4.+Designing+Instruction+and+Assessment+to+Promote+Student+Learning



**Basics of the Equation **

The diagram on the right shows a basic equation. This equation is similar to problems which you may have done in ordinary mathematics such as:

equals 14 or do 30 - 16 to find that stood for an unknown number; in an equation we use variables, or any letter in the alphabet. When written algebraically the problem would be: x + 16 = 30

and the answer should be written: x = 14 **Solving Equations **

These equations can be solved relatively easy and without any formal method. But, as you use equations to solve more complex problems, you will want an easier way to solve them. Pretend you have a scale like the one shown. On the right side there are 45 pennies and on the left side are 23 pennies and an unknown amount of pennies. The scale is balanced, therefore, we know that there must be an equal amount of weight on each side. As long as the same operation (addition, subtraction, multiplication, etc.) is done to both sides of the scale, it will remain balanced. To find the unknown amount of pennies of the left side, remove 23 pennies from each side of the scale. This action keeps the scale balanced and isolates the unknown amount. Since the weight(amount of pennies) on both sides of the scale are still equal and the unknown amount is alone, we now know that the unknown amount of pennies on the left side is the same as the remaining amount (22 pennies) on the right side. Because an equation represents a scale, it can also be manipulated like one. The diagram below shows a simple equation and the steps to solving it. The diagram below shows a more complex equation. This equation has both a constant and a variable on each side. Again, to solve this you must keep both sides of the equation equal; perform the same operation on each side to get the variable "x" alone. The steps to solving the equation are shown below. Take a look at the equation below. As you can see, after the variable is subtracted from the left and the constants are subtracted from the right, you are still left with 2x on one side.
 * Initial Equation / Problem || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x + 23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Subtract 23 from each side || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x + 23 - 23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">45 - 23 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result / Answer || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result / Answer || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result / Answer || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Initial Equation / Problem: || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x + 23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x + 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Subtract x from each side || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x - x + 23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x - x + 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x + 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Subtract 45 from each side || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">23 - 45 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x + 45 - 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Answer || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Answer || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Answer || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Initial Equation / Problem || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x + 23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">3x + 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Subtract x from each side || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">x - x + 23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">3x - x + 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">23 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x + 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Subtract 45 from each side || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">23 - 45 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x + 45 - 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Switch the left and right sides of the equation || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Subtract 45 from each side || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">23 - 45 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x + 45 - 45 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Result || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Switch the left and right sides of the equation || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 ||
 * <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">Switch the left and right sides of the equation || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">2x || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">= || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">-22 ||

= = = = = Equation Basics Worksheet = 

Solve each equation
 * **S1** ||  x + 1 = 9  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||
 * **1** ||||  x + 3 = 5  ||
 * ^  |||| x = ||
 * |||| <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **2** ||  x + -6 = 9  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **3** ||  -32 = x + 3  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **4** ||  29 + -1x = 13  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **5** ||  46 = 47 + -1x  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **6** ||  12 = -1x + 1  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **7** ||  4x = 16  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **8** ||  2x = 10  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **9** ||  10x = 130  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **10** ||  14 = -2x  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **11** ||  -3 + 2x = 11  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **12** ||  4x + 6 = -10  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **13** ||  x + 9 = 18 + -2x  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **14** ||  2x + 6 = 4x + -2  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **15** ||  -1x + -1 = 221 + 2x  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||


 * **16** ||  15 + 5x = 0  ||
 * ^  || x = ||
 * || <span style="color: blue; display: block; font-size: 9.5pt; line-height: 115%; text-align: right; text-decoration: none; textunderline: none;">[|step-by-step] ||