Elementary Mathematics

George Polya Steps: Understand the problem Put together a plan (STRATEGIES) Implement the plan (DO IT)Is it reasonable? Does it make sense? (REFLECT)Strategies to use: Trial and error Draw pictures Algebra Blocks

Base 10 system:Units (5)Tens (7)Hundreds (3)Thousands (2)EX: 2,375 decimal always sits right of the unitEX: 2375.35tenths (3)hundredth (5)Base 5 system Units Fives 25’s 125’sEX: 1225

Base 3 system: Units 3’s9’s27’s 2013This means: there's 1 unit, 0 threes, and 2 nines which when added together equals 19Base 2 system:Units 2’s4’s 8’s16’s 11002(1x23)+(1x22)+(0x21)+(0x20)=12This means: there's 0 units, 0 twos, 1 four, and 1 eight, which added together equals 12

Subtraction: Take away Comparison Missing addend No properties: subtracting 3 is like adding -3Addition:  Putting things togetherIdentity Property (a+0= a)Commutative Property (a+b= b+a)Associative property ((a+b)+c= a+(b+c))Multiplication: (repeated addition)  “Groups of”Identity Property (ax1= a)Commutative Property (axb= bxa)Associative property ((axb)xc= ax(bxc))Zero Property (ax0=0)Distributive Property 3x7=21 3x(5+2)=(3x5)+(3x2) ax(b+c)=(axb)+(axc)Division: Repeated Subtraction

Addition algorithms:1.) American Standard   576 +279 =855Right to left No mention of place value 2.) Partial Sums:  576 +279   15 14 7 =855Right to left No mention of place value 3.) Partial sums w/ emphasis on place value  576 +279   015 140 700 =855Right to left Addresses place value 4.) Left to right  576 +279 700 140     15  =8555.) Expanded notation  576    500+70+6 +279   200+70+9 855 = 800+50+56.) Lattice Subtraction:1.) American Standard 576-289287Right to left No ref to place value 2.) European-Mexican Shortly to explain this one when you need to “borrow” from another number instead of taking one away from the top number you add one to the bottom number3.) Reverse Indian  576 -289 3 29   87 =2874.) Left to right 576 -289 300 200   90    80      7 =2875.) Expanded notation  576  500+70+6 -289 - 200+80+9 =287= 200+80+76.) Integer subtraction  576 -289      -3     -10  +300 300-10+290-3=287Multiplication: 1.) American Standard      23   X14     92 +230  =3222.) Place Value      23        X14 4x3=12 4x20=80 10x3=30 10x20=200           = 3223.) Expanded notation      23        20+3   X14    x 10+4                 90+2          200+30 322 =300+20+24,) Lattice 

STUDY GUIDE:Problem Solving George Polya Steps (name and describe) HW#1Bases- write a number in a different basewrite the base 10 number using expanded notation HW#2 & #3Operations:properties and algorithms HW#4

5 is a factor of 10 2 is a factor of 10 10 is a multiple of 210 is a multiple of 5 5 is a divisor of 10 2 is a divisor of 10 10 is divisible by 2 10 is divisible by 5 A IS DIVISIBLE BY B IF THERE IS A NUMBER C THAT MEETS THE REQUIREMENT C x B= ADivisibility rules *ending By 2: 0,2,4,6,8By 5: 0, 5By 10: 0*sum of digits By 3: sum of digits is div by 3 By 9: sum of digits is div by 9 *last digits By 4: last two digits By 8: last 3 digits *By 6: Div by both 2 and 3 *By 7:Double the last digits 826- 6x2=12Take the remaining number and subtract it by the other number 82-12= 70*By 11Chop off last 2 digits Go to the remaining #Add the remaining # and the chopped off number PRIME NUMBERS TO REMEMBER:  2, 3, 5, 7, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, How to find GCF and LCM List method: GCF24: 1, 2, 3, 4, 6, 8, 12, 2436: 1, 2, 3, 4, 6, 9, 12, 18, 36List Method: LCM 24, 48, 72, 9636, 72, …Prime Factorization Methods 24= 2x2x2x336= 2x2x3x3GCF(24,36)= 2x2x3=12LCM(24,36)= GCF (12)x2x3=72

Fractions: Part of a whole Quotient Ratio Ratios are not fractions when they represent part and part Models (manipulatives) Area Length Set (group of things) *FRACTION PARTS ARE EQUIVALENT PARTS*The more parts we cut our whole into the smaller it gets

1. Jim, Ken, Len, and Max have a bag of miniature candy bars fromtrick-or-treating together. Jim took ¼ of all the bars, and Ken andLen each took 1/3 of all the bars. Max got the remaining 4 bars.How many bars were in the bag originally? How many bars didJim, Ken, and Len each get?¼ +⅓ +⅓ =3/12 + 4/12+ 4/12= 11/12 Max had 1/12= 4 bars4 bars4 bars4 bars 4 bars4 bars4 bars4 bars4 bars4 bars4 bars 4 bars 4 bars(MAX)Whole box= 12x4=48Jim= 4+4+4=12Ken= 4+4+4+4=16Len= 4+4+4+4=162. Jim, Ken, Len, and Max have a bag of miniature candy bars fromtrick-or-treating together. Jim took ¼ of the bars. Then Ken took1/3 of the remaining bars. Next, Len took 1/3 of the remainingbars, and Max took the remaining 8 bars. How many bars were inthe bag originally? How many bars did Jim, Ken, and Len eachget? How is this problem (with regards to fractions) differentfrom Problem 1?66666664448 BARS (MAX)Whole Box= 6+6+4+8=24

1.)  If **** represent 2/7 of the whole, draw whatthe whole looks like.**= 1/7****= 2/7******= 3/7********= 4/7**********= 5/7************= 6/7**************= 7/72.) Janice is preparing a recipe that calls for ¾ ofa cup of oil per serving. If Janice needs toprepare 2 and 2/3 servings, how many cups ofoil will she need?3/4+3/4+2/34/4 and 4/42 CUPS 3.) Marc opened a pizza box. Inside there was ¾of a pizza. Marc ate ½ of what was in the box.How much pizza did Marc eat?3/4= 3/8

3/10=10/100=0.03Smallest to Largest:0.3              0.33            0.03             0.003      0.303           3.33/10             33/100       3/100         3/1000    303/1000   3 3/10300/1000   330/1000     30/1000   0.003-0.03-0.3-0.303-0.33-3.3Fraction to decimal: 2/10=0.22/5=6/10=0.61/8= 1divided by 8= 0.125 - terminating decimal 5/6= division= 0.83333333- repeating decimal Addition: 3.23+ 1.5 3.23+1.50= 4.73Subtraction=3.89-1.52 3.89-1.52=2.37Multiplication:4.3x2 4.21x3.2 4.3 4.21x 2 x 3.2= 8.6 = 13.472Division:3.1/369.63= 31/3696.3- same problem answer will be the same it creates am equivalent problem 

% per cent per 100 out of 100 % --> fraction--> decimal 85% 85/100 0.855% 5/100 0.05 62% 62/100 0.62fraction ---> Decimal ---> %32/100 0.32 32% 2/5=4/10 0.40 40%5/6 0.83(repeating) 83%5/9 0.5 (repeating) 56%

Is: (=)Of: (x)What: (n)(Turn % into decimal)a.) 8 is what % of 22 =36 8=nx22 n=8/22b.) 8% of 22 is what #? = 1.76 0.08x22=nc.) 8% of what # is 22?= 275 0.08xn=22 n=22/0.081.) A students takes a test with 45 questions and gets 37 right. What is their percent on the test? questions: %: 45 100 37 ? 45x=37x100 x= 37x100/45 x= 82%

+ and -"Chip" method Number line "zero" pair= + - = 0Addition:+5 + +1 = 6 -5 + +1= -6 +++++ - - - - - + +Subtraction: +5 - +1 = 4 ++++++Multiplication: 2x3= 2 groups of 3 +3 x +2 = +6++++++Commutative Property: -3 x +2 = +2 x -3 = -6------The opposite of if double negative:-3 x -2 = +6 3 x -2 ------=-6 opposite is +6Division:+3 x +2 = +6+6/+2=+3+6/3=+2

Study Guide for test #3Prime- not prime True of false statements factors, multiples, divisibility rulesDecimals: Comparing, ordering, repeating, terminating Problem Solving w %, LCM, fractions Positive-Negative Integers