STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part I. 2 nd Nine Weeks,

Transcription

1 STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part I 2 nd Nine Weeks,

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 at the end of the document. problems are found 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. 2

3 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 4

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 Transitive Property of Equality Symmetric Property of Equality Any quantity is equal to itself. If one quantity equals a second quantity and the second quantity equals a third, then the first equals the third. 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 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 5

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. Scan this QR code to go to a video tutorial on two-step equations. Example 2: Check your work by plugging your answer back in to the original problem. 6

7 You may have to distribute a constant and combine like terms before solving an equation. Example 3: Scan this QR code to go to a video tutorial on multi-step equations. 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. 7

8 Example 5: You can begin this problem by cross multiplying! Scan this QR code to go to a video tutorial on equations with variables on both sides. Check your work by plugging your answer back in to the original problem. Solving Equations Solve each equation

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: Scan this QR code to go to a video tutorial on transforming formulas. Example 2: Example 3: We will have to un-distribute the a from each term on the left. 9

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

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. Scan this QR code to go to a video tutorial on solving and graphing inequalities. Graph: Example 2: Solve and graph the following inequality. Don t forget to switch the sign direction! Graph:

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 5 th 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. Scan this QR code to get help on setting up and solving inequalities word problems. 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? 12

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! 13

14 Example 1: Example 2: Example 3: Example 4: 14

15 Justifying Steps using Properties List the properties used to justify each step in the problems below

16 Answers to the Properties of Real Numbers problems: 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 Transforming Formulas 1. or or Inequalities 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) 16