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GED® ADULT EDUCATOR MATHEMATICAL REASONING INSTITUTE FOUNDATIONS OF ALGEBRAIC PROBLEM SOLVING TOPIC 2 1 Topic 2 - Foundations of Linear Equations & Inequalities 2 LESSON GOALS A.1.c – Write linear expressions as part of word-tosymbol translations or to represent common settings. A.2.c – Write one-variable and multi-variable linear equations to represent context. A.3.a – Solve linear inequalities in one variable with rational number coefficients. A.3.d – Write linear inequalities in one variable to represent context. A.6.c – Use slope to identify parallel and perpendicular lines and to solve geometric problems. MATHEMATICAL REASONING INSTITUTE 3 WORKING WITH THE CONTENT Think, Pair, Share Think & Work Alone Cooperative Learning In Small Groups MATHEMATICAL REASONING INSTITUTE 4 Symbols to Words Key Phrase Sum, Increase, Add, All together, Total Math Symbols + (Addition) Subtract, Decrease, Difference Minus, Fewer - (Subtraction) Times, Multiply, Product x (Multiplication) Divide, Per, Quotient ÷ (Division) MATHEMATICAL REASONING INSTITUTE 5 Symbols to Words Activity Process: Five posters numbered 1 to 5 with linear expressions written on them Number off 1 to 5 Start at the poster numbered with your number 15 seconds at each poster to write as many word phrases as you can MATHEMATICAL REASONING INSTITUTE 6 Words to algebraic expressions Process: Index card with a word phrase Write the word phrase on your poster Reach consensus on the correct translation to an algebraic expression Record translation on poster MATHEMATICAL REASONING INSTITUTE 7 Try These with a Partner! For each of the following, write an expression in terms of the given variable that represents the indicated quantity. The total cost of a mechanic to repair your car if he spends h hours on the job and charges $39 for parts and $45 per hour for labor. The sum of three consecutive numbers if the first number is n. MATHEMATICAL REASONING INSTITUTE 8 Try These with a Partner! For each of the following, write an expression in terms of the given variable that represents the indicated quantity. The amount of money in Steve’s bank account if he put in d dollars the first year, $600 more the second year than the first year, and twice as much the third year as the second year. The first side of a triangle is s yards long. The second side is 3 yards longer than the first side. The third side is three times as long as the second side. What is the perimeter of the triangle in feet? MATHEMATICAL REASONING INSTITUTE 9 Translating Words to Linear Equations Equations n + 32 = 40 4x = 36 K - 7 = 15 3w = -15 6/x = 2 MATHEMATICAL REASONING INSTITUTE Words A number increased by 32 is equal to 40. Four times a number is 36. Seven less than a number is 15. The product of a number and 3 is -15. Six divided by a number is equal to 2. 10 Context to Linear Equations Context John called a plumber to fix his broken toilet. In addition to a $50 fee for the visit, the plumber charges $22 per hour. Write an equation that models this situation to determine how many hours the plumber took if John’s total bill was $116. MATHEMATICAL REASONING INSTITUTE Equation h = hours the plumber worked 50 + 50 + 22h 50 + 22h = 116 11 Context to Linear Equations Context Jane needs $2100 for a vacation for spring break. She plans to save $350 per month for the trip. Write an equation that represents this situation to help Jane determine how many months it will take her to save for the trip at this rate. MATHEMATICAL REASONING INSTITUTE Equation m = number of months to save for trip 350m 350m = 2100 12 Context to Multi-variable Linear Equations Context A line on a graph represents a ramp that extends from the back of a moving truck to the ground. The line has a slope of -.5 and passes through (8, 0). The y-intercept represents the height of the back of the moving truck. Write an equation with two variables that represents this situation. MATHEMATICAL REASONING INSTITUTE Equation y = mx + b y = -.5x + b 0 = -.5(8) + b 0 = -4 + b 4=b y = -.5x + 4 13 Linear Inequalities https://www.youtube.com/watch ?v=8hhewFQ_K0w 14 Inequalities vs. Equations Activity MATHEMATICAL REASONING INSTITUTE 15 Slopes of Parallel & Perpendicular Lines https://www.khanacademy.org/ math/algebra/linear-equationsand-inequalitie/more-analyticgeometry/v/equations-ofparallel-and-perpendicular-lines 16 Try with a partner! • Describe in your own words the relationship of the slopes of parallel lines. • Describe in your own words the relationship of the slopes of perpendicular lines. MATHEMATICAL REASONING INSTITUTE 17 Try with a partner! Using what you know about parallel and perpendicular lines and the relationships of their slopes and what you know about writing the equations of lines do the following: • Find the equation of the line that is perpendicular to y = -4x + 10 and passes through the point (7, 2). Leave your answer in standard form. • Find the equation of the line that is parallel to y = -4x + 10 and passes through the point (7, 2). Leave your answer in standard form. MATHEMATICAL REASONING INSTITUTE 18 Lunch Time! Please come back on time. 19