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Math
Chapter 1:
1-3 Prime Factoring
- Factor: when two or more numbers are multiplied, each number
is called a factor of the product.
- Prime Number: a whole number that has exactly two unique
factors, 1 and the number itself.
- Composite Number: a number greater than 1 with more than
two factors.
- Neither Prime nor composite: 1 has only one factor and 0 has
an infinite number of factors.
- Prime Factorization: every composite number can be expressed
as a product of prime numbers.
o Example: Prime factorization of 54
54:2
27:3
9:3
3:3
1
1-4 Powers and Exponents
- A product of a prime number can be written using exponents and
a base.
- Base: the number being multiplied.
- Exponent: a shorthand way to show how many times a base is
multiplied by itself.
-The process of using exponents is called raising to a power.
- Powers: Numbers expressed using exponents.
5
Example: 32= 2x2x2x2x2 = 2
- Squared: a number raised to the 2nd power
- Cubed: a number raised to the third power
1-5 Order of Operations
- Numerical Expression: a combination of numbers and operations.
- Example: 4+3x5
- Order of Operations: tells you which operation to perform first.
1. Simplify the expressions inside grouping symbols, like
parenthesis.
2. Find the value of all powers.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
Chapter 3:
3-1 Representing Decimals
- Decimals: Like whole numbers are based on the number ten.
- Standard form: the usual way to write a number.
Example: 54.0007
- Extended form: a sum of the products of each digit and its place
value.
Example: (1x0.1) + (2x0.01)
- Place value chart: - whole number: thousands, hundreds, tens,
and ones.
- less than one: tenths, hundredths, thousandths, and
ten-thousandths.
3-2 Comparing and Ordering Decimals
- > Grater than = 4 > 2
- < Less than = 2 < 4
- Equivalent Decimals: 0.6 = 0.60
3 -3 Rounding Decimals
1. Underline the digit to be rounded
99.96
2. Look at the digit to the right of the digit to be rounded.
3. If the digit to the right is 4 or less, the underlined digit
remains the same.
4. If the digit to the right is 5 or more, add 1 to the
underlined digit.
3-4 Estimating Sums and Differences of Decimals
- To estimate sums and differences of decimals, you can use the
same methods you used for whole numbers.
- Example #1: 60.7 rounds to 60
68.5 rounds to 70
- So, 60+70 = 130
- Example #2: 80.2-72.1 = 80-70 = 10
- Front-End Estimation: add or subtract the front digits; then add
or subtract in the next place position.
- Example #1: 34.6
34.6
+55.3
+55.3
8
89
- Clustering: estimating by rounding a group of close numbers to the
same number.
- Example: 72.50
70
68.50
70
69.75
70
+71.75 +70
280
- ** Multiplication is repeated addition. So, a good estimate is
4x7 = 280
3-5 Addiding and Subtracting Decimals
- To add or subtract decimals, add or subtract digits in the same
place-value position. Be sure to line up the decimal points before
you add or subtract.
Chapter 4
4-1 Multiplying Decimals by Whole Numbers
- When multiplying a decimal by a whole number, multiply as with
whole numbers. Then use estimation to place the decimal point in
the product. You can also count the number of decimal places from
the right to the left.
- Example #1:
14.2
X 6
85.2
- Example #2:
0.83
X 9
7.47
- Scientific Notation: When you express 450 as the product of 4.5
an 102 (a power of ten). You can use the order of operations or
mental math to write numbers like 4.5 x 102 in standard form.
- Example #1: Using order of operations
- Evaluate 102 first. Then multiply
4.5 x 102 = 4.5 x 100 = 450
4.5
X100
450.0
- Example #2: Using mental math. Add two zeros and
move the decimal point 1 place.
4-2 Multiplying Decimals
When multiplying a decimal by a decimal, multiply as with whole
numbers. To place the decimal point, fin the sum of the number of
decimal places in each factor. The product has the same number of
decimal places.
- Example:
4.2
X 6.7
294
252
28.14
4-3 Dividing Decimals by Whole Numbers
- Dividing a decimal by a whole number is similar to dividing whole
numbers.
- Place the decimal point directly above the decimal point
in the dividend.
- Divide as with whole numbers.
- Quotient: the solution in the division.
Note: You need to round the quotient to a specified place-value
position. Always divide to one more place-value position than the
place to which you are rounding.
Chapter 5
5-1 Greatest Common Factor (GCF)
- Venn Diagrams: use overlapping circles to show common elements.
- GCF: the greatest common factor of two or more numbers.
- Example: 42 and 56
Factors of 42: 1,2,3,6,7,14,21,42
Factors of 56: 1,2,4,7,8,14,28,56
42 and 56 GCF = 14
5-2 Simplifying Fractions
- Equivalent Fractions: fractions that name the same number.
- Example: 15
=
3
50
10
- To find equivalent fractions, you can multiply or divide the
numerator and denominator by the same nonzero number.
- Example: 15 : 5
=
3
50 : 5
10
- Simplest Form Fractions: when the GCF of the numerator and
denominator is 1.
- Method 1: divide by common factors
- Method 2: divide by the GCF
5-4 Least Common Multiple (LCM)
- Multiple: is the product of a number and any whole number.
- Common multiple: multiples common to two different numbers.
- LCM: smallest common multiple (other than 0)
o Example: 8 and 4
 Multiples of 8: 0,8,16,24,32,…
 Multiples of 4: 0,4,8,12,16…
 8 and 4 LCM = 8
5-5 Comparing and Ordering Fractions
- Comparing two fractions:
- Find the least common denominator (LCD) – least common
multiple of the denominators.
- Rewrite the each fraction as an equivalent fraction using the
LCD.
- Compare the numerators (<, >, or =).
- Example: 5
>
9
8
16
8 – 16 : 8
1 - 2 : 2
1
LCD = 2 So…
5x2 =
10 >
9
8x2
16
16
- Order Fractions: You can use what you have learned about
comparing fractions to order fractions
- Example: From least to greatest 1, 9, 5, and 3
2 14 7
4
5-6 & 7 Writing Decimals as Fractions and Fractions as Decimals
- Write Decimals as Fractions:
1. Identify the place of the last decimal place.
2. Write the decimal as a fraction using the place value as the
denominator.
3. If necessary, simplify the fraction.
- Example: 0.6 (six tenths) = 6
10
Simplify – 6:2 = 3
10:2
5
- Write Decimals as Mixed Numbers:
- Example: 8.26 = 8 26
100
Simplify – 8 26:2
= 8 13
100:2
50
- Write Fractions as Decimals:
1. Divide the numerator by the denominator.
- Example: 7 or 7:8 = 0.875
8
- Terminating Decimals: are decimals like 0.25 and 0.75 because
the division ends, or terminates, when the remainder is zero.
- Repeating Decimal: a decimal like 0.2222222… because the
digits repeat. Examples: 0.8888888 = 0.8, 2.8787878… = 2.87