Procedure

Pegging Down Area

Procedure

Day One

The first lesson familiarizes students with the use of geoboards, reviews the concept of area of a square in a kinesthetic way. The hope is to demystify the formula for the area of a square.

Anticipatory Set

Begin the first lesson by holding up a 5 by 5 geoboard (borrowed from the first grade teacher). Ask if anyone has used one before [expect most students to recall using them in the primary grades]. Remind students that younger children use geoboards and rubber bands to make shape pictures. Explain that in the younger grades (K, 1, 2) identifying and using shapes is how children learn about geometry. At the sixth-grade level, however, we can use geoboards to examine other geometric properties, such as area.

Modeling and Guided Practice

After a reminder of the proper use of materials (ex. no flying rubber bands), distribute a geoboard and rubber bands to each student. To demonstrate the use of a geoboard, point to adjacent pegs and explain that the distance between the pegs represents one linear unit. Then ask students to trace a path with their finger across the board and find the greatest number of units in a row [4 units].
Affix a rubber band to a peg and manipulate it to correspond to your statements. "When a rubber band is stretched across a row of pegs, we can find the length of the rubber band in these units. When stretched this way the rubber band represents a line segment, therefore the units the rubber band shows are linear units. If we make a square or rectangle with the rubber band, we can count the number of linear units it takes to travel around its shape (its perimeter or its distance)." Count off two minutes so students can use their own materials. Circulate around the room, making sure that the students are counting the spaces between the pegs as units, not the pegs themselves. The statement of linear units expresses the shape's distance. Redirect students as needed.
Call the class to order and direct them to make the smallest possible square on their geoboards. As students finish they hold up their boards to verify they have followed directions. "Okay, now trace the borders of the square with your finger, counting the number of linear units it takes to go around once [4 units]. This is the square's perimeter. Notice that clusters of pegs seem to make individual squares on the geoboard. We call these square units. How many square units are taken up by this square? [1 square unit]"
After clarifying the difference between linear and square units, tell students to find progressively larger squares on the geoboard, pausing to ask the numbers of linear units on a side and the number of square units they can see within each square. As students volunteer this information, enter it on a function table that is drawn on the board (Table 1) and ask…

Square
(side)(area)

S	A
1	1
2	4
3	9
4	16

(Table 1)

"What pattern do you see between the number of units per side and the area of space that square unit covers? [s x s = a]"

Closure

Ask students to remember key words used during the lesson. Start a list on a sheet of chart paper titled "Pegging Down Area".
Pass out index cards and direct students to complete an exit visa before leaving class. The students can use the exit visas to state something they learned from that day's lesson or to ask a question about something they didn't understand. The teacher can use the visas to track students' progress throughout a unit of study and address any student needs in a timely manner.

Post the function table that was generated in class on a bulletin board (ex."Things We Know About Area") for reference.

Day Two

The second lesson reviews the definition for the term rectangle to reinforce the idea that a square is a special type of rectangle. This connects the first lesson on squares to the second lesson dealing with rectangles. The following activities show students that rectangles can have different dimensions but still cover the same area and relate the concept of a rectangle's area to its formula (a = l x w).

Modeling and Guided Practice

After distributing the geoboards and rubber bands, begin the lesson by asking students to tell the properties of a rectangle [has four sides, four right angles, opposite sides are parallel/equal]. Direct students to form shapes on their geoboards that match these properties. Circulate to verify that students are constructing rectangles. Anticipate a disagreement between students if one of them constructs a square. Draw the class' attention to this situation and ask probing questions until the misconception is corrected: "Does Susie's square have four sides? [yes] four right angles? [yes] Are the opposite sides parallel? [yes] equal? [yes] So, can't we say that a square qualifies as a rectangle? [yes]"
Hold up a geoboard and model a review of the previous lesson, in which students counted the square units inside a shape to find the area it enclosed. "So, if I tell you a number of square units, can you make rectangles that match that area? [yes]" Call out a variety of numbers between 2 square units and 16 square units (the upper limit of the 5 x 5 geoboard). As students stretch their rubber bands to accommodate the areas in question, tell students to compare their results with their neighbors. In this way, they should see that there is more than one way to enclose the same number of square units. At this point you want to assess students' strategies for identifying area. "How did you check that the area your rectangle enclosed was correct?" A student may state that he/she can count the individual square units inside the rectangle, OR count the square units in one row/column and just multiply by the number of rows/columns to find the area. The latter response would indicate that the student had made an abstraction that would facilitate his/her understanding of the formula for area.
The limited size of the geoboard does not lend itself to examining rectangles of the dimensions usually presented at the sixth-grade level. To move from the concrete representation to the visual one, pass out grid paper. The squares on the grid paper function as square units in the same way the pegs on the geoboard do. Direct students to draw rectangles on their paper to match the larger areas you call out. Again, encourage students to compare their rectangles and remind them of the two strategies the students stated for checking area in step 9 above.
Project an image of the grid paper onto the classroom's whiteboard using an overhead projector. Tracing over the lines, draw rectangles on the board and label them as you speak. "When we see a rectangle in the textbook, we usually don't have the advantage of seeing the individual square units that make it up." Invite the students to count along with you as you find the number of square units in your example rectangles. Afterwards, turn off the projector so only the outlines of the rectangles and their labels appear. "How do we find the area now? [Multiply the long side by the short side.] Why does this work? [The length tells how many units are in that row, the width tells how many rows.]" Write the formula a = l x w on the board. "Is this what you're telling me to do? [yes]" Compare the formula to that for the area of a square: a = s x s. "They both work the same way because you are multiplying the dimensions of adjacent sides."

Closure

Continue to list key words from the lesson on the sheet of chart paper titled Pegging Down Area. Review frequently in coming lessons.
Pass out index cards and direct students to complete an exit visa (stating something they learned or asking for clarification) before leaving the room. Post the formula for area of a rectangle on the bulletin board for reference.

Day Three

The third lesson is to have students realize that the area of a right triangle is equal to half the area of its corresponding rectangle. You want the students to draw upon their prior knowledge of area to extrapolate the formula for the area of a triangle.

Modeling and Guided Practice

Distribute the geoboards and rubber bands. Direct students to construct a rectangle of any dimensions and note its area. Model this on a geoboard. Take a second rubber band and demonstrate the next step. "Take another rubber band and split your rectangle diagonally, so you make two facing triangles. Now, using the counting method, find the number of square units in each triangle." Students may express frustration; up to this point, they have only dealt with whole units when determining area. Advise those who are having trouble to first count the number of whole units within the triangle, then combine partial units to create whole units. After some recounting, students should come close to the correct areas of their triangles. "How does the area of the rectangle compare to the area of one of its triangles?" Compile student responses on a chart for everyone to see (Table II).

A of rectangle	A of triangle
8	4
6	3
12	6
4	2

(Table II)

After seeing several examples, the students should agree that the area of the triangle is half the size of the area of the rectangle that encloses it. Ask students to restate the formula for the area of a rectangle [a = l x w]. "How could we modify this formula to help us determine the areas of these types of triangles? [Find length x width, then divide that by 2.]" Write the new formula on the board, as stated by the student: a = (l x w) ÷ 2.
Direct the students to determine the area of the same triangle using the new formula instead of the counting method. Students who were confused earlier by the partial units should find the area formula easier to use to arrive at the same answer. However, be prepared for some resistance; some students may prefer the direct measurement approach (counting) to the indirect method (arithmetic).
Ask students what other relationships/similarities they find between the triangle and its corresponding rectangle. A student might point out that the length and width of the rectangle are the same as two sides of the triangle. "So, if you know the measures of those two sides (l and w), can you find the area of the triangle? [maybe]" Point out the new formula generated by the group. Remind the students that the only variables in the formula are l and w; if they can substitute the numbers that l and w stand for, they can calculate area. Suggest that if students encounter such a triangle on a homework or test, they could sketch in the rest of the rectangle if it would help them to remember that the triangle is simply half of a rectangle.
Once again, since the size of the geoboard is restrictive, pass out grid paper. Also pass out plastic right triangles, to be used as tracers (Collect the triangles from tangram sets or from boxes of shapes that a primary grade teacher might have handy. Choose sizes that fit readily onto the grid paper you're using.). Direct the students to trace their triangles onto grid paper and determine the areas by both the counting method and the formula method. Circulate to observe how students tackle this problem and their range of development. To check their own work, suggest students find classmates with identical triangles and compare results. Determine whether discrepancies are due to lack of comprehension or simply due to sloppy tracing.

Independent Practice

Inform the students that they are going to answer another question about area that relates to what they have been practicing in Math class for the last three days. Tell the students: "The last two days I have been asking you to tell me what you know. Now you need to prove what you know on paper." Distribute the Area Assessment Questions worksheet and grid paper for students to show their work. Request that students highlight the outlines of the shapes they draw so the shapes can be seen more easily. Ask students to work quietly and to hand in their work before they leave class.

Closure

Post the chart generated in step 16 and the formula generated in step 17 on the bulletin board for reference. As the learning experience progresses, students should note that the display is growing, as should their knowledge.

Optional Extension (Day Four)

In preparation for concepts to be taught in seventh grade, extend students' knowledge of area to that of complex shapes. In general, students are anxious when they encounter new information. Show them that they can use known information to figure out the unknown.

Modeling and Guided Practice

Using the overhead projector, project the image of the grid paper onto the whiteboard. "Now we know how to find the areas of squares, rectangles, and triangles. We have the tools we need to tackle a new challenge." On the whiteboard draw two complex shapes: a six-sided figure resembling the letter "L" and a trapezoid with two right angles [see Attachment 3]. "How can you find the area of the ‘L' shape? [by counting, by breaking it up into two rectangles and adding their areas] How can you find the area of the quadrilateral? [by counting, by breaking it up into a rectangle and a triangle and adding their areas]"

Independent Practice

Direct the students to find the areas of the shapes using the method with which they are most comfortable. Because the diagonal line makes the counting method problematic, most students attempt to use the known formulas. Allow the students to work cooperatively; the ensuing discussion and reinforcement can increase student confidence and lead to a greater level of success.

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