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Activities for Learning, Inc.

RIGHTSTART™ MATHEMATICS

A HANDS-ON

GEOMETRIC APPROACH

LESSONS

Copyright © 2004 by Joan A. CotterAll rights reserved. No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording, or otherwise, without written permissionof Activities for Learning.

Three-D images are made with Pedagoguery Software, Inc’s Poly (http://www.peda.com/poly)

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www.RightStartMath.com

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Activities for Learning, Inc.PO Box 468; 321 Hill StreetHazelton ND 58544-0468888-272-3291 or 701-782-2002701-782-2007 fax

ISBN 978-1-931980-16-6Febuary 2009

Lesson 1 Getting StartedLesson 2 Drawing DiagonalsLesson 3 Drawing StarsLesson 4 Equilateral Triangles into HalvesLesson 5 Equilateral Triangles into Sixths & ThirdsLesson 6 Equilateral Triangles into Fourths & EighthsLesson 7 Equilateral Triangles into NinthsLesson 8 Hexagrams and Solomon's SealLesson 9 Equilateral Triangles into TwelfthsLesson 10 Measuring Perimeter in CentimetersLesson 11 Drawing Parallelograms in CentimetersLesson 12 Measuring Perimeter in InchesLesson 13 Drawing Parallelograms in InchesLesson 14 Drawing RectanglesLesson 15 Drawing RhombusesLesson 16 Drawing SquaresLesson 17 Classifying QuadrilateralsLesson 18 The Fraction ChartLesson 19 Patterns in FractionsLesson 20 Measuring With SixteenthsLesson 21 A Fraction of Geometry FiguresLesson 22 Making the WholeLesson 23 Ratios and Nested SquaresLesson 24 Square CentimetersLesson 25 Square InchesLesson 26 Area of a RectangleLesson 27 Comparing Areas of RectanglesLesson 28 Product of a Number and Two MoreLesson 29 Area of Consecutive SquaresLesson 30 Perimeter Formula for RectanglesLesson 31 Area of a ParallelogramLesson 32 Comparing Calculated Areas of ParellelogramsLesson 33 Area of a TriangleLesson 34 Comparing Calculated Areas of TrianglesLesson 35 Converting Inches to CentimetersLesson 36 Name that FigureLesson 37 Finding the Areas of More TrianglesLesson 38 Area of TrapezoidsLesson 39 Area of HexagonsLesson 40 Area of OctagonsLesson 41 Ratios of AreasLesson 42 Measuring AnglesLesson 43 Supplementary and Vertical AnglesLesson 44 Measure of the Angles in a PolygonLesson 45 Classifying Triangles by Sides and AnglesLesson 46 External Angles of a TriangleLesson 47 Angles Formed With Parallel Lines

Table of Contents

G: © Joan A. Cotter 2005

Lesson 48 Triangles With Congruent Sides (SSS)Lesson 49 Other Congruent Triangles (SAS, ASA)Lesson 50 Side and Angle Relationships in TrianglesLesson 51 Medians in TrianglesLesson 52 More About Medians in TrianglesLesson 53 Midpoints in a TriangleLesson 54 Rectangles Inscribed in a TriangleLesson 55 Connecting Midpoints in a QuadrilateralLesson 56 Introducing the Pythagorean TheoremLesson 57 Squares on Right TrianglesLesson 58 Proofs of the Pythagorean TheoremLesson 59 Finding Square RootsLesson 60 More Right Angle ProblemsLesson 61 The Square Root SpiralLesson 62 Circle BasicsLesson 63 Ratio of Circumference to DiameterLesson 64 Inscribed PolygonsLesson 65 Tangents to CirclesLesson 66 Circumscribed PolygonsLesson 67 Pi, a Special NumberLesson 68 Circle DesignsLesson 69 Rounding Edges With TangentsLesson 70 Tangent CirclesLesson 71 Bisecting AnglesLesson 72 Perpendicular BisectorsLesson 73 The Amazing Nine-Point CircleLesson 74 Drawing ArcsLesson 75 Angles 'n ArcsLesson 76 Arc LengthLesson 77 Area of a CircleLesson 78 Finding the Area of a CircleLesson 79 Finding More AreaLesson 80 Pizza ProblemsLesson 81 Revisiting TangramsLesson 82 Aligning ObjectsLesson 83 ReflectingLesson 84 RotatingLesson 85 Making Wheel DesignsLesson 86 Identifying Reflections & RotationsLesson 87 TranslationsLesson 88 TransformationsLesson 89 Double ReflectionsLesson 90 Finding the Line of ReflectionLesson 91 Finding the Center of RotationLesson 92 More Double ReflectionsLesson 93 Angles of Incidence and ReflectionLesson 94 Lines of Symmetry


Table of Contents

Lesson 95 Rotation SymmetryLesson 96 Symmetry ConnectionsLesson 97 Frieze PatternsLesson 98 Introduction to TessellationsLesson 99 Two Pentagon TessellationsLesson 100 Regular TessellationsLesson 101 Semiregular TessellationsLesson 102 Demiregular TessellationsLesson 103 Pattern UnitsLesson 104 Dual TessellationsLesson 105 Tartan PlaidsLesson 106 Tessellating TrianglesLesson 107 Tessellating QuadrilateralsLesson 108 Escher TessellationsLesson 109 Tessellation Summary & Mondrian ArtLesson 110 Box FractalLesson 111 Sierpinski TriangleLesson 112 Koch SnowflakeLesson 113 Cotter Tens FractalLesson 114 Similar TrianglesLesson 115 Fractions on the Multiplication TableLesson 116 Cross Multiplying on the Multiplication TableLesson 117 Measuring HeightsLesson 118 Golden RatioLesson 120 Fibonacci SequenceLesson 121 Fibonacci Numbers and PhiLesson 122 Golden Ratios and Other Ratios Around UsLesson 123 Napoleon’s TheoremLesson 124 Pick’s TheoremLesson 125 Pick’s Theorem With the StomachionLesson 126 Pick’s Theorem and Pythagorean TheoremLesson 127 Estimating Area With Pick’s TheoremLesson 128 Distance FormulaLesson 129 Euler PathsLesson 130 Using Ratios to Find Sides of TrianglesLesson 131 Basic TrigonometryLesson 132 Solving Trig ProblemsLesson 133 Comparing CalculatorsLesson 134 Solving Problems With a Scientific CalculatorLesson 135 Angle of ElevationLesson 136 More Angle ProblemsLesson 137 Introduction to Sine WavesLesson 138 Solids and PolyhedronsLesson 139 Nets of CubesLesson 140 Volume of CubesLesson 141 Volume of BoxesLesson 142 Volume of Prisms

1: © Joan A. Cotter 2001

Table of Contents

Lesson 143 Diagonals in a Rectangular PrismLesson 144 CylindersLesson 145 ConesLesson 146 PyramidsLesson 147 Polygons ‘n PolyhedronsLesson 148 Tetrahedron in a CubeLesson 149 Platonic SolidsLesson 150 Views of the Platonic SolidsLesson 151 Duals of the Platonic SolidsLesson 152 Surface Area and Volume of SpheresLesson 153 Plane Symmetry in PolyhedraLesson 154 Rotating Symmetry in PolyhedraLesson 155 Circumscribed Platonic SolidsLesson 156 Cubes in a DodecahedronLesson 157 Stella OctangulaLesson 158 Truncated TetrahedraLesson 159 Truncated OctahedronLesson 160 Truncated IsocahedronLesson 161 CuboctahedronLesson 162 RhombicuboctahedronLesson 163 IcosidodecahedronLesson 164 Snub PolyhedraLesson 165 Archimedean Solids

Table of Contents

Lesson 1 line segments, parallel lines, and intersectionsLesson 2 horizontal, vertical, diagonal, hexagon,Lesson 3 polygon, vertex, vertexes, verticesLesson 4 quadrilateral, equilateral triangleLesson 5 congruentLesson 6 bisect, tick mark, tetrahedronLesson 10 perimeterLesson 13 parallelogramLesson 14 rectangle, right angle, perpendicularLesson 15 rhombusLesson 16 90 degrees, squareLesson 17 trapezoid, Venn diagramLesson 18 fractionLesson 19 numerator, denominatorLesson 21 crosshatchLesson 23 ratioLesson 24 area, square centimeterLesson 25 area, square inchLesson 26 formulaLesson 27 exponentLesson 30 factorLesson 32 millimeter, square millimeterLesson 34 little square, altitudeLesson 36 isoscelesLesson 38 distributive property, straightedgeLesson 42 goniometerLesson 43 supplementary, vertical, complementaryLesson 45 acute, obtuse, scaleneLesson 46 external, internal, adjacent angleLesson 47 corresponding, alternate, interior, exterior anglesLesson 48 SSSLesson 49 similar, SAS, ASALesson 50 vertex angle, base angles, baseLesson 51 median of a triangleLesson 52 centroidLesson 54 inscribedLesson 55 convex, concaveLesson 56 hypotenuse, legLesson 57 obliqueLesson 59 Pythagorean theoremLesson 59 square root, integer, perfect squareLesson 60 Pythagorean tripleLesson 62 point, line, and plane, circumference, diameter, radius, arc, sectorLesson 64 inscribed polygon, regular polygonLesson 65 tangent, tangent segmentLesson 66 circumscribed polygonLesson 67 pi, π

Vocabulary First Introduced

Lesson 68 clockwise, counterclockwiseLesson 69 oblique, concentric, semicircle Lesson 70 internally tangent circles, externally tangent circles, trefoil,

quatrefoilLesson 71 angle bisector, incenterLesson 72 chord, circumcenter*Lesson 73 foot, feetLesson 74 central angleLesson 75 inscribed angle, intercepted arcLesson 76 kilometerLesson 80 per, unit costLesson 81 tangramLesson 83 reflection, image, line of reflection, flip horizontal, flip verticalLesson 86 transformationLesson 87 translation, image, absolute, relativeLesson 88 transformationLesson 93 angle of incidence, angle of reflectionLesson 94 line of symmetry, maximum, minimum, ∞Lesson 95 order of rotation symmetry, point symmetryLesson 97 frieze, cell, tileLesson 98 tessellationLesson 99 pure tessellationLesson 100 nonagon, decagon , dodecagonLesson 101 semiregular tessellationLesson 102 demiregular tessellation, semi-pure tessellationLesson 103 unit, patternLesson 104Lesson 105 tartan, plaid, warp, weft, woofLesson 106Lesson 107Lesson 108 EscherLesson 109 MondrianLesson 110 fractals and the terms iteration and self-similar, exponentLesson 111 Sierpinski TriangleLesson 112 Koch SnowflakeLesson 113 Lesson 114 similar, similar trianglesLesson 115 proportionLesson 116 cross-multiplyingLesson 117Lesson 118 golden rectangle, golden ratio, phi, φ Lesson 119 golden spiral, golden triangleLesson 120 sequence, Fibonacci sequenceLesson 121 Fibonacci spiralLesson 122Lesson 123 generalize


Lesson 124Lesson 125Lesson 126Lesson 127Lesson 128Lesson 129 Euler pathLesson 130Lesson 131 trigonometry, opposite, adjacent, sine, cosine, tangentLesson 132Lesson 133 scientific calculatorLesson 134Lesson 135 angle of elevation, stride, clinometerLesson 136 angle of depressionLesson 137 sine waveLesson 138 solid, polyhedron, polyhedra, face, edge, vertex, net, dimensionLesson 139Lesson 140 volume, cubic centimeter, surface areaLesson 141 decimeter, dmLesson 142 prismLesson 143 short diagonal, long diagonalLesson 144 cylinderLesson 145 coneLesson 146 apex, regular pyramid, right pyramidLesson 147Lesson 148Lesson 149 Platonic solidsLesson 150Lesson 151 dual polyhedraLesson 152 sphere, great circle, small circleLesson 153 planes of symmetryLesson 154 axes of symmetryLesson 155 reciprocalLesson 156Lesson 157 stella octangula, concave polyhedronLesson 158 truncate, semiregular polyhedra, Archimedean solidsLesson 159Lesson 160Lesson 161Lesson 162Lesson 163Lesson 164Lesson 165


Intermediate Level Objectives

NCTM Standards have identified three focal points:Three curriculum focal points are identified and described for each grade level, pre-K–8, along with connections to guide integration of the focal points at thatgrade level and across grade levels, to form a comprehensive mathematics curriculum. To build students strength in the use of mathematical processes,instruction in these content areas should incorporate—

• the use of the mathematics to solve problems;• an application of logical reasoning to justify procedures and solutions; and• an involvement in the design and analysis of multiple representations to learn, make connections among, and communicate about the ideas within and outsideof mathematics.

RightStart™ Mathematics: A Hands-On Geometric Approach is designed for theintermediate student. It employs a hands-on and visual approach through the use of atool set consisting of a drawing board, T-square, triangles, compass, goniometer, andpanels for 3-D constructions. The students explore angles, polygons, area, volume,ratios, Pythagorean theorem, tiling, and so forth. Students will need to read the text and make their own dictionaries. Previously learned concepts are embedded whilemore advanced mathematical topics are gradually introduced.

© by Joan Cotter 2005 • [email protected] • www.ALabacus.com

RightStart™ Mathematics: A Hands-On Geometric ApproachRightStart™ Mathematics: A Hands-On Geometric Approach is an innovative approach for teaching many middle school mathematics topics, including perimeter, area, volume, metric system, decimals, rounding numbers, ratio, and proportion. The student is also introduced to traditional geometric concepts: parallel lines, angles, midpoints, triangle congruence, Pythagorean theorem, as well as some modern topics: golden ratio, Fibonacci numbers, tessellations, Pick’s theorem, and fractals. In this program the student does not write out proofs, although an organized and logical approach is expected.

Understanding mathematics is of prime importance. Since the vast majority of middle schoolstudents are visual learners, approaching mathematics through geometry gives the student anexcellent way to understand and remember concepts. The hands-on activities often createdeeper learning. For example, to find the area of a triangle, the student must first construct thealtitude and then measure it. If possible, students work with a partner and discuss their obser-vations and results.

Much of the work is done with a drawing board, T-square, 30-60 triangle, 45 triangle, atemplate for circles, and goniometer (device for measuring angles). Constructions with thesetools are simpler than the standard Euclid constructions. It is interesting to note that CAD(computer aided design) software is based on the drawing board and tools.

This program incorporates other branches of mathematics, including arithmetic, algebra, andtrigonometry. Some lessons have an art flavor, for example, constructing Gothic arches. Otherlessons have a scientific background, sine waves, and angles of incidence and reflection; or atechnological background, creating a design for car wheels. Still other lessons are purely math-ematical, Napoleon’s theorem and Archimedes stomachion. The history of mathematics iswoven throughout the lessons. Several recent discoveries are discussed to give the student theperspective that mathematics is a growing discipline.

Good study habits are encouraged through asking the student to read the lesson before, during,and following the worksheets. Learning to read a math textbook is a necessary skill for successin advanced math classes. Learning to follow directions is a necessary skill for studying andeveryday life. Occasionally, an activity or lesson refers to previous work making it necessaryfor the student to keep all work organized. The student is asked to maintain a list of new terms.

This text was written with several goals for the student: a) to use mathematics previouslylearned, b) to learn to read math texts, c) to lay a good foundation for more advanced mathe-matics, d) to discover mathematics everywhere, and e) to enjoy mathematics.

About the authorJoan A. Cotter, Ph.D., author of RightStart™ Mathematics: A Hands-On Geometric Approach

and RightStart™ Mathematics elementary program has a degree in electrical engineering, a Montessori diploma, a masters degree in curriculum and instruction, and a doctorate in mathematics education. She taught preschool, children with special needs, and mathematicsto grades 6-8.

© by Joan Cotter 2005 • [email protected] • www.ALabacus.com

Hints on Tutoring RightStart™ Mathematics: A Hands-On Geometric Approach

Before starting a lesson, the student should look over the Materials list andgather all the supplies, including a mechanical pencil or a sharp #2 pencil anda good eraser. Then the student reads over the goals, keeping in mind thatitalicized words will be explained in the lesson. (These new words are to berecorded in the student’s math dictionary.) Next the student begins readingthe Activities, carefully studying any accompanying figures. It is a good habitto summarize the activity after reading it. If a paragraph is unclear, thestudent should reread the paragraph, keeping in mind that sometimes more isexplained in the following paragraph. No one learns mathematics by readingthe text only once.

Each activity needs to be understood before going to the next activity. Makesure the student understands the importance of completing the problems onthe worksheet when called for in the lesson. Sometimes it will be necessary torefer to the lesson while completing the worksheet. All work needs to be keptneatly in a three-ring binder for future reference.

Be careful how you react to the “I don’t get it” plea. Tell the student you needa question to answer. You do not want to get in the habit of reading the textfor your student and then regurgitating to her like a mother robin feeding heryoung. The text is written for students to read for themselves. Learning howto ask questions is an important skill to acquire toward becoming anindependent learner. If questions remain after diligent study, the student cancontact the author at [email protected].

If a child has a serious reading problem, read the text aloud while he followsalong and then ask him to read it aloud. Be sure each word is understood. Forless severe reading problems, you might model aloud the process of readingan activity, commenting on the figure, and summarizing the paragraph. Someof the time, students need encouragement to overcome frustration, which isinherent in the learning process. Occasionally, a student may have aknowledge gap needed for a particular lesson, requiring other resources toresolve. Incidentally, research shows one of the major causes of difficulties inlearning new concepts for this age group is insufficient sleep.

After the student has completed the worksheet, ask her to compare her workwith the solution. If the student has a partner, they can compare and discusstheir work before referring to the solutions. Ask her to explain what shelearned and any discrepancies. Keep in mind that some activities have morethan one solution. You might also ask her to grade her work on some agreedupon scale. It also is a good idea for the student to reread the goals of thelesson to see if they have been met. Encourage discussion on practicalapplications of the topic.

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Lesson 2 Drawing DiagonalsGOALS 1. To review the terms horizontal and vertical

2. To learn the mathematical meaning of diagonal3. To review the term hexagon4. To find the correct edge of the 30-60 triangle to draw diagonals

MATERIALS Worksheet 2Drawing board, T-square, 30-60 triangle

ACTIVITIES Horizontal and vertical. Horizontal refers to the horizon, theintersection between the earth and sky that a person on earth sees (ifthere aren’t too many buildings and trees in the way). Vertical refersto straight up and down, like a flagpole. Sometimes it also meansabove, or overhead.

A horizontal line on paper is a line drawn straight across the paper. Itusually is parallel to the top and bottom of the paper. A vertical lineon paper goes from top to bottom, parallel to the sides of the paper.

Diagonals. In common everyday English, the word diagonal usuallymeans at a slant. It often means a road that runs neither north andsouth nor east and west.

In mathematics, a diagonal is a line connecting points in a closedfigure. For example, the line segments AC and DB drawn in thesquare below on the left are diagonals. If we turn the square, as inthe next figure, diagonal DB is horizontal and diagonal AC isvertical.

A sharppencil, aneraser, andtape areessentialsand will notbe listed infuturelessons.

Don’t forget to add theterms listed in Goals toyour math dictionary.

Worksheet. The worksheet asks you to draw two hexagons and alltheir diagonals. A hexagon is a closed six-sided figure. One way toremember the word is that hexagon and six both have x’s.

Draw all the sides of the hexagon and the diagonals using yourT-square and 30-60 triangle except the horizontal lines, which needonly a T-square. Below are a hexagon and all its diagonals.

Diagonal lines on a building.

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Lesson 3 Drawing StarsGOALS 1. To learn the term polygon

2. To learn the term vertex and its plurals, vertices and vertexes3. To draw stars by following instructions shown in pictures

MATERIALS Worksheets 3-1 and 3-2Drawing board, T-square, 30-60 triangle

ACTIVITIES Polygons. In the row below are examples of figures that arepolygons and figures that are not polygons. Before reading further,think of a good definition for polygon.

Did your definition include a closed figure with straight linesegments?

Vertex and its plurals. In a polygon the point where the linesmeet is call a vertex. You have two choices for the plural of vertex,either vertices (VER-ti-sees) or vertexes. For some reason, eventhough the word vertexes follows the normal English rule forplurals, math books (and tests) prefer vertices.

For example, there are three vertices in a triangle and four verticesin a square.

Worksheets. On the worksheets, you will be drawing stars. Theboldfaced lines in the little figures tell you what to draw. Be sure touse your T-square and, where needed, your triangle to draw all thelines. They will look like the following figures.

These are polygons.

These are NOT polygons.

Star designs inMorocco, where theyare very common.

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Lesson 9 Equilateral Triangle into TwelfthsGOALS 1. To divide an equilateral triangle into twelfths

2. To divide an equilateral triangle into a number of your choosingthat is greater than 12

MATERIALS Worksheet 9 and plain reverse side (or another plain piece of paper)Drawing board, T-square, 30-60 triangle

ACTIVITIES Dividing a triangle into twelfths. How would you divide anequilateral triangle into twelfths, that is, into twelve congruentparts? Think about it for a while before reading further. Would itwork to divide the triangle into thirds and divide each third intofourths? One student even suggested dividing the triangle intotenths and then dividing each tenth in half. Let’s hope he wasjoking!

If you have thought about it, you probably realize you first dividethe triangle into fourths and then each fourth into thirds.

Worksheet. Do the worksheet next, dividing the equilateraltriangle into twelfths.

Dividing a triangle by higher numbers. How would you dividethe triangle into sixteenths? What other numbers could you divide itinto? Two kindergarten girls divided the equilateral triangle into256 equal parts. After hearing about the girls, a teacher learningdrawing board geometry divided his triangle into 432 equal parts.Some divisions are shown below.

Worksheet, reverse side. On the reverse side of yourworksheet or other plain paper, draw a largeequilateral triangle. Choose a number greater than 12that you can divide an equilateral triangle into andthen do the dividing Copy one of the designs, orbetter yet, design your own. You might like tocolor your design.

Sixteenths Eighteenths Eighteenths

Twenty-fourths Twenty-sevenths Twenty-sevenths

Thirty-seconds

Triangle into 432nds byJoseph Hermodson-Olsen, 14.

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21 cm

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Lesson 26 Area of a RectangleGOALS 1. To understand how area is calculated

2. To see the connection between multiplication and area3. To learn and apply the formula for the area of a rectangle

MATERIALS Worksheet 26Drawing board, T-square, 30-60 triangle

ACTIVITIES Worksheet. Begin by completing numbers 1-5 on the worksheet.

Formula for finding the area of a rectangle. A formula is ageneral principle stated in mathematical symbols. The word formulameans “little form.” So, it is a shortcut for stating a mathematicalrelationship. Most of the time you do not need to just memorizeformulas. Rather, they are a logical result you can think through.

In question 5 on the worksheet, you were asked to write the formulafor the area of a rectangle. The number of squares needed to cover afigure is the area, usually written as A. You found the area of allthose rectangles by multiplying the number of squares in a row bythe number of rows. So, if we call the horizontal distance, the widthw, and the vertical distance, the height h, the formula becomes

A = w × h

Actually, in algebra, which has lots of formulas, the operator “×” isnot written. Two letters written together without an operator meansmultiply. This is one of the major differences between arithmeticand algebra. In arithmetic, digits written side-by-side without anoperator mean add; for example, 976 is 900 + 70 + 6 and 3 means3 + . However, computer spreadsheets require operators betweenall numbers and letters.

Symbol, cm2, for square centimeter. The symbol, cm2, is theabbreviation for square centimeter. Just as 32 = 9 and forms a squarewith 3 on a side, so 1 cm2 is a square that is 1 cm on a side. Read1 cm2 as “1 centimeter squared.”

Area problem. How would you find the area of the figure below?Think about several ways and then discuss it with a partner, if youcan, before looking at the solutions on the next page.

h h

ww

In some math text-books (especially oldertexts), the symbol “l”for length is usedrather than “h.”

Today, computersoftware usually useswidth and height.

12

12

What is the area of this figure?

A building in England withmany rectangles.

31


There are several ways to solve this problem.

Solution 1. Make the whole figure into a rectangle and then sub-tract the little rectangle.

A = wh (large rectangle) – wh (square)

A = 21 × 9 – 6 × 6

A = 189 – 36

A = 153 cm2

Solution 2. Divide the rectangle horizontally into two rectangles.

A = wh (upper rectangle) + wh (lower rectangle)

A = 21× 3 + 15 × 6

A = 63 + 90

A = 153 cm2

Solution 3. Divide the rectangle vertically into two rectangles.

A = wh (left rectangle) + wh (right rectangle)

A = 15 × 9 + 6 × 3

A = 135 + 18

A = 153 cm2

21 cm

9 cm

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6 cm

21 cm

9 cm

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3 cm

6 cm

6 cm

21 cm

9 cm

15 cm

3 cm

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© J

oan

A. C

otte

r 20

04

Worksheet 26, Area of a Rectangle

1. T

he p

oint

s, m

arke

d b

y lit

tle

x's,

are

1 c

m a

part

. Sta

rt a

t any

poi

nt. U

se y

our

T-s

quar

e an

d d

raw

the

wid

th o

f the

rec

tang

le b

yd

raw

ing

a ho

rizo

ntal

line

from

the

poin

t to

the

left

ed

ge o

f the

squ

are.

The

n us

e yo

ur T

-squ

are

and

tria

ngle

and

dra

w th

ehe

ight

by

dra

win

g a

vert

ical

line

from

the

poin

t to

the

top

of th

e sq

uare

. Cal

cula

te th

e nu

mbe

r of

squ

are

cent

imet

ers

in th

ere

ctan

gle

and

wri

te th

e nu

mbe

r in

the

low

er r

ight

cor

ner

of th

e re

ctan

gle.

One

is d

one

for

you,

.

12

3. If

a r

ecta

ngle

is 8

cm

wid

e by

9 c

m h

igh,

how

man

y sq

uare

cent

imet

ers

do

you

need

to c

over

it?

____

____

____

____

____

_

4. If

a r

ecta

ngle

is w

cm

wid

e an

d h

cm

hig

h, h

ow m

any

squa

re

cent

imet

ers

do

you

need

to c

over

it?

____

____

____

____

____

_

Rea

d L

esso

n 26

bef

ore

answ

erin

g th

e ne

xt q

uest

ion.

5. W

hat i

s th

e ar

ea o

f the

figu

re b

elow

?

2. Y

ou'v

e pr

obab

ly s

een

this

tabl

e be

fore

. Wha

t is

it?

____

____

____

____

____

____

____

____

____

____

____

___

12 c

m

4 cm

6 cm

16 c

m

2

32


Lesson 27 Comparing Areas of RectanglesGOALS 1. To calculate more areas of rectangles

2. To compare areas of rectangles with constant perimeter

MATERIALS Worksheets 27-1, 27-2Drawing board, T-square, 30-60 triangle4-in-1 ruler

ACTIVITIES Frame problem. Consider the following problem. You have 12 cmof gold edging to place around a rectangular frame. You want themaximum amount of space inside the frame.

First think about the possible dimensions of the rectangles, so theperimeters will be 12 cm. Then study the figures below.

1 cm 2 cm 3 cm 4 cm 5 cm1 cm

2 cm

3 cm

4 cm

5 cm

The areas, which you can do in your head using A = wh, are fromleft to right, 5 cm2, 8 cm2, 9 cm2, 8 cm2, and 5 cm2.

Graphing the frameproblem. It isinteresting to graphthe results as shownbelow. Why is thearea equal to 0 whenthe width is equal to 0or equal to 6? You cansee the greatest areaoccurs when thewidth of the rectangleis to 3. What is theheight when thewidth is 3? Theanswer is at the bottom of the page.

Worksheets. There is a similar problem on Worksheets 27-1 and27-2. Draw the rectangles by measuring with your ruler like you didon Worksheet 11. [Answer: 3]

0 1 2 3 4 5 60123456789

10

The width of the rectangle in cm

Rectangle Areas with Perimeter = 12 cm

cm2

The

are

a in

This type of problem iseasily solved with abranch of mathematicscalled calculus.

The shape of this graphis called a parabola.

Nam

e __

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_

Dat

e __

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__

© J

oan

A. C

otte

r 20

04

If y

ou h

ad 2

0 cm

of e

xpen

sive

trim

to d

ecor

ate

the

edge

of a

rec

tang

ular

bul

leti

n bo

ard

, wha

t sh

ould

the

dim

ensi

ons

of th

ere

ctan

gle

be to

giv

e yo

u th

e m

ost a

rea

for

phot

os a

nd n

otes

? Fo

llow

the

step

s be

low

for

the

solu

tion

.

1. O

n ea

ch o

f the

five

line

s be

low

, dra

w a

rec

tang

le w

ith

a pe

rim

eter

of 2

0 cm

. Wri

te th

e d

imen

sion

s.

2. B

elow

eac

h re

ctan

gle,

cal

cula

te it

s ar

ea in

cm

. W

hich

rec

tang

le g

ives

the

mos

t

area

? __

____

____

____

____

____

____

____

____

Worksheet 27-1 , Comparing Areas of Rectangles

2

Nam

e __

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____

____

____

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____

_

Dat

e __

____

____

____

____

____

____

__

© J

oan

A. C

otte

r 20

04

024681012141618202224262830

Th

e b

ase

of th

e re

ctan

gle

in c

m

Are

a of

Rec

tan

gles

wit

h a

Per

imet

er o

f 20

cm

Areas in sq cm

4. O

n th

e gr

aph

belo

w, p

lace

a p

oint

sho

win

g th

e ar

ea fo

r ea

ch r

ecta

ngle

from

the

prev

ious

pag

e.A

lso

find

the

area

s fo

r th

e re

mai

ning

wid

ths:

0, 6

, 7, 8

, 9, a

nd 1

0. T

hen

conn

ect t

he p

oint

s in

a s

moo

thcu

rve;

do

this

free

hand

(wit

hout

any

dra

win

g to

ols)

.

5. W

hat i

s th

e na

me

of th

e sh

ape

of th

e cu

rve?

___

____

____

____

____

____

____

____

____

___

6. A

ccor

din

g to

the

grap

h, w

hat i

s th

e m

axim

um a

rea?

___

____

____

____

____

____

____

____

____

7. H

ow d

oes

the

grap

h co

mpa

re w

ith

the

exam

ple

in th

e le

sson

? __

____

____

____

____

____

____

____

____

____

____

____

____

____

___

Worksheet 27-2, Comparing Areas of Rectangles

96


12° 90°

180°250°

Lesson 84 RotatingGOALS 1. To learn the mathematical meaning of rotation

2. To construct rotations at various angles

MATERIALS Worksheet 84GoniometerA set of tangramsDrawing board, T-square, 45 triangle

ACTIVITIES Rotating. A clock is a good example of rotation. Both the hour andminute hands rotate about the center of the clock. The hands movein a clockwise direction. However, when we discuss rotationsmathematically, we start with a horizontal ray extending right andmeasure the amount of rotation counterclockwise. So, for a clock tobehave mathematically, the hand would start at the 3 o’clockposition and travel backward.

Rotating the ship. Build the ship shown below in the left figurewith four tangram triangles and tape them together.

Then tape the ship to the upper arm of the goniometer. Hold thelower arm of the goniometer still with your right hand. Use yourleft hand to rotate and upper arm of the goniometer with theattached ship. See the middle figure above.

Keep rotating to 90° as shown in the right figure above. (The seasare getting very rough.) Continue rotating to 180°. (Disaster.) Seethe left figure below.

To set your ship aright, un-tape it, turn your goniometer upsidedown, re-tape it, and continue rotating as in the right figure above.

Worksheet. The first half of the worksheet asks you to constructthe ship at various angles with your tools. You may find it helpfulto set the ship model at the desired angle. Start your construction atthe “×” and draw the first line at the correct angle. Measure only theline for the ship’s bottom (3 cm); construct the other lines.

For the second half, build and rotate the model to the various anglesbefore attempting the constructions. Measure only the 2.5 cm line.

Construct every lineaccurately. Don’t guess.

Star design on the floor.

Name ___________________________________

Date ____________________________

© Joan A. Cotter 2008

Worksheet 84, R

otating

1. 45°2. 90°

3. 135°

4. 180°

5. 45°

6. 90°

7. 180°

8. 270°

2.5

cm

Construct the figures at the angles given withyour geometry tools. Use your ruler only tomeasure the line representing the bottom ofthe ship and the side of the arrow.

The × shows you where to start.

3.0 cm

9. What angle of rotation is

the same turning something

upside down? ______

10. Is a rotation of 180° the

same as reflecting about a

horizontal line? ______

142


Lesson 120 Fibonacci SequenceGOALS 1. To learn the term sequence

2. To learn about the Fibonacci sequence3. To solve some sequence problems

MATERIALS Worksheets 120-1, 120-2, 120-3

ACTIVITIES Fibonacci. Fibonacci (fee-buh-NOT-chee), 1170-1250, was born inPisa (PEE-za), Italy, the city with the Leaning Tower. He waseducated in North Africa (now Algeria). Fibonacci learned othermathematics from talking with merchants while traveling aroundthe Mediterranean region.

Fibonacci became fascinated with the Hindu-Arabic numerals, withdigits 1-9 and a 0. At that time Europe used Roman numerals, whichhad no 0. His book Liber Abaci ("Book of Calculation”) introducedEurope to the numbers we use today. It also showed methods ofcalculating with paper and pencil without an abacus. Conversion tothe new method took time. In 1299 the merchants in Florencerequired using Roman numerals.

He also introduced the fraction bar—the line separating thenumerator and denominator. Before that fractions were written as .

Fibonacci sequence. A sequence is a set of quantities in some typeof order. The answer to one of Fibonacci’s math problems results inthe Fibonacci sequence. It starts as 1, 1, 2, 3, 5. Think about whatcomes next before reading further.

If you think 8 comes next, you’re right. Each number in the sequenceis the sum of the previous two numbers. Thus, 1 + 1 = 2, 1 + 2 = 3, 2+ 3 = 5, and so on.

Problem 1. For Problem 1, you are to practice your addition skillsand calculate the Fibonacci sequence to 26 terms. Incidentally, if youlearned about check numbers, or casting out nines as they’resometimes called, use them to check your work. Fibonacci learnedabout them and explained them in Liber Abaci.

Problem 2. This is the brick wall problem. The sides ofthe brick have a ratio of 2:1. See the figure at the right.You need to make your wall two units high, but withvarious widths.

If your wall is 1 unit wide, there is only one way to make it. If it is 2units wide, there are two ways to make it. See the middle figurebelow. If it is 3 units wide, there are three ways to make. See theright figure.

Continue the process for 4 and 5 units wide. Draw yourarrangements freehand and record the number of arrangements. DoWorksheet 1 before reading any further. Think about your solutions.

12

If you ever played the“Chain” games, you willrecognize the onescolumn as a chain.

i i i i i2 unitswide

1 unitwide

3 unitswide

143


1

2

With a 2 and a 1With three 1s With a 1 and a 2

Discussing Worksheet 1. Notice how quickly the numbersbecome large in the Fibonacci sequence. To be sure you’ve addedcorrectly, the last number in the sequence is 121,393.

Discuss with a partner the number of arrangements you found inProblem 2 and why.

Notice you can make the arrangements for 4 units by copying the 3sarrangement plus a vertical brick and copying the 2s plus twohorizontal bricks. Likewise, the arrangements for 5 are the 4s plus avertical brick and the 3s plus horizontal bricks. Adding the last twoare, of course, what the Fibonacci sequence is all about. (If you needhelp in understanding this, look carefully at the solutions.)

Problem 3. For the next problem, you have two sizes ofcolored rods, the 1s and the 2s. See the figure on the right.You are to make various lengths using these rods. Forexample, the three arrangements for a length of 3 are shownbelow. You can do the worksheet now.

Problems 4-5. For Problem 4, you are climbing stairs. You canclimb either one at a time or two at a time. The left figure belowshows a set of three stairs. The middle figure shows climbing thestairs, one step at a time. The rectangles represent a foot (or shoe);the arcs represent movement.

The right figure shows climbing the stairs by a combination—firsttwo steps at a time and then one at a time. You are to draw allcombinations for 3, 4, and 5 stairs.

For Problem 5, the instructions explain the bee problem. The bee canenter a new cell only if the number is higher. The term 123 means“cell 1, then cell 2, then cell 3.” Do the worksheet.

New problem. Make up your own Fibonacci problem. If you thinkof a good one, let me know at [email protected]. See someproblems at http://www.rightstartgeometry.com.

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© J

oan

A. C

otte

r 20

05

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__

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__

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__

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__

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__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

____

____

__

____

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__

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__

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__

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__

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__1 1 2 3

1. C

onti

nue

the

Fibo

nacc

i seq

uenc

e.2.

A b

rick

wal

l is

2 un

its

high

. The

bri

cks

are

used

two

way

s as

sho

wn.

Find

all

the

dif

fere

nt a

rran

gem

ents

to m

ake

a w

all 3

-5 u

nits

wid

e. R

ecor

dth

e nu

mbe

r of

arr

ange

men

ts o

n th

e lin

es a

t the

left

.

1 2

Wha

t kin

d o

f num

bers

did

you

wri

te?

____

____

____

____

____

____

H

ow m

any

arra

ngem

ents

cou

ld y

ou m

ake

if th

e w

all w

ere

6 un

its

wid

e? _

____

__

Worksheet 120-1, Fibonacci Sequence

Nam

e __

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_

Dat

e __

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____

____

__

© J

oan

A. C

otte

r 20

05

1 2

3. S

ketc

h al

l the

pos

sibi

litie

s fo

r m

akin

g le

ngth

s 1-

6 us

ing

1s a

nd 2

s.W

rite

the

num

ber

of a

rran

gem

ents

at t

he le

ft.

Wha

t kin

d o

f num

bers

did

you

wri

te?

____

____

____

____

____

____

I

n ho

w m

any

dif

fere

nt w

ays

coul

d y

ou m

ake

7? _

____

_

12

Worksheet 120-2, Fibonacci Sequence

Name ___________________________________

Date ____________________________

© Joan A. Cotter 2005

4. You are climbing stairs either one at a timeor two at a time. Draw all the ways you canclimb the stairs either one or two steps at time for stairs with 1-5 steps.

5. A bee is flying to the hive shown and can start only through cell 1or cell 2. It can travel to an adjacent cell only if it has a higher number.List all the ways the bee can travel to the first five cells.

To reach cell #1. __________________________________________________________________

To reach cell #2. __________________________________________________________________

To reach cell #3. __________________________________________________________________

To reach cell #4. __________________________________________________________________

To reach cell #5. __________________________________________________________________

__________________________________________________________________

How many ways can the bee enter cell #7? _________

12 1223

1 3 5 7

2 4 6

Worksheet 120-3, F

ibonacci Sequence

168


Lesson 140 Volume of CubesGOALS 1. To learn the terms volume, cubic centimeter, and surface area

2. To calculate volumes and surface areas of cubes

MATERIALS Worksheet 140Geometry panels and rubber bandsCentimeter cubesRuler

ACTIVITIES Volume. Volume is the amount of space taken up by a solid. Tomeasure this space you need a unit that takes up three-dimensionalspace. Units of volume are called cubic units because they areusually cubes.

Recall that you can measure a line segment with centimeters (cm)and area with square centimeters (cm2). You will measure thevolume of a solid with cubic centimeters (cm3). A cubic centimeter is acube with edges 1 cm long. See these units of measurements below.

Construct a cube with the geometry panels.

Volume of cubes. Use the centimeter cubes to construct a cubethat measures 2 cm on a side. (If you don’t have these cubes, look atthe figures on the worksheet.) How many centimeter cubes do youneed? The number of cubic centimeters you need to fill the solid isthe volume. What is the cube’s volume? The answers are at thebottom of the page.

Cubing a number. With the cube you made, did you notice thatthe total number of cubes is 2 × 2 × 2? You can also write thisexpression with exponents as 23. Read it as “two cubed.”

Surface area of a cube. Surface area is the area of all the surfacesof a solid. For polyhedra, the surface area is the sum of the area ofall the polygons. The symbol for surface area is “S.”

Worksheet. For the first table, you will be consideringcubes of various dimensions. The second table pertainsto a cube made with the panels, but measured withdifferent units. Measure the edges of a panel polygon asshown in the figure on the right. For the inch, use thenearest whole inch. [Answers: 8, 8 cm3]

1 cm

231 cm

1 cm

10 cm

The crystalline structure ofcommon salt. The largerchloride ions form a cubicshape. The smaller sodium ionsfill in the gaps between them.

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e __

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© J

oan

A. C

otte

r 20

06

1 cm

2 cm

4 cm

1. C

ompl

ete

the

tabl

e fo

r va

riou

s cu

bes.

Uni

tA

rea

of a

Fac

e

Cen

tim

eter

Mill

imet

er

Met

er

Inch

Vol

ume

of th

e C

ube

Len

gth

of a

Sid

eSu

rfac

e A

rea

Are

a of

a F

ace

Vol

ume

of th

e C

ube

Len

gth

of a

Sid

eSu

rfac

e A

rea

9 cm

a2 2

2. C

onst

ruct

a c

ube

from

the

pane

ls. M

easu

re a

sid

e us

ing

the

unit

s gi

ven

and

com

plet

e th

e ta

ble.

Worksheet 140, Volume of Cubes

3. H

ow m

any

cent

imet

er c

ubes

will

fit i

n th

e cu

be m

ade

from

the

pane

ls?

____

____

___

4. Im

agin

e a

cube

that

is 1

m o

n an

ed

ge. H

ow m

any

pane

l cub

es w

ould

fit i

n th

e 1-

met

er c

ube?

___

____

____

*5. W

hich

is m

ore,

1 c

m o

r 1

cm ?

Exp

lain

you

r an

swer

. ___

____

____

____

____

____

____

____

____

____

____

____

____

____

____

__3

a

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