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Mathematics 8 Items to Support Formative Assessment Unit 3: Analyzing Functions and Equations 8.EE.C.7 Solve linear equations in one variable. 8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.C.7a (Short Item) Given the equation, determine if the solution is “Always true”, “Sometimes true”, or “Never true”. Show your work and explain why. 1. x - 3 = 3 + x 2. 10x = x 3. 2x (15) = 30x Possible answers: 1. x – 3 = 3 + x x–x–3=3+x–x -3=3 not true; Never true There is never a number that will make this equation true. 2. 10x = x 10x – x = x – x 9x = 0 divide by 9 on both sides Sometimes true; this equation is only true when x = 0. When x = any other number then it isn’t true! 3. 2x(15)=30x multiply 2(15) divide by 30 on both sides Always true; this equation is always true for x. When x= any number it is true on both sides of the equation. 8.EE.C.7 Solve linear equations in one variable. 8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. 8.EE.C.7b Leslie is considering the expressions 5( x + 2) - 6 and - (3x + 9) + 2(4 x + 8) . She wants to know if one expression is greater than the other for all values of x. 1. Which statement about the relationship between the expressions is true? a. The value of the expression 5( x + 2) - 6 is always equal to the value of the expression -(3x + 9) + 2(4 x + 8). b. The value of the expression 5( x + 2) - 6 is always less than the value of the expression -(3x + 9) + 2(4 x + 8). c. The value of the expression 5( x + 2) - 6 is always greater than the value of the expression -(3x + 9) + 2(4 x + 8). d. The value of the expression 5( x + 2) - 6 is always sometimes greater than and sometimes less than value of the expression -(3x + 9) + 2(4 x + 8). 2. Explain how you found your answer to question #1. 3. Write a new expression that always has a greater value than both of these expressions. Solutions: 1. b 5( x + 2) - 6 5x +10 - 6 5x - 4 2. and -(3x + 9) + 2(4 x + 8) -3x - 9 + 8x +16 5x + 7 When I simplified the expressions, I noticed that 5x +7 would always have a greater value than the expression 5x-4 because it has a greater y-intercept even though the slopes are the same. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. 3. Possible Solution: -5(-x+12) 8.EE.C.7a&b (Short item) Part 1: Solve the equation for b. 1 -3(4b -10) = (-24b + 60) 2 Part II: Select the statement that correctly interprets the solution to the equation in part I: a. b. c. d. There is no solution The solution is b=0 There are infinite solutions It can only be solved when b = -12 Solution: Part I: 1 -3(4b -10) = (-24b + 60) 2 -12b + 30 = -12b + 30 30 = 30 Part II: C 8.EE.C.7b Short Task McKala wants to buy a pair of boots. She goes to the mall and sees that Shoe World has the boots on sale for $25 off the original price. She is ready to purchase the boots when Jasmine informs McKala that next door at Shoe Parade the same boots are on sale for 40% off the original price. For what original price will these discounts be equal, if ever? Explain your reasoning. Let x = original price Store A: x - 25 = .60x Store B Setting up the equations and setting them equal to each other we find x - 25 = 0.60x. Like terms on like sides, we have x - 0.60x = 25. Simplify: 0.4x = 25. Divide both sides by 0.40. x = $62.50. There is one price for the boots when they are equal in price at both stores. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. 8.EE.C.7ab (Short item) Classify each equation as having one solution, no solution, or infinitely many solutions by placing a check mark in the appropriate column. No solutions One solution Infinitely many solutions No solutions One solution Infinitely many solutions 2(x +1)+ 3x = 2x +1 2.2x + 6 -.07x = -0.5x + 2(x + 3) -4x +18 = 2(6x + 9) Solution: 2(x +1)+ 3x = 2x +1 2.2x + 6 -.07x = -0.5x + 2(x + 3) -4x +18 = 2(6x + 9) 5(x + 2) - 2 = 9x - 3- 4x Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.