2.1 Integers

Natural (Counting) Numbers: 1, 2, 3, 4, …

Whole Numbers (All Natural Numbers and Zero)0, 1, 2, 3, 4, …

Opposites are the same distance from 0, but on opposite sides of zero.

Integers (Whole Numbers and their opposites): …, -3, -2, -1, 0, 1, 2, 3, …

Negative Integers: Integers that are less than 0.

Positive Integers: Integers that are greater than 0.

2.1 IntegersLess than: 4 is less than 5 4 < 5Greater than: 6 is greater than 1 6 > 1

Absolute value: is the distance from zeroAbsolute value is always positive.

Absolute value of 7 is shown as 7 Absolute value of -7 is shown as -7

7 = 7 -7 = 7 50 = -250 =

Additive inverse rule: the sum of a number and its opposite is zero (0)

5 + -5 = -1,247 + 1,247 = ½ + -½ =

2.2 Adding Integers

To add integers on a number line:

• Start at 0, use sign of number for (-) or (+)

• Move to the left (-) or right (+) the distance of the first addend.• From that point, move to the left (-) or right (+) the distance of the second

addend.

2.2 Adding Integers

Adding Integers with same sign: find the sum of their absolute values, use their common sign.

Different signs: find the difference of their absolute values, use the sign of the number with the greater absolute value.

(Or change to subtraction: -5 + 6 → 6 – 5 = 1)

2.3 Subtracting Integers

To subtract an integer, add its opposite.

Easy:

5 – 2 → 5 + (-2) = |5| - |2| = 3 → +3

4 -1 =

Add it opposite:

12 – (-9) → 12 + 9 = |12| + |9| = 21 → +21

5 – (-2) →

Look at addition rule for opposite signs:

-3 – (-5) → -3+5 = |5| - |3| = 5 – 3 = 2 → +2

-1 – (-8) →

2.3 Subtracting Integers

The highest point in Asia is Mount Everest at 8,850 meters. The shore of the Dead Sea is the lowest point in Asia at about 410 meters below sea level. What is the difference between these elevations?

(Clue: To subtract an integer, add its opposite.)

2.4 Multiplying/Dividing Integers

Multiplication: perform operation (usually adding) a certain number of times.

Division: breaking a collection into a number of equal subsets

2 × 3 = 0 + 2 + 2 + 2 = 6

-3 × 5 = 0 + -3 + -3 + -3 + -3 + -3 = -15

2.4 Multiplying Integers Rules for Multiplying Integers

But what if I multiply a negative by a negative?

Negative x positive

-3 × 5 = 0 + -3 + -3 + -3 + -3 + -3 = -15

negative x negative

-3 × -5 =

0 – (-3) – (-3) – (-3) – (-3) – (-3)

15

2.4 Multiplying/Dividing Integers

Same signs: answer is positive

Different signs: answer is negative

Zero: The product of an integer and 0 is 0

2.4 Multiplying/Dividing Integers

Negative or positive?

-3 × -50 -35 × -50 -55 × 50

43 × 50 43 × -3 -22 × -50

-3 × -5 × -5 -3 × 5 × 5 3 × -5 × -5

3 × 55 × -5 -3 × -5 × 5 -33 × -5 × -5

2.5 Solving Equations Containing Integers

Inverse operation: an operation that undoes another operation. – Addition is the inverse of subtraction.– Subtraction is the inverse of addition.– Multiplication is the inverse of division.– Division is the inverse of multiplication.

Do inverse operations on both sides of the equal sign!

2.5 Solving Equations Containing Integers

Do inverse operations, and pay attention to positive/negative signs:

-3 + y = -5 n + 3 = 10

a/-3 = 9 -120 = 6x

2.5 Solving Equations Containing Integers

p 100 1-7

WB 2.5: 6-20 even, 22, 23

p 102: 1-10 even, 11, 14, 17, 18, 19, 24, 25

2.6 Prime Factorization

• Prime Number: Whole number greater than 1 whose only whole factors are 1 and itself. Examples: 2, 3, 5, , 23…

• Composite Number: Whole number greater than 1 that has whole factors other than 1 and itself. Examples: 4, 6, 8

• Prime Factorization: To factor a whole number as a product of prime numbers. Example: 24 = 4 × 6 = 2 × 2 × 2 × 3

24 = 8 × 3 = 2 × 2 × 2 × 3 = 23 × 3

2.6 Prime Factorization

Factor Tree 252

2 1262 63

3 213 7

2 × 2 × 3 × 3 × 7 = 22 × 32 × 7

2.6 Prime FactorizationStep Diagram

3 252

2 84

3 42

2 14

7

2 × 2 × 3 × 3 × 7 =

• 22 × 32 × 7

2.7 Greatest Common Factor

• Common Factor: A whole number that is a factor of two or more nonzero whole numbers. Example: 3 is a factor of 21 and 30

• Greatest Common Factor (GCF): The largest whole number that is a factor of two or more nonzero whole numbers. Example: 8 is the GCF of 24 and 40.

• Relatively Prime: Two or more nonzero whole numbers are relatively prime if their greatest common factor is 1.

2.7 Greatest Common FactorListing Factors

24: 1, 2, 3, 4, , , ,

36: 1, 2, 3, 4, 6, , , , , ,

48: 1, 2, 3, 4, 6, , , , , ,

2.7 Greatest Common FactorPrime Factorization

Find the GCF of the following by multiplying the common (same) prime factors:

24: 2 × 2 × 2 × 3 36: 2 × 2 × 3 × 348: 2 × 2 × 2 × 2 × 3

GCF: 2 × 2 × 3 = 12

2.7 Greatest Common FactorWord Problem

I am making identical Halloween gift bags using 24 Snickers bars, 80 Tootsie Pops, and 48 giant Sweet Tarts. How many bags can I make?

Snickers (24): 2 × 2 × 2 × 3

Tootsie Pops (80): 2 × 2 × 2 × 2 × 5

Sweet Tarts (48): 2 × 2 × 2 × 3

2.8 Least Common Multiple

Multiple: the product of the number and any non-zero whole number. Examples: for 3: 3, 6, 9, 12; for 15: 15, 30, 45, 60, 75, 90.

Common Multiple: a multiple shared by two or more numbers. Example: 12 is a common multiple of 2 and 3.

Least Common Multiple (LCM): the least of the common multiples. Example: 6 is the LCM of 2 and 3; 12 is the LCM of 3 and 4.

2.8 Least Common Multiple

Finding LCM by listing multiples.

What is the LCM of 12, 15, and 30? (What is the smallest multiple of both 12, 15, and 30?)

12: 12, 24, 36, 48, 60, 72, 84, 96

15: 15, 30, 45, 60, 75, 90, 105

30: 30, 60, 90, 120, 150, 180, 210

2.8 Least Common Multiple

Finding LCM by prime factorization.

1. Do prime factor tree.

2. Multiply any common factors.

3. Then multiply by all uncommon factors.

2.8 Least Common Multiple

What is the LCM of 12, 30 and 15?

12 30

2 6 2 15

2 3 3 5

12: 2 × 2 × 3 15

30: 2 × 3 × 5 3 5

15: 3 × 5

2 × 3 × 5 × 2 = 60 (com x com x com x uncom)

2.8 Least Common Multiple Part II, The Saga Continues

Find the GCF, LCM and product of 12 and 18.

12 18

2 6 2 9

2 3 33

12: 2×2×3

18: 2×3×3

GCF: 2×3 =6 LCM: 2×3×2×3=36

Product 12×18=216

2.8 Least Common Multiple Part II, The Saga Continues

Find the GCF, LCM and product of 12 and 18.

GCF: 6

LCM: 36

Product: 12×18 = 216

What is the relationship of GCF, LCM and product?

2.9 Make Equivalent Fractions

ALL FRACTIONS MUST BE WRITTEN IN SIMPLEST FORM.

To make equivalent fractions, multiply or divide the numerator and denominator by the same number.

Find two fractions equivalent of 14/16.

14/16 ÷ / = /

14/16 × / = /

2.9 Simplest Form1. Find the GCF of the numerator and denominator.2. Divide the numerator and denominator by the GCF.

Write 24/36 in simplest form.

GCF of 24 and 36

24/36 ÷ / = /

2.9 Are Fractions Equivalent?

1. Find LCD (LCM of the denominator) of the fractions.

2. Make equivalent fractions.

3. Compare numerators.

Are 6/8 and 9/12 equivalent?

6/8 × / = /

9/12 × / = /

2.9 Improper to Mixed Fractions

1. Divide numerator by denominator.2. Use quotient, remainder and denominator to write mixed number.

Write 21/4 as a mixed number.

21 ÷ 4 = 5 R 11 → 5 ¼¼

Write 27/8 as a mixed number.

27 ÷ = __ R __ →

2.9 Mixed Fractions to Improper

1. Multiply whole number by denominator.

2. Add product to numerator.

Write 4 2/3 as a mixed number.

4 × 3 = 12 + 2 → 14/3

Write 9 4/5 as a mixed number.

9 × __ = ___ + ___ →

2.10 Equivalent Fractions and Decimals

A fraction a/b is the same as a ÷ b. You can use this relationship to change any fraction to a decimal number.

3 3/4 =

= 3 + (3 ÷ 4)

= 3 + .75

= 3.75

2.10 Equivalent Fractions and Decimals

Terminating Decimal: when a long division problem has a remainder of 0. Example: 4/5 = 0.8

Repeating Decimal: when long division gives a repeating decimal – use a bar over the repeating values. Example

1/3 = 0.3

8/11 = 0.72

Mental Math (make the math easier):

2/5 × 2/2 = 4/10 = 0.4

2.10 Equivalent Fractions and Decimals

Decimal to Fraction: Identify place value of last decimal place, use that as the denominator, and then simplify.

Example: 0.65 =

the 5 is in the 1/100 so = 65/100

65 and 100 can be divided by 5 so = 13/20

2.11 Comparing and Ordering Rational Numbers

Rational Number: any number that can be written as a fraction with integers for its numerator and denominator - a/b, where b≠ 0- (includes terminating and repeating decimals)

Ex: 1/5 (0.2)1/3 (0.3)

Irrational Examples: square roots of prime numbers and some other special numbers

Ex: π (3.14159265359…) √2 (1.4142135….)

Rules: negatives are less than positivelarger denominators mean the pieces are smaller

2.11 Comparing and Ordering Rational Numbers

Comparing fractions: If necessary, rewrite fractions to have the same denominator, and then order by comparing numerators.

Order from least to greatest: 1/7, 3/9, -3/5, 11/14, 5/7, 3/7

Negatives?

Denominators’ sizes?

2.11 Comparing and Ordering Rational Numbers

Comparing Decimals: line up the decimal points and compare digits from left to right.

Compare -0.31, 0.0324, -0.325, -0.36

Comparing Fractions and Decimals: • Convert all to fractions or decimals and compare• Graph them on the number line

Compare 3/5, .77, -0.1, 1 ¼