STANDARDS OF LEARNING

CONTENT REVIEW NOTES

ALGEBRA I – Part I

1st Nine Weeks, 2018-2019

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OVERVIEW

Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a

resource for students and parents. Each nine weeks’ Standards of Learning (SOLs) have been identified and a

detailed explanation of the specific SOL is provided. Specific notes have also been included in this document

to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models

for solving various types of problems. A “ ” section has also been developed to provide students with

the opportunity to solve similar problems and check their answers. The answers to the “ ” problems are

found at the end of the document.

The document is a compilation of information found in the Virginia Department of Education (VDOE)

Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE

information, Prentice Hall textbook series and resources have been used. Finally, information from various

websites is included. The websites are listed with the information as it appears in the document.

Supplemental online information can be accessed by scanning QR codes throughout the document. These will

take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the

document to allow students to check their readiness for the nine-weeks test.

The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of

questions per reporting category, and the corresponding SOLs.

It is the Mathematics Instructors’ desire that students and parents will use this document as a tool toward the

students’ success on the end-of-year assessment.

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VDOE has not released

final version of

Spring 2019 formula

sheet.

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Expressions and Order of Operations A.1 The student will

a) represent verbal quantitative situations algebraically; and b) evaluate algebraic expressions for given replacement values of the

variables. Expression is a word used to designate any symbolic mathematical phrase that may contain numbers and/or variables. An expression can be represented algebraically… Example 1: 6x + 5 Example 2: a – 9b or in written form. Example 1: The sum of a number and eleven Example 2: One half of a number squared minus four Some common words are used to indicate each operation. Many of these are shown in the table below, but there are others.

Add Subtract Multiply Divide Equals

Plus

Sum

More than

Increased by

Total

All together

Add to

And

Difference

Minus

Less than

Decreased by

Take away

How many left

Remaining

Subtracted by

Less

Times

Product

Multiplied By

Doubled (x2)

Tripled (x3)

By

Squared (a·a)

Cubed (a·a·a)

Part

Quotient

Divided by

Each

Half ( ÷ 2)

Split (÷ 2)

Is

Are

Is Equal To

Is equivalent to

Equals

Expressions and Order of Operations Translate the written expressions to algebraic expressions, and algebraic expressions to written

expressions.

1. the difference of eleven and x

2. three times the sum of a number and ten

3. four times the difference of n squared and five

4. 12g ÷ 4

5. a² - b⁴

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Expressions are simplified using the order of operations and the properties for operations with real numbers. The order of operations is as follows: First: Complete all operations within grouping symbols. If there are grouping symbols within other grouping symbols, do the innermost operation first. Grouping symbols

include parentheses (a), brackets [a], radical symbols √𝒂, absolute value bars |𝒂|,

and the fraction bar 𝒂

𝒃.

Second: Evaluate all exponents. Third: Multiply and/or divide from left to right. Fourth: Add and/or subtract from left of right. To evaluate an algebraic expression substitute in the replacement values of the variables and then evaluate using the order of operations.

Example 1: 15 − 9 ÷ (−3)2

Step 1: 15 − 9 ÷ (−3)2 Step 2: 15 − 9 ÷ 9 Step 3: 15 − 1

Step 4: The answer is 14

Example 2: (𝑝 − 3)2 + 2𝑝 − 4, 𝑝 = −7

Step 1: (−7 − 3)2 + 2(−7) − 4

Step 2: (−10)2 + 2(−7) − 4 Step 3: 100 + 2(−7) − 4 Step 4: 100 + (−14) − 4 Step 5: 86 − 4 Step 6: The answer is 82

Scan this QR code to go to an order of operations video tutorial!

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Example 3: 20 − √3 ∙ 10 − 14 + |7 − 5 ∙ 4|

Step 1: 20 − √30 − 14 + |7 − 20|

Step 2: 20 − √16 + |−13| Step 3: 20 − 4 + 13 Step 4: The answer is 29

Expressions and Order of Operations Evaluate each expression. a=2, b=5, x= - 4, and n=10.

6. [𝑎 + 8(𝑏 − 2)]2 ÷ 4

7. (2𝑥)2 + 𝑎𝑛 − 5𝑏

8. 𝑛2 + 3(𝑎 + 4)

9. 𝑏𝑥 − 𝑎𝑥 Evaluate each expression using the order of operations.

10. 63 − √10 ÷ 2 + 2 ∙ 22

11. √75 + (9 + 3)2 − 33

+ 19

12. (7∙3−18)3

√63−14+21

13. |6 − √33 + 22| − 8

Scan this QR code to go to a video for more complicated order

of operations help.

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Properties of Real Numbers A.4EKS The student will solve multistep linear and quadratic equations in two variables,

including justifying steps used in simplifying expressions and solving equations.

Property Definition Examples

Multiplicative Property

of Zero

Any number multiplied by zero

always equals zero.

𝑎 (0) = 0

0 ∙ (−14) = 0

Additive Identity Any number plus zero is equal

to the original number.

𝑎 + 0 = 𝑎

126 + 0 = 126

Multiplicative Identity Any number times one is the

original number.

𝑎 ∙ 1 = 𝑎

1 ∙ 78 = 78

Additive Inverse A number plus its opposite

always equals zero.

𝑎 + (−𝑎) = 0

−21 + 21 = 0

Multiplicative Inverse

A number times its inverse

(reciprocal) is always equal to

one.

𝑎 ∙ 1

𝑎= 1

5

2 ∙

2

5= 1

Associative Property

When adding or multiplying

numbers, the way that they are

grouped does not affect the

outcome.

(𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐)

5 + (3 + 8) = (5 + 3) + 8

(𝑎𝑏)𝑐 = 𝑎(𝑏𝑐)

6(3𝑎) = (6 ∙ 3) 𝑎

Commutative Property

The order that you add or

multiply numbers does not

change the outcome.

𝑎 + 𝑏 = 𝑏 + 𝑎

14 + 6 = 6 + 14

𝑎𝑏 = 𝑏𝑎

8 ∙ 3 = 3 ∙ 8

Distributive Property

For any numbers a, b, and c:

a(b + c) = ab + ac

5 ( 3 − 2) = 5 ∙ 3 − 5 ∙ 2

−3 (𝑎 + 𝑏) = −3𝑎 + (−3𝑏)

𝑜𝑟 − 3 (𝑎 + 𝑏) = −3𝑎 − 3𝑏

Substitution property of

equality

If a = b, then b can replace a.

A quantity may be substituted

for its equal in any expression.

𝐼𝑓 5 + 2 = 7, 𝑡ℎ𝑒𝑛 (5 + 2) ∙ 4 = 7 ∙ 4

𝐼𝑓 𝑎 = 5, 𝑡ℎ𝑒𝑛 11𝑎 = 11 ∙ 5

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Reflexive Property of

Equality Any quantity is equal to itself.

𝑎 = 𝑎

5

3=

5

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Transitive Property of

Equality

If one quantity equals a second

quantity and the second

quantity equals a third, then the

first equals the third.

𝐼𝑓 𝑎 = 𝑏, 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐.

𝐼𝑓 2 + 4 = 6, 𝑎𝑛𝑑 2(3) = 6,

𝑡ℎ𝑒𝑛 2 + 4 = 2(3)

Symmetric Property of

Equality

If one quantity equals a second

quantity, then the second

quantity equals the first.

𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 = 𝑎

𝐼𝑓 25 = 13𝑎 − 1, 𝑡ℎ𝑒𝑛 13𝑎 − 1 = 25

Properties of Real Numbers Match the example on the left to the appropriate property on the right.

1. (𝑥 + 3) + 𝑦 = 𝑥 + (3 + 𝑦)

2. 1 = 1

2𝑥∙ 2𝑥

3. 3𝑥 + 6 = 6 + 3𝑥

4. (5 – 𝑥)2 = 10 – 2𝑥

5. 𝑏5 + 0 = 𝑏5

6. (𝑥 + 3) + 𝑦 = 𝑦 + (𝑥 + 3)

7. 0 ∙ 17𝑛 = 0

8. 5𝑥 + (−5𝑥) = 0

9. 𝑥𝑦𝑧 = 𝑥𝑦𝑧

10. If one dollar is the same as four quarters,

and four quarters is the same as ten dimes,

then ten dimes is the same as one dollar.

A. Multiplicative Property of Zero

B. Additive Identity

C. Multiplicative Identity

D. Additive Inverse

E. Multiplicative Inverse

F. Associative Property

G. Commutative Property

H. Distributive Property

I. Substitution Property of Equality

J. Reflexive Property of Equality

K. Transitive Property of Equality

L. Symmetric Property of Equality

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Solving Equations

A.4 The student will solve a) multistep linear equations in one variable algebraically;

e) practical problems involving equations and systems of equations.

You will solve an equation to find all of the possible values for the variable. In order to solve an equation, you will need to isolate the variable by performing inverse operations (or ‘undoing’ what is done to the variable). Any operation that you perform on one side of the equal sign MUST be performed on the other side as well. Drawing an arrow down from the equal sign may help remind you to do this.

Example 1: 𝑚 − 9 = −3

+9 + 9

𝑚 = 6

Check your work by plugging your answer back in to the original problem.

6 − 9 = −3

Example 2: 𝑥+4

5 = −12

∙ 5 ∙ 5

𝑥 + 4 = −60

−4 − 4

𝑥 = −64

Check your work by plugging your answer back in to the original problem.

−64 +4

5 =

−60

5= −12

Scan this QR code to go to a video tutorial on two-step

equations.

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You may have to distribute a constant and combine like terms before solving an equation.

Example 3: −4(𝑔 − 7) + 2𝑔 = −10

−4𝑔 + 28 + 2𝑔 = −10

−2𝑔 + 28 = −10

−28 − 28

−2𝑔 = −38

÷ (−2) ÷ (−2)

𝑔 = 19

Check your work by plugging your answer back in to the original problem.

−4 (19 − 7) + 2(19) = −10

−4(12) + 2(19) = −10

−48 + 38 = −10

If there are variables on both sides of the equation, you will need to move them all to the same side in the same way that you move numbers.

Example 4: 3𝑝 − 5 = 7(𝑝 − 3)

3𝑝 − 5 = 7𝑝 − 21

−3𝑝 − 3𝑝

−5 = 4𝑝 − 21

+21 + 21

16 = 4𝑝

÷ 4 ÷ 4

𝑝 = 4

Check your work by plugging your answer back in to the original problem.

3(4) − 5 = 7(4) − 21 12 − 5 = 28 − 21

7 = 7

Scan this QR code to go to a video tutorial on multi-step

equations.

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Example 5: 𝑥+10

5𝑥=

−1

5 You can begin this problem by cross multiplying!

5(𝑥 + 10) = −1(5𝑥)

5𝑥 + 50 = −5𝑥

+5𝑥 + 5𝑥

10𝑥 + 50 = 0

−50 − 50

10𝑥 = −50

÷ 10 ÷ 10

𝑥 = −5

Check your work by plugging your answer back in to the original problem.

−5+10

5(−5)=

−1

5

5

−25 =

−1

5

− 1

5 = −

1

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Solving Equations

Solve each equation 1. 𝑘 + 11 = −8

2. 9 − 3𝑥 = 54

3. −17 = 𝑦−6

2

4. 5 (2𝑛 + 6) + 8 = 33

5. 3 − (4𝑘 + 2) = −15

6. 5𝑔 + 4 = −9𝑔 − 10

7. −2(−4𝑚 − 1) + 3𝑚 = 4𝑚 − 8 + 𝑚

8. 𝑥−4

2=

−2(3𝑥−3)

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Scan this QR code to go to a video tutorial on equations with

variables on both sides.

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Answers to the problems: Expressions and Order of Operations 1. 11 − 𝑥

2. 3 (𝑛 + 10)

3. 4 (𝑛² − 5)

4. The product of twelve and a number

divided by four

5. a squared minus b to the fourth power

6. 169

7. 59

8. 118

9. −12

10. 209

11. 25

12. 3

13. −7

Properties of Real Numbers 1. F - Associative 2. E - Multiplicative Inverse 3. G - Commutative 4. H - Distributive 5. B - Additive Identity 6. G - Commutative 7. A - Multiplicative Property of Zero 8. D - Additive Inverse 9. J - Reflexive Property of Equality 10. K - Transitive Property of Equality

Solving Equations

1. 𝑘 = −19 2. 𝑥 = −15 3. 𝑦 = −28

4. 𝑛 = −1

2

5. 𝑘 = 4 6. 𝑔 = −1

7. 𝑚 = −5

3

8. 𝑥 = 2

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