SporkForAll Algebra 1 Lessons- Lesson A

SporkForAll Algebra 1 Lessons

This lesson is for review of Algebra 1 Topics as written by Spork. This will cover all the way from review of PreAlgebra to finding equations for lines on graphs, to substitution, and many more topics! Enjoy and do well in your math courses!!! :)

Lesson A- Review of Algebra

This lesson will cover operations of addition, subtraction, multiplication, and division of rational numbers, and find the Least Common Multiple and the Greatest Common Factor of two or more natural numbers. These topics should be review from a PreAlgebra course.

OPPOSITE OF A NUMBER

• The number -a is called the opposite of the number a.

• The opposite is sometimes also referred to as the negative.

• Example1: The opposite of 5 is -5.

• Example2: The opposite of -3 is -(-3) =3.

ABSOLUTE VALUE OF A NUMBER

• The absolute value |a| of a real number represents the distance between the number a and zero on a number line.

• Therefore, |a| ≥ 0 for any real number a.

• The absolute value of a positive number or zero is the number. |5| = 5

• The absolute value of a negative number is the opposite of the number: |-5| = 5

ABSOLUTE VALUE OF A NUMBER

The opposite numbers 5 and -5 have the same absolute value, because they are the same distance from zero: 

OPERATIONS WITH SIGNED NUMBERS

• When adding a negative number, we rewrite it as subtraction of the opposite: 2 + (-3) = 2 - 3 = -1

• When subtracting a negative number, we rewrite it as addition of the opposite:  5 - (-3) = 5 + 3 = 8

• If we multiply or divide 2 signed numbers, if the signs on both numbers are the same, the answer will be positive: -4 x (-5) = 20, -30/(-6) = 5

• If we multiply or divide 2 signed numbers, if the signs on both numbers are opposite, the answer will be negative: -4 x 5 = -20, 30/(-6) = -5

PRIME vs. COMPOSITE NUMBERS & FACTORS

• If a natural number is divisible by another number, then we say that the second number is a factor of the first number.

• A prime number is a whole number greater than 1 that has exactly two factors: 1 and the number itself.

• A composite number is a whole number that has more than 2 factors.

• Remember this for prime factorization!!!

PRIME FACTORIZATION

Every natural number greater than 1 is either a prime number or it can be factored as a product of primes. Such factorization consists of a unique set of prime factors.

This is sometimes referred to as the Fundamental Theorem of Arithmetic.

LEAST COMMON MULTIPLE (LCM)

• The LCM is the smallest number that is evenly divisible by all of the given numbers in a set.

• Example: LCM (6,8) = 24

• To find the LCM of bigger numbers, we can use prime factorization. The LCM is the product of all of the prime factors occuring in both numbers; each of the factors is raised to the highest power occuring.

LCM Example

Here is an example of finding LCMs of 2 or more bigger numbers in a set. Here we are solving LCM (56,70).

GREATEST COMMON FACTOR (GCF)

• The GCF is the greatest number that is a divisor of all of the given numbers.

• Example: GCF (36,45) = 9

• Just as for finding the LCM, we can use prime factorization to find the GCF. The GCF of given numbers is also the product of the prime factors that both numbers have in common, raised to the smallest power occuring.

GCF Example

Here is an example of finding GCFs of 2 or more bigger numbers in a set. Here we are solving GCF (56,70).

OPERATIONS WITH FRACTIONS

• To reduce a fraction to the lowest term, we divide both the numerator and the denominator by their GCF: (36/45)/(9/9) = 4/5

• To add or subtract two fractions, we first find the LCM of their denominators. We then rewrite both fractions, using the equivalent fractions with our new denominator. Finally, we add or subtract the new numerators, keeping our denominator the same: 1/6 + 3/8 = 4/24 + 9/24 = 13/24

OPERATIONS WITH FRACTIONS

• To multiply two fractions, we multiply the numerators to get the numerator of the product, and we multiply the denominators to get the denominator of the product. We reduce first by canceling, if possible: 4/9 x 15/14 = 2/3 x 5/7 = 10/21

• To divide two fractions, we rewrite the division as a multiplication by the reciprocal. We get the reciprocal of the divisor by writing the number from the numerator in the denominator and vice versa: (3/7)/(1/2) = 3/7 x 2/1 = 6/7

• We can use division to simplify complex fractions. A complex fraction is a fraction where the numerator or denominator, or both, are fractions themselves. The line between the numerator and denominator means that we should divide the top by the bottom. 

ORDER OF OPERATIONS

• 1. Simplify within the symbols of inclusion (parentheses , brackets , etc.).

• 2. Simplify exponents and roots.

• 3. Multiply and divide in order from left to right.

• 4. Add and subtract in order from left to right.

Problem Set A- Lesson Practice

This problem set will have 36 problems reviewing Lesson A topics discussed in this lesson. Good luck and get them correct! :) You will have 300 seconds (5 minutes) to answer each question.

Fill in the Blank

Homework Set 1A:

SIMPLIFY:

-6 + 8

Fill in the Blank

Homework Set 1B:

SIMPLIFY:

6 - 8

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Homework Set 1C:

SIMPLIFY:

-6 + (-8)

Fill in the Blank

Homework Set 2A:

SIMPLIFY:

-15 + 3

Fill in the Blank

Homework Set 2B:

SIMPLIFY:

3 + (-15)

Fill in the Blank

Homework Set 2C:

SIMPLIFY:

-15 - 3

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Homework Set 2D:

SIMPLIFY:

-3 - (-15)

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Homework Set 3:

SIMPLIFY:

5 - 8 + 2 + 5 - 9 - 3 - 7 + 5

Fill in the Blank

Homework Set 4:

SIMPLIFY:

-18 - 53 + 12 - 9 + 44 + 13

Fill in the Blank

Homework Set 5A:

SIMPLIFY:

(-2) x 24

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Homework Set 5B:

SIMPLIFY:

56/(-7)

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Homework Set 5C:

SIMPLIFY:

(-35)/5

Fill in the Blank

Homework Set 5D:

SIMPLIFY:

16 x (-3)

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Homework Set 6A:

SIMPLIFY:

-84/21

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Homework Set 6B:

SIMPLIFY:

(-3)(-3)

Fill in the Blank

Homework Set 6C:

SIMPLIFY:

(-2)(-5)

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Homework Set 6D:

SIMPLIFY:

3(-9)

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Homework Set 7:

SIMPLIFY:

-3 x (-1) x 5 x (-2) x 4

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Homework Set 8:

SIMPLIFY:

16 - 12 x 3 / 18

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Homework Set 9:

SIMPLIFY:

48 / 6 - 3 x 5

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Homework Set 10:

Determine the following absolute value:

|5 - 11|

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Homework Set 11:

SIMPLIFY:

5 - |3 - 7|

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Homework Set 12A:

SIMPLIFY:

(-(-9))

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Homework Set 12B:

SIMPLIFY:

(-(-(-9)))

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Homework Set 13:

SIMPLIFY:

-{-[-(-23)]}

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Homework Set 14:

SIMPLIFY:

-3 + 2(5 - 7) + 4 x [3 - (8/2)]

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Homework Set 15:

List all factors of 91 that are composite numbers.

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Homework Set 16A:

Find the LCM of:

150 and 525

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Homework Set 16B:

Find the LCM of:

12, 15, and 27

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Homework Set 16C:

Find the LCM of:

24 and 24

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Homework Set 17A:

Find the GCF of:

84 and 96

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Homework Set 17B:

Find the GCF of:

15 and 17

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Homework Set 17C:

Find the GCF of:

16 and 32

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Homework Set 18A:

FIND:

LCM (21,35)

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Homework Set 18B:

FIND:

GCF (21,35)

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Homework Set 19:

Reduce 21/39 to its lowest terms.

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Homework Set 20:

ADD:

3 1/5 + 2 3/4

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Homework Set 21:

ADD:

21/14 + 45/60

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Homework Set 22:

ADD:

9 3/17 + 2 15/17

Fill in the Blank

Homework Set 23:

SUBTRACT:

3/7 - 3/14

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Homework Set 24:

SUBTRACT:

12/32 - 15/18

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Homework Set 25:

SIMPLIFY:

4/5 - 3/15 - 1/3

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Homework Set 26:

MULTIPLY:

11/6 x 24/55

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Homework Set 27:

SIMPLIFY:

48/15 x 45/18 x 1/3

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Homework Set 28:

DIVIDE:

18/(6/7)

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Homework Set 29:

DIVIDE:

(5/4)/(2/5)

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Homework Set 30:

SIMPLIFY:

8/35 x 14 / (16/7)

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Homework Set 31:

SIMPLIFY:

(8/21) / (4/3) x 5/6

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Homework Set 32:

SIMPLIFY:

(1/18 + 2/9 - 15/36) x (-48/25)

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Homework Set 33:

SIMPLIFY:

(1/5 - 2/3) / (2 1/6 - 5/8)

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Homework Set 34:

SIMPLIFY:

[(4/5 - 1/3) - (1 1/4 - 1/3)] / [(10-1)/(10 x 4)]

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Homework Set 35:

SIMPLIFY:

(4/7 + 5/6)/(4/7 - 1/3)

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Homework Set 36:

SIMPLIFY:

(-3 1/2) / (-1 2/3)