Download Incoming 8th grade summer packet 2013

Name: ___________________________________
This packet is designed to help you retain the information you learned
in 7th grade and realize what skills are essential for you to have as you
enter 8th grade.
 The most important topics to review for next year are INTEGERS
(know your rules!) and ALGEBRA.
Use websites and online games to help you strengthen your skills in
these areas! (ex. www.math.com or www.mathguide.com/lessons/Integers.html)

Have a Wonderful Summer!
Your eighth grade teachers look forward to working with you next year.
Topic: Integers
Examples:
Addition
Same signs:
Add & keep sign
+6 + +5 = +11
-8 + -2 = -10
Subtraction
+10
-5
Different signs:
Subtract & take sign of
larger value
+9 + -5 = +4
-6 + +1 = -5
Multiplication
Keep–Change-Opposite
- - 8 = +10 + +8 = 18
Division
Same signs:
Positive product
(+7) (+8) = +56
(-2) (-6) = +12
Same signs:
Positive quotient
+42  +6 = +7
-24  -8 = +3
Different signs:
Negative product
(+3) (-9) = - 27
(-5) (+4) = - 20
Different signs:
Negative quotient
+56  -7 = - 8
-50  +2 = - 25
– +12 = -5 + -12 = -17
-20
- -8 = -20 + +8 = - 12
Recall the order of operations:
1 – Parentheses (or grouping symbols)
2 - Exponents
3 - Multiplication / Division (left to right)
4 - Addition/Subtraction (left to right)
Find each answer.
1. - 12 + - 7 = ______
2. - 25 + 18 = ____
3. 2 + - 25 = ____
4. - 28 – - 8 = _____
5. 11 – - 5 = ____
6. - 21 – 4 = ____
7. (- 9) (- 8) = _____
8. ( 2 ) ( - 12) = _____
9. - 35  - 7 = _____
10. - 48  + 8 = _____
11. (- 2) ( + 6) (- 5) = _____
12. - 30 + 24  6 . - 2 = _____
13. 16  4 + 2 . - 8 = _____
14. - 3 (1 – 8) + 23 = _____
** Practice your INTEGER RULES using websites and on-line games!! You really MUST know these!! **
page 1
Scientific Notation
A number written as a number that is at least 1, but less than 10 multiplied by a power of 10.
Ex.
7.16 105  716,000 in standard form
9.2 103  .0092 in standard form
Write each of the following in standard form.
Answers:
15. 8.2 105
15. ___________________
16. 2.45 104
16. ___________________
17. 7.28103
17. ___________________
18. 9.1102
18. ___________________
Write each of the following in scientific notation.
19. 25,900
19. ___________________
20. .039
20. ___________________
21. .0007
21. ___________________
22. 3,207,000,000
22. ___________________
Rounding
Rounding a numerical value means replacing it by another value that is approximately
equal but has a shorter, simpler, or more explicit representation. We round numbers to a
specific place value. UNDERLINE the place value you're rounding to. Then check the
place to the right and decide whether to keep it the same or round up.
Round each of the following decimals to the nearest tenth. Write your answer on the blank.
23. 87.46
_____________
24. 8.862
_____________
25. 13.4395
_____________
26. 654.839
_____________
27. 23.648
_____________
28. 32.971
_____________
Round each of the following decimals to the nearest hundredth. Write your answer on the blank.
29. 26.879
_____________
30. 429.76492
_____________
31. 675.495132
_____________
32. 3.8961
_____________
page 2
Topic: Square roots
The square of 5 is 25.
5 ∙ 5 = 52 = 25
5
5
The square root of 25 is 5 because 5∙5 = 25 OR 52 = 25
The square of an integer is called a perfect square.
The square root of a perfect square is an integer.
Complete the T –Chart below. You must know these perfect squares.
Square Root of a
Perfect Square
Integer Value
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
page 3
To approximate square roots that are not perfect squares, determine the perfect squares which the radicand
lies between.
Example: To approximate 32 , think of the perfect squares 32 is between (25 and 36).
25 <
5
<
32 <
36
<
32
6 therefore
32 is between 5 and 6.
(Because 32 is closer to 36 than 25, we may estimate that
32 ≈ 5.7.)
Find each square root. DO NOT USE A CALCULATOR.
1.
16 =
4.
225 =
____
____
2.
36 =
5.
169 =
____
____
3.
121 =
6.
81 =
____
____
State which integers the square root lies between. Show your work. DO NOT USE A CALCULATOR.
Ex. :
81 <
85 <
100
therefore
85 is between the integers 9 and 10 .
7.
_______ <
37 < _______
therefore
37 is between the integers ______ and ______.
8.
_______ <
26 < _______
therefore
26 is between the integers ______ and ______.
9. _______ <
61 < _______
therefore
10. _______ <
119 < _______
11. _______ <
68 < _______
therefore
12. _______ <
50 < _______
therefore
50 is between the integers ______ and ______.
13. _______ <
40 < _______
therefore
40 is between the integers ______ and ______.
14. _______ <
18 < _______
therefore
18 is between the integers ______ and ______.
15. _______ <
32 < _______
therefore
32 is between the integers ______ and ______.
page 4
therefore
61 is between the integers ______ and ______.
119 is between the integers ______ and ______.
68 is between the integers ______ and ______.
Topic: Polynomials
page 4
Multiplying & Dividing Monomials (powers with the same base)
Recall that exponents are used to show repeated multiplication. You can use the definition of exponent
to help you understand how to multiply or divide powers with the same base.
Examples:
Multiplication:
1. 23
24  (2 2 2) (2 2 2 2)  27
(Note: the values are: 8
2. x3
16, which is equal to 128, which is 2 7 )
x 2  ( x x x) ( x x)  x 5
Product of Powers rule: You can multiply powers with the same base by adding their exponents.
Division:
1.
26
2222 2 2

 2 2 2 2  24
2
2
2 2
2.
x10
xxx x x x x x x x

 x x x  x3
7
x
x x x x x x x
Quotient of Powers rule: You can divide powers with the same base by subtracting their exponents.
Using the power rules above, simplify each expression. Write your answer using exponents.
Answers:
1. 72 75 
2. 38 36 
1. ___________
6 
3. 64
4. x 2
x6 
2. ___________
3. ___________
4. ___________
5. n
6
n

4
6. y
3
y
6

5. ___________
6. ___________
7. m8
m7 
8. x 7
x 
7. ___________
8. ___________
9.
38
33

10.
1010
106

9. ___________
10. __________
11. __________
11.
x11
x7

12.
47
4

13. __________
14. __________
Find the value of n in each equation.
13. 54 5n  512
12. __________
14.
127
12n
 125
page 5
Combining like terms and applying the Distributive Property
In algebraic expressions, like terms are terms that contain the same variables raised to the same power.
Only the coefficients of like terms may be different.
In order to combine like terms, we add or subtract the numerical coefficients
of the like terms using the Distributive Property: ax + bx = (a + b)x .
Examples:
1. 2x + 9x = (2 + 9) x = 11x
2. 12y - 7y = (12 – 7) y = 5y
3. 5x + 8 - 2x + 7 = 3x + 15
Here, the like terms are: 5x and -2x = 3x
and: 8 + 7 = 15
The Distributive Property of multiplication over addition/subtraction is frequently used in Algebra:
Examples:
1. 7 (2x + 9) = 7 2x + 7 9 = 14x + 63
2. 4 (6 – 5x) = 4 (6) - 4(5x) = 24 - 20x
Answers:
Simplify each expression by combining like terms.
15. 8y + 2y
15. ______________
16. 10 – 6y + 4y + 9 =
16. ______________
17. 3x + 7 – 2x =
17. ______________
18. 8n – 7y – 12n + 5 – 3y =
18. ______________
Apply the distributive property and write your answer in simplest form.
19. 7 (x – 4) =
19. ______________
20. 5 (4n – 3) =
20. ______________
21. - 6 (3y + 5) =
21. ______________
22. - 4 (8 – 9x) =
22. ______________
23. 8 (3n + 7) – 10n =
23. ______________
24. – 4 (5 + 7y) + 6 (2y – 9) =
24. ______________
page 6
Topic: Algebra
Solving equations by using the Addition, Subtraction or Multiplication Property of Equality.
Check the solution.
x
 5  9
2
 5  5
2 x
 4 2
1 2
x  8
Ex 1:
Check:
Ex 2: 7x – 6 = 11x – 14
7 x  6  11 x  14
 7x
 7x
 6  4 x  14
+ 14
+ 14
8
4x

x
 5  9
2
4
4
2 = x
8
 5 9
2
4 + 5 = 9
9= 9
Check:
7x - 6 = 11x - 14
7(2) - 6 = 11 ( 2) - 14
14 - 6 = 22 - 14
8 = 8
Translate and evaluate the following equations.
Ex 3: The product of 4 and a number is 28.
Example: 2
4 n  28
Ex 4. The quotient of a number and 3 is 15.
n
 15
3
n  45
4n 28

4
4
n 7
Key words:
Addition: sum, more than , increased by
Subtraction: difference, less than
Multiplication:
product
quotient
Information: Hints,
topics, definitions.
Any kind of Division:
helpful information.
Solve the following equations. Show your work and check your solution.
1. 2 x  5  17
2. x  9  12
3. 5x + 8 = - 12
3
Check:
Check:
Check:
*** Get more practice SOLVING 2-STEP EQUATIONS using websites and on-line games!! ** page 7
(Apply the distributive property first.)
4.  4 x  8  32
5.
Check:
7. 8x – 5 = 6x + 7
Check:
x
 8  20
4
Check:
6. 2 (x – 7) = 8
Check:
8. 10x + 3 = 4x – 15
Check:
9. 6x – 3 = 8x – 11
Check:
Translate each sentence to an algebraic equation. Then use mental math to find the solution.
Equation
Solution
10. One-half of a number is -12.
_____________________
_____________
11. 6 more than 7 times a number is 41.
_____________________
_____________
12. 5 less than three times a number is 10.
_____________________
_____________
13. 16 increased by twice a number is – 24.
_____________________
_____________
page 8
Topic: Angle Relationships… and Algebra
Notation: m  means the " measure of angle "
 means congruent or equal in measure
Vertical Angles
Complementary Angles
Supplementary Angles
1
1
2
4
3
2
Angles that are opposite
each other across two
intersecting lines.
Two angles whose sum
is 90o.
m  1  m  3 and
m  1  m  2  900
m 2  m 4
1
2
Two angles whose sum
is 180o.
m  1  m  2  1800
FOR ALL STUDENTS:
State how the angle labeled x is related to the angle with the given measurement.
Find the value of x in each figure.
1)
xº
49º
3
1) _________________________
x = ___________
2) _________________________
2)
x = ___________
xº
62º
page 9
3)
3) _________________________
xº
123º
x = ___________
Find the measures of each of the angles indicated.
4)
4)
1
2
3
m  1 = ___________
m  2 = ___________
50º
m  3 = ___________
ENRICHMENT: Required for students entering the Enriched Algebra Course
5)
5)
Relationship _____________________
(11x - 4)°
2
(3x + 10)° →1
x = ________________
3
m  1 = ___________
m  2 = ___________
m  3 = ___________
page 10
Topic: Coordinate Geometry
Recall that we can graph coordinates (x, y) on the coordinate plane.
The x-axis is the horizontal axis, and the y-axis is the vertical axis.
To graph a line:
1) make a table of values for the equation
2) choose approx. 5 values for x and solve each for y
3) plot the coordinates on the coordinate plane
4) draw a line through the points (use arrows) and label the line with the equation
Example.
Graph the line y = 2x + 1
x
2x +1
y
(x, y)
-1
2(-1) +1
-1
(-1, -1)
0
2(0) +1
1
(0, 1)
1
2(1) +1
3
(1, 3)
2
2(2) +1
5
(2, 5)
y = 2x + 1
FOR ALL STUDENTS:
Complete the tables then graph the equations.
1. y = 3x - 1
x
3x - 1
y
(x, y)
y
(x, y)
-2
-1
0
1
3
2.
y = -2x+3
x
- 2x + 3
-2
-1
0
1
3
page 11
Complete the tables then graph the equations.
3.
y=½x
x
½x
y
(x, y)
4x – 5
y
(x, y)
-4
-3
0
2
6
4. y = 4x – 5
x
ENRICHMENT: Required for students entering the Enriched Algebra Course
5. Look for a pattern to help you fill in the missing values in the table.
Then try to write an equation that relates x and y.
a)
b)
x
y
x
y
-2
-11
-2
5
-1
-7
-1
3
0
-3
0
1
1
1
1
-1
2
2
3
3
___________________________
___________________________
page 12
Topic: Factors
Factors are numbers that are multiplied to get a product. Factors are numbers which a given number is
divisible by.
Ex. List the factors of 20: 1, 2, 4, 5, 10, 20
20 has 6 different factors
The Greatest Common Factor (GCF) of 2 or more numbers is the largest number that is a divisor of
the given numbers.
Ex. The GCF of 30 and 24 is 6.
Common factors or 30 and 24 are 1, 2, 3 and 6.
The Least Common Multiple (LCM) of 2 or more numbers is the smallest number which is divisible by
each of the given numbers. The LCM is the same as the least common denominator of 2 fractions.
Ex. The LCM of 6 and 10 is 30.
Note: Recall, you also had a method for finding the GCF and LCM by making a Venn diagram with the
prime factorization of the numbers.
List all the factors of each of the following numbers. Then count how many different factors the
number has.
Number of factors:
1. Factors of 28:
______________________________________________
_________
2. Factors of 40:
______________________________________________
_________
3. Factors of 45:
______________________________________________
_________
4. Factors of 36:
______________________________________________
_________
5. Factors of 100:
______________________________________________
_________
Find the Greatest Common Factor (GCF) for each pair of numbers. (The GCF is the greatest
divisor of the numbers.)
6. GCF of 14 and 21 = __________
9. GCF of 60 and 20 = __________
7. GCF of 24 and 16 = __________
10. GCF of 6 and 25 = __________
8. GCF of 45 and 30 = __________
11. GCF of 8 and 60 = __________
Find the Least Common Multiple (LCM) for each pair of numbers.
12. LCM of 4 and 6 = __________
15. LCM of 5 and 7 = __________
13. LCM of 9 and 12 = __________
16. LCM of 20 and 80 = __________
14. LCM of 6 and 10= __________
17. LCM of 40 and 32 = __________
18. Write the prime factorization of each number using exponents (remember… make a factor tree!).
40
72
45
700
page 13
Topic: Geometry
You should know the following formulas and be able to use them to find the area or perimeter of a
geometric figure.
Perimeter of a polygon = the sum of the sides
Rectangle:
Square:
Parallelogram:
Triangle:
Trapezoid:
Circle:
P = 2l + 2w
P = 4s
P = s1 + s2 + s3 + s4
P = s1 + s2 + s3
P = s1 + s2 + s3 + s4
Circumference = πd
A = lw
A = s2
A = bh
A = ½ bh
A = ½ (b1 + b2)h
A = πr2
Find the perimeter and area of each figure. Use π ≈ 3.14 and show your work.
1.
2.
5 cm
4.5 cm
7 cm
3.
4.
7 ½ in
8 cm
10 cm
5 in
4 in
20 cm
9 in
5.
6.
7 cm
(Write answers in terms of pi and as a decimal.)
5 ft
5 cm
4 cm
5 cm
11 cm
page 14