Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

ORDER OF OPERATIONS

1) Work inside the grouping symbols such as parenthesis and brackets.

2) Evaluate the powers.

3) Do the multiplication and/or division in order from left to right.

4) Do the addition and/or subtraction in order from left to right.

Example: 24  4 + 8  2 – 18  (6 + 3) 

work inside grouping symbols = 24  4 + 8  2 – 18  9  evaluate the powers = 24  4 + 8  2 – 18  9  9 multiplication/division = 6 + 16  2  9

= 6 + 16  18 addition/subtraction = 22  18

= 4

Use order of operations to simplify. Show all steps in the space provided below each problem.

1) (4  1)  3 + 6 2) 5  (6  2) + 30  3 3)

36

 1

17  

8

4 

4) 8 + 2  9 – 15  3 5)

 4

11 

10

3

2

INTEGER OPERATIONS

Addition: Subtraction: Multiplication/Division:

If the signs are the same, Subtraction is the same as If you multiply or divide 2 add the absolute values and use adding the opposite. Therefore, numbers with the same sign, the given sign. change the sign of the value the answer is positive.

If the signs are different, being subtracted, then follow If you multiply or divide 2 subtract the absolute values and the rules for addition. numbers with different signs

choose the sign of the larger absolute the answer is negative.

value.

Examples: 1) -6 + -1 = -7 4) -14  15 5) 6  -3 = -18

2) 4 + -5 = -1 - 14 + (-15) 6) -10  -2 = 5

3) -1 + 4 = 3 = -29 7) -1  -9 = 9

Use the rules of integer operations to add, subtract, multiply and divide.



3²



3

PROPERTIES OF REAL NUMBERS

Commutative property of addition  a + b = b + a Commutative property of multiplication  a  b = b  a

1) How does commutative work with subtraction? ____________________________________________

2) Does commutative work for division? Why or why not? _____________________________________

______________________________________________________________________________________

Associative: In certain situations, grouping can be changed.

Associative property of addition  (a + b) + c = a + (b + c) Associative property of multiplication  (a  b)  c = a  (b  c) Distributive: A value is multiplied times each member of a sum or difference.

a(b + c) = ab + ac or a(b  c) = ab  ac

Identity: Applying an identity, doesn’t change the value of the number.

Additive Identity  a + 0 = a (Zero is the additive identity.)

Multiplicative Identity  a  1 = a (One is the identity for multiplication.) Inverse: An inverse changes a number into an identity.

Additive Inverse  a + (-a) = 0 (The additive inverse of a number is its opposite.) Multiplicative Inverse  ¹_1



 





a a (The multiplicative inverse of a number is its reciprocal.)

Match the sample to the property it illustrates.

Column A Column B Answer Column

1) 6  1 = 6 a) Commutative Property of Addition 1)______

2) y  5 = 5y b) Associative Property of Multiplication 2)______

3) m(3 + n) = 3m + mn c) Inverse for Addition 3)______

4) 9 + (41 + 37) = (9 + 41) + 37 d) Identity for Multiplication 4)______

5) 4 + (-4) = 0 e) Distributive Property 5)______

6) 6(3t) + (63)t f) Identity for Addition 6)______

7) a + 0 = 0 + a g) Associative Property of Addition 7)______

8) = 1 h) Commutative Property of Multiplication 8)______

j) Inverse for Multiplication



p q



 

q p



 



SIMPLIFYING ALGEBRAIC EXPRESSIONS

To simplify algebraic expressions, combine like terms, use order of operations, use rules for integer operations and real number properties to simplify the result.

Examples: 4xy – 7xy

= 4xy + (-7xy) = -3xy

Simplify the following algebraic expressions.

1) ax +4ax 2) 2(5 – x) – 4x 3) 5a – 3(7a – 6) 4) 6m – n – 4m – 1( 5n – m) 5) q pq 6 8

EVALUATING ALGEBRAIC EXPRESSIONS

“Evaluate” means replace the variables with given values and simplify completely.

Example: Find the value of Find the value of

when a = 5 and b = 3. xy - 2(yz) when x = -1, y = -2 and z = 3 = (-1)(-2) – 2(-2)(3)

= 2 + 12

= 14

Evaluate the following expressions when a = -2, b = 5, and c = -1. Show every step of work.

1) 10abc 2) 2b - a + 3c 3) 6b(a + c) 4) 5)



2a



2 5

 



10

27



17



3a



3x 4





34x3xyxy 12x3xy1xy 12x2xy



2xy 6y

2xy 23y

 x 32y

2y

 x

3 

2aa 3





 2a a

 

 2a a









 2a3aa²

 5aa²



4b² ac

SOLVING SIMPLE EQUATIONS

“Solving” equations means find the value that makes the sentence true.

•Make sure to keep the equation balanced by doing exactly the same thing to both sides of the equation.

•Ask “what do you want the answer to look like?”

•Remove the numbers that are in the way of the answer one step at a time.

•Use the opposite operation to remove numbers.

•Make sure everything is simplified.

•Complete addition or subtraction steps before the multiplication or division steps.

•Check to make sure your answer makes the original sentence true.

Examples:

1) 5 + x = -2 You want x = a number for the answer so you must remove the 5.

5 + x = -2

-5 -5 Subtract 5 from both sides.

x = -7 Simplify

Check  does 5 + -7 equal -2?

2) 4a – 7 = 1 You want a = a number for the answer so you must remove the 7 (1^st) and the 4.

4a – 7 = 1

+ 7 +7 Add 7 to both sides.

4a = 8

Divide both sides by 4 a = 2 Simplify

Check  does 42 – 7 equal 1?

3)

7 3

5

You want y = a number for the answer so you must remove the 7 and then the 5.

7 7 3 7

5

Subtract 7 from both sides.

4

5

Simplify.

5 4 5

5

Multiply both sides by 5.



4a 4  8

4



y

573

y

20

Check  is -20 divided by 5 plus 7 equal to 3?

Solve the following equations. Show all steps in the space provided and check your work.

1) x – 5 = -1 2) 8y = 56 3) 4) 3 = 4 + x

5) 2 + 3x = -4 6) 8a – 2 = 9 7) 3 – 2c = -5 8)

SOLVING EQUATIONS USING THE DISTRIBUTIVE PROPERTY

Example:

2(x + 3) = 10 Check

2x +6 = 10 Distribute 2 times x and 2 times 3 2(x + 3) = 10 original problem

2x + 6 – 6 = 10 – 6 Subtract 6 from both sides 2(2 + 3) = 10 replace x with the answer 2x = 4 Simplify 2(5) = 10 simplify

Divide both sides by 2 10 = 10

x = 2 Simplify

Solve the following equations and check your answers. Show all steps of work.

1) 4(y + 3) = -4 2) 2(x – 5) = 12 3) 25 = 5(x + 1) 4)

3 1

2 2



 

 y



2 2x4

2



c

60.01



4x 5  3

TRANSLATING WORDS AND SOLVING WORD PROBLEMS

Operation Common Words

Addition more than, sum, increased by, total

Subtraction less than, difference, decreased by, fewer than

Multiplication product, times

Division quotient, divided by

Equals is, is equal to, equals

Example: Translate the words using math symbols. Write an equation and solve. Make sure you answer the question asked and that your answer is reasonable.

1) Seven more than twice a number is 9. Find the number.

•7 more than means +7

•twice a number means multiply a number by 2, or 2n •is 9 means equals 9

7 + 2n = 9 (the equation!)

7 – 7 + 2n = 9 – 7 begin solving by subtracting 7 from both sides 2n = 2 simplify, next you will divide both sides by 2

n = 1 Does this answer make sense? Seven more than twice one is nine?

The answer is 1.

2) Five less than twice a number is 7. Find the number.

•5 less than means 5 is subtracted from something • twice a number means multiply a number by 2, or 2n •is 7 means equals 7

2n – 5 = 7 (the equation!)

2n – 5 + 5 = 7 + 5 begin solving by adding 5 to both sides 2n = 12 simplify, next you will divide both sides by 6

n = 6 Does the answer make sense? Five less than twice 6 is seven?

The answer is 6.

Translate into symbols, solve the equation, and state the correct answer.

1) Sixteen is the sum of three times 2) The product of 6 and 3) The difference between ten a number and one. Find the number. half a number is 10. and four times a number is

Find the number. three. Find the number.

(The answer could be a fraction.)

PERCENT OF CHANGE

The amount of percentage a number increases or decreases from an original number is called the percent of change.

The percent of change increases if the original number is smaller than the new number.

The percent of change decreases if the original number is larger than the new number.

To find the percent of change, use the ratio:

Ex: 1) Find the percent of change if the rent on your apartment increases from $500 to $550 per month.

% increase =

Ex: 2) Find the percent of change if a shirt originally costs $65 and is on sale for $40.

% decrease =

Find the percent of change requested in each problem. Show the ratio you create and work to finish the problem.

1) new number is 15 2) new number is 150 3) original number is 80 original number is 75 original number is 20 new number is 45

RATIOS AND PROPORTIONS

•2 to 3 means

3

2

18

9

2

1

5

1

30

6



amount of increase or decrease original number



50 500 1

1010%

amount of increase original number

=

amount of decrease original number



25

650.3846or38.46%

=

**If a problem uses 2 units for the same type of measure, convert one to the other e.g., change weeks into days, pounds into ounces, etc. to match the way it is written elsewhere in the same problem.

A proportion is an equation with two equal ratios. To solve a proportion, cross multiply and simplify.

Examples: 1) 2)

Solving problems with proportions:

Example: It takes a 4.5 lb. chicken 1.5 hours to roast. How long will it take a 3 lb. chicken to roast?

1) Set up a proportion

2) Cross multiply 3) Solve for x.

4) Answer the question It will take one hour to roast a 3 lb. chicken.

Complete the following ratio/proportion problems. Show all steps.

Write each ratio in simplest form:

1) 13 out of 52 2) 9 : 72 3) 150 to 25 4) 2 days to 3 weeks

Solve the following proportions.

5) 6) 7)

8) If 5 apples cost $1.85, how much will 2 apples cost?



8 4  x

5



85x4 404 x x10



2y3.0 y1.5



2 50.6

y

lbs. of chicken hours





4.5x 3 1.5 4.5x4.5 x1



4.5 1.5 3

x



3

5  15

x



12 a  5

20



8 y 5

3

S.Fischbein

9

GRAPHING ORDERED PAIRS

Ordered pairs, written (x, y) where x represents the horizontal move and y represents the vertical move, represent points on the Cartesian coordinate plane.

(5, -1) means move from the origin: (-2, 0) means move from the origin:

   

horizonally vertically horizontally do not move

to the right 5 down 1 to the left 2

Find the points by writing the correct letter associated with the points.

(1, 3) ________

(0, -4) ________

(-3, -2) ________

(-2, 5) ________

(5, 0) ________

(3, 1) ________

(-2, 0) 

 (5, -1)

x-axis y -axis

C G D

E F A

B

x-axis y -axis

S.Fischbein

10

GRAPHING LINES BY PLOTTING POINTS

You will need to figure out the coordinates of at least 3 points on the given line. Plot those points on an x-y-axis, connect them, and extend the line in both directions.

Every line except horizontal or vertical lines, eventually will run through every value of x and every value of y. So you should choose any x value and figure out the y value that goes with it to complete an ordered pair. Just plug the x of your choice into the equation and solve the equation for y.

Example: Graph 2x + y = 6

Choose 1 for x. 2(1) + y = 6  y = 4  (1, 4) is a point on the line Choose 3 for x. 2(3) + y = 6  y = 0  (3, 0) is a point on the line Choose 4 for x. 2(4) + y = 6  y = -2  (4, -2) is a point on a line If you prefer, you could choose values for y.

Choose 2 for y. 2x + 2 = 6

-2 -2 solve the equation by subtracting 2 from each side 2x = 4

x = 2  (2, 2) is a point on the line.

Select at least 3 ordered pairs and graph the following lines. Show work below each axis.

1) x + y = 5 2) y = 2x – 4 3) y x

2



1



x-axis y -axis

It is common to display your ordered pairs on a T-chart:

x y 1 4 3 0 4 -2 2 2

x-axis y -axis

x-axis y -axis

x-axis y -axis

S.Fischbein

11

SLOPE

The slope of a line is

To find the slope of a line without a grid to count spaces:

1) choose any two points on the line 2) call one point P₁ and the other point P₂ 3) write down the coordinates of the points

4) in the formula, plug in the numbers for x_1,x_2,y₁

, y

Find the slopes of the following lines.

x-axis y -axis



m y₂y₁

x₂x₁ 0 3

 

2 2

 

the rise of the line the run of the line

the change in y values the change in x values

=

x y

rise



y₂y₁

run



x₂x₁



P2x2,y2



P1x1,y1

You can find the rise and the run by counting spaces.

The symbol for slope is “m” which comes from the French verb monter meaning to climb.

slope

m rise

run

change in y change in x



y₂y₁



x₂x₁

= = = =

x y



P2x2,y2



P1x1,y1

(2, 0)

(-2, -3)

x y



P2x2,y2

P1x1,y1

(-3, 2)

x y



P2x2,y2



P1x1,y1

(4, 8)

(3, 2)

x-axis y -axis

1)

2)

3)

4)

VOCABULARY

Please write a brief definition of the following math terms. Select the definition that relates to mathematics.

1. square root

2. system of equations

3. function

4. quadratic equation

5. factor (noun)

6. factor (verb)

x-axis y -axis

7. rational

8. irrational

this line

this line

this line