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ALGEBRA 1 SUMMER PACKET
SHERWOOD HIGH SCHOOL
SUMMER 2013
General Information

The purpose of this packet is to review important skills that you have learned
which are necessary to succeed in Algebra 1:

Calculating Decimals

Calculating Fractions

Evaluating Expressions

Plotting points in the Coordinate Plane

Graphing Equations

Order of Operations

Operations with Signed Numbers (Integers)

Solving Equations

Solving Word Problems

Rounding Numbers
Getting Help:

When completing the problems in this packet, you should show your work and
do your best to complete the problems correctly.

During the summer, the answer key will be posted on the Sherwood website.
Enjoy your summer! We look forward to seeing you in the fall.
(If you have any questions, please contact the math office at (301) 924 – 3253.)
Arithmetic with Decimals
Adding Decimals
Example 1: Add 4.36 and 2.89
1
1 1
Step1 4.36
+2.89
5
Step 2 4.36
+2.89
25
Line up the decimal
points. Add the
hundredths.
Example 2: Add:
1 1
Add the tenths.
Step 3 4.36
+2.89
7.25
Add the ones.
Place a decimal point in the sum in
line with those decimals above it.
4.3 + 12.75 + 0.093
11
Step 1
4.3
12.75
+ 0.093
Step 2 4.300
12.750
+ 0.093
Line up the decimal points.
Place zeros in the
missing digits for
place holders.
Step 3 4.300
12.750
+ 0.093
17.143
Add the the thousandths.
Add the hundredths and ones.
Place a decimal point in the sum in
line with those decimals above it.
Subtracting Decimals
Example 1: 4.72 – 2.65
4.72
- 2.65
2.07
Line up the decimal points.
Subtract hundredths, tenths, and ones.
Place a decimal point in the answer in line with those
decimals above it.
Example 2: 7.6 – 4.362
7.600
- 4.362
3.238
Line up decimal points.
Place zeros in the missing digits for place holders.
Subtract.
Multiplying Decimals
Example 1:
0.90 x 1.2
0.90 ← 2 decimal places
x 1.2 ← 1 decimal place
180
0900
1.080 ← 3 decimal places
Example 2:
.32
x .004
128
0000
.00128
To multiply decimals, multiply as with
whole numbers. The number of decimal
places in the product is the total number
of decimal places in the factors.
.32 x .004
← 2 decimal places
← 3 decimal places
When you multiply decimals, sometimes
you need to write one or more zeros in
the product to equal the total number of
decimal places in the factors.
← 5 decimal places
Dividing decimals
Example 1:
2.48 ÷ 2
1.24
2 2.48
Rewrite 2.48 ÷ 2 as shown. Place the
-2
04
-4
08
-8
0
decimal point in the quotient directly
above the decimal point in the dividend.
Divide as with whole numbers.
Add or subtract the following decimals.
1.
4.2 + 3.7 =
2.
0.09 + 3.6 =
3.
0.72 + 3.921 + 7.5 =
4.
7.41 – 5.63 =
5.
9.4 – 7.25 =
6. 17.365 – 12.19 =
Multiply or divide the following decimals.
7. 1.75 x 2.6 =
8. 0.53 x 0.008 =
10. 45.6 ÷ 8 =
11. 1.3174 ÷ 2 =
9. 2.31 x 0.002 =
12. 2826.6 ÷ 42 =
Arithmetic with Fractions
Adding and Subtracting Fractions
Example 1:
3
1
+
8
8
If the denominators are the same, then add the numerators and
keep the same denominator. Then simplify the answer & reduce, if
possible.
3
1
4
1
+ =
=
8
8
8
2
Example 2:
5
3
–
4
6
The denominators are not the same. You must rename the
fractions to have the same denominator by finding the least
common denominator.
Step 1
Step 2
5 10
=
6 12
5
=
6 12
–
3
=
4
12
–
3
9
=
4 12
Step 3
5 10
=
6 12
–
3
9
=
4 12
=
Find the least common
denominator.
Rename the fractions.
1
12
Subtract the
numerators.
Multiplying Fractions
Example 3:
2
4
x
3
5
Step 1
Multiply the numerators.
2
4 8
x =
3
5
Step 2
Multiply the denominators.
2
4
x
=
3
5
15
Example 2: 5 x
Step 1
3
4
Rename the whole number
5
3
x
1
4
=
15
3
= 3
4
4
using a fraction. Then follow
the example above. Finally
simplify answer by writing
it as a mixed numeral.
Dividing Fractions
To divide fractions, multiply the first fraction (dividend) by the reciprocal of
the second fraction (divisor).
Example 1: 2/3 ÷ 4/5
2
5
10
5
x
=
=
3
4
12
6
Example 2: 2/3 ÷ 4
2
1
2
1
x
=
=
3
4
12
6
Perform the indicated operation with the following fractions. Don’t forget to
simplify (reduce) your final answer to lowest terms!
13.
5 1
–
=
6 6
14.
7 5
+ =
8 8
15.
3 1
+
=
8 4
16.
1 1
– =
2 3
17.
4 3
– =
5 4
18.
3 16
x
=
8 3
19.
2 7
x =
3 8
20.
6x
4
=
3
21.
4
x 30 =
5
22.
7 5
÷ =
3 6
23.
9
3
÷ =
2
8
24.
3
÷2=
4
Evaluating Expressions
Example
Evaluate the following expression when x = 5
Rewrite the expression substituting 5 for the x and simplify.
a.
b.
c.
d.
e.
5x =
-2x =
x + 25 =
5x - 15 =
3x + 4 =
5(5)= 25
-2(5) = -10
5 + 25 = 30
5(5) – 15 = 25 – 15 = 10
3(5) + 4 = 19
Evaluate each expression given that:
x=5
y = -4
z=6
1.
3x
5.
y+4
2.
2x2
6.
5z – 6
3.
3x2 + y
7.
xy + z
4.
2 (x + z) – y
8.
2x + 3y – z
9.
5x – (y + 2z)
13.
5z + (y – x)
10.
xy
2
14.
2x2 + 3
11.
x2 + y2 + z2
15.
4x + 2y – z
12.
2x( y + z)
16.
yz
2
Graphing
Points in a plane are named using 2 numbers, called a coordinate pair. The first
number is called the x-coordinate. The x-coordinate is positive if the point is to the
right of the origin and negative if the point is to the left of the origin. The second
number is called the y-coordinate. The y-coordinate is positive if the point is above
the origin and negative if the point is below the origin.
The x-y plane is divided into 4 quadrants (4 sections) as described below.


Quadrant 2
Quadrant 1













Quadrant 3

Quadrant 4

All points in Quadrant 1 has a positive x-coordinate and a positive y-coordinate (+ x, + y).
All points in Quadrant 2 has a negative x-coordinate and a positive y-coordinate (- x, + y).
All points in Quadrant 3 has a negative x-coordinate and a negative y-coordinate (- x, - y).
All points in Quadrant 4 has a positive x-coordinate and a negative y-coordinate (+ x, - y).
Plot each point on the graph below. Remember, coordinate pairs are labeled (x, y). Label
each point on the graph with the letter given.
1. A(3, 4)
2. B(4, 0)
Example: F(-6, 2)
3. C(-4, 2)
4. D(-3, -1)
5. E(0, 7)
+y







F


-x
        

















 
+x
Determine the coordinates for each point below:
Example. ( 2 , 3 )



6. (____, ____)













































9. (____, ____)










































12. (____, ____)

































13. (____, ____)



10. (____, ____)

11. (____, ____)



8. (____, ____)

7. (____, ____)






















Complete the following tables. Then graph the data on the grid provided.
Example: y = -2x - 3
X
Y
-3
3
-2
1
-1
-1
0
-3
Work:

x = -3
y = -2(-3) – 3 = 6 – 3 = 3
Therefore (x, y) = (-3, 3)
x = -2
y = -2(-2) – 3 = 4 – 3 = 1
Therefore (x, y) = (-2, 1)
x = -1
y = -2(-1) – 3 = 2 – 3 = -1
Therefore (x, y) = (-1, -1)
x=0
y = -2(0) – 3 = 0 – 3 = -3
Therefore (x, y) = (0, -3)
















14. y = x + 2

X
0
Y



1


2






15. y = 2x
X

Y

0

1


2

3






16. y = -x


X
-3
-1
Y








1


3





17. y = 2x - 3

X
Y

0


1




2







3


18. y =
1
x+1
2


X
Y


0

2








4

6
19. y 

3
x 1
2
X


Y

-2

0















2



20. y  
X
2
x 1
3


Y


-3

0
3







Order of Operations
To avoid having different results for the same problem, mathematicians have agreed on an order of
operations when simplifying expressions that contain multiple operations.
1. Perform any operation(s) inside grouping symbols. (Parentheses, brackets
above or below a fraction bar)
2. Simplify any term with exponents.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
One easy way to remember the order of operations process is to remember the acronym PEMDAS or
the old saying, “Please Excuse My Dear Aunt Sally.”
P - Perform operations in grouping symbols
E - Simplify exponents
M - Perform multiplication and division in order from left to right
D
A - Perform addition and subtraction in order from left to right
S
Example 1
Example 2
2 – 32 + (6 + 3 x 2)
2 – 32 + (6 + 6)
2 – 32 + 12
2 – 9 + 12
-7 + 12
=5
-7 + 4 + (23 – 8 ÷ -4)
-7 + 4 + ( 8 – 8 ÷ -4)
-7 + 4 + ( 8 - -2)
-7 + 4 + 10
-3 + 10
=7
Order of Operations
Evaluate each expression. Remember your order of operations process (PEMDAS).
1.
6+4–2∙3=
4.
(-2) ∙ 3 + 5 – 7 =
2.
15 ÷ 3 ∙ 5 – 4 =
5.
29 – 3 ∙ 9 + 4 =
3.
20 – 7 ∙ 4 =
6.
4∙9–9+7=
7.
50 – (17 + 8) =
16.
(12 – 4) ÷ 8 =
8.
12 ∙ 5 + 6 ÷ 6 =
17.
18 – 42 + 7 =
9.
3(2 + 7) – 9 ∙ 7 =
18.
3 + 8 ∙ 22 – 4 =
10.
16 ÷ 2 ∙ 5 ∙ 3 ÷ 6 =
19.
12 ÷ 3 - 6 ∙ 2 – 8 ÷ 4 =
11.
10 ∙ (3 – 62) + 8 ÷ 2 =
20.
6.9 – 3.2 ∙ (10 ÷ 5) =
12.
32 ÷ [16 ÷ (8 ÷ 2)] =
21.
[10 + (2 ∙ 8)] ÷ 2 =
13.
180 ÷ [2 + (12 ÷ 3)] =
22.
¼(3 ∙ 8) + 2 ∙ (-12) =
14.
5 + [30 – (8 – 1)2] =
11 - 22
23.
3[10 – (27 ÷ 9)] =
4–7
15.
5(14 – 39 ÷ 3) + 4 ∙ 1/4 =
24.
[8 ∙ 2 – (3 + 9)] + [8 – 2 ∙ 3] =
Operations with Signed Numbers
Adding and Subtracting Signed Numbers
Adding Signed Numbers
Like Signs
Different Signs
Add the numbers & carry the sign
Subtract the numbers & carry the sign of the
larger number
(+)+(+)=+
( – ) + (– ) = –
( +3 ) + ( +4 ) = +7
(– 2 ) + (– 3 ) = ( – 5 )
( + ) + (– ) = ?
( +3 ) + (–2 ) = +1
(–)+(+)=?
( –5 ) + ( + 3 ) = –2
Subtracting Signed Numbers
Don’t subtract! Change the problem to addition and change the sign of the second number.
Then use the addition rules.
( +9 ) – ( +12 ) = ( +9 ) + ( – 12)
( +4 ) – (–3 ) = ( +4 ) + ( +3 )
( – 5 ) – ( +3 ) = ( – 5 ) + ( – 3 )
( –1 ) – (– 5 ) = ( –1 ) + (+5)
Simplify.
1.
9 + -4 =
7.
20 – - 6 =
2.
-8 + 7 =
8.
7 – 10 =
3.
-14 – 6 =
9.
-6 – -7 =
4.
-30 + -9 =
10.
5–9=
5.
14 – 20 =
11.
-8 – 7 =
6.
-2 + 11 =
12.
1 – -12 =
Multiplying And Dividing Signed Numbers
If the signs are the same,
If the signs are different,
the answer is positive
the answer is is negative
Like Signs
Different Signs
(+ ) ( + ) = +
( +3 ) ( +4 ) = +12
(+)(–)=–
( +2 ) ( – 3 ) = – 6
(– ) (– ) = +
( – 5 ) ( – 3 ) = + 15
(–)(+)=–
( –7 ) ( +1 ) = –7
(+ ) / ( + ) = +
( +3 ) / ( +4 ) = +12
(+)/(–)=–
( +2 ) / ( – 3 ) = – 6
(+ ) / ( + ) = +
( +3 ) / ( +4 ) = +12
(–)/(+)=–
( –7 ) / ( +1 ) = –7
Simplify.
1. (-5)(-3) =
7.
-7 =
-1
2. -6 =
8.
(3) (-4) =
2
3. (2)(4) =
9.
8 =
-4
4. -12 =
10.
(-2)(7) =
11.
-20 =
-4
5. (-1)(-5) =
-1
6.
-16 =
8
12.
(2)(-5) =
Solving Equations
To solve an equation means to find the value of the variable. We solve equations by isolating
the variable using opposite operations.
Opposite Operations:
Addition (+) & Subtraction (–)
Multiplication (x) & Division (  )
Example:
Solve.
3x – 2 = 10
+2
+2
Isolate 3x by adding 2 to each side.
3x
3
=
12
3
Simplify
Isolate x by dividing each side by 3.
x
=
4
Simplify
Check your answer.
3 (4) – 2 = 10
12 – 2 = 10
10 = 10
Please remember…
to do the same step on
each side of the equation.
Always check your
work by substitution!
Substitute the value in for the variable.
Simplify
Is the equation true?
If yes, you solved it correctly!
Try These:
Solve each equation.
1.
x+3 = 5
6.
w – 4 = 10
2.
c – 5 = -8
7.
3p = 9
3.
-7k = 14
8.
–x = -17
4.
h
= 5
3
9.
m
= 7
8
5.
4
d = 12
5
10.
3
j= 6
8
11.
2x – 5 = 11
16.
4n + 1 = 9
12.
5 j – 3 = 12
17.
2x + 11 = 9
13.
-3x + 4 = - 8
18.
-6x + 3 = - 9
14.
f
+ 10 = 15
3
19.
a
– 4 = 2
7
15.
b4
= 5
2
20.
x6
= -3
5
Use substitution to determine whether the solution is correct.
21.
4x – 5 = 7
x=3
23.
-2x + 5 = 13
x=4
22.
6–x=8
x=2
24.
1–x=9
x = -8
Word Problems
Math Dictionary
+
–
X
÷
=
More than
Increased
Greater than
Sum
Older
Less than
Shorter
Decreased
Difference
Reduced
Twice
Times
Of
Product
Double
Divided
Each part
Half
Quotient
One-third
Same as
Total
Is
Equal to
Sum
Use the math dictionary to help you translate words to algebraic symbols.
Matching – Put the letter of the algebraic expression that best matches the phrase.
_____ 1.
_____ 2.
_____ 3.
______4.
_____ 5.
two more than a number
two less than a number
half of a number
twice a number
two decreased by a number
a.
b.
c.
d.
e.
2x
x+ 2
2–x
x–2
x
2
Careful! Pay attention to subtraction. The order makes a difference.
Translate to an algebraic expression, then reread to check!
Use an Equations to Solve Word Problems:
 Translate each word problem into an algebraic equation
Using x for the unknown, write a “let x =” for each unknown, and then write an equation to
represent the relationship in the word problem
 Solve the equation
 Substitute the value for x into the let statement(s) to answer the question asked in the word
problem
 Check your answer by substituting the solution for the unknown in the word problem and for x in
the equation to determine if the solution is correct.
For Example:
Suzi is 6 years older than Marianne. The sum of their ages is 24. How old is each girl?
1. What are you asked to find? Let variables represent what you are asked to find.
How old is each girl?
Let x = Marianne’s age
Let x + 6 = Suzi’s age
2. Write an equation to represent the relationship in the problem, the sum of their ages.
x + (x + 6) = 24
3.
Solve the equation for the unknown.
x + (x + 6) = 24
2x + 6 = 24
-6 -6
2x
= 18
2
2
x = 9
Marianne = x = 9 years old
Suzi = x + 6 = 9 + 6 = 15 years old
Word Problem Practice Set
1. Seven less than a number is 3. What is the number?
2. Janet weighs 20 pounds more than Anna. If the sum of their weights is 250 pounds, how much
does each girl weigh?
3. Three times a number is 2. What is the number?
4. The quotient of a number and 4 is 12. What is the number?
5. Sarah drove 3 hours more than Michael on their trip to Texas. If the trip took 37 hours, how many
hours did Sarah and Michael each drive?
6. The are 125 more 8th graders at Rosa Parks than at Farquhar. If there is a total of 515 8th
graders at the two schools, how many 8th graders are there at each school?
7. The school lunch prices are changing next year. The cost of a hot lunch will increase $0.45 from
the current price. If the next year’s price is $2.60, what did a hot lunch cost this year?
8. Matt spent twice as much as Jim at the mall. If they spent $105 together, how much did each boy
spend?
9. Roberto worked four hours longer than Laura studying for his final exam. If they studied for a
total of 20 hours all together, how long did each student study?
10. Three-fourths of the student body attended the pep rally. If there were 1230 students at the pep
rally, how many students are there in all?
Rounding numbers
Step 1: Underline the place value in which you want to round.
Step 2: Look at the number to the right of that place value.
Step 3: If the number to the right of the place value you want to round is
less than 5, keep the number the same and drop all other numbers.
If the number to the right of the place value you want to round is 5
or more, round up and drop the rest of the numbers.
Example: Round the following numbers to the tenths place.
1. 23.1296
Tenths
23.1
2.
64.2685
64.3
the digit to the right of the tenths place, 6, indicates to
round up
3.
83.9721
84
the digit to the right of the tenths place, 7, indicates to
round up which makes the 9 a zero (10) and the 3 a 4.
the digit to the right of the tenths place, 2, indicates to
drop off
Round the following numbers to the tenths place.
1. 18.6231
_____________
6.
0.2658
______________
2.
25.0543
_____________
7.
100.9158
______________
3.
3.9215
_____________
8.
19.6816
______________
4.
36.9913
_____________
9.
17.1483
______________
_____________
10.
0. 9601
______________
5.
15.9199