Download Algebra I - Chase Collegiate School

AlgebraI-Summer2016
Introduction – PLEASE READ!
The purpose of providing summer work is to keep your skills fresh and strengthen your base knowledge so we
can build on that foundation in Algebra I. All of the topics that are covered in this packet should have been
learned in previous courses. There is background information in some sections, and all sections have links to
websites which can provide more in-depth explanations of concepts that you may have forgotten. You are
expected to complete all of the problems in each section, either on the pages provided or on additional paper.
Please make sure that each problem is clearly identifiable and that your work is well-organized and neat.
It is important that you take this work seriously as it is expected that you will be comfortable with all of these
topics and be able to do additional problems like the problems in this packet when we begin school in
September. You are expected to utilize all available resources, including the links provided here, old books,
notes, additional websites, and/or assistance from your parents. But please remember, you need to make sure
that you know how to do the problems included in the packet. If after using the resources outlined here, you
still have questions, feel free to email me for assistance over the summer ([email protected]) or bring
your questions to class with you when we return. All of the material included in this packet will be assessed
early in the year, so please make sure you clearly understand all of the problems in this packet or ask for help.
Be sure to read all of the directions and information in each individual section to ensure that you get the most
out of the packet.
General Links
As mentioned above, review materials are provided in some sections, and every section has links to websites for
specific topics that you might want to review. Simply click on the link within this document and it will take you
to the associated web page. In addition, here are some links to the Pearson tools for an Algebra 1 book that you
might find helpful. While it is not the exact book that we will use at Chase, the material is the same and the site
does not require a log in. Use this page to review topics from the book:
http://www.phschool.com/webcodes10/index.cfm?area=view&wcprefix=aek&wcsuffix=0099
Use this page to view homework tutorials on specific topics that you might want to review:
http://www.phschool.com/webcodes10/index.cfm?fuseaction=home.gotoWebCode&wcprefix=aee&wcsuffix=0
775
The more you review over the summer, the better prepared you will be for success in the fall!
Name ________________________
PART 1
Find the perimeter.
1.
392 yards
156 yards
156 yards
238 yards
527 yards
2.
75 mi
51 mi
74 mi
46 mi
82 mi
48 mi
Find the area.
3. A = LW
7 cm.
10 cm.
4. A =
1
bh
2
4 in.
12 in.
Online Resources – Perimeter
http://www.khanacademy.org/math/geometry/basic-geometry/perimeter_area_tutorial/v/perimeter-and-areabasics
http://www.khanacademy.org/math/geometry/basic-geometry/perimeter_area_tutorial/v/interesting-perimeterand-area-problems
Find the place value of the 2 in the following numbers.
5. 269,571
6. 793,801,524
Round as indicated.
7. 636,331 to the nearest ten.
8. 504 to the nearest hundred.
Online Resources – Place Value
http://www.khanacademy.org/math/arithmetic/multiplication-division/place_value/v/place-value-3
Write a multiplication sentence that corresponds to the situation.
9. How many months are there in 45 years?
10. Julie can bike 56 miles a day. If she can vacation for 22 days, what is the maximum distance she
can cover?
Write a division sentence that corresponds to the situation. Then carry out the division.
11. A group of 7 people wants to buy a boat. The boat costs $273. If they all pay the same amount,
how much is each person’s share?
12. The city bridge has 9 lanes, all carrying equal number of cars. If 297 cars drive across the bridge,
how many cars cross in each lane?
Write the following in exponential form.
13. 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5
14. 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 7
Find the prime factorization of the number.
15. 70
16. 126
Online Resources – Prime Factorization
http://www.khanacademy.org/math/arithmetic/factors-multiples/prime_factorization/v/prime-factorization
Simplify.
17. (−30) ÷ (5) + 3(4) − 8( −2)
18. 7(5) − 2(4) + ( −42) ÷ ( −7)
19. 98
20. − 78
21. -91 + 97
22. -2 + (-6)
23. 24 - (-9)
24. -11 – (-18)
25. (-5)(-5)(-3)
26. 7 ⋅ ( −10) ⋅ ( −10) ⋅ 6 ⋅ ( −3)
27. − 84 ÷ ( −6)
28.
29. 7( x − 1) − 2 x
30. 8( x + 2) + 9 + 3 x
31.
− −56
32.
80
−5
2 − −7
33.
−42 ÷ 3 ⋅ 2
35.
3 + 10 ⋅ 2
36. 8 ÷ 2 ⋅ 2 − 3 2
38.
6(−3) + 2(−3)
24 − 28
34.
7 − 2 ( 6 − 1)
37. − 7(4 2 ) ÷ 4
39.
3 2 + (7 ⋅ 2 ) − 3
2 2 − (6)(−2) − 12
Online Resources – PEMDAS
http://www.khanacademy.org/math/arithmetic/multiplication-division/order_of_operations/v/order-of-operations-1
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/manipulatingexpressions/v/combining-like-terms-and-the-distributive-property
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/manipulatingexpressions/e/combining_like_terms_2
Multiply.
40. 2 ⋅
42.
1
5
7 7 1
⋅ ⋅
8 8 2
5
41. − ⋅ 7
6
43.
5 5 4
⋅ ⋅
8 12 25
Write the fraction in its lowest terms.
44.
55
190
45.
23
69
Solve.
46. A restaurant has a capacity of 32 patrons. If the restaurant is
7
full, how many patrons are at the
8
restaurant?
47. A recipe calls for
2
1
cup of milk. How much milk should be used to make of the recipe?
3
4
48.
Find the quotient of 6 and −2 .
49.
Subtract 12 from −36 .
50.
Convert 53mm to meters.
51.
How many centimeters are in 0.2km?
Divide and simplify.
51.
3 6
÷
4 7
52.
4 6
÷
9 5
54.
13
8 5
+
−
− 14 21 7
Add and, if possible, simplify.
53.
−2 1
+
5 15
55.
2 5 −2
+ +
3 6
9
Use < or > to write a true sentence.
56.
58.
1
12
5
6
−17
57. −
−4
59.
11
8
12
−
46
40
−20
Compute and, if possible, simplify
60.
5
1
14 + 13
8
4
3
5
61. 7 + 5
4
6
1
5
62. 34 − 12
3
8
63. 23
 1  2 
64.  3  4 
 2  3 
65. 4
66. 20 ÷ 2
3
5
68. − 5( x − 7 ) + 10
5
3
− 16
16
4
7
3
⋅5
10 10
4
1
67. 5 ÷ 2
5
2
69. − 4( x + 2) − 7
Find the LCD.
70.
2 4 2
, ,
3 5 7
71.
72.
8 −15
,
19 76
73.
1 7 7
, ,
6 x 18
−17 61
,
132 165
Online Resources – Fractions
http://www.khanacademy.org/math/arithmetic/fractions/Adding_and_subtracting_fractions/v/adding-andsubtracting-fractions
http://www.khanacademy.org/math/arithmetic/fractions/multiplying_and_dividing_frac/v/multiplying-fractions
http://www.khanacademy.org/math/arithmetic/fractions/multiplying_and_dividing_frac/v/dividing-fractions
PART 2
Algebraic Expressions
In order to model a situation with an algebraic expression, you should:
1. Identify the actions that suggest operations (such as addition, subtraction, multiplication or division).
2. Define one or more variables to represent the unknown values in the situation (be sure to define your
variables clearly).
3. Represent the situation with the appropriate variable and operations.
To evaluate an algebraic expression, substitute the given values for each variable and simplify using order of
operations.
Use the information above to answer the following questions:
Write each as an algebraic expression.
1) 3 squared
2) the difference of 24 and 3
3) the 6th power of n
4) half of a number
5) the difference of 13 and a number
6) twice n
7) 12 times 8
8) half of 6
9) 5 more than 4
10) the 2nd power of 5
11) the quotient of a number and 6
12) the sum of 10 and 7
13) 20 minus a number
14) 23 minus x
15) a number times 7
16) a number increased by 6
Online Resources for This Section:
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/manipulatingexpressions/e/writing_expressions_1
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/manipulatingexpressions/e/writing_expressions_2
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/manipulatingexpressions/e/writing-expressions-3
http://www.khanacademy.org/math/arithmetic/fractions/multiplying_and_dividing_frac/v/multiplying-fractions
http://www.khanacademy.org/math/arithmetic/fractions/multiplying_and_dividing_frac/v/dividing-fractions
An Introduction to Equations
An equation is a mathematical sentence with an equal sign. An equation can be true, false, or open. An
equation is true if the expressions on both sides of the equal sign are equal, for example 2 + 5 = 4 + 3. An
equation is false if the expressions on both sides of the equal sign are not equal, for example 2 + 5 = 4 + 2.
An equation is considered open if it contains one or more variables, for example x + 2 = 8. When a value is
substituted for the variable, you can then decide whether the equation is true or false for that particular value.
If an open sentence is true for a value of the variable, that value is called a solution of the equation. For x + 2
= 8, 6 is a solution because when 6 is substituted in the equation for x, the equation is true: 6 + 2 = 8.
Is the equation true, false, or open? Explain.
a. 15 + 21 = 30 + 6
b. 24 ÷ 8 = 2 · 2
c. 2n + 4 = 12
The equation is true, because both expressions equal 36.
The equation is false, because 24 ÷ 8 = 3 and 2 · 2 = 4; 3 ≠ 4.
The equation is open, because there is a variable in the
expression
on the left side.
Tell whether each equation is true, false, or open. Explain.
1. 2(12) – 3(6) – 12
2. 3x + 12 = –19
3. 14 – 19 = –5
4. 2(–8) + 4 = 12
5. 7 – 9 + 3=x
6. (22 + 12) ÷ –2 = –20 + 3
7. 14 – (–8) – 14 = 8
8. (13 – 16) ÷ 3 = 1
9. 42 ÷ –7 + 3 = (3)(–4) + 9
Is x = –3 a solution of the equation 4x + 5 = –7?
4x + 5 = –7
4(–3) + 5 = –7
– 7 = –7
Substitute –3 for x.
Simplify.
Since –7 = –7, –3 is a solution of the equation 4x + 5 = –7.
Tell whether the given number is a solution of each equation.
10. 4x – 1 = –27; –7
11. 18 – 2n = 14; 2
12. 21 = 3p – 5; 9
13. k = (–6)(–8) – 14; –62
14. 20v + 36 = –156; –6
15. 8y + 13 = 21; 1
16. –24 –17t = –58; 2
17. −26 = m + 5; −7
1
3
1
4
3
2
18. g − 8 = ;38
A table can be used to find or estimate a solution of an open equation. You will have to choose a value to begin
your table. If you choose the value that makes the equation true, you have found the solution and are done. If your
choice is not the solution, make another choice based on the values of both sides of the equation for your first
choice. If you choose one value that makes one side of the equation too high and then another value that makes
that same side too low, you know that the solution must lie between the two values you chose. It may not be
possible to determine an exact solution for each equation; estimating the solution to be between two integers may
be all that is possible in some cases.
What is the solution of 6n + 8 = 28?
If n = 2, then the left side of the equation is 6(2) + 8 or 20, which is too low.
If n = 5, then the left side of the equation is 6(5) + 8 or 38, which is too high.
The solution must lie between 2 and 5, so keep trying values between them.
If n = 3, then the left side of the equation is 6(3) + 8 or 26, which is too low.
If n = 4, then the left side of the equation is 6(4) + 8 or 32, which is too high.
The solution must lie between 3 and 4, but there are no other integers between 3 and 4.
You can give an estimate for the solution of 6n + 8 = 28 as being between the integers 3 and 4.
Write an equation for each sentence.
19. 13 times the sum of a number and 5 is 91.
20. Negative 8 times a number minus 15 is equal to 30.
21. Jared receives $23 for each lawn he mows. What is an equation that relates the number of lawns w
that Jared mows and his pay p?
22. Shariff has been working for a company 2 years longer than Patsy. What is an equation that relates the years of
employment of Shariff S and the years of employment of Patsy P?
Use mental math to find the solution of each equation.
23. h + 6 = 13
27.
z
= −2
5
24. –11 = n + 2
28
j
= 12
−6
25. 6 – k = 14
26. 5 = –8 + t
29. 8c = –48
30. –15a= –45
Use a table to find the solution of each equation.
31. –3b – 12 = 15
32. 15y + 6 = 21
33. –8 = 5y + 22
34. 6t – 1 = –49
Online Resources – Introduction to Solving equations
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/v/addingand-subtracting-the-same-thing-from-both-sides
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/v/intuitionwhy-we-divide-both-sides
http://www.khanacademy.org/math/algebra/solving-linear-equations-andinequalities/equations_beginner/v/solving-one-step-equations-2
Evaluate.
1. P + Prt, for P = 5000, r = 0.06 and t = 5.5
2. 2na+5(d-1) for n = 6, a = 3.5 and d = 9
2
( x + y )2 − y
3. a + (b − a ) for a = 6 and b = 4.
4.
5. − x for x = −95
6. − ( − ( − x ) ) for x = 27
for x = 2 and y = 3
Word Problems.
1. Mr. Lee wanted to keep track of how far he was driving today. He drove to Lodi which was 35.15
miles, then he drove to Merced which was 10.11 miles, then he drove home which was 18.03 miles.
How far did Mr. Lee drive?
2. Mrs. Hernandez prepared her grocery list at home. Her list contained the following items with their
sale prices: cheese - $3.97; crackers - $2.87; soda - $3.01; hamburger - $2.97; and gum - $0.97. She
bought all of the items at the store except the crackers. How much money did she spend?
3. The population of Nowhere, USA was estimated to be 668,100 in 2003, with an expected increase of
5.5% year. At the rate of increase given, what is the expected population in 2004? Round your answer
to the nearest whole number.
4. Suppose that, in 2006, France produced 13.973 million bikes. The total world production that year
totaled 120 million bikes. What percent of the world’s production of bikes is contributed by France?
Round your answer to the nearest tenth of a percent.
Properties of Real Numbers
Equivalent algebraic expressions are expressions that have the same value for all values for the
variable(s). For example x + x and 2x are equivalent expressions since, regardless of what number is
substituted in for x, simplifying each expression will result in the same value. Certain properties of real
numbers lead to the creation of equivalent expressions.
Commutative Properties
The commutative properties of addition and multiplication state that changing the order of the addends
does not change the sum and that changing the order of factors does not change the product.
Addition: a + b = b + a
Multiplication: a · b = b · a
To help you remember the commutative properties, you can think about the root word “commute.”
To commute means to move. If you think about commuting or moving when you think about the
commutative properties, you will remember that the addends or factors move or change order.
Do the following equations illustrate commutative properties?
a. 3
+4=4+3
b. (5 × 3) × 2 = 5 × (3 × 2)
c. 1 – 3 = 3 –1
3 + 4 and 4 + 3 both simplify to 7, so the two sides of the equation in part (a) are equal. Since both sides
have the same two addends but in a different order, this equation illustrates the Commutative Property
of Addition.
The expression on each side of the equation in part (b) simplifies to 30. Both sides contain the same 3
factors. However, this equation does not illustrate the Commutative Property of Multiplication because
the terms are in the same order on each side of the equation.
1 – 3 and 3 – 1 do not have the same value, so the equation in part (c) is not true. There is not
a commutative property for subtraction. Nor is there a commutative property for division.
Associative Properties
The associative properties of addition and multiplication state that changing the grouping of
addends does not change the sum and that changing the grouping of factors does not change the
product.
Addition: (a + b) + c = a + (b + c)
Multiplication: (a · b) · c = a · (b · c)
Do the following equations illustrate associative properties?
a. (1 + 5) + 4 = 1 + (5 + 4)
b. 4 × (2 × 7) = 4 × (7 × 2)
(1 + 5) + 4 and 1 + (5 + 4) both simplify to 10, so the two sides of the equation in part (a) are equal. Since both
sides have the same addends in the same order but grouped differently, this equation illustrates the Associative
Property of Addition.
The expression on each side of the equation in part (b) simplifies to 56. Both sides contain the same 3 factors.
However, the same factors that were grouped together on the left side have been grouped together on the right
side; only the order has changed. This equation does not illustrate the Associative Property of Multiplication.
Other properties of real numbers include:
a. Identity property of addition:
b. Identity property of multiplication:
c. Zero property of multiplication:
d. Multiplicative property of negative one:
a+0=0
a· 1=a
a · 0=0
–1 · a = –a
12 + 0 = 12
32 · 1 = 32
6· 0=0
–1 · 7 = –7
Exercises
What property is illustrated by each statement?
1. (m + 7.3) + 4.1 = m + (7.3 + 4.1)
2. 5p · 1 = 5p
3. 12x + 4y + 0 = 12x + 4y
4. (3r)(2s) = (2s)(3r)
5. 17 + (–2) = (–2) + 17
6. –(–3) = 3
7.
9.
( −12 ) + 15 = 15 + ( −12 )
8. 8 ⋅ ( 3 ⋅ 4 ) = ( 8 ⋅ 3) ⋅ 4
14 + ( x + 3) = ( x + 3) + 14
Simplify each expression. Justify each step.
10. (12 + 8x) + 13
11. (5 · m) · 7
12. (7 – 7) + 12
Online Resources – Properties of Real Numbers
http://www.khanacademy.org/math/arithmetic/order-of-operations/arithmetic_properties/v/commutative-propertyfor-addition
http://www.khanacademy.org/math/arithmetic/order-of-operations/arithmetic_properties/v/commutative-law-ofmultiplication
http://www.khanacademy.org/math/arithmetic/order-of-operations/arithmetic_properties/v/associative-law-ofaddition
http://www.khanacademy.org/math/arithmetic/order-of-operations/arithmetic_properties/v/associative-law-ofmultiplication
http://www.khanacademy.org/math/arithmetic/order-of-operations/arithmetic_properties/v/identity-property-of-1
http://www.khanacademy.org/math/arithmetic/order-of-operations/arithmetic_properties/v/identity-property-of-0
Evaluating Expressions
Exponents are used to represent repeated multiplication of the same number. For example, 4 × 4 × 4 × 4 ×
4 = 45. The number being multiplied by itself is called the base; in this case, the base is 4. The number that
shows how many times the base appears in the product is called the exponent; in this case, the exponent is 5.
45 is read four to the fifth power.
How is 6 × 6 × 6 × 6 × 6 × 6 × 6 written using an exponent?
The number 6 is multiplied by itself 7 times. This means that the base is 6 and the exponent is 7. 6 × 6 × 6
× 6 × 6 × 6 × 6 written using an exponent is 67.
Exercises
Write each repeated multiplication using an exponent.
1. 4 × 4 × 4 × 4 × 4
2. 2 × 2 × 2
3. 1.1 × 1.1 × 1.1 × 1.1 × 1.1
4. 3.4 × 3.4 × 3.4 × 3.4 × 3.4 × 3.4
5. (–7) × (–7) × (–7) × (–7)
6. 11 × 11 × 11
Write each expression as repeated multiplication.
7. 4
3
9. 1.5
8. 5
2
4
2

7
4
12. (5n)
5
10. 
11. x7
13. Trisha wants to determine the volume of a cube with sides of length s. Write an expression that represents
the volume of the cube.
Real Numbers and the Number Line
A number that is the product of some other number with itself, or a number to the second power, such as 9 = 3 × 3
= 32, is called a perfect square. The number that is raised to the second power is called the square root of the
product. In this case, 3 is the square root of 9. This is written in symbols as
Sometimes
square roots are whole numbers, but in other cases, they can be estimated.
What is an estimate for the square root of 150?
There is no whole number that can be multiplied by itself to give the product of 150.
10× 10 = 100
11× 11 = 121
12× 12 = 144
13× 13 = 169
You cannot find the exact value, but you can estimate it by comparing 150 to perfect squares that are close to 150.
150 is between 144 and 169, so the square root of 150 is between 12 and 13. Because 150 is closer to 144 than it
is to 169, we can estimate that the square root of 150 is slightly greater than 12.
Exercises
Find the square root of each number. If the number is not a perfect square, estimate the square root to
the nearest integer.
1. 100
2. 49
3. 9
4. 25
5. 81
6. 169
7. 15
8. 24
9. 40
2
10. A square mat has an area of 225 cm . What is the length of each side of the mat?