MTE 280- Summer Manne

Problem SolvingGeorge Polya designed a 4-step plan for problem solving nearly 100 years ago1.Understandfirst you have to understand the problem before taking action to solve it2.Devise a Planlongest step, figure out how to solve the problem on your own, "what is being asked?"3.Carry out the Planexplain how the problem was solved and show plan, patience is key4.Look back (reflect)Ask, "Is it a reasonable answer?" "Does it make sense?"Everyone learns in a different way (visual, mental. etc)Example Problems:7 people in a roomIf each person shakes each other person's hand only once, how many handshakes will happen?1-2 2-3 3-4 4-5 5-6 6-71-3 2-4 3-5 4-6 5-71-4 2-5 3-6 4-71-5 2-6 3-71-6 2-7 1-7

Number Systems/OperationsBase 10 system 2,3752-thousand3-hundred7-ten5-onesone-to-ten relationship2,375.35.3- tenth.05- thousandthdigits used:0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11Expanded Notation375 = 300 + 70 + 5= (3x100) + (7x10) + (5x1)= (3x10^2) + (7x10^1) + (5x10^0)

Base 5125 = 5^3digits used:0, 1, 2, 3, 4, 10-^5, 11-^5....21-^5122-^5 = 25 + 10 + 2 = 37122^5 = (1x5^2) + (2x5^1) + (2x5^0)= 25 + 10 + 2

Base 5121-^5= (1x5^2) + (2x5^1) + (1x5^0)= (1x25) + (2x5) + (1x1)= 25 + 10 + 1 = 36a. 1075.31 = 1000 + 70 + 5 + 31/100= (1x10^3) + (0x10^2) + (7x10^1) + (5x10^0) + (3x1/10) + (1x1/100)b. 79.003 = (7x10^1) + (9x10^0) + (0x1/10) + (0x1/100) + (3x1/1000)c. 1212-^5 = (1x5^3) + (2x5^2) + (1x5^1) + (2x5^0)= 125 + 50 + 5 + 2= 182d. 32.12-^5 = (3x5^1) + (2x5^0) + (1x1/5) + (2x1/25)= 15 + 2 + 1/2 + 2/25= 17 7/25

Base 3 Digits used: 0, 1, 21's: 3^03's: 3^19's: 3^227's: 3^32122.12 -^3 = (2x3^3) + (1x3^2) + 2x3^1) + (2x3^0) + (1/x1/3) + (2x1/9)= 54 + 9 + 6 + 2 + 1/3 + 2/9= 71 + 3/9 + 2/9= 71 + 5/9= 71 5/9

Base 2ones: 2^0two's: 2^1fours: 2^2eights: 2^3101011 =(1x2^5) + (0x2^4) + (1x2^3) + (0x2^2) + (1x2^1) + (1x2^0)= 32 + 0 + 8 + 0 + 2 + 1= 43101011.11 -^2= (1x1/2) + (1x1/4)= 43 3/4

Multiplication:Identity Property = ax1=aCommutative Property = axb = bxaAssociate Property = (axb)xc = ax(bxc)Zero Property = ax0 = 0Addition: Putting things togetherIdentity Property = a+0 = aCommutative Property = a+b = b+aAssociative Property = (a+b) +c = a + (b+c)Subtraction: take-awayno properties: subtracting3 is like adding -3

Addition AlgorithmsAmerican-Standard 576 R-L+ 279 no place value= 855 29+ 13=3122.Partial SumsR-Lno place value 576+ 279= 15 14 700= 8553.Place Value 576+ 279= 15 140 700= 8554.L-R 576+ 279= 700 140 15= 8555.Expanded Notation 576+ 279= 500 + 70 + 6= 200 + 70 + 9= 800 + 50 +5=8556.Lattice 576+ 279=0 1 1 7 4 5= 855

1.American Standard 576 no reference to place value- 289= 2872.European-Mexican 576- 289= 2873.Reverse-Indian 576- 287= 3 2 9 7 8= 2874.L-R 576- 289= 300 90 200 80 7= 2875.Expanded Notation 576- 289= 500 + 70 + 6-200 + 80 + 9=200 + 80 + 7= 2876.Integer Subtraction 576- 289= -3 -10 +300= 287

1.American-Standard 23x 14= 92 230= 3222.Place Value 23x 144x3 = 124x20 = 8010x3 = 3010x20 = 200= 3223.Expanded Notation 23x 14= 20 +3= 10 + 4= 90 +2= 200 + 30 + 2= 3224.Lattice 2 30 2 0 3 10 8 1 2 43 2 2= 322

1.Standard-Long Division3 _158___ 475 3 17 15__________ 25 24__________ 12.Alternate Algorithm16 _______ 197- 160____________ 37 - 16____________ 21 - 16____________ 5= 12 r5

1st test

Number Theory:-types of numbers-divisibility rules-factors-multiples___________________ready for fractionsDivisibility Rules:Endingby 2: 0, 2, 4, 6. 8by 5: 0, 5by 10: 0Sum of digitsby 3: sum of digits is divided by 3by 9: sum of digits is divided by 1Last digitsby 4: last 2 digitsby 8: last 3 digitsBy 6divided by both 2 & 3By 7double last digittake remaining #subtract from it the # doubledBy 11 (chop-off method)chop off last 2 digitstake remaining # and add # we chopped off10 is divisible by 210 is divisible by 55 is a factor of 102 is a factor of 1010 is a multiple of 510 is a multiple of 25 is a divisor of 102 is a divisor of 10

Factors:28: 1, 2, 4, 7, 14, 2836: 1, 2, 3, 4, 9, 12, 18, 3642: 1, 2, 3, 6, 7, 14, 21, 4260: 1, 2, 3, 4, 5, 6, 10 12, 15, 20, 30, 6091: 1, 7, 13, 91Prime:3: 1, 32: 1, 213: 1, 1311: 1, 111: NO- only 1 factor0: NO- additive identity element

LCM: least common factorList Method:24: 1, 2, 3, 4, 6, 8, 12, 24 GCF = 1236: 1, 2, 3, 4, 6, 9, 12, 18, 36 GCF (24,36) = 12GCF:25/100 = 5/2025/100 = 1/4Prime Factorization Method:24 = 2x2x2x336 = 2x2x3x3GCF (24,36) = 2x2x3 = 12LCM (24,36) = 12x2x3 = 72List: 24: 1, 2, 3, 4, 6, 8, 12, 2430: 1, 2, 3, 5, 6, 10, 15, 3024, 48, 72, 96, 12030, 60, 90, 120GCF (24,30) = 6LCM (24,30) = 120

Fractions:Part-WholeQuotientRatio11 boys 9/11 = ratio9 girls ___________20 studentsboys/whole = 11/20 = ratio & fractionModels (manipulatives)Area (filled in boxes)Length (number line)Set (groups of things)3/3 = 14/4 = 12/2 = 1FRACTIONAL PARTS ARE EQUIVALENT PARTS8a^3b^2c___________24a^2bc = ab/3

1. No, because in order for a number to be divisible by 5, it has to end in a 0 or a 5. In order for a number to be divisible by 10, it has to end in a 0. So if it does not end in a 0, it is not divisible by 5, or by 10. 2. Yes, a number that is not divisible by 10 can still be divisible by 5. This is because if it were divisible by 10, it would have to end in a 0. Numbers that are divisible by 5 can end in a 0 or a 5. 3. No, two numbers cannot have a "greatest common multiple" because the greatest common multiple of any two numbers is always infinity since any multiple of both numbers would be a common multiple. 4. I would say that she is wrong, her prime numbers will be the same at Tom’s. This is because it doesn’t matter what factors you start with, as long as it equals the same number. 5. You have to find the LCM in order to find the first caller to receive both. The LCM (12,13,20) = 780, so the first caller to get a coupon and a ticket is the 780th caller

1.Jim=1/4 = 3/12 = 12 barsKen-1/3 = 4/12 = 16 barsLen-1/3 = 4/12 = 16 barsMax-4 Bars = 1/12 = 4 bars2.Len- 4 barsKen- 6 barsJim-6 barsMax- 8 barsTotal = 24 barsHW #6

Practice Problems:3/82/7

3.23 + 1.50__________ 4.73 4.39+ 2.37__________ 6.76 3.89- 1.52_________ 2.37 5.00- 2.37_________ 2.63

3.2x 2.2_________ 6.46 4. 0_________7. 0 4 4.3x 2________ 8. 6369.63/3= 123.21369.63/31= 119.2

3/11 = 0.27 repeating5/6 = 0.83 repeating5/9 = 0.5 repeatingPractice Problems:a. 8 = nx22n = 8/22= 0.36= 36%b. 0.08x22 = nn = 1.76c. 0.08 x n = 22n = 22/0.08= 275

7/8 = 0.8755/3 = 1.6 repeating5/6 = 0.83 repeating = 83%5/9 = 0.5 repeating = 56%a. 24 = n x 180= 0.13 = 13%b. 0.30 x n = 21 = 70

Number Line"chip method""zero pair"Addition:5+1=65+-1= 4-5+-1= -6Subtraction:5-+1= 4-5--1= -4-5-+1= -6

2x3= 6 = 2 groups of 33x-2= -6 = 3 groups of -2-3x-2 = opposite -6 | 6