A Program of The Actuarial Foundation. Aligned with Common Core State and NCTM Standards.

What is an actuary? An actuary is an expert in statistics who works with businesses, governments, and organizations to help them plan for the future. Actuarial science is the discipline that applies math and statistical methods to assess risk.

6-8

DURATION
20

COLLECTION
Setting the Stage with Geometry

# Lesson 1: Perimeter and Area of 2D Shapes

In this lesson, students build knowledge of perimeter and area, and complete related worksheet problems related to building a stage for a concert.

STANDARDS (CCSS AND NCTM)

• Grade 7: Geometry (CCSS 7.G.B.4 and 6)
• Grades 6–8: Make Sense of Problems, Reason Abstractly and Quantitatively, Construct Viable Arguments, Model with Mathematics, and Attend to Precision (CCSS MP1-4 and 6); NCTM Geometry

OBJECTIVE
In ”Geometry Works! The Stage Takes Shape,” students will understand the formulas that measure the perimeter and area of these basic two-dimensional shapes: rectangles, circles, and triangles.

MATERIALS

Introduction to Formulas for Perimeter and Area

DIRECTIONS

Time Required: 40 minutes, plus additional time for worksheets, which may be split over two days.

1. On the board, draw a rectangle labeled with a length of 4 feet and width of 3 feet. Then draw a right triangle with a base of 4 feet, height of 3 feet, and hypotenuse (the side opposite the right angle) of 5 feet. Ask the class to determine the distance around each shape (14 feet for the rectangle, 12 feet for the triangle). Explain that perimeter is the distance around a polygon (a closed figure made up of line segments). To measure the perimeter of any polygon (a closed figure made up of line segments), you add together the lengths of the sides. Show that a quicker way to calculate perimeter for rectangles is to add the lengths of two adjacent sides and multiply by 2, i.e., 2(l + w).

2. Guided Practice: Either individually or in pairs, ask students to complete the following problems:

• What is the perimeter of a rectangle with length = 10 meters and width = 7 meters? (34 meters)
• What is the perimeter of a triangle with side lengths of 3 feet, 7 feet, and 9 feet? (19 feet)
• Using a ruler, draw and label a rectangle with a length of 5 inches and width of 7 inches. Calculate the perimeter (24 inches).
• Using a ruler, draw and label a right triangle with side lengths of 6 centimeters, 8 centimeters, and 10 centimeters. Calculate the perimeter (24 centimeters).
• Does the formula for calculating the perimeter of a rectangle apply to squares? Explain your thinking. (The formula does apply, but it is quicker to simply multiply the length of one side by 4 to find the perimeter. Make sure students understand that a square is also a rectangle, but a special type of rectangle.)

3. Draw a circle on the board. Draw a line from the center of the circle to the edge and mark it as 3 feet. Tell students that this is the radius. Point out that any line from the center of a circle to the edge is also a radius and would measure 3 feet. Indicate that the diameter of a circle is the distance from one edge of a circle to the other, passing through the center. In this case, the diameter is 6 feet. Then explain that, while circles do not have a perimeter, the distance around a circle is called the circumference. The formula for calculating the circumference: C (circumference) = π • d (diameter). Explain that π is a number equal to 3.14159... The decimal continues on infinitely, but to solve most math problems, 3.14 (pi taken to two decimal places) is acceptable.

4. Guided Practice: In groups or in pairs, ask students to calculate:

• The circumference of the circle you have drawn. Ask them to provide the answer rounded to the nearest half foot. (3.14 • 6 = 18.84 feet; the answer is 19 feet.)
• The diameter for a circle with a circumference of 15.7 centimeters (5 centimeters = 15.7 centimeters/3.14).

5. Point out the definition of area on the poster: the measure of a bounded region of a two-dimensional shape expressed in square units,e.g., square inches or square feet. Show your students the formula for the area of a rectangle: A(area) =l • w. Explain that the area of the rectangle you had drawn earlier is 12 square feet (4 • 3 = 12 square feet).

6. Now point out the formula for the area of a triangle on the poster: A= 1/2 • [b(base) h (height)]. Refer to your drawing of a right triangle with a base of 4 feet and height of 3 feet and demonstrate how to calculate the area of 6 square feet (1/2 (4 • 3) = 6 square feet).

7. Finally, go over the area formula for circles. Again, refer to the poster: A= π • r2, where r2 means radius squared, or r • r. Show that the area of a circle with a radius of 3 feet = 28.26 square feet (32 x 3.14 = 28.26 square feet).

8. Guided Practice: Either in pairs or individually, ask students to:

• Determine the area of a rectangle with a length of 7 feet and a width of 3 feet (3 x 7 = 21 square feet).
• Determine the area of a triangle with a base of 10 centimeters and a height of 7 centimeters (1/2 x 7 x 10 = 35 square centimeters).
• Determine the area of a circle with a diameter of 10 centimeters. (The trick is that, to find the radius of 5 cm, divide the diameter by 2.) 52 x 3.14 = 78.5 square centimeters.
• Maria determines that the area of a triangle with a base of 3 inches and a height of 7 inches is 10.5 inches. Is she correct? Explain your thinking. (Not correct. 10.5 is the correct number (1/2 x 3 x 7) but the unit of measure that should be used is square inches, not inches.)
• Consider the following: Bob is trying to calculate the area of a triangle with a base of 5 meters and a height of 7 meters. He mistakenly switches the height and the base. How will this affect his answer? Explain your thinking. (No effect. 1/2 x 3 x 7 = 1/2 x 7 x 3)

9. Independent Practice: Distribute Worksheet 1 Printable (PDF). Tell students they should complete all the questions.

Check for Understanding: Go over correct answers as a class using the Worksheet Answer Key (PDF).

10. Use the following printables as extensions to Lesson 1:

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