9th GRADE SUMMER PRACTICE ASSIGNMENT

This document contains vocabulary you will need, examples and practice problems. Use these to practice your skills before you complete the actual SUMMER ASSIGNMENT. This document should be kept. It will be needed when you begin 9th grade at Raritan High School.

Algebra Glossary Order of Operations: Rules for evaluating an expression involving more than one operation. Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction Ex.

Prime Number: A number which has one and itself as it’s only factors. Ex. 2, 3, 5, 7, 11, …

Composite Number: A number which is not prime and has other factors besides one and itself. Ex. 4, 6, 8, 9, 10, 12, …

Integers: Positive and Negative whole numbers. Ex. …, -3, -2, -1, 0, 1, 2, 3, …

Absolute Value: The absolute value of a number a is the distance between a and 0 on a number line. The symbol ǀaǀ represents the absolute value of a. Ex. ǀ5ǀ=5

Ex. ǀ-3ǀ=3

Variable: A letter that is used to represent one or more numbers.

Coefficient: The number part of a term with a variable part. Ex. 3xy (3 is the coefficient)

Ex. -5abc (-5 is the coefficient)

Constant term: A term with a number part and no variable part. Ex. 2x + 3 (3 is the constant term)

Ex. -3y – 7

(-7 is the constant term)

Solution: A number that produces a true statement when substituted for the variable in an equation. Ex. X = 2

is a solution to

Expression: A math phrase that contains numbers, variables and operations. Ex.

2x + 3y – 7

Ex. 4a + 6

Equation: A mathematical sentence formed by placing the symbol = between two expressions. Ex.

Ex.

Rational Numbers: A number that can be written as where a and b are integers and b≠0. The decimal form of a rational number terminates or repeats. Ex. 2, -5, ,

, 2.5, -3.44444…

Irrational Numbers: A number that cannot be written as the quotient of two integers. The decimal form of an irrational number neither terminates or repeats. Ex. 4.567238…, Factors: One of two or more numbers, algebraic expressions, or the like, that when multiplied together produce a given product; a divisor: Ex. 6 and 3 are factors of 18 Multiples: The result of multiplying a number by an integer ( not a fraction). Ex. 12 is a multiple of 3 because 3 x 4 = 12

Integer Rules

Integer Rules for Multiplication and Division (Positive Number)(Positive Number) = Positive Number (Positive Number)(Negative Number) = Negative Number (Negative Number)(Positive Number) = Negative Number (Negative Number)(Negative Number) = Positive Number

EXAMPLES: Simplify the expressions. (8)(9) = 72

(3)(-4) = -12

(-7)(6) = -42

(-2)(-5) = 10

PRACTICE: Simplify the expressions. (4)(9) =

(3)(-8) =

(-12)(7) =

(-15)(-11) =

(-4)(16) =

(20)(-3) =

(-25)(-6) =

(9)(-6) =

Integer Rules for Addition and Subtraction First, identify each number in the expression as positive or negative.

If the signs of the two numbers are the same  add them and assign the same sign. If the signs of the two numbers are different  subtract the two numbers and assign the sign of the number with the greater absolute value.

EXAMPLES: Simplify the expressions. 4 + 9 = 13

-5 – 7 = -12

12 – 16 = -4

-5 + 11 = 6

23 + 18 = 41

-40 – 9 = -49

15 – 29 = -14

-37 + 52 = 15

PRACTICE: Simplify the expressions. 2+9=

-8 – 9 =

20 – 25 =

-3 + 6 =

13 + 0 =

-12 – 0 =

-13 – 5 =

18 – 25 =

-20 – 30 =

53 – 67 =

-89 – 13 =

-31 + 17 =

-121 + 22 =

43 – 32 =

16 – 43 =

-12 – 29 =

Order of Operations/Evaluating Expressions Rules

Order of Operations 1.) 2.) 3.) 4.)

Parenthesis Exponents Multiplication and Division from Left to Right Addition and Subtraction from Left to Right

Examples: Evaluate the expressions below for the given value(s). 5x – 11

when x = -6

-18 – 6y

5(-6) – 11

-18 – 6(-9)

-30 – 11

-18 + 54

-41

36

4x2 – 5x + 11

when x = -3

-5(t – 1)2 + 7 when t = 6

4(-3)2 – 5(-3) + 11

-5((6) – 1)2 + 7

4(9) – 5(-3) + 11

-5(5)2 + 7

36 + 15 + 11

-5(25) + 7

51 + 11

-125 + 7

62

-118

when x = -8 and y = -4

when y = -9

when x = -2 and y = 4

Practice: Evaluate the expressions below for the given value(s). 7x – 19

3m2 – 8m + 17

when x = -5

when m = -4

when a = -10 and y = -6

-24 – 8y

when y = -7

-4(p – 6)2 + 17 when p = 4

when x = -3 and y = 5

Adding/Subtracting Fractions When Adding or Subtracting Fractions, it is important to keep all fractional terms relative to their denominator. Fractions must have a common denominator before they can be added or subtracted from each other. Please look below to the following examples: Example 1 (addition)

=

Example 2 (subtraction)

=2

Try these: Simplify 1.

2.

3.

4.

When fractions do not have a relative denominator and one can’t be found after reducing, then the solver must CREATE a common denominator. The common denominator must be a multiple of both numbers! The least common multiple or LCM is the most efficient way of creating a common denominator, it makes the arithmetic much more manageable. Let’s look at the example below in regards to the subtraction of the two reduced, lowest term fractions. Example = ? Our first objective is to find the least common multiple of both denominators, to do this you can create a list adding the number to itself:

Multiples of 3: 3, 6, 9, 12, 15 etc. Multiples of 2: 2, 4, 6, 8, 10 etc.

Both numbers, 2 and 3, share a multiple of 6, this is the LCM and thus shall be our common denominator. We then must take each fraction and rewrite their equivalent fraction relative to our new denominator, to do this we must multiple the entire fraction, both numerator and denominator, by the number that creates our LCM. This is the equivalent of multiplying by one and will not change the value of our fraction.

So, , notice that

both reduce back to the original fractions.

Another way of creating a common denominator is not through the establishment of an LCM, but through a process called “The Rocking Horse Method.” This method will manufacture equivalent fractions while establishing a common denominator through a series of steps. Let’s say we use the following variables to represent two fractions being subtracted:

If

let’s use our example from above to further investigate this process.

then it is equal to

Example:

=

, the term “Rocking Horse Method” stems from the diagram that

is represented when the sum of the numerator, created by the products of the numerator and denominator of the opposite fraction, are put over the product of the denominators in the resulting denominator, displaying what appears to be the bottom of an old fashioned rocking horse or rocking chair.

=

Try these 1.

2.

3.

4.

5.

6.

Using the same rules, how would you handle the following equation?

+

=

Using “The Rocking Horse Method” all of our arithmetic rules stay

=

=

Try these:

1.

2.

Keep going to next page

3.

Translating into Algebra: In this area you will be translating words and phrases from English into Algebra. Listed below are the mathematical terms that you will need to use for these translations. You may already be familiar with some or all of these terms. *Important notes:

is not used for multiplication and

multiplication of three times a number is shown as

is not used for division. In Algebra,

Division of three and a number will be shown as a

fraction              

Product – multiply Difference – subtraction Sum – addition Quotient – division More than – addition Less than – subtraction Is less than – the less than symbol < Is greater than – the greater than symbol > Increased by – addition Decreased by – subtraction Quantity – this is not a specific operation, but it means to use parentheses Squared – to multiply a number by itself. It also means raised to the 2nd power. A Number – whenever this is written you can translate this into any variable that you choose Is the same as – equal to

Examples:

English:

Algebra:

1) The product of nine and a number.

2) The difference between a number and eight.

3) The sum of a eleven and a number

or

4) Two times the quantity of four decreased by six times a number.

5) Seven more than 6 times a number is the same as five times the sum of three and a number.

Examples: (continued)

6) Seven less than eight times a number.

7) Ten is less than nine times the quotient of one and a number

Practice:

Translate each of the following into Algebra.

1) Eight less than six times a number.

2) Eleven more than four times a number

3) Nine decreased by five times a number is the same as a number squared.

4) The product of sixteen and a number is greater than five more than a number.

5) A number decreased by three times the sum of a number and seven.

6) Nineteen times the difference between a number and six is the same as fourteen.

7) Eight times the quantity of twice a number increased by nine is the same as the number squared.

The Distributive Property: In this section you will be using the distributive property. You will also be practicing your integer rules and fraction rules. *Important notes: In Algebra 1 you should not have two signs between numbers. No longer will you be writing 8 + -3. Instead we want you to write it as 8 – 3. There should not be more than one sign in between terms. Using the distributive property means to use multiplication. You take the number that is multiplying the parentheses and use it to multiply everything inside the parentheses.

Examples: 1)

This means to multiply

and multiply

to get 15.

The answer is then

2)

3)

This next example has two sets of parentheses meaning that you are using the distributive property more than once in the same problem. You will also have like terms that need to be put together. Like terms are terms that have the same variable. 4) When you distribute with the

you get the following:

When you distribute with the

you get the following:

This then gives you

. This is not your answer though because you have like terms.

The two terms with the variables: The two constant terms (terms with no variable):

The final answer is then

are added together to get are added together to get 44.

Practice problems: 1)

3)

5)

7)

2)

4)

6)

hint: multiply the second set of parentheses by