PDF viewing archiving 300 dpi - The Math Learning CenterActions 4. Ask the students to construct the polygon shown below. Point out that this polygon has six sides. Ask them to make

Unit VI Math and the Mind's Eye Activities

Looking at Geometry Geoboard Figures The geoboard is used as a means of representing geometric figures and as a

medium for geometric explorations.

Geoboard Areas Regions arc formed on a geoboard and their areas are determined using fllrmula-free methods.

Areas of Silhouettes The area of a region is found by determining rhe number of unit squares

needed to cover ir.

Geoboard Triangles The size, shape and area of cerrain geoboard triangles are investigated.

Geoboard Squares The lengths of geoboard segments are determined by viewing them as the sides

of squares.

Pythagorean Theorem The Pythagorean relationship is developed visually with rhc usc of dot paper

drawings.

Geoboard Perimeters The perimeters of gcoboard polygons are determined and the relationship

between area and perimeter is explored.

An Introduction to Surface Area and Volume Solids of a given volume arc formed with cubes and thc.:ir surface areas dcn.:r

mined by constructing grid paper coverings.

Shape and Surface Area The effect of shape on surface area is investigated.

Areas of Irregular Shapes Basic area concepts arc used to estimate the areas of irregularly shaped regions.

ath and the Mind's Eye materials

are intended for use in grades 4-9.

They are written so teachers can adapt

them to fit student backgrounds and

grade levels. A single activity can be ex

tended over several days or used in part.

A catalog of Math and the Mind's Eye

materials and reaching supplies is avail

able from The Math Learning Center,

PO Box 3226, Salem, OR 97302, I 800

575-8130 or (503) 370-8130. Fax: (503)

370-7961.

Learn more about The i'vlarh Learning

Center at: www.mlc.pclx.edu

Math and the Mind's Eye

Copyright {(;I 1 '187 The J\tnh L.e:1rning Center.

The !VIath Lcaming Center grants penniv;ion to cb<,_l

room tcadH:r.\ w reproduce the _\tudenr Ktil'ity page'>

in appropriate qu;mririe; fl1r their cLt1.1tuom usc.

The-;e lll.Hcrial; were prepared with dw suppun of

NJtionJI Science f-ouml:uiun (_;rant MDH.-tl:iOJ/1.

ISBN l-8Wil31-17-1

Unit V • Activity 1 Geoboard Figures

Actions

1. Put rubber bands on a geoboard as shown below. Show the geoboard to the students and ask them to form these six segments on their geoboards.

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1 Unit V • Activity 1

Prerequisite Activity None

Materials A geoboard, rubber bands and geoboard recording paper for each student; a transparent geoboard or a transparency of geoboard recording paper for the overhead projector.

Comments

1. A transparent geoboard on an overhead works well for demonstration purposes. Alternatively, the segments may be drawn on a transparency of geoboard paper and displayed on the overhead. A master for geoboard paper is attached.

Some students may have difficulty reproducing the segments on their geoboards. It may be helpful to make statements like "segment e is formed by coming down 3 spaces and over 1 space."

©Copyright 1986, Math Learning Center

Actions

2. Ask the students which segment is longer: (a) a ore? (b) c orb? (c) b ord? (d) d ore? (e) d orf?

Ask the students how they arrived at their conclusions.

3. Point out that these four segments have different lengths. Ask the students to find as many geoboard segments, all of different lengths, as they can, and to record the segments on geoboard paper.

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Comments

2. Encourage students to fmd their own methods of comparison. For some, the length relationships may be obvious. Others may need to compare lengths of segments by measuring with rulers or by making marks on paper. Some may recognize that "3 down and 1 over is longer than 2 down and 1 over."

For those students who think segment b is the same length as segments a and c because they connect two adjacent points, the following figure may help. There are two paths from P to Q. Half the length of the shorter path must be less than half the length of the longer path. So segment a is shorter than segment b.

a • • Y. • • • • • PMQ •

b b • • • • • • • • •

3. There are 14 different lengths that are possible for geoboard segments. One collection of 14 geoboard segments, all with different lengths, is shown below.

Students can be asked to plot segments on the overhead one at a time. As a segment is plotted, the class can decide if it is a segment of new length.

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Actions

4. Ask the students to construct the polygon shown below. Point out that this polygon has six sides. Ask them to make and record polygons with more than six sides by altering side AB while leaving the remaining sides fixed. Discuss the results.

A

7 Sides

10 Sides 11 Sides

5. The students may enjoy other geoboard explorations. Here are two that can be done as individual or small group activities:

(a) Find and record geoboard polygons with differing numbers of sides. What is the greatest number of sides possible?


22 Sides

Comments

4. In this activity, a figure will be called a polygon if its sides are segments and they enclose one interior r-------, region. Figures of • •

• this type are also called simple polygons while figures whose sides are segments, but enclose • more than one interior ..._ ___ _.

region, such as the one shown, are sometimes called non-simple polygons.

Side"AB can be altered to form (simple) polygons of 7, 8, 9, 10, 11, 12 and 13 sides. One example of each is shown.

8Sides 9 Sides

12 Sides 13 Sides

5. (a) Here are examples of polygons of 22, 23 and 24 sides.

23 Sides 24 Sides


Actions

5. (b) Put bands around the edge of a geoboard to form a square as shown on geoboard I. On geoboard IT this square has been divided into two congruent parts. Find and record other ways to divide the square into two congruent parts.

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Comments

. 5. (b) Two figures are congruent if they have the same size and shape. If there is any doubt that the two parts of the square are congruent, record them on geoboard paper, cut them out, and see if one can be made to fit exactly on top of the other.

There are many different ways to divide the square into two congruent parts. The students may want to make a bulletin board display of different solutions. Here are a few:

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Geoboard Recording Paper (Separated Boards)

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Unit V • Activity 2 Geoboard Areas 0 v E R v E w Prerequisite Activity

=~ None, but it may be helpful to .. do Ceo

Actions

1. Form the two geoboard regions shown below. Show the geoboard to the students and ask them to make the same regions on their geoboards.

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2. Ask the students to use additional rubber bands to subdivide the large region into squares the size of the small square. Remind them that the number of squares covering a region is called its area. Discuss.


I

board Figures (Unit VI Activity 1) first.

Materials A geoboard and rubber bands for each student; a transparent geoboard or a transparency of geoboard recording paper.

Comments

1. A transparent overhead geoboard works well for demonstrating the formation of regions. The large regions require several rubber bands.

2. The area is 12 square units. The small square will be the unit of area throughout this activity.

D


Actions

3. Ask the students to form each figure from Activity Sheet A on their geoboards and determine its area. Discuss.

4. Have the students • • • • • form this region on their geoboard and ask them to • determine its area. Dis-cuss.

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Comments

3. The last four regions on sheet A involve both squares and half-squares.

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areaS (4 squares and 2 half-squares)

area 14 (12 squares and 4 half-squares)

area2 (4 half-squares)

area 12 ( 1 0 squares and 4 half-squares)

4. The region can be subdivided into 4 halfsquares. .----------,

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Some students are able to obtain the area by mentally manipulating parts of the region into a convenient shape. (One possible sequence of manipulations is given below.) This ability should be acknowledged and encouraged.

. . . . . . . . . . . . . . ..


Actions

5. Ask the students to form each figure from Activity Sheet B on their geoboards and determine its area. Discuss.

6. Tell the students to form this triangle on their geoboards and find its area. Discuss.

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Comments

5. Each of these regions can be partitioned into squares and half-squares. As you monitor this activity, acknowledge different approaches.

6. This triangle doesn't divide nicely into squares and half-squares, so it is interesting to see how students find its area. A common method is to cut off the right side of the triangle and fit it under the left side to make a rectangle of area 6.

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Another method is to enclose the whole triangle in a rectangle of area 12 and notice that the triangle is half of this rectangle.

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0 i -----Discuss these approaches and others that students have devised.


Actions

7. Ask the students to find the areas of the triangles on Activity Sheet C. Discuss.


Comments

. 7. There is more than one way to fmd the area of each of these triangles. However, the "enclosing method" works for each.

One way to fmd the area of the following triangle is to divide it in two and fmd the area of each half by enclosing it in a rectangle. Each half has area 4 since it is half of a rectangle of area 8 which encloses it Thus, the triangle has area 8.

Before After

When the triangle below is enclosed in a rectangle of area 16 one must subtract 8 (for unwanted region a) and subtract 2 (for unwanted region b), which leaves area 6 (16-8 - 2) for the triangle .

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Before After

Look for and acknowledge other approaches to fmding areas.


8. (Optional.) Ask each student to find the areas of geoboard regions selected from Activity Sheets D and E as appropriate for the level of the student.


8. Sheet D uses a combination of methods that have been discussed previously.

Several different techniques may work to find the . .area of a region. For example, the area of this region may be visualized as:

a) the sum of the areas of two triangles (4 + 4 = 8),

b) the sum of the areas of four triangles (1 + 3 + 1 + 3 = 8),

c) what remains when the areas of four triangles are subtracted from the area of the whole geoboard (16 - 1 - 3 - 1 - 3 =

8).

Sheet E has challenge problems. However, the area of each region can be obtained by the methods that have been discussed. You may wish to have the students solve and discuss these problems one at a time.


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Activity Sheet V-2-A Math and the Mind's Eye

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Find Shaded Areas

Activity Sheet V-2-B Math and the Mind'sEye

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Activity Sheet V-2-C Math and the Mind's Eye

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Activity Sheet V-2-D Math and the Mind'sEye

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Find Shaded Area

Activity Sheet V-2-E Math and the Mind's Eye

Unit V • Activity 3 Areas of Silhouettes

Actions

1. Distribute Activity Sheet A (attached) and a transparent grid sheet to each student.

2. Have the students place their grids on figure B to determine the number of squares needed to exactly cover this figure. Tell (or remind) them that the number of squares exactly covering a figure is called its area.

Figure B

3. Ask the students to use their grids to determine the area of each of the figures on Activity Sheet A.



Materials Activity sheets A. B, C, and D and a transparent centimeter grid for each student (see Comment 1)

Comments

1. One transparency of the attached grid sheet cut along the indicated lines will supply transparent grids for six students.

2. Figure B can be covered by 8 squares. These squares are actually centimeter squares but you may wish to refer to the area as "8 square units" rather than "8 square centimeters".

3. You may want students to work individually and then compare answers in groups of two or three.

The number of units for each figure is: A, 1; B, 8; C, 12; D, 6; E, 12; F, 6; G, 7; H, 20; I, 15; J, 13; K, 12; L, 13.


I Actions

4. Distribute Activity Sheet B. Have the students use their grids to determine the area of figure M. Discuss.

Figure M

5. Discuss the "guide lines" on the activity sheets. Have students use their grids to determine the area of figures N through U on Activity Sheet B.


Comments

4. Figure M can be covered with 3 squares and 3 half-squares, or 4 lfz squares altogether. There are other ways to find the area of figure M. (See Comment 6.) If a student uses one of these methods, it can be discussed now.

5. To correctly place the grid on a figure, one line of the grid should be aligned with the guide line, as illustrated, and each vertex of the figure should coincide with a point of intersection on the grid sheet.

The areas of figures N through U are: N, 8; 0, 6; P, 9; Q, 9; R, 6 lfz; s, 10 lfz; T, 12; u, 10.


Actions

6. Ask the students to determine the area of figure W. Discuss.

Figure W

7. Have the students to use their grids to find the areas of figures V, X, Y, and Z.


Comments

· 6. Figures 1 and 2 illustrate two solutions. In figure 1, the top part of the triangle is moved to fill in the bottom 2 by 3 rectangle. Thus the area of the triangle is the area of this rectangle which is 6. In figure 2, the triangle is enclosed in a.~ by 6 rectangle. The area of this rectangle is 12. Because the triangle covers half the rectangle, the area of the triangle is 6.

Figure 1 Figure 2

7. The areas of V, X, Y and Z are 3, 10, 6, and 10 lfz respectively.


Actions

8. Distribute Activity Sheet C and ask the students to use their grids to determine the area of figure A. Discuss.

Figure A


Comments

8. There are several ways to discover that the area of figure A is 9 lfz square units. One way is to subdivide figure A into three triangles and a rectangle as shown in figure 3. The areas of these parts are 1 lfz, 1, 3, and 4. (Some students may n~ to redraw the subregions, as shown in figure 4, to fmd the area of each part.)

Figure 3

Figure 4

Another way is to enclose figure A in a rectangle and then subtract away unwanted areas from the area of the rectangle. The area of the rectangle is 15 and the areas of unwanted regions I, II, and ill are 1 lfz, 1, and 3, respectively. These unwanted areas totalS lfz. Hence the desired area is 15 - 5 lfz or 9 lfz.

Figure 5


9. Ask the students to find the areas of figures B through N.

10. Distribute Activity Sheet D and ask the students to use their grids to determine the areas of figures 0 through Z.


9. The areas of these figures are: B, 6; C, 6; D, 3 lfz; E, 11; F, 5; G, 6; H, 10; I, 4; J, 10; K, 11 lfz; L, 8; M, 17; N, 10.

10. If necessary, remind students of the correct placement of the guide lines.

All of these areas can be obtained by enclosing the figures in rectangles and subtracting unwanted areas from the areas of the rectangle, as illustrated in Comment 8. Look for and discuss other methods students use.

The areas of the figures are: 0, 6; P, 2; Q, 8; R, 4; S, 5 lfz; T, 12; u, 14 Ifz; v, 10; w, 3; X, 4; Y, 4; Z. 4.


Name ____________________________ __ Areas of Silhouettes A

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Name ____________________________ __ Areas of Silhouettes 8

Activity Sheet V-3-8 Math and the Mind's Eye

Name ____________________________ _ Areas of Silhouettes C

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Activity Sheet V-3-C Math and the Mind's Eye

Name __________________________ __ Areas of Silhouettes D

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Activity Sheet V-3-D Math and the Mind's Eye

CENTIMETER GRID SHEET

Unit V • Activity 4 Geoboard Triangles

Actions

1. Distribute a geoboard, rubber bands and Activity Sheet A to each student. Place the segment illustrated in figure 1 on your demonstration geoboard and form a triangle on that base, as in figure 2. Review, as necessary, methods for finding the area of this triangle (see Unit V/Activity 2, Geoboard Areas).

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Fig. 1 Fig. 2

2. Ask the students to find the number of ways they can make a triangle using the base in figure 1. Tell them to record each triangle and its area on the record sheet.

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When all triangles, with their areas, have been recorded, ask the students to cut out each geoboard square that has a triangle drawn on it.


Prerequisite Activity GeoboardAreas (Unit V/Activity 2)

Materials A geoboard, rubber bands, Activity Sheet A, scissors and geoboard recording paper for each student; a transparent geoboard or a transparency of geoboard recording paper; and a transparency of page 4 for thet~cher.

Comments

1. A master of Activity Sheet A is attached.

The area of the triangle is 4 square units.

2. The students should record only one triangle per geoboard on record sheet A. The area may be written inside the triangle. The 20 triangles that can be formed are shown on page 4 of this activity.

Make sure that students cut around each geoboard and not around each triangle.


Actions

3. Have the students use their geoboard cutouts and do the following.

A. Group the triangles by area. Then find a characteristic that triangles of the same area have in common. Discuss.

B. Match the triangles which have the same size and shape.

C. Identify all triangles that have a square comer (right angle).

D. Find all triangles which have two sides of equal length.

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Comments

3. Demonstration cutouts for the overhead can be made from a transparency of page 4.

A Besides having the same base, the triangles of equal area have the same height. Students might say, for example, that all triangles of area 4 have their top vertex in the second row of dots from the bottom, those with area 6 have their top vertex on the third row from the bottom, etc.

B. Figures which have the same size and shape are called congruent. There are 8 pairs of congruent triangles and 4 triangles that have no match. Pairs like the following can be seen to match by turning one over and placing it on top of the other.

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C. There are 9 right triangles. This one is sometimes overlooked.

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D. Triangles which have two sides of equal length are called isosceles. There are 6 isosceles triangles.

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Actions

4. (Optional.) Ask the students to conduct the following investigations.

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A. Find and record geoboard triangles with areas 1, 2, 3, 4, 5, 6, 7 and 8 square units.

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B. Find and record as many isosceles geoboard triangles as possible. Do not record a triangle if it is congruent to one that has been previously recorded.

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4. A. Here are examples of each.

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B. There are 22 non-congruent isosceles triangles.

The following triangles were found by systematically considering bases of different length. There are other strategies for systematically finding these triangles.

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4 Unit V • Activity 4 Math and the Mind's Eye

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Unit V • Activity 5 Geoboard Squares

Actions

1. Distribute geoboards, rubber bands, and geoboard recording paper. Ask the students to form geoboard squares of different sizes and to record each square on different sections of the geoboard recording paper.

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2. Ask for volunteers to sketch the squares on the overhead projector.


Prerequisite Activity Geoboard Areas, Unit V, Activity 2

Materials Geoboards, rubber bands, geoboard recording paper with centimeter spacing and metric rulers for the students. A demonstration geoboard and transparencies of geoboard recording paper for the instructor. ·Calculators are optional.

Comments 1. It may be necessary to tell the students that some of the squares are "slanted" Here are the eight different geoboard squares that have nails at the comers:

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2. A transparency of geoboard recording paper may be used for this purpose. Many students enjoy putting their squares on the overhead. Students can add squares one-byone until all have been drawn. If anyone thinks two recorded squares are congruent, a paper pattern made for one can be compared to the other.


5. Display these squares on an overhead transparency and ask students how the length of the side of a square appears to be related to its area. Discuss.

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6. Examine the relationship side x side = area for these squares.

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7. Tell the students that in every square side X side =area. Ask them why they think it is, that for some of their geoboard squares side X side does not exactly equal the area.

8. Tell the students that the symbol...f5 is used to represent the exact length of the side of a square of area 5. So ...f5 is the number such that

...J5 x...J5 = 5 Label the side lengths of these squares using this new notation.

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5. The students should become aware that the length of a side, multiplied by itself, gives the area of the square. Or, for any square:

side x side = area .

6. Calculators are useful for fast computation here. Collect and display several of the students measures for the side lengths of a

square. For example, the students measures for the side of the square with area 5 might be 2.2, 2.25, 2.3, and · 2.4. Squaring each of those side lengths yields 4.84, 5.0625, 5.29 and 5.76, respectively.

7. For squares of areas 1, 4, 9 and 16 a measuring instrument was not necessary to determine the length of a side. For the other four squares the readings from the rulers gave approximate lengths, as all ruler readings do.

8. The side lengths are ~2, ~10 and ~8. These symbols represent numbers which when multiplied by themselves give 2, 10 and 8, respectively. We can approximate these numbers using a calculator with a ~ key.

~2 = 1.4142135 (approximately) ..J10 = 3.1622776 (approximately) ..J8 = 2.8284271 (approximately)

Calculators which display more digits will give a better approximation. One can never write down all the digits in the decimal name for these numbers because they continue forever. So it is

convenient to give a length as ..f8 or approximately, 2.8.


Actions

9. (Optional) Ask the students to use their calculators to obtain ..J2 = 1.4142135 and then multiply

1.4142135 X 1.4142135 to see if they get exactly 2 or close to 2. Let the students experiment with the ..J key on their calculators. Point out that most calculator computations are approximate.

10. Display this geoboard segment: • • • • • • • • • •

Ask the students to: • • • • • a) Represent this segment AB on

~8 their geo-recording paper. (Fig. ~ . . . . I A)

b) Draw a square with AB as one side. (Fig. B)

c) Determine the area of the square. (Fig. C) d) Determine the length of side AB. (Fig. D)

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

~ • • • • • • • • • •

• • Fig. A • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

17 • • • • • • • • • • • Fig. C • • • • • • • • • • • • • • • • • • • • • • • • • Fig. D • • •

17 • • • • • • • • • .Y17 • • • • • • • • • • • • • • • • • • • • •


Comments

9. For perfect squares ( 1, 4, 9, 16, 25, 36, .... ) the ...J key will produce an exact number. For all other numbers (2, 3, 5, 6, 7, 8, 10, 11, .... ) the ...J key will give an approximation.

10. You may wish to go through each step on t:Pe overhead requesting student help in the process.


Actions

3. Ask the students to find the area of each square and to record it inside the diagram of that square on their record sheets.

[] •

• • • •

• • • • •

• • • • •

4. Furnish students with a metric ruler. Ask the students to determine the length of a side of each square. Record the lengths at the side of each square.

• • •


• • •

• • • • • •

Comments

3. Methods of determining these areas are discussed in Unit N, Activity 2, Geoboard Areas . The areas of the eight squares are 1, 2, 4, 5, 8, 9, 10 and 16 square units.

OJ • • •

[] • •

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

0. • •

• • • • • • 16 • • • • • • • • • • • • •

• • • • •

4. Have the students compare the subdivisions on their metric rulers with the spacing between the dots on the geoboard paper. The horizontal and vertical distance between dots is 1 centimeter The squares with areas 1, 4, 9 and 16 have side lengths 1, 2, 3 and 4 units, repectively. A metric ruler with millimeter marks is convenient for measuring the sides of slanted squares to the nearest tenth of a centimeter. The squares of areas 2, 5, 8, and 10 have side lengths of approximately 1.5, 2.2, 2.8 and 3.2 units, respectively .


Actions Comments

11. Ask the students to find all segments of different length 11. Below are the 14 geoboard segments of

that can be constructed on a geoboard and then determine the different length. Each can be thought of as

length of each segment. the side of a square as in Action #10.

• • • • • • • • • • • • • • • 4

• • • • • • • • • • 3• • • • • • • • • • •

2• • • • • /. • • ~ - 1 • • • • • • • • • •

-./2 -./5

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

/: /: •

• ~ :.:.--:-:--: • • • • • • • • •

• • • • • • • • • • • • • • • •

-./10 -./17 -./a -./13

• • • • • • • • • • • • • • • • • • • •

/ •

• • • • • • • •

• • • •

-./20 -./18 -./25 -./32


Geoboard Recording Paper · (Adjoining Boards)

• • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Unit V • Activity 6 Pythagorean Theorem 0 v E R v E

Actions

1. Give the students dot paper and ask them to draw this right triangle.

• • • • • • • • • • • • • • • • • • • • • • • • • • • •

2. Ask the students to draw a square on each side of the triangle.


w Prerequisite Activity Geoboard Areas (Unit VI Activity 2) or Areas of Silhouettes (Unit V/ Activity 3), Geoboard Triangles (Unit VI Activity 4) and Geoboard Squares (Unit V/ Activity 5)

Materials Dot paper for students and overhead transparencies as noted (see Actions 1, 10 and 12).

Comments 1. Model this on an overhead dot-paper transparency. Attached to this activity is a dot paper master from which a transparency can be made.

2. Illustrate this by drawing the square on the shortest side of the triangle.

• • • • • • • • • • • • • • • I ~ • • • • • • • • • • • • • • • • •


Actions

3. Have a volunteer draw the squares on the sides of the triangle on the overhead transparency.

4. Ask the students to compute the area of each square and write it inside the square.


Comments 3. Make sure that the quadrilateral drawn on the hypotenuse is a square.

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ·-• • • • • • • •

4. The area of the square on the hypotenuse may be determined by the enclosing method (see Unit VI Activity 2, Geoboard Areas). Someone may notice that the sum of the areas of the squares on the legs of the triangle is equal to the area of the square on the hypotenuse. If so, they will get a chance to check their conjecture in Action 5.

• • • • • • • • • • • • • • • • • • • • • • • •

9 • • • • • • • • • • • • • • • • • •


Actions Comments

5. Ask the students to draw the following right triangles on 5. In each case, the sum of the areas of the

dot paper, construct squares on the sides of the triangles, and squares on the legs of the triangles is the same as the area of the square on the hypot-

compute the areas of all squares. Discuss the results. en use.

~ • • • • • • • • • • • • • • • • • • • •

1 '''12 • • ~

• ~· • ~ • •

~. . . • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • . . • • • • • .2!:2 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

•


Actions

6. Draw this oblique (non-right) triangle on an overhead transparency. Ask the students to construct a square on each side and determine the areas of these squares.

• • • • • • • • • • • • • • • • • • • •

7. Ask the students to make conjectures about the areas of squares on the sides of triangles. Write these conjectures on the overhead or chalkboard.


Comments

6. You may wish to have students create their own oblique triangles and compute the areas of the squares on the sides. In no case, for oblique triangles, will the areas of two of the squares sum to the area of the third square .

7. Some possible conjectures are:

The sum of the areas of the squares on the shorter sides of a right triangle equals the area of the square on the longest side.

The sum of the areas of two squares on the sides of an oblique triangle never equals the area of the third square.

The sum of the areas of any two squares on the sides of an oblique triangle is greater than the area of the remaining square.

The last conjecture turns out to be false (see the figure below), but that happens with some conjectures. A visual proof of the first conjecture is given in Action 8.


Actions

8. On an overhead dot-paper transparency, demonstrate a visual proof of the Pythagorean Theorem as follows:

a. Draw congruent right triangles at the top and bottom of the dot paper (A).

b. Construct squares on the legs of the top triangle. Construct a square on the hypotenuse of the bottom triangle (B) ..

c. Enclose both top and bottom figures in the smallest square possible (C). Ask the students to determine the area of the enclosing squares.

d. Add the dotted diagonal to the top figure in D. Pose questions like: How do the areas of the triangles in the top and bottom figures compare? If the four triangles in the top and bottom figures of D are cut away, what can be concluded about the areas of the regions that remain?

A B c

Comments . 8 This demonstration at the overhead

focuses attention on each step and allows the teacher to pose questions to the whole class.

The .enclosing squares have the same area. In this case, 100 square units.

Triangles a, b, c and d in the top figure of D and triangles a, b, c and d in the bottom figure of D are all congruent to one another. Each is a right triangle with legs measuring 3 and 7 units, and has an area of 10 lf2 square units.

The portions that remain have the same area, that is, the sum of the areas of the two squares in the top figure of D is the same as the area of the square in the bottom figure of D.

D

llli~

. . . . . . . . . . . . .


Actions Comments

9. Have the students repeat Action 8 using a right triangle of their own design. Discuss their conclusions.

9. The important observation here is that no matter what size or shape right triangle is chosen, the area of the squo.re on the hypotenuse is equal to the sum of the areas of the squo.res on the legs.

10. Put a transparency of page 8 on the overhead. For the first triangle, draw squares on the legs and ask the students to tell you the area of the square on the hypotenuse without drawing it. Then ask them to imagine drawing squares on

10. Record the answers in square boxes so students don't confuse the area of the square on the hypotenuse with the length of the hypotenuse.

the legs of the other right triangles. .---------------------., This will aid them in computing the area of the square on the hypo ten-use. ::: ·~· :[§]:::::::::::::: :: ~4·1·:::::::

0 • • • • • • • • • • • • • • • • • • • • • • • t.::!:..!.J ••••••• . . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . . . . . . . . . . . . . . . . . . . . . . ...... .

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8 UnivV•Aclivily6 Malh and lhe Mind's Eye


Actions Comments

11. Review area of squares with the students: first, if the length of a side of a square is known then the area of the square is (length of side) x (length of side); second, if the area of a square is known then the length of the side is the square root of the area.

11. These ideas were covered in Unit V, Activity 5, Geoboard Squares. Before proceeding to Action 12, students should be able to find the length of the hypotenuse if the area of the square on the hypotenuse is known.

12. Put a transparency of page 9 on the overhead. For the first triangle, compute the area of the square of the hypotenuse and then the length of the hypotenuse. Ask the students to do the same for the remaining triangles.

12. These right triangles all have sides with whole number lengths. Depending on the level of your class, you may wish to introduce examples in which the hypotenuse has non-integral lengths. All the triangles in Action 10 have hypotenuse lengths which are non-integral .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rtt+iliU13 ........ . • • • • • • • 0 • • • • • • • • • • • • • • • • • ••••••••

·····~·s········s ···· ··········· ::: 3 . : .. :::::::: .. ::::::.: .. :::::: •• 0 • • • • • • • • • • • • • •••

. . . . . . 4 ................ 12 .......... . 0 • • • • • • • • • • • • • • • • • • • • • • ••••••••••

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: :: : : : : : : : 16: : : ::: ::: : : : :: : : :: :: ::: :: r------------;- ................ . . . . . . . . . . . . . . . . ................ . ::::::::::: .. ::::::::::: 6::::::: ·····················~···· . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . ........ . ::::::.:20:::::::::: 8 :: .. ::::: . . . . . . . . . . . . . . . . . . . . . . .. 10 .... . . . . . . . . . . . . . . . . . . . . . . . ........ . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . . . . . . . . . . . . . . . . . . . . ........ .

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. . . . . . . . . . . . . . . ................... .

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...... 8 ............... 21 ........... . 9 Unil V • Aclivily G Math and the Mind's Eye


• • • • • • • • •

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8 Univ V • Activity 6

• •

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• • • •


Unit V • Activity 7 Geoboard Perimeters 0 v E R v E w

Actions

1. Have the students form these two polygons on their geoboards. Ask them to imagine that these polygons represent fenced enclosures. Have them compute and compare the areas of the enclosures. Then ask them to compare the length of fencing needed for each enclosure.

• •

• • • •

2. Tell the students that the sum of the lengths of the sides of a polygon is called its perimeter. Ask the students to compute the perimeter of the two enclosures in Action 1.


Prerequisite Activity Pythagorean Theorem (Unit . V, Activity 6)

Materials A geoboard, rubber bands and geoboard recording paper for each student. An overhead geoboard or geoboard transparency for the teacher.

Comments

1. The enclosures each have an area of 4 square units.

There are several ways to see that the nonrectangular enclosure takes more fencing than the other. The short ends of both regions are the same. However, the "slanted" sides of the one are longer than the vertical sides of the other. This can be seen by i) directly measuring, ii) recalling from past experiences that the shortest distance from the ceiling to the floor is the plumb-line distance, or iii) by using the Pythagorean theorem to determine that the length of the slanted side is .,J 17, which is greater than 4, the length of the vertical side.

2. The perimeter of the rectangular enclosure is 1 + 1 + 4 + 4 = 10 units. The perimeter of the other is 1 + 1 + .,J17 + .,J17, which can also be written as 2 + 2-,J 17. Note that v 17 > 4 because the side of a square with area 17 is greater that the side of a square with area 16 (See Geoboard Squares, Unit V, Activity 5). This is also a good place to use calculators to approximate .,J17 as 4.1. Then one can write, 2 + 2-,J17"" 10.2, where the symbol, ""• means approximately equal.


Actions

3. Show the students the following pairs of geoboard poly-gons.

1 • • 2 • •

2J • • • a

• • • • •

3

Ask them to:

i) Look at each pair of figures and make a mental comparison of their perimeters. Conjecture which figure in each pair has the greater perimeter, or if their perimeters are equal.

~ •

• •

4

5

ii) Find the actual perimeters to check the conjectures.

Discuss their results.


• •

Comments

3. A visual inspection of the polygons in #1 will reveal that the sides of each contain 4 unit lengths and 2 diagonal segments of the same length, so their perimeters are equal. In #5, both polygons have two sides of unit length. The remaining 4 sides of polygon a have the same length. This length is longer than the length of each of the remaining 4 sides of polygon b. Hence polygon a has the greater perimeter. Students may have other ways of making mental comparisons of perimeter.

Below are the actual perimeters of the polygons. Notice that there are different ways of expressing perimeters. For example, the hypotenuse of triangle a in #4 can be computed in two ways: Since each of the 2 legs has length 3, by the Pythagorean Theorem, the length of the hypotenuse is ...f(32 + 32) = ...J18. On the other hand, the hypotenuse can be viewed as being comprised of 3 segments, each of which is the diagonal of a unit square. Since the length of the diagonal of a unit square is ...J2, the length of the hypotenuse is 3...f2.

1 a) 4 + 2...J2 = 6.8 b) 4 + 2...J2"" 6.8

2 a) 6 + 2...J2 = 8.8 b) 8

3 a) ...J8 + ...J5 + ...f17 = 2...J2 + ...J5 + ...J17"" 9.2

b) 2 + 6...J2"" 10.5

4 a) 6 + ...J18 = 6 + 3...f2 = 10.2 b) 2 + ...J13 + ...J25 = 2 + ...Jn + 5 =

10.6

5 a) 2 + 4...J10 = 14.6 b) 2 + 2...Js + 2...J2 = 2 + 6...J2 = 1o.s


Actions

4. Point out that 7 of the 10 polygons in Action 3 have an area of 3 square units, and each of these polygons has a different perimeter. Ask the students to draw other geoboard polygons with area 3 and perimeters differ-ent from the above. :lj :. :

• • • • • • • • • • •

• • • • •

5. i) Ask the students to find geoboard polygons with perimeter 14 but different areas.

ii) Ask the students to find geoboard polygons with perimeter 2 + 2-/2 + 2-/5 but different areas.

c:=;-1 : : : u • • • • •

Comments 4. In Action 3 polygons la, 2a, 2b, 3a, 3b, 4b and 5b have area 3 but different perimeters. Here are some additional polygons with area 3:

• • • • •

:<::::I • • • • • • • • • •

• • • • •

: :~ -~ • • • • •

: :8:: • • • • • • • • • • • •

5. i) It is possible to get geoboard poly-gons of area 6, 7, 8, 9, 10, 11, and 12 square units which have perimeter 14. Here is one diagram of each. There are other possible diagrams.

• • • • • • • • • •

A=7

r:--;=:J LJ:: • • • • •

g SJ .. CJ·. o·. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Continued next page.


Actions

6. i) Show the students this polygon on the overhead projector and, with their help, compute its perimeter.

ii) Exhibit this polygon on the overhead and ask the students to compute its perimeter. Discuss.

iii) Ask the students to find a geoboard polygon with greater perimeter than the ones just discussed.


Comments

5. ii) With perimeter 2 + 2...f2 + 2...f5 it is also possible to get polygons with different areas. Here are some possibilities:

• • • • •

• • • • •

~ :-v.-: • • • • • • • • • •

Y :. ·: ·v· ·: ·o· ·: • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

6. i) One way to avoid confusion when computing perimeters is to select a place to begin and then list the length of each side in order. If we start at the side indicated below, and proceed around the polygon clockwise, the perimeter will be: 1 + ...f2 + 1 + ...f2 + 1 + 1 + 1 + ...f2 + 1 + 1 + ...Js + 2...J2 + ...Js + ...J2 + 3 + ...J2 + 1 + ...J2 + 1 + ...f2 + 1 + 1 = 14 + cr/2 + 2...f5"" 31.2

Starting Here

ii) The perimeter of this polygon is: 2 + 1 + ...J2 + ...J1o + ...Js + 3 + ...Jn + 2...J2 + 2 + 1 +...f2 + ...Jw + ...Js + 3 +...f13 + 2...J2 =

12 + 6...f2 + 2...f5 + 2...f10 + 2...f13 "" 38.5



5 Unit V ·Activity 7

11 + 9->/2 + 2"-/5 + 3-.f13 "'39.0

iii) Here are a few of the many polygons of larger perimeter. Some students enjoy trying to fmd the geoboard polygon of greatest perimeter.

11 + 5-.f2 + 7-.fS + 2"-113 "'40.9

13 + -.f2 + 7-.fS + 4-.f1 0 "'42.7

Here is the polygon with the greatest perimeter found to date.

11 + 3-.f2 + 7-.fS + 2"-/10 + 2"-113 "' 44.4


Unit V • Activity 8 An Introduction to Surface Area and Volume

0 v

Actions

1. Distribute one cube, a supply of grid paper and a pair of scissors to each student.

2. Show students an example of a jacket for a cube. Ask them to cut out other jackets.

3. Collect and display various jackets students have made.



Materials Cubes, grid paper, scissors

Comments 1. Use grid paper that has squares which are the same size or slightly larger than the face of the cube. Three-fourths inch cubes and 2 centimeter grid paper work well. A master for 2 centimeter grid paper is attached.

2. Here is one jacket:

I

When folded around the cube it covers each face of the cube exactly once. (The jacket shown is a hexominoe. There are 10 other hexominoes which are jackets for a cube.)

Jackets need not be cut on grid lines:

3. Jackets can be silhouetted on an overhead projector. Students may cut out shapes that do not cover all faces of the cube or cover a face.more than once. These can be called quasi-jackets or almostjackets; you can discuss why they are not jackets.

© Copyright 1985, Math Learning Center

Actions

4. Distribute another cube to each student. Ask the students to suppose the cubes have been glued together to form a single rectangular solid. Have them cut out jackets for this solid. Collect and display a variety of the jackets produced.

5. Ask students what they have discovered about the number of squares in jackets for one- and two-cube rectangular solids. Describe what is meant by the volume and suiface area of a rectangular solid.

6. Display a three-cube rectangluar solid. Ask students to determine how many squares in a jacket for this solid. Ask for volunteers to explain how they arrived at an answer.


Comments 4. Here is one jacket:

I

Other names for a rectangular solid are rectangular prism or rectangular parallelpiped.

5. Jackets for one cube contain 6 squares. Those for two-cube rectangular solids contain 10 squares. The volume of a rectangular solid is the number of cubes it contains; its surface area is the number of squares in a jacket which covers each face of the solid exactly once.

6. A jacket for this three-cube rectangular solid will contain 14 squares. Following are some of the ways students reported they arrived at this answer:

• "A three-cube solid is like a two-cube solid with another cube in the middle. This cube has 4 faces showing, so a jacket for 3 cubes will be 4 more than a jacket for 2 cubes. Since a two-cube solid has 10 squares in a jacket, a three-cube solid will have 14."

• "Each cube has 6 sides so it will take 18 squares to cover 3 cubes. But in the threecube solid, 4 sides don't show. So a jacket will have 14 squares."

• '1t will take 3 squares to cover each of the 4 sides. That's 12 squares. It will take 1 square to cover each end. So altogether it takes 14 squares."

Other students will make a jacket for the solid and count the number of squares in the jacket.


Actions

7. Summarize the results that have been discovered so far.

8. Distribute additional cubes so each student has 4. Ask the students to determine the number of squares in a jacket for a four-cube rectangular solid.

9. Build and display a 2 X 3 X 4 rectangular solid. Ask students to determine (a) the number of cubes in the solid, and (b) the number of squares in a jacket for the solid. Ask several students to explain their reasoning.

10. (Optional) Ask students to cut out jackets for other cube structures.


Comments

7. This can be done in a series of statements:

• A one-cube rectangular solid has a volume of 1 and surface area of 6.

• A two-cube rectangular solid has a volume of 2 and surface area of 10.

• A three-cube rectangular solid has a volume of 3 and surface area of 14.

8. Some students may claim there are 16 squares and some may claim there are 18. Both are correct; it depends on whether the cubes are arranged in this way:

orin this warfJ

Thus, a four-cube rectangular solid has a volume of 4 and a surface area of 16 or 18. This shows that solids with the same volume can have different surface areas.

9. There are 24 cubes in the solid. A jacket will contain 52 squares. Students may use a variety of methods to arrive at their answers. Some students may have a difficult time explaining how they arrived at answers. Encourage the students to help each other clarify their reasoning processes.

10. Some possibilities are:


Unit V • Activity 9 Shape and Surface Area 0 v E R v E w

Actions

1. Divide the students into groups of two or three. Distribute 24 cubes to each group. Ask each group to construct a rectangular solid, or box, using all 24 cubes.

2. Ask each group to find the surface area of the box they constructed. Ask them to record this surface area as well as the dimensions of their box.

Surface Area= 52 Volume=24

3. Move among the groups verifying their results and discussing how they computed the surface area. Once a group has successfully found the surface area of a box, ask them to construct another box, again using all 24 cubes, that has a larger (or smaller) surface area. Have them record the dimensions and surface area of this box.


Prerequisite Activity Unit VI Activity 8, An Introduction to Surface Area and Volume

Materials 24 cubes for each group of students (see Action 1).

Comments 1. A rectangular solid is a solid whose faces are all rectangles. In this activity, the word box will be used to refer to a rectangular solid. Note that all boxes are solid, with no empty spaces in their interiors.

2. Remind the students that the surface area of a box is the number of squares in a jacket for the box (see Unit 5, Activity 8, An Introduction to Surface Area and Volume).

The box shown has dimensions 2 by 3 by 4, usually written 2x3x4, and surface area 52. Some students may not be familiar with the concept of dimensions and it may be necessary to discuss this with them.

The dimensions and surface area of all possible boxes are listed in Comment 4.

3. Students will use various methods to compute the surface area. Some may wish to have graph paper to make or sketch a jacket for their box.

Unless a group built a lxlx24 box, there will be a box constructed of 24 cubes that has a larger surface area.


Actions

4. Ask the various groups of students to report the dimensions of the boxes they constructed and the surface area of each box. Tabulate this information on the chalkboard or overhead.

5. Ask the students' assistance in constructing all possible boxes containing 24 cubes. Arrange these in a row from largest to smallest surface area. Do this on a desk, table, the floor or other location where all students can see the row of boxes.

f''F'r r r r r ,r 'F''''t r' r r: r,,,r t' r r 12"" ,z,r F''F r, ~ 1 x1 x24

Surface Area= 98

1 x 2x 12 Surface Area= 76

1 x3x 8 Surface Area= 70

Comments

· 4. There are 6 different boxes, each containing 24 cubes, that can be built. A table might look like the following, although it is not necessary that entries appear in any particular order.

Dimensions lxlx24 lx2x12 lx3x8 lx4x6 2x2x6 2x3x4

Surface Area 98 76 70 68 56 52

Volume 24 24 24 24 24 24

5. Each group can be asked to construct a different box. The row of boxes is shown below.

1 x4x 6 Surface Area= 68


2x2x 6 Surface Area= 56

2x3x 4 Surface Area= 52


6. Ask the students to look at the row of boxes and make observations about what they see. Discuss.

7. Tell students to imagine the 2x3x4 box is made of clay. Discuss with them how the clay might be rearranged to decrease its surface area.

8. (Optional) Ask the students to find the surface area of a cube of volume 24.


Volume=24 Edge = 3'V24 = 2.88

Area of Each Face= 8.29 Surface Area= 49.77

6. Some possible observations are:

• "As the blocks get skinnier, the smface area increases."

• "As the dimensions of the bOxes get closer to each other, the smface area decreases."

• "The more compact the box, the smaller the smface area."

7. Some students may suggest the clay be formed into a cube, others may suggest it be made into a ball. Both will produce a smaller smface area. The smface area of a ball is the smallest smface area possible for a given volume.

8. To find the surface area, one must first find the length of an edge of the cube. Since the volume of the cube is obtained by multiplying edge x edge x edge, the edge of a cube of volume 24 satisfies edge x edge x edge = 24 (i.e., the edge is the cube root of 24). Since 2x2x2 = 8 and 3x3x3 = 27, the edge will be between 2 and 3. One can use a calculator and trial and error to find that the edge, to 2 decimal places, is 2.88. Hence the area of one face of the cube, is 2.88x2.88, or 8.29, and the total smface · area is 6 times this, or 49.77.

Some students may wish to know the surface area of a ball of volume 24. Using formulas relating the volume and smface area of a ball, one finds that the smface area of a ball of volume 24 is about 40.24.


Actions

9. Call the students attention to the lxlx24 box. Tell them to suppose that the cubes comprising the box may be cut. Discuss with them how a solid could be formed whose volume is 24 and has a larger surface area than the lxlx24 box.

B

10. Ask the students for their thoughts about the largest possible surface area for a volume of 24.


Comments

9. One way to obtain such a solid is to cut

the 1x1x24 box in half from A to B. Then, imagining a hinge at B, lift the top half at A and unfold.

The result is a row of 48 half-blocks.

The swface area of the new solid is 145. One way to see this is to imagine a jacket around the original1x1x24 solid. It will contain 98 squares. Now slice the solid as above. Remove the one square of the jacket at end B (since none of this end will be part of the swface of the new solid). Now unfold. The resulting solid will be covered by a jacket on all sides except the top. The top has an area of 48 squares. Thus a jacket for the new solid can be obtained from a jacket for the original solid by deleting one square (at end B) and adding 48 squares (along the top) for a net gain of 47 squares. Hence the new jacket will contain 98 + 47 or 145 squares.

It may be helpful to show the above illustrations to the students. A transparency of these figures can be made from the last page of this activity.

10. By repeating the process in Comment 9 (i.e., splitting a box from end to end along the greatest dimension and unfolding), one can always construct a series of boxes, all of the same volume, whose surface areas become as large as one pleases. Hence, theoretically, there is no upper limit for the swface area possible for a given volume.


Actions

11. (Optional.) Give the students examples of occurrences of the general principle: For a given volume, the more compact a solid is, the smaller its surface area or, equivalently, the less compact a solid is, the greater its surface area. Ask the students for other examples that may occur to them.


Comments

11. Some examples are:

• The thinner a coat of paint is applied, the greater the surface area covered.

• In cold weather, animals curL up to conserve body heat by reducing the surface area exposed to air.

• Wood is chopped into kindling to increase the amount of surface area exposed to air so that it is easier to ignite.


8

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Actions

1. Place a loop of string on the overhead. Ask the students for suggestions on how the area enclosed by the loop might be determined. Discuss.


Unit V, Activity 2, Geoboard.Areas or Unit V, Activity 3, Areas of Silhouettes.

Materials Irregularly shaped area pieces (see Comment 3) and centimeter grid paper for each student; loop of string for teacher's use ..

Comments

1. Encourage the students to express their ideas, including those about which they have misgivings or do not have clearly formulated. Other students may have observations which help clarify these ideas.

A student may suggest a procedure for fmding the area which changes the shape of the loop. The student may be assuming that figures with the same perimeter have the same area. To show this assumption is false, the loop of string can be stretched out until there is virtually zero area enclosed. Action 2 further addresses this issue.

The same loop of string can enclose regions with different areas.



Actions

2. (Optional) Give each student a sheet of grid paper. Ask the students to draw several rectangles whose perimeter is 20. Then have them compute the area of each of their rectangles.


Comments

. 1. (Continued.) If no one suggests selecting a unit square and then approximating the number of squares inside the loop, remind the students of the meaning of area.

The area of the region enclosed by the loop is the number of unit squares that

exactly cover it.

2. This Action is intended to help students see that shapes with the same perimeter can have different areas. If your students understand this, you may want to omit this Action; it's not essential for the remainder of the Activity.

11111111

Perimeter: 20 Area: 21

Perimeter: 20 Area: 24

Here is a list of the dimensions and areas of rectangles whose perimeters are 20 and have sides whose lengths are whole numbers:

Dimensions Area 1 X 9 9 2 X 8 16 3 X 7 21 4x6 24 5 x5 25

Note that the more "squarelike" the rectangle, the greater its area.


Actions

3. Distribute an irregular shape and grid paper to each student. Ask each student to approximate the area of their shape. After the students have made their approximations, form the students into groups so that students with shapes from the same set are in the same group.

Comments

3. The six pages following page 7 of the activity are masters for irregular shapes. Shapes can be prepared by making copies of the masters on cardstock and cutting on the heavy lines. There should be one shape for each student

There are six different masters for shapes. On three of them, a set of 4 shapes forms a rectangle and, on the other three, a set of 3 shapes forms a rectangle. In the next Action, each group of students forms a rec~gle with its pieces. So each group has a complete rectangle, the number of sets of 3 shapes and the number of sets of 4 shapes that are distributed can be adjusted to fit the number of students in the class. For example, if there are 29 students present, one choice is to distribute 5 sets of 4 shapes and 3 sets of 3 shapes.

A set of 4 shapes which forms a rectangle.

A set of 3 shapes which forms a rectangle.



'Actions


Comments

3. (Continued.) To facilitate forming groups so that students with shapes from the same set are placed in the same group, sets can be identified by recording the same numeral on the back of each shape in the set Thus all students with 1 on the back of their shapes form a group, those with 2 form another group, etc. Also, making copies of different masters on different colors of cardstock helps distinguish sets.

The students with these shapes form a group.

The masters for shapes have been designed so that the areas of the shapes, and the rectangles they form, are best measured in square centimeters. A master for centimeter grid paper is attached.

An approximation, in square centimeters, of the area of a shape can be obtained by tracing an outline of the shape on a sheet of centimeter grid paper and approximating the number of squares inside the outline.

The outline of this shape contains about 63 square units. There are 48 whole squares in the shaded portion. The remaining partial squares in the outline comprise about 15 additional squares.


Actions

4. Have the students in each group form a rectangle with their group's shapes. Ask them to find the area of the completed rectangle. Then ask them to sum their estimates and compare the total with the area of the rectangle.

5. Discuss with the students the methods they used for estimating the areas.

Comments

4. The area of a group's rectangle can be found by multiplying its dimensions. These can be measured with a centimeter ruler or sheet of centimeter grid paper. For purposes of this activity, measurements to the nearest centimeter are sufficiently accurate.

Having the students sum their estimates and compare it with the total area of the rectangle gives them a check on the accuracy of their estimates.

5. The students will use various methods for counting the unit squares in an outline of their shape. A common procedure is to count the whole squares in the outline and then approximate the total area of the partial squares as shown in Comment 3.

Another way to approximate the area of a shape is to fmd the number of squares in an "inner" covering of the shape and the number in an "outer" covering and then avemge these two numbers, as shown in the following example.

Inner Covering = 48 sq. units Outer Covering = 81 sq. units

Area Estimate = 48 ~ 81 = 64.5 sq. units


~ctions Comments

6. Discuss with the students how the accuracy of an area estimate can be improved.

6. One way to increase the accuracy of an approximation is to subdivide unit squares. In the example shown below, the area of a region is approximated first by fmding the average of the number of square units in an inner and outer covering. Then' an approximation is made by averaging the number of quarter-squares in an inner and outer covering. Better approximations could be made by further subdividing the grid.

Inner Covering = 45 sq. units Outer Covering = 91 sq. units

Area Estimate = 45 ; 91 = 68 sq. units


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Inner Covering = 212 quarter-squares Outer Covering = 301 quarter-squares

Area Estimate = 212 ; 301 = 256 1/2 quarter-squares = 64 1/8 sq. units

The last two pages of this Activity are masters for making overhead transparencies of the two figures shown above.


"Actions Comments

7. (Optional) Ask each student to approximate the area of a leaf.

7. One way to approximate the area of a leaf is to draw an outline of the leaf on grid paper and then approximate the number of unit squares within the outline.

7 Unit V ·Activity 10

T 20cm

Another method involves weighing: A piece of cardboard of known area is weighed on a sensitive scale. An outline of the leaf is traced on the cardboard and then cut out. This cutout of the leaf is weighed. An approximation of the area of the cutout is then obtained by comparing its weight to that_ of the original piece of cardboard.

For example, a 20 x 25 em rectangle, cut from the backing of a writing pad, has an area of 500 sq ems and weighs 28.2 grams. The cutout of a maple leaf traced on it weighs 9.1 grams. This is 32.2% of the weight of the rectangle. Hence, the area of the cutout is 32.2 % of 500 square ems, or 161 square ems.

~~------25cm------~~

l.__________ This 500 sq em cardboard rectangle

weighs 28.2 grams. This cutout weighs 9.1 grams. Hence its area is (9.1 I 28.2 ) X 500""

161 sq ems.


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Unit VI Math and the Mind's Eye

·~ ansparencJes for Loo~g at Geometry

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PDF viewing archiving 300 dpi - The Math Learning CenterActions 4. Ask the students to construct the polygon shown below. Point out that this polygon has six sides. Ask them to make