Expressions and equations
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The document discusses translating verbal phrases into algebraic expressions and using verbal models to write mathematical equations and inequalities. It provides examples of common verbal phrases involving numbers, operations, and variables and their corresponding algebraic translations. It also outlines a three-step process for writing a mathematical model from a word problem: 1) write a verbal model, 2) assign labels, and 3) write the algebraic model. Finally, it demonstrates this process with a sample word problem about the cost of dim sum plates after tax.
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Expressions and equations
- 1. TRANSLATING VERBAL PHRASES Verbal Phrase Expression The sum of six and a number 6 + xsum +six 6number x Eight more than a number y + 8more than + A number plus five n + 5plus + A number increased by seven x + 7+increased A number decreased by nine n – 9–decreased Ten times a number Seven divided by a number 10 • n 7 x times divided • The sum of six and a number 6 + x Eight numberEight more than a number fivenumberA number plus five sevennumberA number increased by seven ninenumberA number decreased by nine Ten numberTen times a number 8yy + 8 1010 • nn 5nn + 5 7xx + 7 9nn – 9 or ( )10n10n( )n(10 )n10(10n) Seven 7 x number
- 2. USING A VERBAL MODEL 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
- 3. USING A VERBAL MODEL Phrase Expression The sum of six and a number The sum of six and a number is 6 + x 6 + x =is = Sentence Equation The sum of six and a number is twelve. 6 + x = 12is twelve. = 12 Sentence Inequality 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 number 6 + 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
- 4. USING A VERBAL MODEL Use three steps to write a mathematical model. WRITE A VERBAL MODEL. ASSIGN LABELS. WRITE 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.
- 5. 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. SOLUTION
- 6. LABELS VERBA L MODEL Writing an Algebraic Model Cost per plate • Number of plates = Bill Tax– Cost per plate = 2 Number of plates = p Amount of bill = 25.20 Tax = 1.20 (dollars) (dollars) (dollars) (plates) 25.20 1.20–2 =p 2p = 24.00 p = 12 Your group ordered 12 plates of food costing $24.00. ALGEBRAIC MODEL
- 7. 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. VERBAL 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. Check that your answer is reasonable. ALGEBRAIC MODEL LABELS SOLVE CHECK
- 8. Using a Verbal Model JET PILOT 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.
- 9. LABELS VERBA L MODEL 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 ALGEBRAIC MODEL a. At what speed would the jet have to fly to arrive in Chicago in 2 hours? 2 x SOLUTION 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.
- 10. 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.