Using 3” x 5” Index Cards to Develop “Area” Formulas for Polygons

Card 1

Area of Rectangle: A = bh

1. Label base (b) and height (h) of rectangle. (Discuss you’re moving from length/width vocabulary previously used to base/height and length and height need to be perpendicular). 2. Use graph paper and cubes to draw or create a rectangle to verify A = bh.

Card 2

Area of Parallelogram: A = bh

1. Label base (b) and height (h) on index card (rectangle). 2. Draw a line from “upper right” vertex to point on the base (do not draw a diagonal line from vertex to vertex). Cut along that line. You will have a triangle and a trapezoid. 3. Slide triangle to opposite side of trapezoid and tape into place forming a parallelogram. 4. Notice that base and height length have not changed. Area of original rectangle (index card) and this parallelogram have same area. Note: discussion may lead into the height of a parallelogram may be outside of the figure.

Card 3

Area of Triangle (non-right): A = 1/2bh

1. Label and create a parallelogram like you did with Card 2. 2. Draw a diagonal from vertex to vertex. Cut along the diagonal. Discuss the fact that you have two congruent triangles. Verify that the two triangles are congruent (Note: students may have 2 congruent acute triangles or 2 congruent obtuse triangles depending on which two vertices they used to draw the diagonal). 3. Discuss that it takes two triangles to make the parallelogram (or rectangle) and one of them is equal to half the area of the original parallelogram or rectangle. Note: Question may arise about scalene triangle’s area formula or you may ask “Does this formula work for any triangle?” now or after the right triangle (Card 4).

Card 4

Area of Triangle (right): A = 1/2bh

1. Label an index card (rectangle) with base (b) and height (h). 2. Draw a diagonal from vertex to vertex. Cut along the diagonal. Discuss the fact that you have two congruent right triangles. Verify that the two triangles are congruent. 3. Discuss that it takes two triangles to make the parallelogram (or rectangle) and one of them is equal to half the area of the original parallelogram. Area of a triangle: A = 1/2bh. 4. Ask the question if not discussed: Does the formula A = 1/2bh work for all triangles? Card 5

Area of Trapezoid (isosceles): A = ½ (b1 + b2) h

1. Draw a diagonal from vertex to a point along the base. Cut along that line. Take the triangle and slide it to the opposite side of the trapezoid and flip the triangle and tape it into place. An isosceles trapezoid has been created. 2. Have students discuss their observations about the height and bases of the new figure. Label the height (h) and the bases (b1 and b2). Discuss that the base lengths are different. Finding average of the two bases will relate the formula to A = bh of rectangle or parallelogram (are of figure remains the same since the entire index card is used to make an isosceles trapezoid). 3. A =1/2 (b1+ b2 )h is formula for an isosceles trapezoid. Note: formula can be written in several forms. It is important to let students share their variations of this formula.

Card 6

Area of Trapezoid (right): A = ½ (b1 + b2) h

1. Draw a diagonal from vertex to a point along the base. Cut along that line. A right triangle and a right trapezoid have been created. Label the bases (b1 and b2) and height (h) of the right trapezoid. 2. Put the right triangle aside and discuss that all trapezoids have the area formula: A = ½ (b1 + b2) h.

Card 7

Area of a square: A = bh = s2

1. You may opt to just have the discussion with students that a square is a special rectangle, a rectangle with four congruent sides and A = bh = s2. If yes, skip to Card 8 directions for rhombus. 2. Fold in vertex of upper base to lower base of index card (rectangle). 3. Draw vertical line along the edge perpendicular to bases. Cut along that perpendicular line. Put the smaller shape aside. 4. Label the base (b) and height (h) of square. The b = h = s, therefore A = bh = s2.

Card 8 Area of a rhombus: A = ½ d1 d2

1. Fold the index card in half both horizontally and vertically and pinch the midpoints of the rectangle’s sides. Unfold the index card. 2. Discuss that you have two congruent rhombi (plural for rhombus). 3. Arrange the four right triangles and tape together to make a rhombus congruent to the other rhombus. It takes two of these to make the original rectangle. 4. Label the diagonals of the rhombus d1 and d2. 5. The b and h of the original rectangle represent d1 and d2 of the rhombus. 6. The area of rhombus: A = ½ bh = 1/2d1d2.

Extensions and follow-up Questions:

Will these formulas work for all polynomials? Why or why not?

If you know one formula, can one derive the others?

Use pattern blocks to visualize breaking down original figure into squares, triangles, trapezoids, rectangles and find area of entire figure. (I.e. find total area by finding area of parts)

Have students explore to find the area formula of a circle using a circle that is divided into 16 2 equal wedges, cut apart, and taped together alternately to form a rectangle. A = r . Use derived formulas to find area of shaded region, example below:

Adaptation of Glencoe Math activity and Others.