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A process standard, an essential component in every classroom.
Problem Solving Steps:
1. Understand/Communicate
2. Devise a Plan/Strategize
3. Implement Plan
4. Look Back/Check your answer.
Is this reasonable?
Invented by George Polya (Einstein's Son's advisor)
A numeration system is a system for representing quantity of a number in a consistent manner
There are 10 digits (numerals): 0,1,2,3,4,5,6,7,8,9
Using the ten digits every number can be written.
Base 10 means using the ten different digits to write numbers. Digits (numerals): 0,1,2,3,4,5,6,7,8,9.
Base 10 = Decimal System
Decimal System is positional which means that the placement of digits (or numerals) in the number specifies the value of the number.
Base-5 (Quinary /ˈkwaɪnəri/ or pental) is a numeral system with five as the base.
5^0 = Ones = five to the 0
5^1 = Fives = five to the one
5^2 = 25's = five to the two
5^3 = 125's = five to the 3
Digits used in numeration system:
In Base 10 - 0,1,2,3,4,5,6,7,8,9
In Base 2 - 0,1
In Base 3 - 0,1,2
In Base 4 - 0,1,3
In Base 5 - 0,1,2,3,4
Expanded Notation:
Example:
723 = 700+20+3
723 = (7x100) + (2x10) + (3x1)
723 = (7x10^2) + (2x10^1) + (3x10^0)
IMPORTANT: Exponent operations do not have a commutative property. AND, the ^ symbol is used to represent an exponent.
A fraction is a symbol that shows the relationship between a part and a whole.
Which is more:
3/7 or 1/7 (3/7 is more)
4/5 or 4/9 (4/5 is more)
3/7 or 5/8 (5/8 is more)
3/4 or 9/10 (9/10 is more)
7/8 or 8/7 (8/7 is more)
Meanings of Fractions:
Models: of fractions:
-area: pattern black shading, pattern blocks, pie pieces, shading paper.
-length: number line, line segments, fraction bars, folding paper
-sets: groups of things
Whole: when the numerator and denominator are the same.
Addition and subtraction solve by keeping the same denominator. Use the least common multiple to solve an improper addition and subtraction fraction problem.
Divisibility Rules:
a is divisible by b, if there is a number c that meets the requirement.
A number is divisible by a second number if the second number divides into the first number with no remainder.
A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
A number is divisible by 3 if the sum of its digits is also divided by 3. Add each digit and check the result, is it divisible by 3.
A number is divisible by 4 if the last two digits of the number are divisible by 4. Look at the number. Ask yourself, are the last 2 digits divisible by 4.
A number is divisible by 5 if it ends in 5 or 0.
A number is divisible by 6 if it is divisible by BOTH 2 and 3.
A number is divisible by 7 if the following works:
Double the last digit. Then subtract it from the remaining number. If the result is divisible by 7, so is the original number. Apply this rule as many times as needed.
A number is divisible by 8 if the last three digits are divisible by 8.
A number is divisible by 9 if the sum of its digits is divisible by 9.
A number is divisible by 10 if it ends with 0.
A number is divisible by 11 if the following rule works.
Chop off the last two digits of your number. Add them to what it is left of the original number. Repeat until you have two digits. If the digits are the same, then your original number is divided by 11.
A prime number has exactly two factors: one and itself.
The prime numbers through 60 are; 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.
A composite number is a number with more than two factors.
Zero and one are very special numbers in out number system. 0 is the identity additive element. 1 is the multiplicative identity element. Zero and one are neither prime nor composite numbers.
Fractions:
A fraction is a part of something.
A penny is a fraction of a dollar.
The two numbers in a fraction are called the numerator and denominator.
Numerator is over the denominator:
Numerator - which tells how many parts you have
Denominator - which tells how many parts in the whole
Forms of Fractions:
Proper fraction The top number is less than the bottom number.
Improper fraction The top number is equal to or larger than the bottom number.
Mixed number A whole number is written next to a proper fraction
Reducing a fraction means writing it an easier way - with smaller numbers. When you reduce a fraction the value does not change. A reduced fraction is equal to the original fraction. When you have reduced a fraction as much as possible, the fraction is then in lowest terms.
Reduce: 20
30
In multiplication, when both the top and bottom numbers end with 0's, cross out the 0's. Then check to see if you can continue to reduce.
Raising fractions to higher terms is the opposite of reducing a fraction to lowest terms. Steps follow:
Step 1: Divide the old bottom number into the new one (the number you want to raise to). Keep the bottom number as the number you want to raise to ( the new one).
Step 2: Multiply the answer by the old top number. Place it on the top.
Check: Reduce the new fraction to see if you get the original fraction.
Changing improper fraction to whole or mixed numbers.
An improper fraction is a fraction with the top number that is big or bigger than the bottom number. An improper fraction is equal to or larger than one whole.
Change an improper fraction by dividing the bottom number into the top number and writing the remainder. Steps follow:
Step 1: Divide the bottom into the top
Step 2: Write the remainder as a fraction over the original bottom.
Step 3: Reduce the remaining fraction.
Changing mixed numbers to improper fractions. Steps follow:
Step 1: Multipy the bottom number by the whole number.
Step 2: Add the result to the top number.
Step 3: Place the total over the bottom number.
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
A natural number greater than 1 that is not prime is called a composite number.
Composite numbers have lots of factors.
1 is neither prime or composite because it is the identity element of multiplication. It is the multiplicative identity element.
0 is neither prime or composite because 0 is the identity element of addition. It is the additive identity element.
Prime factorizaton of 24 is as follows:
24: 1, 2, 3, 4, 6, 8, 12, 24
Prime factorization of 28 is as follows:
28: 1, 2, 4, 7, 14, 28
Prime factorization of 30 is as follows:
30: 1, 2, 3, 5, 6, 10, 15, 30
Prime factorization of 36 is as follows:
36: 1, 2, 3, 4, 6, 9, 12,18, 36
Prime factorization of 90 is as follows:
91: 1, 7, 13, 91
Prime factorization of 42 is as follows:
42: 1, 2, 3, 6, 7, 14, 21, 42
The prime numbers of 60 are as follows:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 54, 59
Prime factorization:
If I take any of the factors of a number, I can create a factorization tree.
GCF: Greatest Common Factor.
We use greatest common factors when we simplify fractions.
GCF is used when we simplify fractions.
LCM: Least Common Multiple.
Multiples reminds you of "skip" counting.
LCM is used to find a common denominator to create equivalent fractions.
LCD: Lowest (least) Common Denominator
If the fractions in an addition problem do not have the same bottom numbers (denominators), you must rewrite the problem so that all of the fractions have the same bottom number (called a common denominator).
This will mean raising at least one of the fractions to higher terms.
A common denominator is a number that can be divided evenly by all of the denominators in the problem.
The smallest number that can be divided evenly by all of the denominators in the problem is called the lowest common denominator or LCD. Sometimes, the largest denominator in the problem will work as the LCD.
PROPERTIES:
Properties of Multiplication:
Division does not have properties. Division, in a sense is the opposite of multiplication.
Properties of Addition:
Subtraction does not have properties. Subtraction in a sense is the opposite of addition.
EXAMPLE of Addition Expanded Notation:
478 + 394 =
Solve using expanded notation:
400 + 70 + 8 =478
300 + 90 + 4 = 394
800 + 70 + 2 = 872
EXAMPLE of Subtraction Expanded Notation:
645 - 279 = 366
Solve using Expanded Notation of Subtraction:
_ 600 + 40 + 5 =
200 + 70 + 9 =
300 + 60 + 6 = 366
Type in the name of your subject.
Divisibility rules:
When a # is divisible by a number without remainders?
a is divisible by another, b if there is a number c. that meets the requirement.
A number that divides a number are also factors of that number.
Factor: a number or quantity that when multiplied with another produces a given number or expression.
Divisibility rules for 2, 3, 4, 5, 6, 7, 8, 9, 10 & 11:
Divisibility rule for 2.
2: 0, 2, 4, 6, 8
meaning look for even numbers. So if the number has a 0, 2, 4, 6, or 8, then be assured it is divisible by 2.
Divisibility rule for 3.
3: if the sum of the digits is divisible by 3.
We will add up all the digits and divide the sum by 3.
Divisibility rule for 4.
4:
We need to pick the last two digits, and divide it by 4, if it leaves no remainders, the number is divisible by 4.
If last 2 digits are divisible by 4.
Divisibility rule for 5.
5: 0, 5
A number is divisible by 5 if it has 0 or 5 it its ones place (at the end of the number).
Divisibility rule for 6.
6:
If it's divisible by both 2 and 3.
If a number is divisible by 2 and 3, then it is divisible by 6 as well.
If it is divisible by both 2 and 3. Lets look at divisibility rule for 2: If a number has 0, 2, 4, 6, or 8 as its last digit, then be assured it is divisible by 2. Lets look at dvisibility rule for 3: We'll add up all the digits, and divide the sum by 3. Passes the divisibility test if "no remainders."
Divisibility rule for 7.
7:
There are 3 steps to determine the divisiblity:
Step 1: double the last digit, meaning 2 x the last digit = this number.
Step 2: Remove that last number, the last digit. Then subtract the result, from that number on the left of the last number. Take the other number, that is doubled, and subtract it from the number to the left.
Step 3: if the difference is divisible by 7, then the number is divisible by 7.
Double last digit, (2 x itself), then subtract that number from remaining number, and repeat.
Divisibility rule for 8.
8:
If the last three digits are divisible by 8, then the number is divisible by 8.
If the last 3 digits are divisible by 8.
Divisibility rule for 9.
9:
Add up all the digits and divide the number by 9.
If the sum of digits is divisible by 9.
Divisibility rule for 10.
10: 0
If the number ends with 0, then it is divisible by 10.
Divisibility rule for 11.
11:
The "chop off" method.
Algorithm is a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer.
Addition Algorithms:
Subtraction Algorithms:
Multiplication Algorithms:
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