Elementary Mathematics

Overview of what this class involves and introductions of everyone within the classroom. We went over the syllabus which included assignments and how they are weighted, the amount of tests, and what is required every week for homework.

Problems need to be understood in order to be solved. There is a five step process one should take when trying to solve a problem.1) Understand the problem 2) Devise a plan3) Implement plan4) Check Work5) Is the solution reasonble Problem solving is a process that is continuous Example Problem: A game of tug of war takes place:Round one: 4 acrobats vs 5 grandmas, comes to a drawRound two: 1 dog vs 2 grandmas and 1 acrobat, comes to a drawRound 3: 1 dog and 3 grandmas vs 4 acrobats, who wins?Solution: 4 acrobats=5 grandmas 1 dog=2 grandmas and 1 acrobat 1 dog and 3 grandmas>4 acrobatsWe know this because 1 dog is equivalent to 2 grandmas and one acrobat, therefor 1 dog and 3 grandmas is really 5 grandmas and 1 acrobat

Builds off of knowledge from week oneExample Problem: Seven people who don't know each other are being introduced, if everyone shakes everyone's hand one time, how many handshake will take place?Solution: Label each person (A-G) and write down who's hand they would shake while making sure there aren't any repeatsA) B, C, D, E, F, GB) C, D, E, F, GC) D, E, F, GD) E, F, GE) F, GF) GG)21 handshakes will take place

Symbols are used to represent a quantity, and in our base 10 system (positional system) we use the symbols 0-9Base 10 (0-9) vs Base 5 (0-4)Ones=10^0 Ones=5^0Tens=10^1 Fives=5^1Hundreds=10^2 25s=5^2Thousands=10^3 125s=5^3Example of one-to-ten relationship:375.25The 3 is in the hundreds place, the 7 is in the tens place, the 5 to the left of the decimal is in the ones place, the 2 is in the tenths place, and the 5 on the far right is in the hundredths place. As you move to the left you get 10x bigger, and as you move to the right you get 10x smallerExpanded Notation in Base 10 Example:931= 900+30+1 = 9(100)+3(10)=1(1) =(9x10^2)+(3x10^1)+(1x10^0)Expanded Notation in Base 5 Example:111 base 5=(1x5^2)+(1x5^1)+(1x5^0) =(1x25)+(1x5)+(1x1) =25+5+1 =31

Principal of bases stays the same Base 10 (0-9):Ones=10^0Tens=10^1Hundreds=10^2Thousands=10^3Base 5 (0-4):Ones=5^0Fives=5^125s=5^2125s=5^3Base 3 (0-2)Ones=3^03s=3^19s=3^227s=3^3What is 28 in different bases?Base 10= two 10s and eight 1s =28Base 3=one 27, zero 9s, zero 3s, one 1 =1001 base 3Base 5=one 25, zero 5s, three 1s =103 base 5

Build off of prior knowledge and apply that to new basesBase 2 (0-1)Ones=2^0Twos=2^1Fours=2^2Eights=2^316s=2^4Example of converting a base 2 number into base 10:1111base 2=(1x2^3)+(1x2^2)+(1x2^1)+(1x2^0) =8+4+2+1 =15Comparing Numbers:34base 5<34base6 (19<22)4base 5=4base 6 (both just 4)23base 6>23 base 5 (15>13)Give the Base 10 Numeral Examples:31.2base 4=(3x4^1)+(1x4^0)+(2/4) =12+1+2/4 =13 2/4221.2base 3=(2x3^2)+(2x3^1)=(1x3^0)+(2/3) =18+6+1+2/3 =25 2/337base 9=(3x9^1)+(7x9^0) =27+7 =34Converting Base 10 Numbers to Other Bases:Base 5) 135= One 125, zero 25s, two 5s, zero 1s =1020base 5Base 3) 135=One 81, two 27s, zero 9s, zero 3s, zero 1s =12000base 3Base 2) 135=One 128, zero 64s, zero 32s, zero 16s, zero 8s, one 4, one 2, one 1 =10000111base 2

Addition: Putting together, joiningAddition has three properties:1) Identity: When you add zero, the identity of the number does not change. 2+0=22) Commutative: The order of two numbers being added does not matter. 2+1=1+23) Associative: The way you group numbers does not matter. (3+2)+1=3+(2+1)Subtraction: 1) Take Away: 4-3=12) Comparison: Zack has 5 cookies and Emma has 2, how many more cookies does Zack have?3) Missing Addend: Emily had 3 cookies, Zoey gave her more cookies and now she has 7. How many did Zoey give her? 3+__=7Multiplication: Groups of something, repeated addition1) Identity: When you multiply a number by 1, the identity does not change. 8x1=82) Commutative: The order of two numbers being multiplied does not matter. 8x2=2x83) Associative: The grouping of numbers does not matter. (5x5)x2=5x(5x2)4) Zero: When you multiply a number by zero, it will always be zero. 7x0=05) Distributive Property: Repeated addition. 3x7=7+7+7, 3x7=3(5+2)

Division: 3 different symbols used in division 4 key components to a division problem when using the Standard American Algorithm: Divisor, dividend, quotient, and remainderThere are no properties for division or subtraction

Addition Algorithms:1) American Standard: Right to left, no mention of place value2) Partial Sums: Right to left, no mention of place value, uses columns, doesn't carry numbers above the equation3) Partial Sums with Place Value: Right to left, uses place value by making numbers be read as their correct value. 375+240 would be read 300+200, 70+40, and 5+04)Left to right: Start on the left side and work your way over. 576+279 would be 700+140+155) Expanded Notation: 576+279 would be 500+70+6 +200+70+96) Lattice Method: uses a box with the same amount of columns as numbers and the columns are slashed in half, right to left additionSubtraction Algorithms:1) American Standard: Right to left, no mention of place value2) Reverse India: Left to right 3) Left to Right: Shows place value 4) Expanded Notation: 576-289= 500+70+6 -200+80+95) Integer Algorithm: Uses negative numbers, right to leftMultiplication Algorithms:1) American Standard: Right to left, place value isn't well explained2) Place Value: 23x14= 4x3=12 4x20=80 10x3=30 10x20=20012+80+30+200=3223) Expanded Notation: 23x14 take 23 fourteen times, 14 groups of 23. 23x14= 20+3 x10+4 90+2 200+30+0 200+120+2=3224) Lattice Method: Uses a box with the same amount of columns as numbers which are then slashed in half