Elementary Mathematics (Valentina Bitsoie)^

Discussion: What was surprising to you about the number systems? That discovery of one and zero made a big impact to numeral system, but to our future as well. Example: technology was created by pattern of one and zero.

Explored different bases; such as unit(cube) is one, long (a line of cubes) is ten, and a flat (single big square) is hundred. Then explored how it would look in different bases.

Assignment introduced how to convert base ten into other various bases. For example 100 base six converted to base ten is 36.

Order of operation: Solved from left to right whether its1) Parenthesis ()2) Exponents (9^2) 3) Multiplication/Division4) Addition/Subtraction

The YouTube video helped me understand how to do standard algorithm of addition in different bases. Standard Algorithm: 521+371 892Partial Sums: 435+389400+300=70030+80=1109+5=14=824Lattice Algorithm:similar to expanded algorithm and uses diagonals to achieve regrouping.Column Addition: 4 9 7+ 3 6 5 7 15 12 8 5 12 862Adding with bases and using different algorithm 35 base nine+57 base nine 0 1 8 3103 base nine

Subtraction Algorithms:Expanded Form: 56-23=56-20=3636-3=33Equal Addends: 524-158=534+42=576-158+42=-200 376Problems in Different Bases:53 bases 6 -14 base six=algorithms: equal addends53 six -14 six53 six+ 2= 55 six14 six +2= -20 six 35 six

Multiplication Methods:Method 1) Build upThis method is build up from what we know or considered mental math.Ex. 12(11)start with 12(10)= 120then add 12 more which equal 132Method 2) Multiplication with base 10 blocksThis method is using base ten blocks as a table. https://www.coolmath4kids.com/manipulatives/base-ten-blocksMethod 3) Multiplication arrays This is the creating array of numbers to visualize the product. Like, https://quizizz.com/_media/quizzes/b7e8e498-427c-4f08-850b-81ef6ddbb233_900_900

Divisibility Rule:2: must be even & must end in 0,2,4,6,83: sum must be divisible by 34: sum must divisible by four especially the last two digits5: the last digit that end with 0 and 5 are divisible 6: the number must be divisible by 2 or 39: sum must divisible by 910: last digit with 0 are divisible11: sum of the digits in odd-numbered places will equal to the sum of the digits in even numbered places or will differ Although there are some special rules and could be divisible by different numbersinteger n is divisible by m if n=km ex) 25 is divisible by 525=5x5

Greatest Common Divisor (GCD)/Greatest Common Factor: is of two whole numbers known as a and b not both 0 is the greatest whole number that is divisible by both a and b.Three Methods to finding GCD / GCF:1) Colored Rods: is a manipulative tool such as the base 10 blocks, hands on by comparing two rodsExample:Finding the Greatest Common Factor of Two Numbers The rods shown below show the factors of 6 and 8. Look at the rods and write down the factors of 6 and the factors of 8.The 3 red rods build up to 6 but not 8Then 4 red rods build up to 8 but not 6Therefore the white rods are in 2 and build both so the GCD (6,8) =22) Prime Factorization Method: is the finding the GCD of two or more by identifying each common prime number. Then the product of common factors of it lowest prime factors together. Example:3) The Intersection of Sets Method: list all the member of whole number divisors of both numbers. Then pick the greatest element in that set.Example:Let set A = {2, 3, 4, 5, 6}and set B = {3, 5, 7, 9}In this two sets, the elements 3 and 5 are common

Least Common Multiple (LCM): as the smallest multiple that two or more numbers have in common.Methods: 1) Colored Rods: is using the same manipulative tool of colored rods to compare the same length of blocks between the two numbers.(Picture used from the slides)LCM (3,4)=122) Intersection of Sets: set of all multiples of each number and then intersect these sets to find the common multiples. The smallest element of the intersection is the least common multiple.M6=6,12,18,24,30,36,42,48.54,60..M8=8,16,24,32,40,48,56,64..LCM(6,8)=243) Prime Factorization: find the prime factorization of both numbers. The LCM is the product of all of the primes in either number, raised to the greatest power that shows up in either prime factorization.Example:2⋅2⋅2⋅2⋅324⋅3= 484) Number Line: using a number line to find the where both numbers land on an exact point on the number line. Example: LCM (3,4)= 125) Division by Primes: Divide the given numbers the least prime numberExample:

Negative integer= -n and are opposites of the positive integers.Positive integer= +n and are opposites of the negative integers.Absolute Value: for any integer xlxl=x, if x > 0lxl=-x, if x < 0Example:l20l= 20l-20l= 20Chip models: a yellow chip will represent a positive integer and a red chip will represent a negative integer.Example:https://www.basic-mathematics.com/modeling-integers.htmlUsing a number line: -3-4=-7

Real (R): Any number found on the positive and negative number line{...-3.4, …-1, … 0 …10, …, 2304.5…}Integers (Z): Positive and negative whole numbers including 0{..., -3, -2, -1, 0, 1, 2, 3…]Rational (Q): Any number that can be expressed as fraction of two integers  a & bIrrational (P): Numbers that can not be written as a fraction of two integers, non-repeating decimals such as π (pi), Τ (tao), eNatural (N): Counting numbers {1, 2, 3, 4, …}imaginary (i): Square root of -1

Integers-multiplication rulesRULE 1: The product of a positive integer and a negative integer is negative.RULE 2: The product of two positive integers is positive.RULE 3: The product of two negative integers is positive.Examples:Rule 1: 1. (+4) x (-2) = -8 2. (-2) x (+5) = -10Rule 2: 1. (+6) x (+8) = +48 2. (+6) x (+2) = +12Rule 3: 1. (-6) x (-8) = +48 2. (-2) x (-4) = +8Properties of multiplication:Commutative Property of Multiplication: is the states that the answer remains the same when multiplying each numbers, changing the order will not change the product.Example:3x5=155x3=15Associative Property of Multiplication: states that we multiply three numbers together, the product will be the same irrespective of the order in which we multiply the numbers.Example:2×(3×4)2×(3×4)=2×12=24Distributive Property of Multiplication: states that multiplication can be distributed over addition, as well as subtraction.Example:2×(3+1)2×(3+1)=2×4=82×(3+1)=2×3+2×1=6+2=8both cases, the answer we get is the same, hence, multiplication is distributive.Identity Property of Multiplication: states that if you multiply any number by 1, the answer will always be the same number.Example:3×1=37×1=7– If you multiply any number by 0, the answer will always be zero.Integers using counters and chip model:Example: 4x -2+-8

Division of real numbers:Inverse Property of Multiplication: every real number besides zero has a reciprocal and the product of any number is the reciprocal of one.Dividing by a number is the same as multiplying by its reciprocal Dividing by zero is impossible or no solution.Rules:Positive ÷ Positive =positive Negative ÷ Negative = positive Positive ÷ Negative = negativeNegative ÷ Positive = negativeExamples:-56÷-7=8

Real (R): Any number found on the positive and negative number line{...-3.4, …-1, … 0 …10, …, 2304.5…}Integers (Z): Positive and negative whole numbers including 0{..., -3, -2, -1, 0, 1, 2, 3…]Rational (Q): Any number that can be expressed as fraction of two integers a & bIrrational (P): Numbers that can not be written as a fraction of two integers, non-repeating decimals such as π (pi), Τ (tao), eNatural (N): Counting numbers {1, 2, 3, 4, …}imaginary (i): Square root of -1

Objective: to compare and order decimals and to round numbersPlace Value:Using base 10 blocks as place value:Flat=1 Long=0.1 Cube=0.01  Example: 0.06

able to use this as a classroom resource and visual.

Multiplying decimals using base 10 blocks:Example: 3 x 1.4 = 4.2 Dividing decimals using base 10 blocks:Example: 3.24/8=0.4

Decimals, fraction, and percentage:Example:0.125 by 100%0.125 × 100% = 12.5%Answer 0.125 = 12.5%Converting a decimal to percent: move the decimal two places to the right.Example:0.35 → 3.5 → 35%Answer 0.35 = 35%Another example:0.25 (a quarter) of the box is green.25 of the 100 squares are green, 25 100 of the box is green,25% of the box is green.

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Decimals and able to convert into percentage0.26/2=0.13Convert into percent= 13% is shadedFind the percentage of the shaded circle:Divide evenly by 20 by 100 which equals 5therefore it is count the shaded area by 5 which equals 75 so that is 75% or 3/4 after to simplify.Division of decimals using base ten:https://www.mathsisfun.com/numbers/percentage-change.html

Percent Change (Three Types of Problem)Increase: the amount has grownexample: 30 to 69Percent change= 59-30 30Percent change= 0.9666Percent change= 96% (increase)Decrease: the amount has reducedexample: 39 to 25Percent change= 39-25 25Percent change=0.56Percent change= 56%Formula:Percent change= NEW AMOUNT- original amount original amount

Introduction to Fraction: (Chapter 6-6.1)Q={a/b where b cannot 0}a as the numeratorb as the denominatorExample:Division/multiplication problemPortion/part/wholeRatioProbabilityIllustration of fractions:Bar model:Number-line model:Set Model:Fractions:Two Types:Proper fraction:example:4/7 is properImproper fraction:example:9/7 is improperFundamental Law of FractionsIf a/b is a fraction and n a non-zero number, then a/b=an/baExample: 2/-19=-2/19 because 2/-19=2(-1)/-19(-1)=-2/19This is important when working with integers. Simplifying fractions Improper fraction (Mixed number):24/3=8 or 26/3= 8 2/3 Proper Fraction:2/12=1/6 or 12/24=12 Fractions can be converted to decimals as well. Example:2/5=2/5=0.4=40%

-      Equivalent Fraction Models:Area Model of how equivalent fraction models -Equivalent Fraction Models:According to shows a whole that is divide into equal parts. Which could be a good visual for students to use to understand fractions. https://mathigon.org/polypad

Various real number properties:-      Associative propertyExample:-      Commutative propertyExample:-      Identity Properties of MultiplicationExample:  Models:Example of adding using fraction bars Using pattern blocks:example you could use the pattern block to add. Definition of Addition of Rational Numbers with alike denominatorsif a/b and c/d are rational numbers, then a/b+c/b=a+c/bDefinition of Addition of Rational Numbers with unlike denominatorsif a/b and c/d are rational numbers, then a/b+c/b=ad+bc/bdExample (denominators alike):2/15+7/15=9/15Example (denominators unlike):4/14+5/7=?Step 1) Find the LCMwhich equals 14Step 2) Convert the fraction into the LCMso, 5/7=10/14Step 3) Then add them togetherso, 4/14+10/14=14/14=1

Subtraction of Rational Numbersif a/b and c/d are any rational numbers, then a/b-c/d is the unique rational number e/f such that a/b=c/d+e/fSubtraction of rational numbers:if a/b and c/d are any rational numbers, thena/b-c/d=a/b+-c/dIf a/b and c/d are any rational numbers, thena/b-c/d=ad-bc/bdFinding each difference:7/8-1/4=xx=5/8I was able to solve this using https://mathigon.org/polypadwithout converting and was able find the difference without finding the LCM.

Multiplying fractions:Properties with the same basesa^m*a^n=a^m+ntherefore,property (a/b)^m=a^m/b^mexample:(1/7)^3 *(1/7)^9= (1/7)^12= 1/7^12Properties with different basesexample:(2/3)^12*(4/9)^2Find the LCM which, (4/9)= (2/3)^2Rewrite equation: (2/3)^12*((2/3)^2)^3) Simplify/Rewrite: (2/3)^12*(2/3)^6=(2/3)^18Multiplying using models/manipulative tools:Area Model:Pattern Blocks:

Dividing Fractions:Properties:Commutative property: the order of the numbers does not matterex) a/b=b/a, where a and b are rational numbers. a and b does not equal 0ex) 12/3 does not equal 3/12Associative property: the operation can be applied regardless of how the numbers are groupedex) (a/b)/c=a/(b/c), where a, b, c are rational numbers, b and c can not equal 0ex) (24/6)/2 does not equal 24/(6/2)Property of division (same base):a^m/a^n=a^m-nExample:(1/3)^11/(1/3)^4=(1/3)^6=1/3^6Another Example:(11/2)^3/(11/2)^3=(11/2)^0=1Dividing fractions with models:Example:Area Models:Rods:

Manipulative tools:number linecuisenaire rodsfraction stripsblocksbase ten blocksinterlocking linking cubesconstruction setscolored tilespattern blocksResourceshttps://mathigon.org/polypadhttps://nrich.maths.org/1249https://play.google.com/store/apps/details?id=com.desmos.calculator&hl=en_US&gl=US&pli=1https://www.mathway.com/Algebra