtracting Decimals: We can show adding and subtracting decimals like we showed with whole numbers, by using base ten blocksWe can also show addition and subtraction of decimals by making the decimal a fraction. Place value: https://docs.google.com/document/d/1S4bXL5Z-HmbsCGJdQraK1y0Q3kPEWUmUH4btkhkQdNo/edit?usp=sharinghttps://docs.google.com/document/d/1CAs47cZZaLVXRZXQkw00b9gu_vzbbaMEFg51Y-5TK3M/edit?usp=sharing

Multiplying and Dividing Decimals: algorithm for multiplying decimals: if there are x digits to the right of the decimal in number 1 and y digits to the right in number 2 then the product = x+y digits to the right of the decimalWe can also use base ten blocks to show multiplication and division Dividing decimals can be shown with: Repeated subtraction equal groups or partition model 10 by 10 grid Estimation can be useful when multiplying and dividing decimals. Example: estimate then multiply 2.3 * 3.3: we can estimate 2* 3 and then know that our answer should be more than 6. Our answer is 7.59More notes here: https://docs.google.com/document/d/1pgvIDQaQ2F8UxN4MjBkFlWJcADyTA8qt6hmRn0uAnvg/edit?usp=sharing