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Verbal PhraseExpression The sum of six and a number 6 + x sum + six 6 number x Eight more than a number y + 8 more than + A number plus five n + 5 plus + A number increased by seven x increased A number decreased by nine n – 9– decreased Ten times a number Seven divided by a number 10 n 7 x timesdivided The sum of six and a number 6 + x Eight number Eight more than a number five number A number plus five seven number A number increased by seven nine number A number decreased by nine Ten number Ten times a number 8 y y n n 5n n x x n n – 9 or ( )10n ( ) n (10 ) n 10 (10n) Seven 7 x number T RANSLATING V ERBAL P HRASES

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M ODEL AND S OLVE R EAL-LIFE P ROBLEMS In Mathematics there is a difference between a phrase and a sentence. Phrases translate into expressions; sentences translate into equations or inequalities. Expressions Phrases Equations or Inequalities Sentences

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M ODEL AND S OLVE R EAL-LIFE P ROBLEMS PhraseExpression The sum of six and a number The sum of six and a number is 6 + x 6 + x = is = SentenceEquation The sum of six and a number is twelve.6 + x = 12is twelve. = 12 SentenceInequality The sum of six and a number is less than twelve. 6 + x < 12 is less than twelve. < 12 The sum of six and a number6 + x The sum of six and a number is 6 + x = The sum of six and a number is twelve.6 + x = 12 In this sentence, “is” says that one quantity is equal to one another. In this sentence, the words “is less than” indicate an inequality. The sum of six and a number is less than twelve. 6 + x < 12 number is twelve. 6 + x = 12 is less than twelve. 6 + x < 12

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M ODEL AND S OLVE R EAL-LIFE P ROBLEMS Use three steps to write a mathematical model. W RITE A VERBAL MODEL. A SSIGN LABELS. W RITE AN ALGEBRAIC MODEL. Writing algebraic expressions, equations, or inequalities that represent real-life situations is called modeling. The expression, equation, or inequality is a mathematical model.

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Writing an Algebraic Model You and three friends are having a dim sum lunch at a Chinese restaurant that charges $2 per plate. You order lots of plates. The waiter gives you a bill for $25.20, which includes tax of $1.20. Use mental math to solve the equation for how many plates your group ordered. Understand the problem situation before you begin. For example, notice that tax is added after the total cost of the dim sum plates is figured. S OLUTION

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L ABELS V ERBAL M ODEL Writing an Algebraic Model Cost per plate Number of plates = Bill Tax – Cost per plate =2 Number of plates = p Amount of bill = Tax = 1.20 (dollars) (plates) – 2 = p 2p2p = p = 12 Your group ordered 12 plates of food costing $ A LGEBRAIC M ODEL

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A PROBLEM SOLVING PLAN USING MODELS Writing an Algebraic Model Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know. Assign labels to each part of your verbal problem. Use the labels to write an algebraic model based on your verbal model. Solve the algebraic model and answer the original question. V ERBAL M ODEL Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know. Assign labels to each part of your verbal problem. Use the labels to write an algebraic model based on your verbal model. Solve the algebraic model and answer the original question. Check that your answer is reasonable. A LGEBRAIC M ODEL L ABELS S OLVE C HECK

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Using a Verbal Model J ET P ILOT A jet pilot is flying from Los Angeles, CA to Chicago, IL at a speed of 500 miles per hour. When the plane is 600 miles from Chicago, an air traffic controller tells the pilot that it will be 2 hours before the plane can get clearance to land. The pilot knows the speed of the jet must be greater then 322 miles per hour or the jet could stall. a. At what speed would the jet have to fly to arrive in Chicago in 2 hours? Is it reasonable for the pilot to fly directly to Chicago at the reduced speed from part (a) or must the pilot take some other action? b.

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L ABELS V ERBAL M ODEL Using a Verbal Model Speed of jet Time = Distance to travel Speed of jet = x Time = 2 Distance to travel = 600 (miles per hour) (miles) (hours) 600 = x = 300 A LGEBRAIC M ODEL a. At what speed would the jet have to fly to arrive in Chicago in 2 hours? 2 x S OLUTION To arrive in 2 hours, the pilot would have to slow the jet down to 300 miles per hour. You can use the formula (rate)(time) = (distance) to write a verbal model.

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Using a Verbal Model It is not reasonable for the pilot to fly at 300 miles per hour, because the jet could stall. The pilot should take some other action, such as circling in a holding pattern, to use some of the time. Is it reasonable for the pilot to fly directly to Chicago at 300 miles per hour or must the pilot take some other action? b.

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