Purpose: T his activity is designed to help develop the idea of pi and how this relates to circles.

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1 What is Pi? Grade Competencies 5 th 1: d, e 2: a, b, c, f, g 6 th 4: a, b, c 7 th 3: c 5: g 8 th 1: a, c, d 6: c Materials: paper, pencil, calculator and tape measure Purpose: T his activity is designed to help develop the idea of pi and how this relates to circles. Time: 1 hour Find as many round objects of various sizes as you can. Measure the objects across the end to determine the diameter. Next measure the distance around the object. Record the information in a table as follows. Name of Object Distance Across Distance Around Ratio Day T wo Page 1

2 After the data has been collected, determine the mean of the ratios. You may want to make as many measurements as you can, but 10 should be a minimum with some of the objects being fairly large in diameter. After these measurements are made and the ratios determined, compare your results with the exact answer pi. For greater accuracy use the smallest units possible. Day T wo Page 2

3 What is area? Grade Competencies 5 th 1: c 2: b, c, e 6 th 3: a, c 7 th 5: f 8 th 6: c Materials: Handouts, calculators, pencils Purpose: T o build an u nderstanding of area and how this ties into formulas for standard figures. Time: 2 hours The first question that needs to be addressed is, What is area? In lay terms, area is the number of area units required to completely cover a simple closed curve with no overlaps or gaps. This definition contains a minor problem: what is an area unit? An area unit is a unit used to cover a simple closed curve. Normally, we use a square of predetermined size to cover the simple closed curve. This does not need to be the only shape that we use. On the next several pages, you will find several figures. The smaller simple closed curves are to be used as area units. Use the area unit to cover the figure to the left. You are not to have a unit go outside of the main figure nor are they to overlap. T his will allow you to determine an underestimate of the actual area of the figure. Use a pen or some other writing instrument to produce more units to finish covering the object by going outside of the boundaries. This will produce an overestimate of the actual area. Day T wo Page 3

4 Use the file named area figure notebook1 with these figures. Day T wo Page 4

5 Use the file named area figure notebook2 with these figures. Day T wo Page 5

6 What are you going to do with this understanding of area, and secondly how does this apply to standard geometric shapes? First, let us begin by developing an understanding of the relationships between shapes. Only after developing a relationship will the formulas be developed. Use the file named tri between parallels for the first investigation. The purpose of the demonstration is to illustrate that between two parallels the area of a triangle is constant so long as the base remains fixed. Use the file named tri to para for this investigation. The purpose of this activity is to determine a ratio between the area of triangles and parallelograms. The participants should be able to state a ratio between the areas of these figures, but not necessarily the formulas, that will come later. Use the file named para to rect for this investigation. The purpose is to determine the ratio between the area of parallelograms and rectangles. The participants may not be able to write a formula for the area of either figure this will come later. Use the file named trap to para for this investigation. The purpose is to determine a ratio between the area of a trapezoid and a parallelogram. Now is the time for formulas! From the investigation entitled tri to para, the participant should understand that for every triangle there exists an associated parallelogram that has twice the area. Hence, 2 triangles = 1 parallelogram From the investigation entitled para to rect, the participant should understand that for every parallelogram there exists an associated rectangle that has the same area. Hence, 1 parallelogram = 1 rectangle. From the investigation entitled trap to para, the participant should understand that for every trapezoid there exists an associated parallelogram which has twice the area. Day T wo Page 6

7 The easiest formula to u nderstand is the formula for the area of a rectangle. By using tiles it is easy to understand that the area of a rectangle is the length times the width (A = l * w). Since we have a one to one relationship between the area of the rectangle and the parallelogram, it is safe to conclude that the formula for the area of a parallelogram is some how related to length times width. Since the parallelogram leans to one side or the other, it follows that instead of width (slant height would be a better descriptor) we need to use height (A = l* h). Thus by back tracking we can see that the formula for the area of a triangle is one-half of the formula we just figured out for the parallelogram ( A = (l*h)/2). From the investigation on the trapezoid we see that the associated parallelogram has as its base the sum of the top and the bottom of the trapezoid. Also, it has taken two trapezoids to equal the parallelogram, therefore we will need to divide the answer by two to produce the area of a trapezoid. Area = {(top + bottom) * height }/ 2, which is the correct formula for the area of a trapezoid. Question: Does the formula for the area of a trapezoid produce a correct answer when applied to a rectangle or square? Why is this so? Day T wo Page 7

8 None of these topics explain the one question that plagues students to this day. Why is the formula for the area of a circle pi*r 2? To show students how this comes about is relatively easy. Cut out the circle. Then, cut the circle into pie pieces along the dotted lines. Do this by first cutting it into qua rters. Then, cut each piece in half. Day T wo Page 8

9 What is volume? Grade Competencies 5 th 2: b 6 th 3: d 4: b, c 7 th 8 th Materials: Power solids, calculators, water Purpose: T o develop an understanding of volume. To determine the volume of various 3 - dimensional figures. Time: 1 hour In order to discuss the concept of volume, it is necessary to completely understand what a volumetric unit is. A volumetric unit is like an area unit. Once a volumetric unit is decided upon, then the volume, the number of volumetric units to fill an object, can be determined. The most common unit is that of a cube. T he reason for this is the general understanding of the shape of a cube. One way to describe volume is area pushed through space. To see an illustration of this explanation go to the file named basic volume and basic volume2. For any object, volume can be thought of as one face of the object being pushed to meet another face of the object, provided these two faces are exact copies of one another. Also, the face cannot be rotated bu t must be pu shed evenly throu gh space. Think of this as toothpaste being pushed ou t of a tube. The circle at the opening of the tube has an area, and as the tube is squeezed, a cylinder is formed. The cylinder can be thought of as a circle pushed, evenly, through space. T he push distance is referred to as height by most accounts. Recall the illustrations seen in the two files viewed earlier. This means that all formulas for the volume of an object can be thought of as the area of a base times the distance needed to push the base to the opposite side. Sometimes this product may need to be multiplied by a non-zero constant, but more on this later. From the explanation above it is easy to see that the volume of a box is relatively easy to compute, and in fact this is true. Any object that can be thought of as a box has a simple formula for volume. The formula can be thought of as area or base times height. In the case of a rectangular box, length times width times the height, since length times width is area. Determining the volume of cones and pyramids presents a challenge. These two cases can be handled with ease when a student understands that for every cone there is an associated cylinder, and for every pyramid there exists an associated prism. To see an example of this idea, look at the Power Solids provided. For each cone or pyramid, there is an associated cylinder or prism. The Day T wo Page 9

10 volume of prisms and cylinders are easy to determine simply by finding the area of the base and then multiplying by the height of the associated prism or cylinder. Determine if there is a relationship between the volumes of these associated object by filling them with water and comparing the amounts. The participant should be looking for some constant that could be used to help determine the volume of any object. Day Two Page 10

11 Using Area and Volume Grade Competencies 5 th 1: c 2: b, c, e 6 th 3: a, c, d 4: b, c 7 th 5: f 8 th 6: c Materials: Handouts, calculators, pencils Purpose: T o utilize the principles learned in the previous activities. Time: 1 hour As seen earlier, the constant used with determining the area is one-half (triangles and trapezoids); and likewise, the constant used with determining volume is one-third (cones and pyramids). The area and volume of many figures can be found using these two constants. It may be necessary to use the Pythagorean Theorem when working these problems. More attention will be paid to the Pythagorean Theorem later in this document. Day Two Page 11

12 Determine the area of the figure below. Determine the distance from point G to point F. Day Two Page 12

13 Determine the area of the patio described below. Day Two Page 13

14 Determine the volume of the figure described below. Day Two Page 14

15 Determine the volume and surface area of the box described below. Day Two Page 15

16 Determine the volume of the figure provided. Day Two Page 16