作者：Christina Robbins 3 年以前
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When multiplying fractions the part will get smaller
When multiplying fractions - multiply across
3 groups of 2 = 3 x 2
1/2 of 1/2 = 1/2 x 1/2 = 1/4
1/2 x 1/4 = 1/8
1/3 x 1/8 = 1/24
Objection: Adding fraction with the same denominator
(easy to add and subtract)
1/4 + 2/4 = 3/4
3/12 + 4/12 = 7/12
1/6 + 4/6 = 5/6
Least Common Multiple:
1/4 + 1/6 - 3/12 + 2/12 = 5/12
1/2 + 1/6 - 3/6 + 1/6 = 4/6
3/12 + 1/4 - 3/12 + 3/12 = 6/12 - 1/2
#3 - Correct answer
No...Multiples are not factors and multiples of numbers continue forever
Numbers List:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59
Definition: A prime number is a number greater than 1 with only two factors themselves and 1
(LCM)
Definition: The least common multiple is the first common multiple for the given numbers. For 36 and 48, the number 144 is the LCM. The dimension of using LCM of two numbers starts with basic math operations such as addition and subtraction on fractional numbers.
How do you calculate least common multiple?
One way to find the least common multiple of two numbers is to first list the prime factors of each number. Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.
LCM:
(24) (36)
2: 24 36
2: 12 18
2: 6 9
3: 3 9
3: 1 3
1 1
LCM: 2x2x2x3x3=72
LCM=72
(GCF)
Definition: The greatest number that is a factor of two (or more) other numbers. When we find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.
How do you calculate the greatest common factor?
To find the greatest common factor (GCF) between numbers, take each number and write it's prime factorization. Then, identify the factors common to each number and multiply those common factors together.
GCF:
24: 1,2,3,4,6,8,12,24
36: 1,2,3,4,6,9,12,18,36
(The greatest common factors of 24 & 36 is 12)
A number A is divisible by a second number B, if there is a third number C that meets this requirement:
CxB=A - 2x5=10 (2&5 are factors) - 10/5=2
American Standard (R-L)
(Last Step) (no emphasis on place value)
European/Mexican (R-L)
(no emphasis on place value)
Reverse Indian (L-R)
(no emphasis on place value)
Left to Right (L-R)
*(place value)*
Expanded Notation
*(place value)*
Integer Subtraction
Subtract: 645-279
Standard American Algorithm (last step)
(R-L) (no reference to place value)
Partial Sums (R-L)
Problem Solving (emphasis on place value) (R-L)
Left to Right (L-R)
Expanded notation
Lattice Algorithm
Homework #4 (Addition)
Add: 478+394
Answer all questions and show your work.
1. Write each of these numbers:
a) 29 in base 3=
b) 69 in base 2=
c) 115 in base 5=
2. How do you know there is an error in each statement?
a) 10=243 and b) 13 3/4=25.34
*2 the answer: YOU CAN NOT HAVE A NUMBER BIGGER THAN THE BASE
#2
1. Give the Base-10 numeral for each given number. Use expanded notation to explain your answer:
a) 41.58=
b) 13415=
2. Write the number 12 in each given base:
a) Base 9
b) Base 8
c) Base 7
Base-5
ones, fives, 25s, ect
The first digit after the point is a fifth of the unit
The second digit after the point is a 25th of the unit etc.
(Example)
ones, fives, 25s, 125s, 625s
5^0, 5^1, 5^2, 5^3, 5^4..
Expanded Notation:
23 in base 5 = 13 in base 10
(2x5^1)+(3x5^0)
(10)+(3) = 13
Digits Used:
0,1,2,3,4
The same concept is used in all of the other bases
The number system we use in our schools and society is the Base-10 system. There is a consistent one-to-ten relationship between the digits of any number in the Base-10 system.
•One unit
•Ten units make a ten
•Ten tens make a hundred
•Ten hundreds make a thousand, etc.
•A tenth of a unit is a tenth
•A tenth of a tenth is a hundredth
•A tenth of a hundredth is a thousandth, etc.
Digits used
In Base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Expanded Notation
723=700+20+3
723=(7x100)+(2x10)+(3x1)
723=(7x10^2)+(2x10^1)+(3x10^0)
Decimal Point
23.14
•A tenth of a unit is a tenth
•A tenth of a tenth is a hundredth
•A tenth of a hundredth is a thousandth, etc.
The same idea applies to the other bases!
Numerations systems have been developed over the centuries as a way of recording quantity.
In essence a numeration system is a system for representing quantity or a number in a consistent manner.
For example, the symbol “11” may represent a variety of numbers depending on the numeration system being used.
Some number systems are positional which means that the placement of digits (or numerals) in the number specifies the value of the number. An example of such a number system is the decimal system that we use (Base-10)
Solve both of the following problems. Show enough details, draw pictures, and give written explanation, so I can tell how you solved the problem.
1. There are 12 basketball teams in a league. If each of the teams plays each of the other teams once and only once, how many games take place?
2. I have four 3-cent stamps and three 7-cent stamps. Using one or more of these stamps, how many different amounts of postage can I make?
There are seven people in a room. If each person shakes every other person’s hand, how many handshakes will take place? One way to solve this problem is to make a chart.
Person 1 shakes hands with: 2, 3, 4, 5, 6, and 7. 6 handshakes
Person 2 shakes hands with: 3, 4, 5, 6, and 7.
5 handshakes
Person 3 shakes hands with: 4, 5, 6, and 7.
4 handshakes
Person 4 shakes hands with: 5, 6, and 7.
3 handshakes
Person 5 shakes hands with: 6, and 7.
2 handshakes
Person 6 shakes hands with: 7.
1 handshake
Person 7 has already shaken hands with everyone.
0 handshakes
21 handshakes in total
Polya's 4 steps to problem solving
Understand the problem
1.What are you asked to find out or show?
2.Can you restate the problem in your own words?
3.Can you draw a picture or diagram to help you understand the problem?
Devise a plan
1.What problem solving strategy are you going to use
2.Guess and check/Trial and error
3.Draw a picture or a diagram
4.Make a table
5.Act it out
Carry out the plan
1.Carrying out the plan is usually easier than devising the plan
2.Be patient ~ most problems are not solved quickly nor on the first attempt
3.If a plan does not work immediately, be persistent
Look back (reflect)
1.Does your answer make sense?
2.Is it a reasonable one?
3.Did you answer all of the questions?
4.Could you have solved this problem in a different way?
