User:Susan D. Fall/Books/Foundations in Mathematics
Foundations in Mathematics
A Guide for Non-math Majors
How to Study Math When I think about learning math, one word comes to mind, perseverance. As Americans, we tend to believe that all first-rate math students have an innate mathematical ability. On the other hand, in Japan, where students generally do quite well in math, mathematical success is more likely to be attributed to hard work. My personal experience and observation have taught me that diligence is the key to mastering mathematical concepts. Keep in mind that everyone, even mathematicians, occasionally struggle with math problems. What are the things you have to do to be deemed persevering? First, you have to read the text, section by section, and after you read through the examples, you need to work them out with pencil and paper. Next, write out the definitions of new terms. Write out any new formulas that are mentioned and what they are used for along with the definition of each component of the formulas. For example, you might write down that the formula F=1.8C+32 is used to convert degrees Celsius to degrees Fahrenheit where the component F represents degrees Fahrenheit and the component C represents degrees Celsius. So, when you are doing your homework, and you encounter a problem relating to Fahrenheit and Celsius, you have the formula right before you. Make sure to keep up with your homework. Also, write down what the directions mean. For example, Simplify means to perform any addition, subtraction, multiplication or division that appear in the problem set. You can’t do the problems unless you know what you are being directed to do. It’s true that there is much to be learned from your mistakes. So, when you are working through your homework, and get a wrong answer, take the time to go through your work to find the error. This will help you to correct any misconceptions that might have led you astray or will keep you from repeating careless mistakes in the future. Learning Objective: Combine like terms If we were sitting at a table with a pile of paper clips, pencils, markers, rubber bands and sticky notes in the center, and If I were to ask you to tell me how many items were in the pile, you would automatically count the things that are alike. For example, you might tell me that there were 5 paper clips, plus 2 markers, plus 9 rubber bands, plus 12 sticky notes. Now, if I were to add two Xs made out of stenciled paper, your answer would become: 5 paper clips, plus 2 markers, plus 9 rubber bands, plus 12 sticky notes, plus 2 Xs. Now, if I were to add 10 Ys made out of stenciled paper, your answer would become: 5 paper clips, plus 2 markers, plus 9 rubber bands, plus 12 sticky notes, plus 2 Xs, plus 10 Ys. The best information we can give about a collection of items is to combine the items that are alike. When we take away items, we also take them away from items that are the same. So If I were to take away or subtract 5 paper clips, 2 markers, 2 rubber bands, and 7 sticky notes. We would be left with 7 rubber bands, plus 5 sticky notes, plus 2Xs, plus 10Ys. Next, take away 7 rubber bands and 5 sticky notes. We would have 2Xs and 10Ys which we can write in shorthand as 2X+10Y. Suppose I put these paper letters in a pile in the center of the table: X, X, Y, Y, Y,Y,Z, Z, Z, Z, Z, Z. We have 2Xs plus 4Ys, plus 6Zs which we write as 2X + 4Y + 6Z. Similarly, X, Y, Y , Z, X, Y, Z, Z, Y, Z, Z, Z is 2X + 4Y + 6Z. If I were to add 2 Zs to the collection, it would look like X, Y , Y , Z, Z, Z, X, Y, Z, Y, Z, Z which gives our original “shorthand” 2X + 4Y + 6Z plus 2Z which would give us 2X + 4Y +8Z. Taking two Zs away from the original set leaves X, Y , Y , Z, Z, Z, X, Y, Z, Y. The 6Zs minus 2Zs leaves 4Zs, written as 2X +4Y +6Z – 2Z, becomes 2X + 4Y + 4Z. All we are doing, then, is counting, or combining, terms that are alike. Terms are easy to recognize, they are separated by a plus or minus sign. The directions for combining like terms is to Simplify. (Actually, Simplify means to add, subtract, multiply or divide, but for now, we are simply combining with addition and subtraction) Examples: Simplify the following: (Combine terms that are alike) 1. X + X + Y + Y + Y = 2x + 3Y 2. 3X + 2X + 5Y = 5X + 5Y 3. X + 5Y + 2Y + 3X + 8Z = X + 7Y + 11Z 4. 5X – 2X +7Y + Y + 9Z – 4Z = 3X + 8Y + 10Z Now, lets jumble them up like they would be in the center of the table. You look for like terms and combine them just the same. 5. 3Z + 4X + 2Z + 8Y + 2X + Y = 6X + 9Y + 5Z Now lets take some off of the table and combine what we have left. 6. 3Z +4X + 2Z + 8Y +2X +Y -3X -4Y – Z = 3X +5Y + 4 Z
Introduction to Equations It is important to understand the significance of the equal sign in an equation. The equal sign tells us that the left side of the equation is equal to the right side of the equation. And, no matter what, that must always be true. Therefore, if you make a change to one side of the equal sign, you must make the same change to the other side. For example, in the equation 6 + 5 = 11, if I add 2 to the left side, I must also add 2 to the right side for the = to be true. 6 + 5 + 2 = 11 + 2 13 = 13 Picture a ruler balancing on my finger. The left side equals the right side when the ruler is in balance. However, if I put an eraser on one side, the ruler is out of balance. To put it back in equilibrium, I must also put an eraser of equal size on the other side. The balanced ruler is like an equation where my finger is the equal sign. In solving equations, it will become necessary to perform calculations (add, subtract, multiply, or divide) on the equation. Just remember, what you do to one side of the equation, you must do to the other side. Lets subtract 2 from the equation 6 + 5 = 11 so we get 6 + 5 -2 = 11 – 2 9 = 9 Now, multiplying both sides by 2 6 + 5 = 11 2(6+5) = 11*2 (Remember Please Excuse My Dear Aunt Sally.) You do what’s inside the parentheses first. 2*11 = 11*2 In the equation, 2*3 = 6, if we divide one side by 2, we must divide the other side by 2 also. 2*3 = 6
2*3/2 = 6/2
3 = 3
