Unit Essential Questions. Can equations that appear to be different be equivalent? How can you solve equations?

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1 Unit Essential Questions Can equations that appear to be different be equivalent? How can you solve equations? What kinds of relationships can proportions represent? Williams Math Lessons

2 TARGET ONE-STEP EQUATIONS MACC.9.A-CED.A.: Create equations and inequalities in one variable and use them to solve problems. MACC.9.A-REI.A.: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. RATING LEARNING SCALE use one-step equations to solve real-world applications or more challenging problems that I have never previously attempted solve one-step equations solve one-step equations with help understand that I can solve problems by creating equations WARM UP Tell whether the ordered pair is a solution of each equation. ) y = x + 5; (,8) ) y = (x + ); ( 6,0) Yes No KEY CONCEPTS AND VOCABULARY Equivalent Equations have the same solutions and are the result of balancing an equation (whatever is done to one side of the equal sign has to be done to the other side). Isolating a variable means to have a variable with a coefficient of by itself on one side of the equal sign. Inverse Operations are operations which undo each other. Subtraction and addition undo each other as well as multiplication and division. Addition/ Subtraction Property of Equality adding /subtracting the same number to/from each side of an equation produces an equivalent equation. Multiplication/ Division Property of Equality multiplying/dividing each side of an equation by the same number produces an equivalent equation. EXAMPLES EXAMPLE : SOLVING EQUATIONS BY ADDITION OR SUBTRACTION Solve the following equations. a) = +w b) + r = 7 w = 5 r = c) 5 d) 6 + j = 6 t 5 = j = t = 5-8-

3 EXAMPLE : SOLVING EQUATIONS BY MULTIPLICATION OR DIVISION Solve the following equations. a) 5k = 5 b) x = k = 9 x = 6 c) 6g = d) g = c 7 = c = 77 EXAMPLE : SOLVING EQUATIONS WITH FRACTIONS Solve the following equations. a) b) 5 p = 5 p = 5 y 7 = y = 9 6 c) d) = k 5 k = 6 7 k = 6 k = 00 6 EXAMPLE : SOLVING EQUATIONS FOR A REAL WORLD SITUATION Shannon and Kelly spent $5 on gift certificates for their friends during the holidays. If this amount is 5/7 of their total spending money, how much spending money did they originally have? Set up an equation and solve. Let x = spending money 5 7 x = 5 x = $89 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: -9-

4 TARGET TWO-STEP EQUATIONS MACC.9.A-CED.A.: Create equations and inequalities in one variable and use them to solve problems. MACC.9.A-REI.A.: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. RATING WARM UP LEARNING SCALE use two-steps equations to solve real-world applications or more challenging problems that I have never previously attempted solve two-step equations in one variable solve two-step equations in one variable with help undo operations by working in reverse order from the order of operations Simplify each expression. ) [ ( ) ] ) KEY CONCEPTS AND VOCABULARY Steps for Solving Two-Step Equations (ax + b = c ) Add or subtract (undo) the b value (to both sides) Multiply or divide (undo) the a value (to both sides) Check answer Steps for Solving Two-Step Equations ( x + a = c ) b Multiply (undo) the b value (to both sides) Add or subtract (undo) the a value (to both sides) Check answer EXAMPLES EXAMPLE : SOLVING TWO-STEP EQUATIONS: (ax + b = c) Solve. a) 5q = 7 b) x + = 8 q = 0 x = 5 c) a 5 = 9 d) 7 + 5p = 5 a = p = 9-0-

5 e) t 9 = f) 0.n + = 8.6 t = n = 8 EXAMPLE : SOLVING TWO-STEP EQUATIONS: ( x + a = c ) b Solve. x + 8 a) b) = h 5 = 0 x = h = 5 EXAMPLE : WRITING AND SOLVING TWO-STEP EQUATIONS Write an equation and solve. Eight more than five times a number is negative x = 6 x = EXAMPLE : SOLVING EQUATIONS FOR REAL WORLD SITUATIONS A calling plan charges $0.0 per minute and a monthly fee of $5.99. How many minutes can a customer talk if they want the bill to equal $7.99? 0 Minutes RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: --

6 TARGET MULTI-STEP EQUATIONS MACC.9.A-CED.A.: Create equations and inequalities in one variable and use them to solve problems. MACC.9.A-REI.A.: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. RATING WARM UP LEARNING SCALE use multi-step equations to solve real-world applications or more challenging problems that I have never previously attempted solve multi-step equations in one variable solve multi-step equations in one variable with help undo operations by working in reverse order from the order of operations Simplify each expression. ) 8( x) ) ( x ) 6 8x x + ) ( x + ) ) 6( x) 8x + 8 x KEY CONCEPTS AND VOCABULARY Steps for Solving Multi-Step Equations Simplify each side of the equation if possible (distribute, combine like terms) Add or subtract (undo) the b value (to both sides) Multiply or divide (undo) the a value (to both sides) Isolate variable Check answer EXAMPLES EXAMPLE : SOLVING MULTI-STEP EQUATIONS Solve. a) 5 + 7t t = 9 b) 7p + 8p 6 = 59 t = 7 p = 5 --

7 c) f + (f + ) = d) (x + 9) = 8 f = 5 x = 7 EXAMPLE : SOLVING MULTI-STEP EQUATIONS WITH FRACTIONS AND DECIMALS Solve. a) h 5 h = b) m 7 m = h = 5 m = c).5 5d =.5 d). =. 0.6x d =.75 x = EXAMPLE : SOLVING EQUATIONS FOR REAL WORLD SITUATIONS There is a -foot fence on one side of a rectangular garden. The gardener has feet of fencing to enclose the other three sides. What is the length of the garden s longer dimension? 6 feet RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: --

8 TARGET SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES MACC.9.A-REI.B.: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MACC.9.A-REI.A.: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. RATING LEARNING SCALE solve equations with variables on both sides that apply to real-world situations or more challenging problems that I have never previously attempted solve equations with variables on both sides solve equations with variables on both sides with help undo operations by working in reverse order from the order of operations WARM UP Describe and correct the error. x 5 = x 5 = x 5 = x = 6 x = Every term in the rd line should have been multiplied by Should be x 5 = x = 8 KEY CONCEPTS AND VOCABULARY Steps for with Variables on Both Sides Simplify each side of the equation if possible (distribute, combine like terms). If fractions exist, multiply by the LCD or distribute. Add or subtract to get variables on one side and numbers without variables on the other side of the equation Multiply or divide to isolate the variable Check answer Identity - an equation is true for any value (Always) No Solution an equation where there is no value to satisfy an equation (Never) EXAMPLES EXAMPLE : SOLVING EQUATIONS WITH VARIABLES ON EACH SIDE Solve. a) 8 + 5y = 7y b) c = 5 + c y = 5 c = 7 --

9 c) x + = 7 + x d) x x = 6 + x x = 9 x = EXAMPLE : SOLVING EQUATIONS WITH GROUPING SYMBOLS Solve. a) 5(y ) = 7(y + ) b) (g 7) = g + (g + ) y = g = 7 ( ) = 6( t 7) d) c) 8 + t ( b ) = ( 6 b + ) t = 6 b = 0 EXAMPLE : FINDING SPECIAL SOLUTIONS Solve. a) x + 8 = 5(x 7) x b) 6(q ) 0 = 6q 8 No Solution Identity, All real numbers EXAMPLE : WRITING EQUATIONS Five times the sum of a number and is the same as multiplied by less than twice the number. What is the number? Write an equation and solve. 5(x + ) = (x ) x = 8 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: -5-

10 TARGET LITERAL EQUATIONS MACC.9.A-CED.A.: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. RATING LEARNING SCALE use formulas to solve real-world problems or to solve more challenging problems that I have never previously attempted solve equations for given variables solve equations for given variables with help understand that equations may contain more than one variable. WARM UP Evaluate each expression for the given values of the variables. ) x + y; x =, y = ) x y + xy; x =, y = 0 0 KEY CONCEPTS AND VOCABULARY Literal Equation - an equation that uses at least letters as variables. You can solve for any variable in terms of the other variables. EXAMPLES EXAMPLE : REWRITING A LITERAL EQUATION Solve the following for the variable specified. a) x + y =, for y b) x h = g, for x y = x + x = h + g c) x + y = 6, for y d) Hg = j + a for a. y = x + a = Hg j -6-

11 e) v = ⅓whl, for w f) + x g = y for g w = V hl g = + x y EXAMPLE : LITERAL EQUATIONS FOR REAL WORLD SITUATIONS A car s fuel economy E (miles per gallon) is given by the formula E =m/g, where m is the number of miles driven and g is the number of gallons of fuel used. a) Solve the formula for m. m = Eg b) If Claudia s car has an average fuel consumption of 0 miles per gallon and she used 9.5 gallons, how far did she drive? 85 miles EXAMPLE : REWRITING A GEOMETRIC FORMULA The formula for the volume of a cylinder is V = πr h, where r is the radius of the cylinder and h is the height. a) Solve the formula for h. h = V πr b) What is the height of a cylindrical swimming pool that has a radius of feet and a volume of 80 cubic feet? Use. for pi. feet RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: -7-

12 TARGET RATIOS AND PROPORTIONS MACC.9.A-REI.B.: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MACC.9.N-Q.A.: Define appropriate quantities for the purpose of descriptive modeling. RATING LEARNING SCALE WARM UP Simplify each product. ) compare ratios and solve proportions in real-world situations or to solve more challenging problems that I have never previously attempted compare ratios solve proportions compare ratios with help solve proportions with help understand the definition of a ratio, rate, and proportion ) 66 ) / 6 KEY CONCEPTS AND VOCABULARY A Ratio is a comparison of two numbers by division. The ratio x to y can be expressed as: x to y x:y x/y A Rate is a ratio with different units of measure, such as price per pound, miles per hour, etc. A rate with a denominator of is called a. Unit Rate A Proportion is when two ratios are equal. In order to check this, reduce both fractions to lowest form and check if they are the same. Another way to check this is to use cross products. If the cross products are equal, then a proportion exists. If a b = c d, then ad = bc EXAMPLES EXAMPLE : DETERMINING WHETHER RATIOS FORM PROPORTIONS Determine whether the ratios form proportions. a) 7 8, 9 56 b) ,.5 c) 5, 6 0 Yes No Yes -8-

13 EXAMPLE : SOLVING A PROPORTION Solve. a) x = 8 b) x = 7 0 x = 9 x = 5 c) h = h d) x x + = x = 5 x = 7 EXAMPLE : SOLVING A REAL-WORLD SITUATION INVOLVING A PROPORTION Jeff rides a 0-mile trail every Saturday. It takes him hours. At this rate, how far can he ride in 7 hours? Set up a proportion and solve. (Let m represent miles) 0 = x 7 5 miles EXAMPLE : SOLVING WITH SCALE MODELS In a road atlas, the scale for the map of Connecticut is 5 inches = miles. The scale for the map of Texas is 5 inches = miles. What are the distances in miles represented by inches on each map? Set up proportions and solve..9 miles in Connecticut and 76.8 miles in Texas RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: -9-

14 TARGET PERCENTAGES MACC.9.N-Q.A.: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. RATING WARM UP LEARNING SCALE Solve each equation. solve percent problems in real-world situations or more challenging problems that I have never previously attempted solve percent problems solve percent problems with help understand that one can solve a percent problem in a variety of ways ) x (x ) = x ) + 6c = 6 c No Solution c = 5 KEY CONCEPTS AND VOCABULARY THE PERCENT PROPORTION A is P percent of B can be represented by A B = P 00 and B 0 where A is a part of the whole and B is the whole and also called the base. P is the percent. THE PERCENT EQUATION A is P percent of B can be represented by A = P % B and B 0 where A is a part of the whole and B is the whole and also called the base. P is the percent as decimal. SOLVING PERCENT PROBLEMS Problem Type Example Proportion Equation Find a percent What percent of is.5? Find a part What is 5% of 00? Find a base 0% of what number is?.5 = p 00 a 00 = 5 00 b = = p% a = 5% 00 = 0% b -0-

15 EXAMPLES EXAMPLE : FINDING A PERCENT Find each percent. a) What percent of 50 is 8? b) What percent of 6 is 8? 8% 9% c) What percent of 80 is 5? d) What percent of 00 is 0? 65% 05% EXAMPLE : FINDING A PART Find each percent. a) What is % of 80? b) What is 78% of 5? c) What is 98% of 00? d) What is % of 5? 96.5 EXAMPLE : FINDING A BASE Find each base. a) 5% of what number is.? b) 56% of what number is.8? c) 5% of what number is? d) % of what number is 0? EXAMPLE : SOLVING A REAL-WORLD SITUATION INVOLVING A PERCENTAGE $88 tickets to the Miami Dolphins game are offered at a 5% discount. What is the amount of the discount? $.0 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: --

16 TARGET PERCENT OF CHANGE MACC.9.N-Q.A.: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. RATING WARM UP LEARNING SCALE find the percent of change in real-world situations or more challenging problems that I have never previously attempted find the percent of change find the percent of change with help understand the definition of percent of change Solve each percent problem. ) What percent of is 00? ) What is 0% of 5? ) 5 is what percent of 5? 769%.5 % KEY CONCEPTS AND VOCABULARY Percent Change - expresses an amount of change as a percent of an original amount. If the new amount is greater than the original, then the percent change is called a. Percent Increase If the new amount is less than the original, then the percent change is called a. Percent Decrease PERCENT CHANGE P % = amount of increase or decrease original amount amount of increase = new amount original amount of decrease = original amount new amount EXAMPLES EXAMPLE : DETERMINING WHETHER EACH PERCENT CHANGE IS INCREASE OR DECREASE Determine whether each percent change is an increase or decrease. Then find the percent change. a) original amount: 6 b) original amount: 05 new amount: 0 new amount: 95 Increase; 67% Decrease; 9.5% c) original amount: 5 d) original amount: 5.5 new amount: 7 new amount: 65 Decrease; % Increase; % --

17 EXAMPLE : FINDING A PERCENT DECREASE a) A dress is on sale. The original price is $0. The sale price is $ What is the discount expressed as a percent change? 5% discount b) A video game is on sale for $50. The original price was $65. What is the discount expressed as a percent change? % discount EXAMPLE : FINDING A PERCENT INCREASE a) A music store buys a saxophone for $800. The store then marks up the price of the saxophone to $500. What is the mark up expressed as a percent change? 5% mark up b) There were 75 students that attended University High School last year. This year, 95 students are attending. What is the percent of increase? 0% increase EXAMPLE : CALCULATING SALES TAX Emma is purchasing a new car for $,595 before tax. If the tax is 7% of the total sale, what is the final cost of the car? $5,6.65 EXAMPLE 5: DETERMINING FINAL PRICE Lucius clipped a 5% off coupon in the Sunday paper for a new suit. The original price for the suit is $50 and sale tax is 6.5% of the discounted price. What is the final cost of the suit? $.5 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: --