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STANDARDS OF LEARNING
CONTENT REVIEW NOTES
ALGEBRA I Part I
2
nd Nine Weeks, 2016-2017
2
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.
Algebra I Blueprint Summary Table
Reporting Categories No. of Items SOL
Expressions & Operations 12 A.1
A.2a – c
A.3
Equations & Inequalities 18 A.4a – f
A.5a – d
A.6a – b
Functions & Statistics 20 A.7a – f
A.8
A.9
A.10
A.11
Total Number of Operational Items 50
Field-Test Items* 10
Total Number of Items 60
* These field-test items will not be used to compute the students’ scores on the test.
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.
3
4
Properties of Real Numbers A.4 The student will solve multistep linear and quadratic equations in two variables,
including b) justifying steps used in simplifying expressions and solving equations, using field
properties and axioms of equality that are valid for the set of real numbers and its subsets;
A.5 The student will solve multistep linear inequalities in two variables, including
b) justifying steps used in solving inequalities, using axioms of inequality and properties
Property Definition Examples
Multiplicative Property
of Zero
Any number multiplied by zero
always equals zero.
Additive Identity Any number plus zero is equal
to the original number.
Multiplicative Identity Any number times one is the
original number.
Additive Inverse A number plus its opposite
always equals zero.
Multiplicative Inverse
A number times its inverse
(reciprocal) is always equal to
one.
Associative Property
When adding or multiplying
numbers, the way that they are
grouped does not affect the
outcome.
Commutative Property
The order that you add or
multiply numbers does not
change the outcome.
Distributive Property
For any numbers a, b, and c:
a(b + c) = ab + ac
5
Substitution property of
equality
If a = b, then b can replace a.
A quantity may be substituted
for its equal in any expression.
Reflexive Property of
Equality Any quantity is equal to itself.
Transitive Property of
Equality
If one quantity equals a second
quantity and the second
quantity equals a third, then the
first equals the third.
Symmetric Property of
Equality
If one quantity equals a second
quantity, then the second
quantity equals the first.
Properties of Real Numbers Match the example on the left to the appropriate property on the right.
1.
2.
3.
4. – –
5.
6.
7.
8.
9.
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
6
Solving Equations
A.4 The student will solve multistep linear and quadratic equations in two variables,
including d) solving multistep linear equations algebraically and graphically; f) solving real-world 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:
Check your work by plugging your answer back in to the original problem.
Example 2:
Check your work by plugging your answer back in to the original problem.
Scan this QR code to go to a video tutorial on two-step
equations.
7
You may have to distribute a constant and combine like terms before solving an equation. Example 3:
Check your work by plugging your answer back in to the original problem.
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:
Check your work by plugging your answer back in to the original problem.
Scan this QR code to go to a video tutorial on multi-step
equations.
8
Example 5:
You can begin this problem by cross multiplying!
Check your work by plugging your answer back in to the original problem.
Solving Equations
Solve each equation 1.
2.
3.
4.
5.
6.
7.
8.
Scan this QR code to go to a video tutorial on equations with
variables on both sides.
9
Transforming Formulas A.4 The student will solve multistep linear and quadratic equations in two variables,
including a) solving literal equations (formulas) for a given variable;
Transforming Formulas is done the same way as solving equations. Treat the variables the same way that you treat numbers, being sure to combine like terms when you can. Remember that in order to be like terms, both terms need to have the same variables, and those variables have to have the same exponent.
Example 1:
Example 2:
Example 3:
We will have to “un-distribute” the a from each term on the left.
Scan this QR code to go to a video tutorial on transforming
formulas.
10
Example 4:
You can cross multiply to rewrite this problem without fractions.
Don’t forget to simplify your fractions!
Example 5:
To divide by
, you can multiply by the reciprocal, which is
, or just 2.
Transforming Formulas Solve each equation for the stated variable.
1.
2.
3.
4.
5.
11
Inequalities A.5 The student will solve multistep linear inequalities in two variables, including
a) solving multistep linear inequalities algebraically and graphically; c) solving real-world problems involving inequalities
An inequality is solved the same way as an equation. The only important thing to remember is that if you multiply or divide by a negative number, you need to switch the direction of the inequality sign. A proof of this is included in the online video tutorials or on the top of page 179 in your text book. You will also need to know how to graph inequalities on the number line. If the inequality has a greater than or equal to ( ) or less than or equal to ( ) sign, then you will use a closed point to mark the spot on the number line. This closed point indicates that the number that the point is on IS included in the solution. For a greater than ( ) or less than ( ) sign, you will use an open point on the number line. This open point indicates that the number that the point is on is NOT included in the solution. Example 1: Solve and graph the following inequality.
Graph:
Example 2: Solve and graph the following inequality.
Don’t forget to switch the sign direction!
Graph:
0 3 6 9 120–3–6–9
0 3 6 90–3–6–9
Scan this QR code to go to a video tutorial on solving and
graphing inequalities.
12
Example 3:
Example 4: Dan’s math quiz scores are 88, 91, 87, and 85. What is the minimum score he would need on his 5th quiz to have a quiz average of at least 90?
The average of his 5 quiz scores must be greater than or equal to 90.
Dan needs to score a 99 or better on his final quiz to have a 90% quiz average.
Inequalities 1. Solve and graph: 2. Solve and graph: 3. Solve: 4. Solve: 5. A salesman earns $410 per week plus 10% commission on sales. How many dollars in sales will the salesman need in order to make more than $600 for the week?
Scan this QR code to get help on setting up and solving
inequalities word problems.
13
Justifying Steps using Properties A.4 The student will solve multistep linear and quadratic equations in two variables, including
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets;
You are using the properties of real numbers to solve equations and inequalities, and to simplify expressions. You will need to be able to identify the property that you are using in each step of the simplification or solution. When you solve an equation or inequality and perform the same operation on both sides of the equal sign this is a special property of equality.
Property Equation Example Inequality Example
Addition Property of
Equality and
Inequality
Subtraction Property
of Equality and
Inequality
Multiplication
Property of Equality
and Inequality
*Don’t forget to switch the sign if you multiply or divide
by a negative!
Division Property of
Equality and
Inequality
*Don’t forget to switch the sign if you multiply or divide
by a negative!
14
Example 1: –
Example 2:
Example 3:
Example 4: –
15
Justifying Steps using Properties List the properties used to justify each step in the problems below. 1.
2.
16
Answers to the problems: 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. 2. 3.
4.
5. 6.
7.
8.
Transforming Formulas
1.
or
2.
3.
4.
5.
or
Inequalities 1.
2.
3. 4. 5. Justifying Steps using Properties 1. Distributive Commutative Subtraction (Substitution) Subtraction (Substitution) 2. Associative Addition (Substitution) Subtraction Property of Inequality Subtraction (Substitution) Division Property of Inequality Divide (Substitution)
0 3 6 9 120–3–6–9
0 3 6 90–3–6–9