Raeghen Hawkins- MTE 280

Base 10 #'s- 0,1,2,3,4,5,6,7,8,910 is where the double digits come in they have 2 places valuesin base 10 number get their value by where they sitEX: 375The 3=300 it is in the hundred placeThe 7=70 it is in the tens placeThe 5=5 it is in the ones placethere is also a thousands place Ones place = a singular cubetens place= a stick of ten cubeshundreds= a flat square thousands= a large cubeExpanded notation in base-10:375+ 300+70+5= (3x100) + (7x10) + (5x1)= (3x10^2) + (7x10^1) + (5x10^0)any # to the zero power is oneanother example of expanded notation:492= 400+ 90+ 2= (4x100) + (9x10) +(2x1)= (4x10^2) +(9x10^1)+ (2x10^0)

Base-5 #'s: 0,1,2,3,4,10,11,12,13,14,20 (0-4)Base-5 palace values:Ones: 5^0Fives: 5^125's: 5^2125's: 5^3 Base-5 conversion to Base-10Example:212_5= (2x5^2)+(1x5^1)+(2x5^0)= (2x25)+5+2= 50+5+2= 57Another example:2304_5= (2x5^3)+(3x5^2)+(0x5^1)+(4x5^0)= (2x125)+(3x25)+(0x5)+(4x1)=250+75+0+4=392Base 10|Base5 1 | 1_5 2 | 2_5 3 | 3_5 4 | 4_5 5 | 10_5 6 | 11_5 7 | 12_5 8 | 13_5 9 | 14_510 | 20_5 11 | 21_5 12 | 22_5There is no 5's in base-5 No(5,15,25,35,45,50) goes from 144-200

Base-3: (0,1,2)Base-3 place values:Ones:3^0Threes: 3^1Nines: 3^2Twenty-sevens: 3^3Base-8: (0,1,2,3,4,5,6,7)Base-8 place values:Ones: 8^0Eights: 8^1Sixty-fours: 8^2Base-2: (0,1)Base-2 place values:Ones: 2^0 Twos: 2^1 Fours: 2^2 Eights: 2^3 sixteens: 2^4 Example:A) 111_2= (1x2^2)+(1x2^1)+(1x2^0)= 4+2+1=7B) 31.2_4= (3x4^1)+(1x4^0)+(2x 1/4)= 12+1+2/4=13 1/2Expanded notation:738= (7x10^2)+(3x10^1)+(8x10^0)=(7x100)+(3x10)+(8x1)= 700+ 30+ 8= 736

Division Operations:meaning: repeated subtraction (it is sharing between a certain # ex:2)8/2: division sign8: - is the division fraction bar22|- 8: this is the vinculumn2|-8= 4 4 is the quotient2 is the divisor8 is the dividendEx:11/311-3=88-3=55-3=2

Division algorithms:American standard algorithm 1583|-475 -3-1715-2524---------1How many times does 3 go into 4multiply 3x1 then subtract 4-3=1bring down the 7how many times does 3 go into 17Place value explicit 1583|-475 -300 ------ 175 -150 ------ 25 -24 ------ 1Alternating algorithm197 cookies and there are 16 in each box how many boxes will we need? 1216|-197 -160 -----> 10 boxes --------- 37 -32 -----> 2 boxes --------- 5 10 +2 ------- 12 boxes will be needed

Multiplication algorithms1)American standard 1 23X 14-------- 921230------3224x3=12 put 2 carry the 14x2=8 +1=9put a 0 to be back in the 10's1x3=31x2=2add 92+2322) Expanded notation place v.explicit 23x 14--------1020+310+4---------100+90+2200+30+0-------------300+20+2---------------3223)Place value 23x14------4x3=12 } 924x20-80 }10x3=30 } 230 92 and 230 are partial products10x20=200 }----------------3224)Lattice (x) 23 X / 1/ 1 0 / 0/ 4 2/3/ ------ 0/1/ 8/2/3 2 2------322

Multiplication operations:meaning: repeated addition3x2=63 & 2 are the factors 6 is the product 3x2 could be 3 groups of 2Group 1: has 2 cookiesGroup 2: has 2 cookiesGroup 3: has 2 cookies2+2+2=6Identify property Ax1=A/ Ax0=0Commutative property: AxB=BxAassociative property: (AxB)xC=Ax(BxC)Distribution property of multiplication:3x7=3x(5+2)= (3x5)+(3x2)(5+2) & (3x5) are partial products3x7=3x(5+2) = (3x5) + (3+2)= 15+6= 21= 3x(3+4)= 9+12=21= 3x(6+1)= 3x6+ (3x1)= 19+2=21

Problem solving: 1) understand- tell me what the problem says 2) plan- whatever strategy makes sense (trial and error/ work backwards) 3) Implement- do you plan test it out (make the problem simpler/find a patter/be patient don't give up)4) look back and is it reasonable- explain how? (does it make sense/ what did you learn/ did you answer all the questions?/ could it be done differently)Problem #1:12 basketball teams in a league if each team plays each other once how many games take place?Teams: T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12# of games: 0 1 3 6 10 15 21 28 36 45 55 66 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 The amount of games that would be played if each of the 12 teams played each other once time there would be 66 games played. I found this out by finding a pattern in a table for each team that was added the amount of games played would go up by 1 from the previous number.Problem #2: 7 9 centX5 6 cent---------35 7+5+35=47

George Polya steps:1) Understand - tell me what the problem says/ what are you asked to find2)Plan- Devise a plan/trial and error/make a table/ try different ways3)Implement- do your plan try it out/make the problem simpler/be patient/explain how?4)Look back is it reasonable- does it make sense/ what did you learn/did you answer all questions?/ could it have been a different way?

Addition meaning: Combine/join groups together3+4=73 & 4 are the addends and 7 is the sumIdentify property: 9+0=9ORDER:commutative property: A+B=B+A 4+0=4 -3+0=-3The order in which I add the #'s don't matterGROUPING: associative property: (a+b)+c=a+(b+c) (2+3)+5=2+(3+5)

Subtraction Operationsmeaning: take away7-3=47 is the minuend 3 is the subtrahend4 is the differenceComparison ( I read 3 books Sara read 5 how many more books did Sara read than me?) = 2 more books used blocks to solvemissing addened: 3+ _=7 ( take 7 and take away 3)

Subtraction Algorithms:1) American standard: 135 3 15645-279-------3662) Europe/Mexican 14 15 645-279 2 become a 3 and 7 becomes 8 so 4 and 5 become 14 and 15 -------3663) Reverse Indian 14 15 645-279-------437 6 6--------3664)Left-to-right 14 15645-279------400300 70 60 6 =3665) Expanded notation 130 500 30 15 645= 600-40-5-279= 200-70-9----------------------366= 300+60+66) Integer subtraction 645-279------- -4 -30 400-30-4+400--------366Addition Algorithms:1) American Standard 1 1 478 + 394 ------ 8722) Partial Sums 4|7|8 +3|9|4 ---------- |1|2 1|6| 7| |------------- 8723) partial sums with place value 4|7|8 +3|9|4 ---------- |1|2 1|6|0 + 7|0|0------------- 8724)Left-to-right 478+394---------700160 12-------8725)Expanded notation 100 10 478= 400+70+8+394= 300+90+4-------------------------872= 800+70+26)Lattice 478+394---------0/ 1/ 1/ 7/ 6/ 2----------- 8/7/2------->

Test Review:20 questionsG. Polya steps (name 4 of them and describe) Xstamp problem homeworkNumeration systems (convert from base 10 or a # convert to base 5 or another #)123_5 or 45_6 which is bigger convert both to base 10 (123_5 is bigger)different operations (names and translate) AB=BA communitive property of multiplication the order in which I multiply my number does not matterAlgorithms (find the sum using any algorithm besides...)Know all descriptions and name of +, -, X, / propertiesmemorize algorithms place values -----> know themhow you would reply to a student if you were a teacher