by Sam Meyer 6 hours ago
4
Elementary Mathematics
Various methods for performing basic mathematical operations such as addition, subtraction, multiplication, and division are outlined, each with unique techniques and considerations for place value.
by Sam Meyer 6 hours ago
4
More like this
Addition Algorithms:
1) American Standard: Right to left, no mention of place value
2) Partial Sums: Right to left, no mention of place value, uses columns, doesn't carry numbers above the equation
3) 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+0
4)Left to right: Start on the left side and work your way over. 576+279 would be 700+140+15
5) Expanded Notation: 576+279 would be 500+70+6
+200+70+9
6) Lattice Method: uses a box with the same amount of columns as numbers and the columns are slashed in half, right to left addition
Subtraction Algorithms:
1) American Standard: Right to left, no mention of place value
2) Reverse India: Left to right
3) Left to Right: Shows place value
4) Expanded Notation: 576-289= 500+70+6
-200+80+9
5) Integer Algorithm: Uses negative numbers, right to left
Multiplication Algorithms:
1) American Standard: Right to left, place value isn't well explained
2) Place Value: 23x14= 4x3=12
4x20=80
10x3=30
10x20=200
12+80+30+200=322
3) Expanded Notation: 23x14 take 23 fourteen times, 14 groups of 23. 23x14= 20+3
x10+4
90+2
200+30+0
200+120+2=322
4) Lattice Method: Uses a box with the same amount of columns as numbers which are then slashed in half
Division:
3 different symbols used in division
4 key components to a division problem when using the Standard American Algorithm: Divisor, dividend, quotient, and remainder
There are no properties for division or subtraction
Addition: Putting together, joining
Addition has three properties:
1) Identity: When you add zero, the identity of the number does not change. 2+0=2
2) Commutative: The order of two numbers being added does not matter. 2+1=1+2
3) Associative: The way you group numbers does not matter. (3+2)+1=3+(2+1)
Subtraction:
1) Take Away: 4-3=1
2) 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+__=7
Multiplication: Groups of something, repeated addition
1) Identity: When you multiply a number by 1, the identity does not change. 8x1=8
2) Commutative: The order of two numbers being multiplied does not matter. 8x2=2x8
3) 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=0
5) Distributive Property: Repeated addition. 3x7=7+7+7, 3x7=3(5+2)
Build off of prior knowledge and apply that to new bases
Base 2 (0-1)
Ones=2^0
Twos=2^1
Fours=2^2
Eights=2^3
16s=2^4
Example of converting a base 2 number into base 10:
1111base 2=(1x2^3)+(1x2^2)+(1x2^1)+(1x2^0)
=8+4+2+1
=15
Comparing 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 2/4
221.2base 3=(2x3^2)+(2x3^1)=(1x3^0)+(2/3)
=18+6+1+2/3
=25 2/3
37base 9=(3x9^1)+(7x9^0)
=27+7
=34
Converting Base 10 Numbers to Other Bases:
Base 5) 135= One 125, zero 25s, two 5s, zero 1s
=1020base 5
Base 3) 135=One 81, two 27s, zero 9s, zero 3s, zero 1s
=12000base 3
Base 2) 135=One 128, zero 64s, zero 32s, zero 16s, zero 8s, one 4, one 2, one 1
=10000111base 2
Principal of bases stays the same
Base 10 (0-9):
Ones=10^0
Tens=10^1
Hundreds=10^2
Thousands=10^3
Base 5 (0-4):
Ones=5^0
Fives=5^1
25s=5^2
125s=5^3
Base 3 (0-2)
Ones=3^0
3s=3^1
9s=3^2
27s=3^3
What is 28 in different bases?
Base 10= two 10s and eight 1s
=28
Base 3=one 27, zero 9s, zero 3s, one 1
=1001 base 3
Base 5=one 25, zero 5s, three 1s
=103 base 5
Symbols are used to represent a quantity, and in our base 10 system (positional system) we use the symbols 0-9
Base 10 (0-9) vs Base 5 (0-4)
Ones=10^0 Ones=5^0
Tens=10^1 Fives=5^1
Hundreds=10^2 25s=5^2
Thousands=10^3 125s=5^3
Example of one-to-ten relationship:
375.25
The 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 smaller
Expanded 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
Builds off of knowledge from week one
Example 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 repeats
A) B, C, D, E, F, G
B) C, D, E, F, G
C) D, E, F, G
D) E, F, G
E) F, G
F) G
G)
21 handshakes will take place
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 plan
3) Implement plan
4) Check Work
5) 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 draw
Round two: 1 dog vs 2 grandmas and 1 acrobat, comes to a draw
Round 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 acrobats
We 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
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.
