math and the minds eye activities 5

Ask the students to find as many geoboard segments, all of different lengths, as they can, and to record the segments on 2 down and 1 over." For those students who think segment b is t

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

fllr-mula-free methods

Areas of Silhouettes

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

needed to cover ir

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

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 ducing the segments on their geoboards It may be helpful to make statements like

repro-"segment e is formed by coming down 3 spaces and over 1 space."

©Copyright 1986, Math Learning Center

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

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

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

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

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

great-est number of sides possible?

3 Unit V • Activity 1

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 - - - ,

•

this type are also called simple poly- gons while figures whose sides are seg-ments, but enclose • more than one interior ._ _ _ _ _ region, such as the one shown, are some-times 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

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

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

-2 Ask the students to use additional rubber bands to

sub-divide 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

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

©Copyright 1986, Math Learning Center

Actions

3 Ask the students to form each figure from Activity Sheet

A on their geoboards and determine its area Discuss

form this region on their

geoboard and ask them to

2 half-squares)

area 14 (12 squares and

4 half-squares)

area2 (4 half-squares)

area 12 ( 1 0 squares and

Some students are able to obtain the area

by mentally manipulating parts of the region into a convenient shape (One pos-sible sequence of manipulations is given below.) This ability should be acknow-ledged and encouraged

Math and the Mind's Eye

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

6 This triangle doesn't divide nicely into squares and half-squares, so it is interesting

to see how students find its area A mon 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

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

Actions

7 Ask the students to find the areas of the triangles on

Activ-ity Sheet C Discuss

4 Unit V • Activity 2

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 rec-tangle 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 un-wanted region b), which leaves area 6 (16-

8 - 2) for the triangle

8 (Optional.) Ask each student to find the areas of

geo-board regions selected from Activity Sheets D and E as

appropriate for the level of the student

tech-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 subtrac-

ted 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

Math and the Mind's Eye

Find Shaded Areas

Find Shaded Area

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

deter-mine the number of squares needed to exactly cover this

fig-ure Tell (or remind) them that the number of squares

ex-actly 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

Comments

1 One transparency of the attached grid sheet cut along the indicated lines will sup-ply 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

©Copyright 1985, Math Learning Center

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

stu-dents use their grids to determine the area of figures N

through U on Activity Sheet B

2 Unit V • Activity 3

Comments

4 Figure M can be covered with 3 squares and 3 half-squares, or 4 lfz squares alto-gether 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 ver-tex 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

Math and the Mind's Eye

· 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 tangle Thus the area of the triangle is the area of this rectangle which is 6 In figure

rec-2, the triangle is enclosed in ｡ Ｎ ｾ by 6 tangle The area of this rectangle is 12 Because the triangle covers half the rec-tangle, the area of the triangle is 6

Actions

8 Distribute Activity Sheet C and ask the students to use

their grids to determine the area of figure A Discuss

3 The areas of these parts are 1 lfz, 1, 3,

and 4 (Some students may ｮ ｾ 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

Math and the Mind's Eye

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

Name Areas of Silhouettes A

Name Areas of Silhouettes 8

Name _ Areas of Silhouettes C

Name Areas of Silhouettes D

p

v

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,

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

•

•

When all triangles, with their areas, have been recorded, ask

the students to cut out each geoboard square that has a

tri-angle drawn on it

1 Unit V • Activity 4

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

ｴ ｨ ･ ｴ ｾ ｣ ｨ ･ ｲ Ｎ

Comments

1 A master of Activity Sheet A is tached

at-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

©Copyright 1986, Math Learning Center

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

tri-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 follow-ing can be seen to match by turning one over and placing it on top of the other

Actions

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

investigations

A Find and record geoboard

triangles with areas 1, 2, 3,

4, 5, 6, 7 and 8 square units

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

Unit V • Activity 5

Geoboard Squares

Actions

1 Distribute geoboards, rubber bands, and geoboard

record-ing paper Ask the students to form geoboard squares of

different sizes and to record each square on different

sec-tions of the geoboard recording paper

record-tor ·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:

to the other

©Copyright 1986, Math Learning Center

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

7 Tell the students that in every square side X side =area

Ask them why they think it is, that for some of their

geo-board 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

side x side = area

6 Calculators are useful for fast tion 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

computa-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 ｾ Ｒ Ｌ ｾ Ｑ Ｐ and ｾ Ｘ Ｎ

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 ｾ

key

ｾ Ｒ = 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 imately, 2.8

approx-Math and the Mind's Eye

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

ex-periment with the J key on their calculators Point out that

most calculator computations are approximate

10 Display this geoboard segment: • • • • •

a) Represent this segment AB on

c) Determine the area of the square (Fig C)

d) Determine the length of side AB (Fig D)

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

Math and the Mind's Eye

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

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

2, 4, 5, 8, 9, 10 and 16 square units

subdivi-1, 4, 9 and 16 have side lengths subdivi-1, 2, 3 and

4 units, repectively A metric ruler with millimeter marks is convenient for measur-ing the sides of slanted squares to the near-est 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

Math and the Mind's Eye

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

Geoboard Recording Paper

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

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

2 Unit V • Activity 6

Comments

3 Make sure that the quadrilateral drawn

on the hypotenuse is a square

hypote-of the areas hypote-of the squares on the legs hypote-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