Keeping the Meaning in Mathematics: The Craft of Word Problems

Introduction

Word problems are a problem. Students of all levels continually struggle with word problems; however, there is a solution to this "problem". The objective for this curriculum unit is to show students the multiple ways that system of equations can be used to solve real-world problems. In order for students to be engaged and interested in learning, they need to see the real-world practicality behind the math concept being learned. Without a lot of practice, System of Equations can be difficult for students to compute correctly and efficiently. When difficulty is combined with a lack of interest or a question of relevance, the student can become frustrated and disengaged. Therefore, in order to capture the attention and interest of my students, I will first discuss what a system of equations is and then explain how to solve one using three different methods. Utilizing the substitution method, addition/subtraction elimination and multiplication/elimination methods, we will review how to solve a system of equations.

Once the students understand those concepts, I will discuss and review I = Prt and d = rt. Using the simple interest formula, I = Prt, the students will be exposed to how their money can grow in a year. This will undoubtedly be of interest to any student. The students will also be able to see the advantage and real-world practicality of calculating the distance an object can travel, given the rate of speed and time using the d = rt formula. Then I will combine the distance and interest formulas together with a system of equations and demonstrate how to solve real-world scenarios and exciting problems. I will also provide practical examples of problems involving wind and water currents, chemistry problems involving mixtures and solutions, more interest problems involving the time value of money, and other practical examples that involve systems of equations. I will spend two days on each of the three methods. Next, I will spend another three or four days introducing the interest and distance formulas, incorporating them into system of equations. It is critical that the students get a feel for solving the problems before I introduce the more complicated word problems. The distance formulas and simple interest formulas are not extremely complicated, and my students typically pick up on them quickly. I have also introduced them earlier in the year and use them as warm-up problems several times per week.

The first system of equations I will introduce is one that will require the substitution method to solve. For example;

y = 4x 3x + 4y = 36

"Teddy has 8 more songs on his iPod than Tyler. Together they have 112 songs. How many songs does each person have?"

Another example of a system of equations solvable by substitution is;

x + 3y = 9 2x - 5y = 27

2x + 4y = 33 2x + 6y = 54

"Teddy cuts grass in the summer for $ 20 a lawn. His neighbor, Gavin is very competitive and decides to cut grass as well but only charges $ 18 a lawn. Their total combined revenue one summer was $ 348. Teddy made $ 132 more than Gavin. How many lawns did they each cut?"

The last type of system of equations that I will speak of is one that requires the multiplication elimination method. An example of this type of system is;

3x + 6y = -6, 5x - 2y = 14.

"The air-mail rate for letters to Europe is 45 cents per half-ounce and to Africa as 65 cents per ounce. If Shirley paid $ 18.55 to send 35 half-ounce letters to Europe and Africa, how many did she send to Africa?"

This is just a brief overview of the types of problems that I will cover with my class in this unit.