Elementary Mathematics

Four OperationsAddition:Identity Property: a, a+0=aCommutative/Order Property: a, b, a+b=b+aAssociative Property: a, b, c, (a+b)+c=a+(b+c)Subtraction:Take Away: 4-1=3Comparison: 4-1=3ex. comparing amount of books to readMissing Addend: 4+___=7ex. Yvonne has 4 cookies. Her mom gave her some more cookies and now she has 7. How many cookies did her mom give her?It has no properties because -3 is the same as +(-3)Multiplication:repeated additionIdentity Property: a, a*1=aCommutative/Order Property: a, b, a*b=b*aAssociative Property: (a*b)*c=a*(b*c)Zero Property: a, a*0=0Distributive Property: a, b, c, a*(b+c)=(a*b)+(a*c)Division:repeated subtractionYou can put 15 cookies in 3 boxes. How many cookies are in each box? _______|xxxxx| Box 1__________ _______|xxxxx| Box 2___________ _______|xxxxx| Box 3___________Answer: 5 cookies per boxIf you have 15 cookies and can put 5 cookies in each box, how many boxes will you have?15 cookies-5 Box 1___10-5 Box 2___ 5 Box 3Answer: 3 boxes

Addition AlgorithmsAmerican Standardright->leftno place valueLast Step51 71 6+2 7 9 ____________8 5 5Partial Sumsright->leftshows place value5 | 7 | 6+2 | 7 | 9 __________ |1 | 51 | 4| 7 | |_________ 8 5 5Partial Sums (b)right->leftemphasis on place value5 | 7 | 6+2 | 7 | 9__________ | 1 | 5 1 | 4 | 07 | 0| 0_________ 8 5 5Left-to-Rightleft->rightshows place value (biggest->smallest)5 7 6+2 7 9__________7 0 01 4 0 1 5_________8 5 5Expanded Notationright->leftemphasis on place value 100 105 7 6 = 500+70+6+2 7 9 = 200+70+9 __________ _____________ 8 5 5 = 800+50+5Latticemake a diagram that looks like a lattice5 7 6+2 7 9__________|0|1|1||/|/|/||7|4|5|_________/8 /5 / 5/

Four Basic Operations in Mathhttps://www.youtube.com/watch?v=JpJOW8L-IsQ

AlgorithmsSubtraction:American StandardLast Step547166-2 8 9________2 8 7European/Mexican51716 6-8-23899 7-8 ________ 8-6=9-7 2 8 7Reverse Indian51716-2 8 9________3____2 9__ 8 7_______2 8 7 Left-to-RightLeft->rightplace value517016-2 8 9________3  0 02 0 0 9 0 8 0 7______ 2 8 7Expanded Notationplace value explicit 400 60576=500+70+16-289=200+80+9________=________287=200+80+7Integer Subtraction5 7 6-2 8 9________ -3 -1 0+ 3 0 0______2 8 7Multiplication:2*5=10 ^ ^factor productStandard113173 6x 2 5________1 8 8 03 5 3 0________4 4 0 0Place Value1 7 6x 2 5________4 4 0 05x5=305x70=3505x100=50020x6=12020x70=140020x100=2000________________4 4 0 0Lattice1 7 6x 2 5________4 4 0 0 1 7 6 x |0/|1/ |1/| 2 |/2| / 4|/2| 5 ------------------ |0/|3/|3/0| |/5|/5|/0 |____________________/4/4 /0 /0 |Division: ___3 ->quotient3|10 ->dividend 9 ________ 1 ->remainder11 cookies/3 plates=3r2 -3. ____ 8 -3____ 5 -3_____ 23 plates:______|xxx|______________|xxx|______________|xxx|________xx=2 cookies leftLong DivisionStandard 158r1 _____3|475 -3↓↓________ 17↓ -15↓________ 25 -24________ 1Place Value 158 _____3|475 -300________ 175 150________ 25 -24________ 1Alternate Algorithm 12r5 _____16|197 160 ->10 boxes ________ 37 -32 ->2 boxes _________ 5 cookies

Subtracting Using Standard Algorithmhttps://youtu.be/WoufCsWLLsQ

Divisibility and FractionsDivisibility: a is divisible by b if there is a number c that meets this requirement -> c+b=a 10 is divisible by 5...2*5=10Terms: 10 is divisible by 5 10 is a multiple of 5 5 is a divisor of 10 5 is a factor of 10Divisibility Rules Ending: -by 2: 0, 2, 4, 6, 8 -by 5: 0, 5 -by 10: 0 Sum of Digits: -by 3: if sum of digits is divisible by 3 -by 9: if sum of digits is divisible by 9 Last Digits: -by 4: if the last 2 digits of a number form a number divisible by 4 344->divisible by 4 -by 7: double last digit subtract number from remaining number see if final number is divisible by 7 ex) 826 2*6=12 -12 -------- 70 -> divisible by 7 -by 8: if the last 3 digits form a number divisible by b -by 11: "chop method" -chop off last 2 digits -add it to remaining number -see if it's divisible by 11 -repeat if needed ex) 29,194 291 +94 -------- 385 +85 -------- 88 -> divisible by 11What number are they divisible by?a) 770: 2, 5, 7, 10, 11b) 136: 2, 4, 8 770 +70 ------ 77 136 6*2=12 -12 ----- 1Factors: 24: 1,2,3,6,8,12,24 42: 1,2,3,6,7,14,21,42 91: 1,7,13,91Prime numbers have 2 factors: 2 and itselfNon-prime numbers are composite numbers0: additive identity element1: multiplicative identity elementPrime numbers 0-60:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59

Prime Factorization List:Factors:24: 1,2,3,4,10,8,12,2436: 1,2,3,4,6,9,12,18,36GCF=1224, 48, 72...36, 72...LCM=72Why we need them:Fractions GCF for simplifying25 5 1--- = --- = ---100 20 425 1--- = ---100 4GCF allows you to simplify with only one step used for simplifying fractionsLCM for finding common denominator2 1 16 6 22-- + -- = ---- + ---- = ----3 4 24 24 243=3,6,9,12,15,18,21,24...4=4,8,12,16,20,24...2 1 8 3 11-- + -- = ---- + ---- = ----3 4 12 12 1222 11---- and --- are equivalent fractions24 12no need for simplificationsmaller numbers -> less mistakes for kidsExamples:GCF and LCM24 and 3024: 1,2,3,4,6,8,12,24...30: 1,2,3,5,6,10,15,30GCF=624: 24,48,72,96,120...30: 30,60,90,120...LCM=120

Prime Factorizationhttps://youtu.be/XBnUWjo3TgM

Problem Solving With Fractions

Problem Solving Using Drawing Diagramshttps://youtube.com/watch?v=g0W3ciRUYkA&feature=share

Integers Notes

How to Add Positive and Negative Integershttps://youtu.be/CfkaifC7tGY

Problem SolvingPolya's Four Steps to Problem Solving:Understand the problem.Devise a plan.Carry out a plan.Look back (reflect).Understand the problem:What are you asked to find/show?Can you restate the problem in your own words?Can draw a problem or diagram to help?Devise a plan:What problem solving strategy are you going to use?guess and checkmake a tablelook for a patternetc.Carry out the plan:usually easier than devising the planbe patient-usually not solved quickly or on the first attemptbe persistentdon't get discouragedif on strategy doesn't work, try anotherLook back:Does the answer make sense? Is it reasonable?Did you answer all the questions?Could you have solved differently? Easier? What did you learn?

Polya's 4 steps Problem Solving Processhttps://youtu.be/aMlVcGEn7EE

Problem SolvingExample Problems:If there are 7 people shaking hands and everyone only shakes each person's hand once, how many handshakes are there in all?The people are each represented by a number. 1 2 3 4 5 6 7 6 handshakes2 3 4 5 6 7 5 handshakes3 4 5 6 7 4 handshakes4 5 6 7 3 handshakes5 6 7 2 handshakes6 7 1 handshake7 0 handshakesThe number goes down by one each time because each person has already shook the previous person's hand.Answer: 21 handshakesI have four 3-cent stamps and three 7-cent stamps. Using one or more of these stamps, how many different amounts of postage can I make? | 7 | 77 | 7773 |37 |377 |377733 |337 |3377 |33777333 |3337 |33377 |3337773333 |33337 |333377 |333777I used a multiplication chart to write out each combination possible for the postage.Answer: 19 postage

Problem Solving - How to Use the Four-Step Methodhttps://youtu.be/HTp55ozYdQA

Numeration Systemsways to record quantityOur system: Base Ten (One-to-Ten)decimalpositional3 7 5hundredstensonesExpanded Notation:375=300+70+5 =(3*100)+(7*10)+(5*1) =(3*102)+(7*101)+(5*100)Base-5:2 3 225sfivesones2325=(2*25)+(3*5)+(2*1) =(2*52)+(3*51)+(2*50) =50+15+2=67Digits Used in Base Ten:0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10Digits Used in Base-5:0, 1, 2, 3, 4, 105, 115...205Base 10ones 100tens 101hundreds 102thousands 103Base 5ones 50fives 5125s 52125s 53Base 3ones 30threes 31nines 3227s 33Base 10305=(3*100)+0+(5*1) =(3*102)+0+(5*100)203=(2*100)+0+(3*1) =(2*102)+(3*100)Base 53335=(3*25)+(3*5)+(3*1) =(3*52)+(3*51)+(3*50) =75+15+3=93435=0+(4*5)+(3*1) =0+(4*51)+(3*50) =0+20+3=23Base 32213=(2*9)+(2*3)+(1*1) =(2*32)+(2*31)+(1*30) =18+6+1=25Comparing Numbers:a) 2113____31015Base 31s 303s 319s 3227s 332113=18+3+1=22Base 51s 505s 5125s 52125s 5331015=375+25+0+1=401Answer: 2113<31015Finding a Number in a Different Base from Base 10130 in Base 5:Base 51s5s25s125s130-125______ 5- 5______ 0Answer: 130=10105

Base 10 Number Systemhttps://youtu.be/iGjOubria74

What is a Fraction?

Fractions

Fractions for Kidshttps://youtube.com/watch?v=p33BYf1NDAE&feature=share

Problem Solving With Fractions

Fraction Word Problemshttps://youtube.com/watch?v=F0EOkIFAyN4&feature=share

Decimals Notesunderstand well if they know place holders and fractions375.323hundredstensonestenthshundredthsthousandthsdecimal point: another way/symbol to represent parts of a wholefunctions: separates whole from the part and sits to the right of the unit$375.32375 dollars and 32 cents0.3>0.033/10 has bigger parts than 3/100Practice Problems:72/100=0.7231/1000=0.0310.83=83/1000.01=1/1002/5=4/10=0.4find equivalent fraction of 10, 100, 1000 if possibleif not possible, divideRepeating Decimal {photo}Terminating Decimal{photo}

DecimalsPut in order from smallest to greatest0.10.010.1010.0010.0110.001<0.01<0.011<0.1<0.101Percentsper one hundred or out of one hundred9/100=0.09=9%Practice Problems: percents->fractions->decimals85%=85/100=0.855%=5/100=0.05120%=120/100=1 and 20/100=1.2Practice Problems: fractions->percents4/5=8/10=80/100=80%5/6=0.8333=0.83 line over the 3=83%Practice Problems: decimals->percents0.65=65/100=65%0.07=7/100=7%4=400%Practice Problems: fraction->percent{picture}

Four OperationsAddition:Subtraction:Multiplication:

Place Value With Decimalshttps://www.youtube.com/watch?v=wtrrr15mbvQ

Practice Problems With Decimals

Percent Problems8 is what percent of 22?8% of 22 is what number?8% of what number is 22?'is' means ='of' means to multiply'what' means the variable nchange % into decimals'what percent' write the decimal as a percenta) 8=n*22n=8/22=0.3636%b) 0.08*22=nn=1.76c) 0.08*n=22n=22/0.08=2200/8n=275

Percent Practice Problemsa) 13/20=75/10065%b) 2/100=n/700n=14 studentsc) 9/25=36/10036%d) 30/100=n/50n=15 gramse) 20/100=10/nn=50 beatsf) 4/50=n/1008%g) 30/100=n/32000n=9600

Solving Basic Percent Problemshttps://youtu.be/NIuaIY9YqnE