In this lesson, students practice the essential skill

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1 LEVEL NATIONAL MATH + SCIENCE INITIATIVE a b c a b b c b a c b a c b a c b Since a 0, a c b Algebra 1 or Math 1 in a unit on solving multi-step equations Mathematics Literal Equations Geometric Formulas ABOUT THIS LESSON In this lesson, students practice the essential skill of solving literal equations for a specific variable by showing the algebraic steps of their solution. In addition, students also must explain the meaning of the new equation. OBJECTIVES Students will solve literal equations. explain the meaning of geometric formulas. MODULE/CONNECTIONS TO AP* Rate of Change: Related Rates *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. G N P A V P Physical V Verbal A Analytical N Numerical G Graphical Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at i

2 COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. Targeted Standards A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V IR to highlight resistance R. See questions 1- COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.6: Attend to Precision. Students use the defined variables to explain the meaning of each re-expressed formula. ii Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at

3 FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Solve multi-step equations Solve equations involving radicals ASSESSMENTS The following formative assessment is embedded in this lesson: Students engage in independent practice. The following additional assessments are located on our website: Rate of Change: Related Rates Algebra 1 Free Response Questions Rate of Change: Related Rates Algebra 1 Multiple Choice Questions MATERIALS AND RESOURCES Student Activity Pages Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at iii

4 TEACHING SUGGESTIONS Students in mathematics and science must be able to manipulate literal equations to solve for a particular variable. Most students seem to manage reasonably well when solving for one variable in an equation that involves only that variable along with other numbers. When the equation involves several variables, however, students often struggle to isolate the required variable. One strategy for helping students decide on the appropriate solution steps is to create a second, similar equation by replacing all of the variables except the variable to be isolated with numbers. Tell students that they may not simplify the numbers by performing the indicated operations (addition, multiplication, etc.). Next have students solve the numerical equation and the literal equation side by side. Other possible strategies include: (1) write the variable to be isolated in a different color or draw a circle or box around it to set it apart from the other variables, and/or () have students write in any implied operations or groupings. Experience and practice with this skill will benefit students at all levels. This lesson can be used as a review or as an assessment. In order to reach the lesson s full potential, students need to not only perform the algebraic manipulations, but also to consider what each revised formula means. Suggested modifications for additional scaffolding include the following: 1 Provide a template explaining the meaning of the new equation. For example: 1a 1b 1c 1f The of the rectangle is the divided by the of the rectangle. The of the triangle is the divided by the height of the. The value of pi is the of a circle by the radius. Provide a template which presents the formula, solving for the required variable, and simplifying the new formula. Provide space for the student to explain the reasoning. For example, S Ph Ph + B Subtract Ph from each Ph side of the equation. S Ph B Simplify. S Ph B Divide each side of the equation by. S Ph B Simplify. S Ph B If preferred, write equation with variable solved for on the left. The area of the base of the prism is one-half of the difference between the total surface area of a prism and the product of the perimeter of its base and the height of the prism. iv Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at

5 NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from sixth grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 6th Grade Skills/Objectives 7th Grade Skills/Objectives Algebra 1 Skills/Objectives Geometry Skills/Objectives Algebra Skills/Objectives Pre-Calculus Skills/Objectives Solve literal equations (perimeter, area, and volume). Solve literal equations (perimeter, area, and volume). Solve literal equations (perimeter, area, and volume). Solve literal equations (perimeter, area, and volume). Solve literal equations (perimeter, area, and volume). Solve literal equations (perimeter, area, and volume). Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at v

7 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Literal Equations Geometric Formulas Answers 1. a. A lw l l A w l A w l The width of the rectangle is the area divided by the length of the rectangle. e. S Ph h h S P h S P h The perimeter of the base of the prism is the lateral surface area of a prism divided by its height. bh b. A A bh h h A b h A b h The base of the triangle is twice the area divided by the height of the triangle. c. f. S Ph + B Ph Ph S Ph B S Ph B S Ph B S Ph B The area of the base of the prism is one-half of the difference between the total surface area of a prism and the product of the perimeter of its base and the height of the prism. The value of pi is the circumference of a circle divided by twice the radius. d. V Bh h h V B h V B h The area of the base of the prism is the volume of a prism divided by the height of the prism. g. ( b + b ) h A ( b1+ b) h h h A b1+ b h b b A b b 1 h A 1 A b1 b h The length of one of the bases of a trapezoid is twice the area of the trapezoid divided by its height minus the second base of the trapezoid. Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at vii

8 Mathematics h. P ( l + ) w P l+ w l l P l w P w l The width of a rectangle is one-half the perimeter of the rectangle minus its length. 1 i. 3 V 3 Bh 3 3V Bh h h 3V B h 3V B h The area of the base of the pyramid is three times the volume of the pyramid divided by its height. j. k.. a. The height of the cylinder is the difference between the total surface area of the cylinder and twice the product of pi and the radius of the cylinder squared divided by twice the product of pi and its radius. S 6s 6 6 S s 6 S 6 S 6 s s 6 S s s > 0 because it is a length The length of the side of a cube is the square root of the quotient of the surface area of the cube and six. The value of pi is the area of a circle divided by the radius squared. viii Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at

9 Mathematics b. a + b c b b a c b a c b a c b But a> 0 so a c b The leg of a right triangle is the square root of the difference between the square of the hypotenuse and the square of the other leg of the same right triangle c. V πr π 3V π π r 4 3V π 3 r 4 3 r V 4π 3V r 3 4π The radius of the sphere is the cube root of the quotient of three times the volume of the sphere and four times pi. Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at ix

11 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Literal Equations Geometric Formulas 1. For each equation, solve for the given variable. Explain the meaning of the new equation. Example: Solve P 4s, for s, where P the perimeter of a square and s the side of a square. To receive full credit for this section, first show the steps needed to solve for the given variable. In order to solve P 4s for s, divide both sides by 4. The result is s 4 P. Second, explain the meaning of the answer. In this case, an acceptable answer would be: The side of a square is the perimeter of the square divided by 4. a. Solve A lw for w, where A area of a rectangle, l length of the rectangle, and w width of the rectangle. bh b. Solve A for b, where A area of a triangle, b base of the triangle, and h height of the triangle. c. Solve C πr for π, where C circumference of a circle and r the radius of the circle. d. Solve V Bh for B, where V volume of a prism, B the area of the base of the prism, and h the height of the prism. Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at 1

12 e. Solve S Ph for P, where S the lateral surface area of a prism, P the perimeter of the base of the prism, and h the height of the prism. f. Solve S Ph + B for B, where S the total surface area of a prism, P the perimeter of the base of the prism, h the height of the prism, and B the area of the base of the prism. ( b1+ b) h g. Solve A for b, where A area of a trapezoid, h height of the trapezoid, and b 1 1 and b are the lengths of the two bases of the trapezoid. h. Solve P (l + w) for w, where P is a rectangle s perimeter, l is the length, and w is the width. 1 i. Solve V Bh for B, where V the volume of a pyramid, B the area of the base of the pyramid, 3 and h the height of the pyramid. j. Solve A πr² for π where A the area of a circle and r the radius of the circle. Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at

13 k. Solve S πrh + πr² for h where S the total surface area of a cylinder, r the radius of the cylinder, and h the height of the cylinder.. The formula to determine the area of a square is is the side of a square. A s A s A s A s where A represents the area of the square and s Since s is the side of the square, s > 0, s A. When appropriate, use this process to solve the following questions. Remember to show the steps needed to solve for the given variable and to explain the meaning of the answer. a. Solve S 6s² for s, where S the surface area of a cube and s the length of the side of a cube. b. Solve a + b c for a, where a and b are the legs of a right triangle and c is the hypotenuse of the same right triangle. 4 c. Solve V πr 3 for r, where V is the volume of a sphere and r is the radius of the sphere. 3 Copyright 014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at 3