Trainer/Instructor Notes: Area What Is Area? Unit 5 − Area What Is Area? Overview: Participants determine the area of a rectangle by counting the number of square units needed to cover the region. Group discussion deepens participants’ understanding of area (number of square units needed to cover a given region) and connects the formula for the area of a rectangle to the underlying array structure. Objective: TExES Mathematics Competencies III.013.D. The beginning teacher computes the perimeter, area, and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc length, area of sectors). V.018.D. The beginning teacher uses formal and informal reasoning to justify mathematical ideas. Geometry TEKS b.3.D. The student uses inductive reasoning to formulate a conjecture. e.1.A. The student finds area of regular polygons and composite figures. Background: No prerequisite knowledge is necessary for this activity. Materials: index cards, patty paper, straightedge New Terms: area Procedures: Background information: In order to accurately count the units of a given space, it is necessary to mentally organize the space in a structured manner. However, research referenced in Schifter, Bastable, and Russell (2002), on how children learn the concept of area supports the theory that the structure of the rectangular array is not intuitively obvious to children. When asked to cover a rectangular region, children progress from incomplete or unsystematic coverings to individually drawn units to the use of a row or column iteration. Gradually they will rely less on drawing and move towards multiplication or repeated addition. Covering a rectangular region with unit squares helps children understand area measure, but they must ultimately be able to formally connect area, linear measurement, and multiplication in order to truly understand the area formula A=b⋅h. According to Schifter, Bastable, and Russell (2002), the drawing, filling, and counting that children use in this developmental process are both motor and mental actions that coordinate to organize spatial structuring. What implications does this research have for secondary teachers? Many of our students come to us with an understanding of area at the Visual Level of the van Hiele model of Geometry Module 5-1 Trainer/Instructor Notes: Area What Is Area? geometric development. They recognize figures by their shape and understand area as the blank space within the boundaries. Others have moved to the Descriptive Level, which implies that they can mentally see the rectangular array which overlays the shape. This activity asks participants to draw the grid, rather than giving them a pre-structured grid in order for them to experience the type of activity necessary to move a student from the Visual to the Descriptive Level. Secondary students functioning on the Relational Level are able to compare linear dimensions with grid areas and apply formulas with understanding. Teachers must be aware that although students may become proficient at rote application of formulas, they may not be functioning at the Relational Level. If they have not been given sufficient opportunity to understand the principles behind the formulas, they will have difficulty modifying a procedure to fit a particular situation as is necessary to find areas of composite figures and shaded regions. Distribute index cards to participants and ask them to write a response to the question “What is area?”. Indicate that they will have an opportunity to share and revise their responses at the conclusion of this activity. Then, allow time for participants, working independently or in pairs, to complete the activity using the patty paper or straight edge to determine the number of square units needed to completely cover the rectangular region. Note that the given square units are not convenient measures, such as 1 cm2 or 1 in.2. Consequently, participants will be less likely to simply measure the rectangle and use the area formula without having the experience of drawing the units. Whether participants mark off the units on two adjacent sides of the rectangle and multiply or actually draw in one or more rows and columns of units, they will be counting the units by considering how many rows of squares are needed to cover the region. 1. Determine the number of square units needed to cover this rectangular region. 1 square unit The rectangle measures 6 ·8 square units. Therefore, it will take 48 square units to cover the rectangle. Geometry Module 5-2 Trainer/Instructor Notes: Area What Is Area? 2. Determine the number of square units needed to cover this rectangular region. (Same rectangle, different square unit) 1 square unit The rectangle measures 9 · 12 square units. Therefore, it will take 108 square units to cover the rectangle. Was it necessary to draw all 108 square units to determine that it would take 108 units to cover the rectangular region? No. After drawing one row and one column of square units the total number of squares can be obtained by considering how many rows and columns of squares will be needed to cover the entire region. Has the activity caused you to reconsider your definition of area? Some participants may have responded to the question “What is area?” by stating that area is the amount of space covered by a particular region. It is important to make the distinction that area is the number of square units needed to completely cover a particular region. If the same figure is measured in different units, the number representing the area of the region will be different, but the area will remain constant. A more abstract definition of area, provided by Michael Serra (Serra, 2003) states that area is a function that assigns to each two-dimensional geometric shape a nonnegative real number so that (1) the area of every point is zero, (2) the areas of congruent figures are equal, and (3) if a shape is partitioned into sub regions, then the sum of the areas of those sub regions equals the area of the shape. If a figure is rotated so that a different side is considered the base, will the area formula necessarily give the same result? Yes. Surprisingly, the answer to this question is not evident to all students. According to the work of Clements and Battista referenced in Schifter, Bastable, and Russell (2002), orientation, the position of objects in space in relation to an external frame of reference, is for some children a part of their definition of a particular shape. If secondary students have not had adequate experience manipulating by rotating, flipping or sliding shapes, they may be working at the Visual Level with an inadequate understanding of shape. In developing an understanding of area, students should observe that rotations, reflections and translations preserve area while dilations do not. Geometry Module 5-3 Activity Page: Area What is Area? What Is Area? 1. Determine the number of square units needed to cover this rectangular region. 1 square unit Geometry Module 5-4 Activity Page: Area What is Area? 2. Determine the number of square units needed to cover this rectangular region. (Same rectangle, different square unit) 1 square unit Geometry Module 5-5 Trainer/Instructor Notes: Area Investigating Area Formulas Investigating Area Formulas Overview: Participants cut and rearrange two triangles, a parallelogram, and a kite to form rectangles with the same areas. Examination of the points at which figures must be cut will lead to a deeper understanding of the formula for the area of each figure. Objective: TExES Mathematics Competencies III.011.A. The beginning teacher applies dimensional analysis to derive units and formulas in a variety of situations (e.g., rates of change of one variable with respect to another and to find and evaluate solutions to problems. III.013.C. The beginning teacher uses geometric patterns and properties (e.g., similarity, congruence) to make generalizations about two- and three-dimensional figures and shapes (e.g., relationships of sides, angles). V.018.D. The beginning teacher uses formal and informal reasoning to justify mathematical ideas. Geometry TEKS b.2.A. The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. b.3.C. The student demonstrates what it means to prove mathematically that statements are true. d.2.C. The student develops and uses formulas including distance and midpoint. Background: Participants should know the formula for the area of a rectangle and be able to identify the base and altitude of a triangle, the base and altitude of a parallelogram, and the diagonals of a kite. Materials: transparency sheets, colored pencils, glue or tape, patty paper, scissors New Terms: Procedures: Participants sit in groups of 3-4 to work collaboratively. However, each participant should cut and glue his/her own figures. Can any parallelogram, triangle or kite be cut and rearranged to form a rectangle of the same area? Allow time for discussion within groups and then ask two or three groups to share their responses with the entire group. It is fairly obvious that any parallelogram can be cut to Geometry Module 5-6 Trainer/Instructor Notes: Area Investigating Area Formulas form a rectangle of equal area, but the group may or may not be able to reach a consensus on the triangle and kite. Briefly describe the activity. Each of the figures on the activity sheet has the same area as the rectangle. Using a colored pencil, trace the parallelogram on patty paper. Then, using the least number of cuts possible, cut the parallelogram and rearrange the pieces to form a rectangle of equal area. The rectangle will help you determine where to cut. Lay the patty paper tracing over the rectangle and slide it around to decide where to cut. Using a different colored pencil, draw the cut line on the patty paper figure and then cut. Assemble the pieces to form a rectangle and glue it next to the original parallelogram. Repeat the process for each of the figures. While the groups are working, assign each of the figures to a different participant to draw on a transparency for use during the group discussion. When most participants have completed the task, reconvene as a large group for discussion. Parallelogram: b h Can we express the dimensions of the rectangle in terms of the dimensions of the parallelogram? Yes. The base of the rectangle is the base of the parallelogram. The height of the rectangle is the height of the parallelogram. What does this tell us about the formula for the area of the parallelogram? Since the area of the parallelogram is equal to the area of the rectangle, the area of the parallelogram is b⋅h. Geometry Module 5-7 Trainer/Instructor Notes: Area Investigating Area Formulas Obtuse Triangle: b= m h 2b How would you describe the location of the cut lines on the obtuse triangle? The cuts must pass through the midpoints of the sides of the triangle as shown. What do we call the segment that connects the midpoints of the sides of a triangle? The midsegment What do we know about the midsegment of a triangle? The midsegment is parallel to the base and one half the length of the base of the triangle. Can we express the dimensions of the rectangle in terms of the dimensions of the triangle? Yes. The length of the base of the rectangle is equal to the length of the midsegment of the triangle, m. The height of the rectangle, h, is the height of the triangle. Can we use this information to derive the formula for the area of a triangle? Since we know Area of triangle = Area of rectangle = b · h By substitution, Area of triangle = m⋅h By the definition of a midsegment, 1 Area of triangle = (2b)⋅h 2 1 = (base of triangle) (height) 2 Geometry Module 5-8 Trainer/Instructor Notes: Area Investigating Area Formulas Acute Triangle: 2h h b How would you describe the location of the cut lines on the acute triangle? One cut line goes through the midsegment of the triangle and one cut line is the altitude joining the midsegment to the opposite vertex of the triangle. Can we use this information to derive the formula for the area of a triangle? Since we know Area of triangle = Area of rectangle = b⋅h 1 =b⋅ (2h) 2 1 = (base of triangle) (height of triangle) 2 Geometry Module 5-9 Trainer/Instructor Notes: Area Investigating Area Formulas Kite: h d 2 b d 1 How can we use the formula we have derived for the area of a triangle to derive the formula for the area of the kite? d lies on the line of symmetry for the kite. The two triangles formed by the line of 1 symmetry, d are congruent. 1, 1 1 (d ) is the length of the altitude of each triangle, and (d )=h 2 2 2 2 1 ⎛1⎞ ⎛1⎞ Area of one triangle = (d )⋅h= (d ) (d ) 2 1 ⎜⎝2⎟⎠ 1 ⎜⎝2⎟⎠ 2 1 1 1 Area of kite = 2( )(d )( )(d )= (d ⋅d ) 2 1 2 2 2 1 2 1 = d ⋅d 2 1 2 Success in this activity indicates that participants are working at the Relational Level because they must discover the relationship between the area rule for a rectangle and the area rule for a parallelogram, triangle, or kite. While a participant at the Descriptive Level will be able to cut the figures to form the rectangle of the same area, he/she will need prompting to explain “why it works” using informal deductive arguments. Geometry Module 5-10