Solve Literal Equations Student Probe At a Glance Erin knows the area of several circles with differing radii. She would like to determine the radius of each circle by re-­‐writing the formula for the area of a circle. Solve for r. Answer: Lesson Description In this lesson students will re-­‐write formulas and equations in terms of a specified variable. The lesson builds on students’ previous understanding and expertise in solving equations in one variable. Rationale Formulas used to solve real world problems are usually given, and memorized, in a prescribed form. However, that form may not be in the most useful format when solving a particular problem or set of problems. Consider the question in the student probe. Suppose Erin knows the area of a circle is and needs to determine the radius of the circle. She can solve the problem two ways: or What: Solve literal equations for a specified variable Common Core State Standards: CC.9-­‐
12.A.REI.3 Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Matched Arkansas Standard: AR.9-­‐
12.SEI.AI.2.3 (SEI.2.AI.3) Solve linear formulas and literal equations for a specified variable. Mathematical Practices: Make sense of problems and persevere in solving them. Look for and make use of structure. Who: Students who cannot solve literal equations Grade Level: Algebra 1 Prerequisite Vocabulary: literal equation, variable, equation Prerequisite Skills: solve equations, order of operations Delivery Format: individual, small group, whole group Lesson Length: 30-­‐45 minutes Materials, Resources, Technology: graphing calculator (optional) Student Worksheets: none If Erin only needs to find the radius of one circle the first method is easier. But if she needs the radii of several circles, she can reduce the amount of calculation required by using the “new” formula . Now, if she knows the area of a circle is , the radius is easily found by . The ability to work with literal equations will be especially useful in geometry and science courses, and when using technology such as a calculator or spreadsheet software. Preparation None Lesson The teacher says or does… 1. Solve this equation for h. How did you solve it? 2. Keep in mind how you solved it. Now I want you to solve this equation for h. How is this like the first problem? How is it different? (Emphasize that the algebra was the same.) 3. Debbie commutes 40 miles to work each day. She is wondering what her speed is on her commute. She knows that distance = rate x time or . Solve for r. (Ask for a student volunteer to show how they worked the problem.) Expect students to say or do… If students do not, then the teacher says or does… Refer to Solving Equations. Prompt students. We divided. We used letters instead of numbers. Monitor students. If students continue to have difficulty, they may need more time solving equations before attempting this lesson. The teacher says or does… 4. We can make the new formula easier by substituting . What does the formula look like now? 5. Here are Debbie’s travel times (in hours) for five days. Find her rate for each day. 0.73, 0.78, 0.80, 0.68, 1.14 Which units go with our answers? Expect students to say or do… If students do not, then the teacher says or does… Monitor students Distance is measured in miles. Time is measured in hours. Therefore the rate is miles/hour 6. Garrett has a water trough for his cattle that is shaped like a rectangular prism. He wants to know how deep the water is in the trough, based on the volume of water it contains. He knows that Volume = length x width x height. Solve for h. 7. The length of Garrett’s trough is 9 feet, and its width is 4 feet. Use this data to simplify your new formula. What are the units? 8. If Garrett’ trough contains of water, how deep is the water? What is ? 9 feet x 4 feet = 36 feet2 . The teacher says or does… 9. Repeat with similar problems. Expect students to say or do… If students do not, then the teacher says or does… Teacher Notes 1. When solving literal equations formulas are rearranged into more convenient forms so values can be substituted for a variable later. Area, force, volume, and distance formulas are examples that can be starting points of solving a literal equation. 2. Emphasize that the algebra used when solving literal equations is the same as the algebra used when solving “normal” equations. However, because students are using letters (variables) rather than numbers, the results may not be as compact as they are used to seeing. 3. Understanding and applying the order of operations correctly is critical when solving literal equations. If students continue to have difficulty, make sure they are applying order of operations correctly. 4. Students will be more likely to see the value of solving literal equations if they are situated in real world examples, or if they are required to use the new formula numerous times. Variations None Formative Assessment The area of a triangle is given by the formula Answer: . Solve this formula for b. References Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on March 11, 2011, from http://iris.peabody.vanderbilt.edu/case_studies/ICS-­‐010.pdf