Warm-Up, 3.1.1, P.7 Sydney subscribes to an online company that allows her to download electronic books. Her subscription costs a flat fee of $30 for up to 10 downloads each month. For each download over 10, there is an additional charge per download. Common Core State Standard: A–REI.1 3.1.1: Properties of Equality

Each additional download costs $3.75. During the month of September, Sydney downloaded 22 books and was charged $75. Set up an equation to find the charge for each downloaded book over 10. 30 + x(22 – 10) = 75 Solve the equation. 30 + 12x = 75 Simplify the equation. 12x = 45 Divide both sides by 12. x = 3.75 Each additional download costs $3.75. 3.1.1: Properties of Equality

The total correct cost is 30 + 3.75(x – 10). In October, Sydney was incorrectly charged $67.50 for 18 books. How much should she have been charged? Set up an equation to find the amount that Sydney should have been charged. The cost for 10 books is $30. Each book over 10 costs an additional $3.75. The total correct cost is 30 + 3.75(x – 10). 3.1.1: Properties of Equality

The total amount Sydney should have been charged for 18 books is $60. Total cost = 30 + 3.75(x – 10) Total cost = 30 + 3.75(18 – 10) Substitute the number of books she ordered for x. Total cost = 30 + 3.75(8) Multiply. Total cost = 30 + 30 Total cost = 60 The total amount Sydney should have been charged for 18 books is $60. 3.1.1: Properties of Equality

If Sydney received a bill for $101.25, how many books did she download? Use the equation for total cost found earlier to determine the number of books Sydney downloaded. Total cost = 30 + 3.75(x – 10) 101.25 = 30 + 3.75(x – 10) Substitute the value for total cost. 101.25 = 30 + 3.75x – 37.5 Distribute 3.75 over (x – 10). 101.25 = 3.75x – 7.5 Combine like terms. 3.1.1: Properties of Equality

The total number of books Sydney downloaded was 29. 108.75 = 3.75x Add 7.5 to both sides of the equation. 29 = x Divide both sides of the equation by 3.75. The total number of books Sydney downloaded was 29. Connection to the Lesson Students will continue to use their knowledge of solving equations, but will be asked to justify the steps used in the process. 3.1.1: Properties of Equality

Example #1: Don’t copy!!!!!! Solve for the following variable: –7x + 22 = 50? 3.1.1: Properties of Equality

3.1.1: Introduction Equations are mathematical sentences that state two expressions are equal. In order to solve equations in algebra, you must perform operations that maintain equality on both sides of the equation using the properties of equality. Properties of equality: rules that allow you to balance, manipulate, and solve equations. Properties of operations: explain the effect that the operations of addition, subtraction, multiplication, and division have on equations. 3.1.1: Properties of Equality

3.1.1: Key Concepts In mathematics, it is important to follow the rules when solving equations, but it is also necessary to justify, or prove that the steps we are following to solve problems are correct and allowed. The following table summarizes some of these rules. 3.1.1: Properties of Equality

Key Concepts, continued Properties of Equality Property In symbols In words Reflexive property of equality a = a A number is equal to itself. Symmetric property If a = b, then b = a. If numbers are equal, they will still be equal if the order is changed. Transitive property If a = b and b = c, then a = c. If numbers are equal to the same number, then they are equal to each other. Addition property If a = b, then a + c = b + c. Adding the same number to both sides of an equation does not change the equality of the equation. 3.1.1: Properties of Equality

Key Concepts, continued Properties of Equality, continued Property In symbols In words Subtraction property of equality If a = b, then a – c = b – c. Subtracting the same number from both sides of an equation does not change the equality of the equation. Multiplication If a = b and c ≠ 0, then a • c = b • c. Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. Division property of equality a ÷ c = b ÷ c. Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation. 3.1.1: Properties of Equality

Key Concepts, continued Properties of Equality, continued Property In symbols In words Substitution property of equality If a = b, then b may be substituted for a in any expression containing a. If two numbers are equal, then substituting one in for another does not change the equality of the equation. 3.1.1: Properties of Equality

Key Concepts, continued Properties of operations: explain the effect that the operations of addition, subtraction, multiplication, and division have on equations. The following table describes some of those properties. 3.1.1: Properties of Equality

Key Concepts, continued Properties of Operations Property General rule Specific example Commutative property of addition a + b = b + a 3 + 8 = 8 + 3 Associative property of addition (a + b) + c = a + (b + c) (3 + 8) + 2 = 3 + (8 + 2) Commutative property of multiplication a • b = b • a 3 • 8 = 8 • 3 Associative property of (a • b) • c = a • (b • c) (3 • 8) • 2 = 3 • (8 • 2) Distributive property of multiplication over addition a • (b + c) = a • b + a • c 3 • (8 + 2) = 3 • 8 + 3 • 2 3.1.1: Properties of Equality

Key Concepts, continued When we solve an equation, we are rewriting it into a simpler, equivalent equation that helps us find the unknown value. When solving an equation that contains parentheses, apply the properties of operations and perform the operation that’s in the parentheses first. The properties of equality, as well as the properties of operations, not only justify our reasoning, but also help us to understand our own thinking. 3.1.1: Properties of Equality

Key Concepts, continued When identifying which step is being used, it helps to review each step in the sequence and make note of what operation was performed, and whether it was done to one side of the equation or both. (What changed and where?) When operations are performed on one side of the equation, the properties of operations are generally followed. 3.1.1: Properties of Equality

Key Concepts, continued When an operation is performed on both sides of the equation, the properties of equality are generally followed. Once you have noted which steps were taken, match them to the properties listed in the tables. If a step being taken can’t be justified, then the step shouldn’t be done. 3.1.1: Properties of Equality

Common Errors/Misconceptions incorrectly identifying operations incorrectly identifying properties performing a step that is not justifiable or does not follow the properties of equality and/or the properties of operations 3.1.1: Properties of Equality

Guided Practice Example #1: Which property of equality is missing in the steps to solve the equation –7x + 22 = 50? Equation Steps 1) –7x + 22 = 50 1) Original equation(Given) 2) –7x = 28 2) Subtraction Property of Equality 3) x = –4 3) Division property of equality 3.1.1: Properties of Equality

Guided Practice Example #2: Which property of equality is missing in the steps to solve the equation ? Equation Steps 1) Original equation(Given) 2) Addition property of equality 3) –x = 42 3) Multiplication Property of Equality 4) x = –42 4) Division property of equality 3.1.1: Properties of Equality

Guided Practice Example #3: Which property of equality is missing in the steps to solve the equation 76 = 5x – 15 + 2x ? Equation Steps 1) 76 = 5x – 15 + 2x 1) Original Equation(Given) 2) 76 = 5x + 2x - 15 2) Commutative Property of Addition 3) 76 = 7x - 15 3) Combine Like Terms(Simplify) 4) 91 = 7x 4) Addition Property of Equality 5) 13 = x 5) Division Property of Equality 6) x = 13 6) Symmetric Property of Equality 3.1.1: Properties of Equality

Guided Practice Example #4: Which property of equality is missing in the steps to solve the equation 5x + 3(x + 4) = 28 ? Equation Steps 1) 5x + 3(x+4) = 28 1) Original Equation(Given) 2) 5x + 3x + 12 = 28 2) Distributive Property 3) 8x + 12 = 28 3) Combine Like Terms(Simplify) 4) 8x = 16 4) Subtraction Property of Equality 5) x = 2 5) Division Property of Equality 3.1.1: Properties of Equality

b) Key Concepts: Copy 1 Chart(U3-11) 4/28/2017: Homework Workbook: P.13,14 #1-8 2) 3.1.2 Notes(U3-11) a) Intro: read only b) Key Concepts: Copy 1 Chart(U3-11) c) ?’s and Summary http://www.walch.com/ei/00004