Developing the Area Formulas for Regular Polygons

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Objective

SWBAT develop area formulas.

Big Idea

Students discover the relationships between regular polygons.

Do Now

10 minutes

The Do Now will serve as a quick assessment of students' prior knowledge of polygons.  Students will have 10 minutes to brainstorm with their group.

Do Now

Draw a picture of each polygon and describe its characteristics.

Square

Rectangle

Trapezoid

Triangle

After 10 minutes, groups will share out their ideas.  We will compile a list of characteristics.  We will discuss that these shapes are polygons that share common characteristics.

Square 

  • 4 equal sides
  • 4 right angles
  • 2 pairs of parallel sides

Rectangle

  • 4 sides
  • 4 right angles
  • opposite sides are parallel
  • opposite sides are congruent

Trapezoid

  • 4 sides
  • one pair of parallel sides

Triangle

  • 3 sides

Polygons

  • 3 or more sides
  • NO curved sides
  • NO intersecting lines
  • Closed Figure

 

 

Mini Lesson

30 minutes

Rather than give students a list of area formulas to memorize, students will develop the other formulas.  This will help students remember the formulas.  Each student will be given a numbered coordinate plane and a ruler to use throughout this lesson.  This lesson incorporates plotting coordinates, a former lesson, with finding the area of polygons.

What is area?

Students may know that area is the "space" inside a shape.  I will share a formal definition of area.

Area - The number of square units (units2) it takes to cover the surface of a figure.

Square

We will start with developing the formula for area of a square.  Students will be given the following coordinates to plot and connect:  (3,3)  (3,7)  (7,7)  (7,3).

What type of polygon is this?  Why?

Students should be able to identify that this is a square because all the sides are equal.

If area is the number of square units inside the figure, how can we find the area of this square?

By counting, students should be able to determine the squares units to be 16.

What are the lengths of the sides of this square?  How can we use these to determine the area?

Students should identify the lengths to be 4 and realize that if we multiply two of the sides we will have 16.

Rather than count the squares inside the shape, what would be another way to find the area of a square?

Area = side x side    or Area = s2

Rectangle

 Students will be given the following coordinates to plot and connect: (2,-4)  (9,-4)  (9,-8) (2,-8).

What type of polygon is this?  Why?

Students should be able to identify that this is a rectangle, because opposite sides are parallel and equal.

If area is the number of square units inside the figure, how can we find the area of this rectangle?

By counting, students should be able to determine the squares units to be 28.

What are the lengths of the sides of the rectangle?  How can we use these to determine the area?

Students should identify the lengths to be 7 and 4 and realize that if we multiply two of the sides we will have 28.

Rather than count the squares inside the shape, what would be another way to find the area of a rectangle?

Area = length x width

Triangle

Students will draw a diagonal inside the rectangle.

What shapes do we have now?

Students should be able to identify 2 triangles.

Can we find the area of one of these triangles by counting the square units?

Students should realize that it is difficult to do this because there is only a fraction of some of the squares.

What did the diagonal line do to the rectangle?

Students should verify that the diagonal divides the rectangle in half.

How can we use this to find the area of a triangle?

A = 1/2base x height  or A = 1/2bh

I will explain to students that rather than use the words length and width, we will use base and height.  

What do you notice about the angle formed by the base and height of the triangle?

Students should identify that the angle is a right angle.  It is important for students to note that the base and height must always be perpendicular.

Parallelogram

Students will be given the following coordinates to plot and connect: (-4,-3)  (-9,-3)  (-7,-6) (-2,-6).

What shape is this?  What are its characteristics?

Students should be able to identify this shape as a parallelogram.  They should verify that the angles are not right angles.

Does it have a base and height?  How can we determine the height?

Students may be confused regarding the height.  It is important to remind them that the base and height should form a perpendicular angle.

What is the formula for the area of a parallelogram?

Area = base x height     or     A = bh

Trapezoid

Students will be given the following coordinates to plot and connect: (-2,2)  (-8,2)  (-7,5) (-3,5).

What shape is this?  How is it different from a rectangle and parallelogram?

Students should identify it as a trapezoid with only one pair of parallel sides.

Does it have a base and height?  How can we determine the height? 

Group Work

15 minutes

The final formula that students need to know is for a trapezoid.  This is usually the most difficult for students to understand.  For the group work, students will develop the area formula for a trapezoid.

Trapezoid

Students will be given the following coordinates to plot and connect: (-2,2)  (-8,2)  (-7,5) (-3,5).

What shape is this?  What are its characteristics?

Students should identify it as a trapezoid with only one pair of parallel sides.  Students will be instructed to find the area of this trapezoid by dividing it into two triangles.  Groups will have 10 minutes to determine the area and develop a formula for the area of a trapezoid.

After 10 minutes, we will discuss the problem as a class.  I will ask groups to share their answers and formula.

Groups should have calculated the area of the trapezoid to be 15.  Most groups will have come with the formula:  Area =1/2bh + 1/2bh

Do both triangles have the same base?

Students should verify that they have different bases.  We will distinguish between the bases by using b1 and b2 and rewrite the formula:

Area = 1/2b1h + 1/2b2h

How can we use factoring to rewrite this formula?  What is the GCF?

Students should identify that 1/2h is the GCF and therefore we can factor it out.  This will give us the formula for area of a trapezoid.

Area = 1/2h(b1 + b2)

It is important for students to note that the trapezoid has two bases and the height is perpendicular to one of the bases. 

 

Lesson Summary

5 minutes

For the lesson summary I will review the categories and types of regular polygons with students.

What do a square, rectangle, parallelogram, and trapezoid have in common?

Students should recognize that they all have four sides.

What do we call a regular polygon with four sides?

Students may know that they are identified as quadrilaterals.

What category do rectangles, squares and rhombuses belong in, but a trapezoid does not?

Students should identify that rectangles, squares, and rhombuses are parallelograms, however trapezoids are not.