Connecting Middle Grades to Advanced Placement* Mathematics A Resource and Strategy Guide

Working with Formulas New: 09/08/08 Objective: Students will manipulate formulas for area, volume, perimeter and circumference and interpret the meanings of the new versions of these formulas. Connections to Previous Learning: Students should be able to solve one step equations. Connections to AP*: AP Calculus Topics: Areas and Volumes Materials: Student Activity pages Teacher Notes: The skill of manipulating literal equations to solve for a particular variable is essential in mathematics and science. Students seem to manage reasonably well with solving for a variable in an equation that involves only one variable along with other “numbers.” But when the equation involves several variables, students often struggle to isolate the required variable. Experience and practice with this skill will benefit students at all levels. In this lesson, students are asked to solve one-step equations using familiar formulas for area, volume, perimeter, and circumference. Using the revised version of the formula, students are asked to interpret its meaning in the context of the original formula. The last several questions give students new versions of familiar formulas and ask them to explain their meanings.

*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. ®

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Student Activity

Working with Formulas The formulas below are used with various geometric shapes to determine area, volume, perimeter, or circumference. For each formula below, solve for the specified variable. Show all the steps and explain the meaning of the “new” equation using complete sentences. For example, the distance formula is d = rt . Solve d this equation for r by dividing both sides of the equation by t. The result is r = which means that t rate is equal to distance divided by time. 1. Solve A = lw for l, where A is the area of a rectangle, l is the length of the rectangle, and w is the width of the rectangle.

2. Solve C = π d for d, where C is the circumference of a circle and d is the diameter of the circle.

3. Solve C = 2π r for π , where C is the circumference of a circle and r is the radius of the circle

4. Solve P = 4s for s, where P is the perimeter of a square and s is the length of one of the sides.

5. Solve V = lwh for h, where V is the volume of a rectangular prism, l is the length of the rectangular prism, w is the width of the rectangular prism, and h is the height of the rectangular prism.

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Student Activity

6. Solve A = bh for b, where A is the area of a rectangle, b is the length of the base of the rectangle, and h is the height of the rectangle.

7. Solve A = π r 2 for π , where A is the area of a circle and r is the radius of the circle.

8. Solve V = Bh for h, where V is the volume of a cylinder, B is the area of the base of the cylinder, and h is the height of the cylinder.

2A is the result of solving the formula for the area of a triangle for the height. b Explain the meaning of this formula using a complete sentence.

9. The equation h =

10. Interpret the meaning of the formula h =

2A . Which area formula was used to solve for (b1 + b2 )

the height, h?

P − l has been solved for the width from which formula? Explain the new 2 meaning of this formula.

11. The equation w =

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Connecting Middle Grades to Advanced Placement* Mathematics A Resource and Strategy Guide

Working with Formulas Answers: A lw 1. = w w A =l w A l= w The length of the rectangle is the area divided by the width of the rectangle.

2.

C

π C

π

=

πd π

=d

d=

C

π The diameter of a circle is the circumference divided by the value of pi. 3.

4.

C 2π r = 2r 2r C =π 2r C π= 2r The value of pi is the circumference of a circle divided by twice the radius. P 4s = 4 4 P =s 4 s=

P 4

The side of a square is the perimeter of the square divided by four.

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Answers

5.

V lwh = lw lw V =h lw h=

V lw

The height of a rectangular prism is the volume divided by the product of the length and width.

6.

A bh = h h A =b h b=

A h

The base of a rectangle is the area divided by the height.

7.

8.

A π r2 = 2 r2 r A =π r2 A π= 2 r The value of pi is the area of a circle divided by the radius squared. V Bh = h h V =h B V h= B The height of the prism is the volume of a prism divided by the area of the base of the prism.

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Answers

9. The height of the triangle is twice the area divided by the base of the triangle. 10. The height of a trapezoid is twice the area divided by the sum of the bases. The formula for area of a trapezoid was used to solve for the height. 11. Perimeter of a rectangle; the width of a rectangle is one-half the perimeter of the rectangle minus its length.

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