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Activities for Learning, Inc.
RIGHTSTART™
MATHEMATICS
by Joan A Cotter Ph D
A HANDS-ON
GEOMETRIC APPROACH
LESSONS
Copyright © 2004 by Joan A. Cotter
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of Activities for Learning.
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Activities for Learning, Inc.
PO Box 468; 321 Hill Street
Hazelton ND 58544-0468
888-272-3291 or 701-782-2002
701-782-2007 fax
ISBN 978-1-931980-16-6
Febuary 2009
Table of Contents
Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Lesson 11
Lesson 12
Lesson 13
Lesson 14
Lesson 15
Lesson 16
Lesson 17
Lesson 18
Lesson 19
Lesson 20
Lesson 21
Lesson 22
Lesson 23
Lesson 24
Lesson 25
Lesson 26
Lesson 27
Lesson 28
Lesson 29
Lesson 30
Lesson 31
Lesson 32
Lesson 33
Lesson 34
Lesson 35
Lesson 36
Lesson 37
Lesson 38
Lesson 39
Lesson 40
Lesson 41
Lesson 42
Lesson 43
Lesson 44
Lesson 45
Lesson 46
Lesson 47
Getting Started
Drawing Diagonals
Drawing Stars
Equilateral Triangles into Halves
Equilateral Triangles into Sixths & Thirds
Equilateral Triangles into Fourths & Eighths
Equilateral Triangles into Ninths
Hexagrams and Solomon's Seal
Equilateral Triangles into Twelfths
Measuring Perimeter in Centimeters
Drawing Parallelograms in Centimeters
Measuring Perimeter in Inches
Drawing Parallelograms in Inches
Drawing Rectangles
Drawing Rhombuses
Drawing Squares
Classifying Quadrilaterals
The Fraction Chart
Patterns in Fractions
Measuring With Sixteenths
A Fraction of Geometry Figures
Making the Whole
Ratios and Nested Squares
Square Centimeters
Square Inches
Area of a Rectangle
Comparing Areas of Rectangles
Product of a Number and Two More
Area of Consecutive Squares
Perimeter Formula for Rectangles
Area of a Parallelogram
Comparing Calculated Areas of Parellelograms
Area of a Triangle
Comparing Calculated Areas of Triangles
Converting Inches to Centimeters
Name that Figure
Finding the Areas of More Triangles
Area of Trapezoids
Area of Hexagons
Area of Octagons
Ratios of Areas
Measuring Angles
Supplementary and Vertical Angles
Measure of the Angles in a Polygon
Classifying Triangles by Sides and Angles
External Angles of a Triangle
Angles Formed With Parallel Lines
G: © Joan A. Cotter 2005
Table of Contents
Lesson 48
Lesson 49
Lesson 50
Lesson 51
Lesson 52
Lesson 53
Lesson 54
Lesson 55
Lesson 56
Lesson 57
Lesson 58
Lesson 59
Lesson 60
Lesson 61
Lesson 62
Lesson 63
Lesson 64
Lesson 65
Lesson 66
Lesson 67
Lesson 68
Lesson 69
Lesson 70
Lesson 71
Lesson 72
Lesson 73
Lesson 74
Lesson 75
Lesson 76
Lesson 77
Lesson 78
Lesson 79
Lesson 80
Lesson 81
Lesson 82
Lesson 83
Lesson 84
Lesson 85
Lesson 86
Lesson 87
Lesson 88
Lesson 89
Lesson 90
Lesson 91
Lesson 92
Lesson 93
Lesson 94
Triangles With Congruent Sides (SSS)
Other Congruent Triangles (SAS, ASA)
Side and Angle Relationships in Triangles
Medians in Triangles
More About Medians in Triangles
Midpoints in a Triangle
Rectangles Inscribed in a Triangle
Connecting Midpoints in a Quadrilateral
Introducing the Pythagorean Theorem
Squares on Right Triangles
Proofs of the Pythagorean Theorem
Finding Square Roots
More Right Angle Problems
The Square Root Spiral
Circle Basics
Ratio of Circumference to Diameter
Inscribed Polygons
Tangents to Circles
Circumscribed Polygons
Pi, a Special Number
Circle Designs
Rounding Edges With Tangents
Tangent Circles
Bisecting Angles
Perpendicular Bisectors
The Amazing Nine-Point Circle
Drawing Arcs
Angles 'n Arcs
Arc Length
Area of a Circle
Finding the Area of a Circle
Finding More Area
Pizza Problems
Revisiting Tangrams
Aligning Objects
Reflecting
Rotating
Making Wheel Designs
Identifying Reflections & Rotations
Translations
Transformations
Double Reflections
Finding the Line of Reflection
Finding the Center of Rotation
More Double Reflections
Angles of Incidence and Reflection
Lines of Symmetry
G: © Joan A. Cotter 2001
Table of Contents
Lesson 95
Lesson 96
Lesson 97
Lesson 98
Lesson 99
Lesson 100
Lesson 101
Lesson 102
Lesson 103
Lesson 104
Lesson 105
Lesson 106
Lesson 107
Lesson 108
Lesson 109
Lesson 110
Lesson 111
Lesson 112
Lesson 113
Lesson 114
Lesson 115
Lesson 116
Lesson 117
Lesson 118
Lesson 120
Lesson 121
Lesson 122
Lesson 123
Lesson 124
Lesson 125
Lesson 126
Lesson 127
Lesson 128
Lesson 129
Lesson 130
Lesson 131
Lesson 132
Lesson 133
Lesson 134
Lesson 135
Lesson 136
Lesson 137
Lesson 138
Lesson 139
Lesson 140
Lesson 141
Lesson 142
1: © Joan A. Cotter 2001
Rotation Symmetry
Symmetry Connections
Frieze Patterns
Introduction to Tessellations
Two Pentagon Tessellations
Regular Tessellations
Semiregular Tessellations
Demiregular Tessellations
Pattern Units
Dual Tessellations
Tartan Plaids
Tessellating Triangles
Tessellating Quadrilaterals
Escher Tessellations
Tessellation Summary & Mondrian Art
Box Fractal
Sierpinski Triangle
Koch Snowflake
Cotter Tens Fractal
Similar Triangles
Fractions on the Multiplication Table
Cross Multiplying on the Multiplication Table
Measuring Heights
Golden Ratio
Fibonacci Sequence
Fibonacci Numbers and Phi
Golden Ratios and Other Ratios Around Us
Napoleon’s Theorem
Pick’s Theorem
Pick’s Theorem With the Stomachion
Pick’s Theorem and Pythagorean Theorem
Estimating Area With Pick’s Theorem
Distance Formula
Euler Paths
Using Ratios to Find Sides of Triangles
Basic Trigonometry
Solving Trig Problems
Comparing Calculators
Solving Problems With a Scientific Calculator
Angle of Elevation
More Angle Problems
Introduction to Sine Waves
Solids and Polyhedrons
Nets of Cubes
Volume of Cubes
Volume of Boxes
Volume of Prisms
Table of Contents
Lesson 143
Lesson 144
Lesson 145
Lesson 146
Lesson 147
Lesson 148
Lesson 149
Lesson 150
Lesson 151
Lesson 152
Lesson 153
Lesson 154
Lesson 155
Lesson 156
Lesson 157
Lesson 158
Lesson 159
Lesson 160
Lesson 161
Lesson 162
Lesson 163
Lesson 164
Lesson 165
Diagonals in a Rectangular Prism
Cylinders
Cones
Pyramids
Polygons ‘n Polyhedrons
Tetrahedron in a Cube
Platonic Solids
Views of the Platonic Solids
Duals of the Platonic Solids
Surface Area and Volume of Spheres
Plane Symmetry in Polyhedra
Rotating Symmetry in Polyhedra
Circumscribed Platonic Solids
Cubes in a Dodecahedron
Stella Octangula
Truncated Tetrahedra
Truncated Octahedron
Truncated Isocahedron
Cuboctahedron
Rhombicuboctahedron
Icosidodecahedron
Snub Polyhedra
Archimedean Solids
Vocabulary First Introduced
Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 10
Lesson 13
Lesson 14
Lesson 15
Lesson 16
Lesson 17
Lesson 18
Lesson 19
Lesson 21
Lesson 23
Lesson 24
Lesson 25
Lesson 26
Lesson 27
Lesson 30
Lesson 32
Lesson 34
Lesson 36
Lesson 38
Lesson 42
Lesson 43
Lesson 45
Lesson 46
Lesson 47
Lesson 48
Lesson 49
Lesson 50
Lesson 51
Lesson 52
Lesson 54
Lesson 55
Lesson 56
Lesson 57
Lesson 59
Lesson 59
Lesson 60
Lesson 62
Lesson 64
Lesson 65
Lesson 66
Lesson 67
line segments, parallel lines, and intersections
horizontal, vertical, diagonal, hexagon,
polygon, vertex, vertexes, vertices
quadrilateral, equilateral triangle
congruent
bisect, tick mark, tetrahedron
perimeter
parallelogram
rectangle, right angle, perpendicular
rhombus
90 degrees, square
trapezoid, Venn diagram
fraction
numerator, denominator
crosshatch
ratio
area, square centimeter
area, square inch
formula
exponent
factor
millimeter, square millimeter
little square, altitude
isosceles
distributive property, straightedge
goniometer
supplementary, vertical, complementary
acute, obtuse, scalene
external, internal, adjacent angle
corresponding, alternate, interior, exterior angles
SSS
similar, SAS, ASA
vertex angle, base angles, base
median of a triangle
centroid
inscribed
convex, concave
hypotenuse, leg
oblique
Pythagorean theorem
square root, integer, perfect square
Pythagorean triple
point, line, and plane, circumference, diameter, radius, arc, sector
inscribed polygon, regular polygon
tangent, tangent segment
circumscribed polygon
pi, π
Vocabulary First Introduced
Lesson 68
Lesson 69
Lesson 70
Lesson 71
Lesson 72
Lesson 73
Lesson 74
Lesson 75
Lesson 76
Lesson 80
Lesson 81
Lesson 83
Lesson 86
Lesson 87
Lesson 88
Lesson 93
Lesson 94
Lesson 95
Lesson 97
Lesson 98
Lesson 99
Lesson 100
Lesson 101
Lesson 102
Lesson 103
Lesson 104
Lesson 105
Lesson 106
Lesson 107
Lesson 108
Lesson 109
Lesson 110
Lesson 111
Lesson 112
Lesson 113
Lesson 114
Lesson 115
Lesson 116
Lesson 117
Lesson 118
Lesson 119
Lesson 120
Lesson 121
Lesson 122
Lesson 123
clockwise, counterclockwise
oblique, concentric, semicircle
internally tangent circles, externally tangent circles, trefoil,
quatrefoil
angle bisector, incenter
chord, circumcenter*
foot, feet
central angle
inscribed angle, intercepted arc
kilometer
per, unit cost
tangram
reflection, image, line of reflection, flip horizontal, flip vertical
transformation
translation, image, absolute, relative
transformation
angle of incidence, angle of reflection
line of symmetry, maximum, minimum, ∞
order of rotation symmetry, point symmetry
frieze, cell, tile
tessellation
pure tessellation
nonagon, decagon , dodecagon
semiregular tessellation
demiregular tessellation, semi-pure tessellation
unit, pattern
tartan, plaid, warp, weft, woof
Escher
Mondrian
fractals and the terms iteration and self-similar, exponent
Sierpinski Triangle
Koch Snowflake
similar, similar triangles
proportion
cross-multiplying
golden rectangle, golden ratio, phi, φ
golden spiral, golden triangle
sequence, Fibonacci sequence
Fibonacci spiral
generalize
Vocabulary First Introduced
Lesson 124
Lesson 125
Lesson 126
Lesson 127
Lesson 128
Lesson 129
Lesson 130
Lesson 131
Lesson 132
Lesson 133
Lesson 134
Lesson 135
Lesson 136
Lesson 137
Lesson 138
Lesson 139
Lesson 140
Lesson 141
Lesson 142
Lesson 143
Lesson 144
Lesson 145
Lesson 146
Lesson 147
Lesson 148
Lesson 149
Lesson 150
Lesson 151
Lesson 152
Lesson 153
Lesson 154
Lesson 155
Lesson 156
Lesson 157
Lesson 158
Lesson 159
Lesson 160
Lesson 161
Lesson 162
Lesson 163
Lesson 164
Lesson 165
Euler path
trigonometry, opposite, adjacent, sine, cosine, tangent
scientific calculator
angle of elevation, stride, clinometer
angle of depression
sine wave
solid, polyhedron, polyhedra, face, edge, vertex, net, dimension
volume, cubic centimeter, surface area
decimeter, dm
prism
short diagonal, long diagonal
cylinder
cone
apex, regular pyramid, right pyramid
Platonic solids
dual polyhedra
sphere, great circle, small circle
planes of symmetry
axes of symmetry
reciprocal
stella octangula, concave polyhedron
truncate, semiregular polyhedra, Archimedean solids
Intermediate Level Objectives
RightStart™ Mathematics: A Hands-On Geometric Approach is designed for the
intermediate student. It employs a hands-on and visual approach through the use of a
tool set consisting of a drawing board, T-square, triangles, compass, goniometer, and
panels 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 while
more advanced mathematical topics are gradually introduced.
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 that
grade 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 outside
of mathematics.
RightStart™ Mathematics: A Hands-On Geometric Approach
RightStart™ 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 school
students are visual learners, approaching mathematics through geometry gives the student an
excellent way to understand and remember concepts. The hands-on activities often create
deeper learning. For example, to find the area of a triangle, the student must first construct the
altitude and then measure it. If possible, students work with a partner and discuss their observations and results.
Much of the work is done with a drawing board, T-square, 30-60 triangle, 45 triangle, a
template for circles, and goniometer (device for measuring angles). Constructions with these
tools 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, and
trigonometry. Some lessons have an art flavor, for example, constructing Gothic arches. Other
lessons have a scientific background, sine waves, and angles of incidence and reflection; or a
technological background, creating a design for car wheels. Still other lessons are purely mathematical, Napoleon’s theorem and Archimedes stomachion. The history of mathematics is
woven throughout the lessons. Several recent discoveries are discussed to give the student the
perspective 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 success
in advanced math classes. Learning to follow directions is a necessary skill for studying and
everyday life. Occasionally, an activity or lesson refers to previous work making it necessary
for 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 previously
learned, b) to learn to read math texts, c) to lay a good foundation for more advanced mathematics, d) to discover mathematics everywhere, and e) to enjoy mathematics.
About the author
Joan 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 mathematics
to grades 6-8.
© by Joan Cotter 2005
• info@ALabacus.com •
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 and
gather all the supplies, including a mechanical pencil or a sharp #2 pencil and
a good eraser. Then the student reads over the goals, keeping in mind that
italicized words will be explained in the lesson. (These new words are to be
recorded in the student’s math dictionary.) Next the student begins reading
the Activities, carefully studying any accompanying figures. It is a good habit
to summarize the activity after reading it. If a paragraph is unclear, the
student should reread the paragraph, keeping in mind that sometimes more is
explained in the following paragraph. No one learns mathematics by reading
the text only once.
Each activity needs to be understood before going to the next activity. Make
sure the student understands the importance of completing the problems on
the worksheet when called for in the lesson. Sometimes it will be necessary to
refer to the lesson while completing the worksheet. All work needs to be kept
neatly 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 need
a question to answer. You do not want to get in the habit of reading the text
for your student and then regurgitating to her like a mother robin feeding her
young. The text is written for students to read for themselves. Learning how
to ask questions is an important skill to acquire toward becoming an
independent learner. If questions remain after diligent study, the student can
contact the author at JoanCotter@ALabacus.com.
If a child has a serious reading problem, read the text aloud while he follows
along and then ask him to read it aloud. Be sure each word is understood. For
less severe reading problems, you might model aloud the process of reading
an activity, commenting on the figure, and summarizing the paragraph. Some
of the time, students need encouragement to overcome frustration, which is
inherent in the learning process. Occasionally, a student may have a
knowledge gap needed for a particular lesson, requiring other resources to
resolve. Incidentally, research shows one of the major causes of difficulties in
learning new concepts for this age group is insufficient sleep.
After the student has completed the worksheet, ask her to compare her work
with the solution. If the student has a partner, they can compare and discuss
their work before referring to the solutions. Ask her to explain what she
learned and any discrepancies. Keep in mind that some activities have more
than one solution. You might also ask her to grade her work on some agreed
upon scale. It also is a good idea for the student to reread the goals of the
lesson to see if they have been met. Encourage discussion on practical
applications of the topic.
© by Joan Cotter 2005
• info@ALabacus.com •
www.ALabacus.com
8/06
4
Drawing Diagonals
Lesson 2
GOALS
A sharp
pencil, an
eraser, and
tape are
essentials
and will not
be listed in
future
lessons.
MATERIALS
ACTIVITIES
1. To review the terms horizontal and vertical
2. To learn the mathematical meaning of diagonal
3. To review the term hexagon
4. To find the correct edge of the 30-60 triangle to draw diagonals
Worksheet 2
Drawing board, T-square, 30-60 triangle
Horizontal and vertical. Horizontal refers to the horizon, the
intersection between the earth and sky that a person on earth sees (if
there aren’t too many buildings and trees in the way). Vertical refers
to straight up and down, like a flagpole. Sometimes it also means
above, or overhead.
A horizontal line on paper is a line drawn straight across the paper. It
usually is parallel to the top and bottom of the paper. A vertical line
on paper goes from top to bottom, parallel to the sides of the paper.
Don’t forget to add the
terms listed in Goals to
your math dictionary.
Diagonals. In common everyday English, the word diagonal usually
means at a slant. It often means a road that runs neither north and
south nor east and west.
In mathematics, a diagonal is a line connecting points in a closed
figure. For example, the line segments AC and DB drawn in the
square below on the left are diagonals. If we turn the square, as in
the next figure, diagonal DB is horizontal and diagonal AC is
vertical.
A
A
B
D
Diagonal lines on a building.
D
C
B
C
Worksheet. The worksheet asks you to draw two hexagons and all
their diagonals. A hexagon is a closed six-sided figure. One way to
remember the word is that hexagon and six both have x’s.
Draw all the sides of the hexagon and the diagonals using your
T-square and 30-60 triangle except the horizontal lines, which need
only a T-square. Below are a hexagon and all its diagonals.
G: © Joan A. Cotter 2008
© Joan A. Cotter 2009
6. How many diagonals at each vertex are not horizontal or vertical? ________
5. How many diagonals at each vertex are either horizontal or vertical? ________
4. How many diagonals are vertical? ________
Include both hexagons:
3. How many diagonals are horizontal? ________
2. Next, draw all the diagonals in the hexagons, using your drawing tools. There
are 3 diagonals at each vertex.
1. First, trace the dotted lines forming the two hexagons. Use your T-square for
drawing all lines. Use your 30-60 triangle for all lines except horizontal lines.
Date ____________________________
Name ___________________________________
Worksheet 2, Drawing Diagonals
5
Drawing Stars
Lesson 3
GOALS
MATERIALS
ACTIVITIES
1. To learn the term polygon
2. To learn the term vertex and its plurals, vertices and vertexes
3. To draw stars by following instructions shown in pictures
Worksheets 3-1 and 3-2
Drawing board, T-square, 30-60 triangle
Polygons. In the row below are examples of figures that are
polygons and figures that are not polygons. Before reading further,
think of a good definition for polygon.
These are polygons.
These are NOT polygons.
Did your definition include a closed figure with straight line
segments?
Vertex and its plurals. In a polygon the point where the lines
meet is call a vertex. You have two choices for the plural of vertex,
either vertices (VER-ti-sees) or vertexes. For some reason, even
though the word vertexes follows the normal English rule for
plurals, math books (and tests) prefer vertices.
For example, there are three vertices in a triangle and four vertices
in a square.
Worksheets. On the worksheets, you will be drawing stars. The
boldfaced lines in the little figures tell you what to draw. Be sure to
use your T-square and, where needed, your triangle to draw all the
lines. They will look like the following figures.
Star designs in
Morocco, where they
are very common.
G: © Joan A. Cotter 2008
3. How many vertices does a hexagon have? _____
___________________ or ___________________
2. What is the plural of vertex?
1. A hexagon is a six-sided _____________________ .
Draw a star by using your drawing tools and
following the instructions below.
© Joan A. Cotter 2009
Date ____________________________
Name ___________________________________
Worksheet 3-1, Drawing Stars
the previous page? __________________________
4. How does this star compare in size to the star on
3. How many vertices does it have? _____
2. How many sides does it have? _____
1. Is the star that you drew a polygon? _______
Draw a star by using your drawing tools and
following the instructions below.
© Joan A. Cotter 2008
Date ____________________________
Name ___________________________________
Worksheet 3-2, Drawing Stars
12
Equilateral Triangle into Twelfths
Lesson 9
GOALS
MATERIALS
ACTIVITIES
1. To divide an equilateral triangle into twelfths
2. To divide an equilateral triangle into a number of your choosing
that is greater than 12
Worksheet 9 and plain reverse side (or another plain piece of paper)
Drawing board, T-square, 30-60 triangle
Dividing a triangle into twelfths. How would you divide an
equilateral triangle into twelfths, that is, into twelve congruent
parts? Think about it for a while before reading further. Would it
work to divide the triangle into thirds and divide each third into
fourths? One student even suggested dividing the triangle into
tenths and then dividing each tenth in half. Let’s hope he was
joking!
If you have thought about it, you probably realize you first divide
the triangle into fourths and then each fourth into thirds.
Worksheet. Do the worksheet next, dividing the equilateral
triangle into twelfths.
Dividing a triangle by higher numbers. How would you divide
the triangle into sixteenths? What other numbers could you divide it
into? Two kindergarten girls divided the equilateral triangle into
256 equal parts. After hearing about the girls, a teacher learning
drawing board geometry divided his triangle into 432 equal parts.
Some divisions are shown below.
Sixteenths
Eighteenths
Eighteenths
Twenty-fourths
Twenty-sevenths
Twenty-sevenths
Triangle into 432nds by
Joseph Hermodson-Olsen, 14.
Worksheet, reverse side. On the reverse side of your
worksheet or other plain paper, draw a large
equilateral triangle. Choose a number greater than 12
that you can divide an equilateral triangle into and
then do the dividing Copy one of the designs, or
better yet, design your own. You might like to
color your design.
Thirty-seconds
G: © Joan A. Cotter 2008
1. Draw an equilateral triangle. Divide it
into fourths. Then divide each fourth into
thirds, as shown.
© Joan A. Cotter 2009
48
27
24
18
4
9
9
2
4
4
12
16
4
8
4
Number
First
Second
Third
of Pieces Division Division Division
2. Fill in the chart.
Date ____________________________
Name ___________________________________
Worksheet 9-1, Equilateral Triangle into Twelfths and More
____________________________________
____________________________________
____________________________________
4. Describe how you did it.
3. Draw an equilateral triangle. Divide it into
more than 12 equal parts.
© Joan A. Cotter 2009
Date ____________________________
Name ___________________________________
Worksheet 9-2, Equilateral Triangle into Twelfths and More
30
Area of a Rectangle
Lesson 26
GOALS
MATERIALS
ACTIVITIES
1. To understand how area is calculated
2. To see the connection between multiplication and area
3. To learn and apply the formula for the area of a rectangle
Worksheet 26
Drawing board, T-square, 30-60 triangle
Worksheet. Begin by completing numbers 1-5 on the worksheet.
Formula for finding the area of a rectangle. A formula is a
general principle stated in mathematical symbols. The word formula
means “little form.” So, it is a shortcut for stating a mathematical
relationship. Most of the time you do not need to just memorize
formulas. Rather, they are a logical result you can think through.
In some math textbooks (especially older
texts), the symbol “l”
for length is used
rather than “h.”
In question 5 on the worksheet, you were asked to write the formula
for the area of a rectangle. The number of squares needed to cover a
figure is the area, usually written as A. You found the area of all
those rectangles by multiplying the number of squares in a row by
the number of rows. So, if we call the horizontal distance, the width
w, and the vertical distance, the height h, the formula becomes
A=w×h
h
Today, computer
software usually uses
width and height.
h
w
w
Actually, in algebra, which has lots of formulas, the operator “×” is
not written. Two letters written together without an operator means
multiply. This is one of the major differences between arithmetic
and algebra. In arithmetic, digits written side-by-side without an
1
operator mean add; for example, 976 is 900 + 70 + 6 and 3 2 means
1
3 + 2 . However, computer spreadsheets require operators between
all numbers and letters.
Symbol, cm2, for square centimeter. The symbol, cm2, is the
abbreviation for square centimeter. Just as 32 = 9 and forms a square
with 3 on a side, so 1 cm2 is a square that is 1 cm on a side. Read
1 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 you
can, before looking at the solutions on the next page.
21 cm
3 cm
9 cm
What is the area of this figure?
A building in England with
many rectangles.
15 cm
G: © Joan A. Cotter 2008
31
There are several ways to solve this problem.
Solution 1. Make the whole figure into a rectangle and then subtract the little rectangle.
21 cm
3 cm
9 cm
6 cm
6 cm
15 cm
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.
21 cm
3 cm
9 cm
6 cm
15 cm
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.
21 cm
3 cm
9 cm
6 cm
15 cm
A = wh (left rectangle) + wh (right rectangle)
A = 15 × 9 + 6 × 3
A = 135 + 18
A = 153 cm2
G: © Joan A. Cotter 2008
© Joan A. Cotter 2004
_______________________________________________
2. You've probably seen this table before. What is it?
12
12 cm
16 cm
4 cm
5. What is the area of the figure below?
6 cm
Read Lesson 26 before answering the next question.
2
centimeters
do you need to cover it? _____________________
4. If a rectangle is w cm wide and h cm high, how many square
centimeters do you need to cover it? _____________________
3. If a rectangle is 8 cm wide by 9 cm high, how many square
1. The points, marked by little x's, are 1 cm apart. Start at any point. Use your T-square and draw the width of the rectangle by
drawing a horizontal line from the point to the left edge of the square. Then use your T-square and triangle and draw the
height by drawing a vertical line from the point to the top of the square. Calculate the number of square centimeters in the
rectangle and write the number in the lower right corner of the rectangle. One is done for you,.
Date ____________________________
Name ___________________________________
Worksheet 26, Area of a Rectangle
32
Comparing Areas of Rectangles
Lesson 27
GOALS
MATERIALS
ACTIVITIES
1. To calculate more areas of rectangles
2. To compare areas of rectangles with constant perimeter
Worksheets 27-1, 27-2
Drawing board, T-square, 30-60 triangle
4-in-1 ruler
Frame problem. Consider the following problem. You have 12 cm
of gold edging to place around a rectangular frame. You want the
maximum amount of space inside the frame.
First think about the possible dimensions of the rectangles, so the
perimeters will be 12 cm. Then study the figures below.
2 cm
3 cm
4 cm
5 cm
1 cm
3 cm
2 cm
4 cm
1 cm
5 cm
The areas, which you can do in your head using A = wh, are from
left to right, 5 cm2, 8 cm2, 9 cm2, 8 cm2, and 5 cm2.
The shape of this graph
is called a parabola.
Graphing the frame
Rectangle Areas with Perimeter = 12 cm
problem. It is
interesting to graph
10
9
the results as shown
8
below. Why is the
7
area equal to 0 when
6
the width is equal to 0
5
4
or equal to 6? You can
3
see the greatest area
2
occurs when the
1
width of the rectangle
0
0
1
2
3
4
5
6
is to 3. What is the
T
he
width
of
the
rectangle
in
cm
height when the
width is 3? The
answer is at the bottom of the page.
The area in cm2
This type of problem is
easily solved with a
branch of mathematics
called calculus.
Worksheets. There is a similar problem on Worksheets 27-1 and
27-2. Draw the rectangles by measuring with your ruler like you did
on Worksheet 11. [Answer: 3]
G: © Joan A. Cotter 2008
© Joan A. Cotter 2004
area? __________________________________
2. Below each rectangle, calculate its area in cm2. Which rectangle gives the most
1. On each of the five lines below, draw a rectangle with a perimeter of 20 cm. Write the dimensions.
If you had 20 cm of expensive trim to decorate the edge of a rectangular bulletin board, what should the dimensions of the
rectangle be to give you the most area for photos and notes? Follow the steps below for the solution.
Date ____________________________
Name ___________________________________
Worksheet 27-1 , Comparing Areas of Rectangles
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
The base of the rectangle in cm
Area of Rectangles with a Perimeter of 20 cm
© Joan A. Cotter 2004
7. How does the graph compare with the example in the lesson? _________________________________________________________
6. According to the graph, what is the maximum area? ___________________________________
5. What is the name of the shape of the curve? ______________________________________
Areas in sq cm
4. On the graph below, place a point showing the area for each rectangle from the previous page.
Also find the areas for the remaining widths: 0, 6, 7, 8, 9, and 10. Then connect the points in a smooth
curve; do this freehand (without any drawing tools).
Date ____________________________
Name ___________________________________
Worksheet 27-2, Comparing Areas of Rectangles
96
Rotating
Lesson 84
GOALS
MATERIALS
ACTIVITIES
1. To learn the mathematical meaning of rotation
2. To construct rotations at various angles
Worksheet 84
Goniometer
A set of tangrams
Drawing board, T-square, 45 triangle
Rotating. A clock is a good example of rotation. Both the hour and
minute hands rotate about the center of the clock. The hands move
in a clockwise direction. However, when we discuss rotations
mathematically, we start with a horizontal ray extending right and
measure the amount of rotation counterclockwise. So, for a clock to
behave mathematically, the hand would start at the 3 o’clock
position and travel backward.
Rotating the ship. Build the ship shown below in the left figure
with four tangram triangles and tape them together.
12°
90°
Then tape the ship to the upper arm of the goniometer. Hold the
lower arm of the goniometer still with your right hand. Use your
left hand to rotate and upper arm of the goniometer with the
attached ship. See the middle figure above.
Keep rotating to 90° as shown in the right figure above. (The seas
are getting very rough.) Continue rotating to 180°. (Disaster.) See
the left figure below.
250°
180°
Star design on the floor.
Construct every line
accurately. Don’t guess.
To set your ship aright, un-tape it, turn your goniometer upside
down, re-tape it, and continue rotating as in the right figure above.
Worksheet. The first half of the worksheet asks you to construct
the ship at various angles with your tools. You may find it helpful
to set the ship model at the desired angle. Start your construction at
the “×” and draw the first line at the correct angle. Measure only the
line for the ship’s bottom (3 cm); construct the other lines.
For the second half, build and rotate the model to the various angles
before attempting the constructions. Measure only the 2.5 cm line.
G: © Joan A. Cotter 2008
Name ___________________________________
Date ____________________________
The × shows you where to start.
1. 45°
3.0 cm
2. 90°
4. 180°
3. 135°
cm
5. 45°
2.
5
6. 90°
9. What angle of rotation is
the same turning something
upside down? ______
8. 270°
10. Is a rotation of 180° the
same as reflecting about a
7. 180°
horizontal line? ______
© Joan A. Cotter 2008
Worksheet 84, Rotating
Construct the figures at the angles given with
your geometry tools. Use your ruler only to
measure the line representing the bottom of
the ship and the side of the arrow.
142
Fibonacci Sequence
Lesson 120
GOALS
MATERIALS
ACTIVITIES
1. To learn the term sequence
2. To learn about the Fibonacci sequence
3. To solve some sequence problems
Worksheets 120-1, 120-2, 120-3
Fibonacci. Fibonacci (fee-buh-NOT-chee), 1170-1250, was born in
Pisa (PEE-za), Italy, the city with the Leaning Tower. He was
educated in North Africa (now Algeria). Fibonacci learned other
mathematics from talking with merchants while traveling around
the Mediterranean region.
Fibonacci became fascinated with the Hindu-Arabic numerals, with
digits 1-9 and a 0. At that time Europe used Roman numerals, which
had no 0. His book Liber Abaci ("Book of Calculation”) introduced
Europe to the numbers we use today. It also showed methods of
calculating with paper and pencil without an abacus. Conversion to
the new method took time. In 1299 the merchants in Florence
required using Roman numerals.
He also introduced the fraction bar—the line separating the
numerator and denominator. Before that fractions were written as 12 .
Fibonacci sequence. A sequence is a set of quantities in some type
of order. The answer to one of Fibonacci’s math problems results in
the Fibonacci sequence. It starts as 1, 1, 2, 3, 5. Think about what
comes next before reading further.
If you think 8 comes next, you’re right. Each number in the sequence
is the sum of the previous two numbers. Thus, 1 + 1 = 2, 1 + 2 = 3, 2
+ 3 = 5, and so on.
If you ever played the
“Chain” games, you will
recognize the ones
column as a chain.
Problem 1. For Problem 1, you are to practice your addition skills
and calculate the Fibonacci sequence to 26 terms. Incidentally, if you
learned about check numbers, or casting out nines as they’re
sometimes called, use them to check your work. Fibonacci learned
about them and explained them in Liber Abaci.
Problem 2. This is the brick wall problem. The sides of
the brick have a ratio of 2:1. See the figure at the right.
You need to make your wall two units high, but with
various widths.
If your wall is 1 unit wide, there is only one way to make it. If it is 2
units wide, there are two ways to make it. See the middle figure
below. If it is 3 units wide, there are three ways to make. See the
right figure.
1 uniti
wide
2 units
i
wide
i
3 units
i
wide
i
Continue the process for 4 and 5 units wide. Draw your
arrangements freehand and record the number of arrangements. Do
Worksheet 1 before reading any further. Think about your solutions.
G: © Joan A. Cotter 2008
143
Discussing Worksheet 1. Notice how quickly the numbers
become large in the Fibonacci sequence. To be sure you’ve added
correctly, the last number in the sequence is 121,393.
Discuss with a partner the number of arrangements you found in
Problem 2 and why.
Notice you can make the arrangements for 4 units by copying the 3s
arrangement plus a vertical brick and copying the 2s plus two
horizontal bricks. Likewise, the arrangements for 5 are the 4s plus a
vertical brick and the 3s plus horizontal bricks. Adding the last two
are, of course, what the Fibonacci sequence is all about. (If you need
help in understanding this, look carefully at the solutions.)
Problem 3. For the next problem, you have two sizes of
colored rods, the 1s and the 2s. See the figure on the right.
You are to make various lengths using these rods. For
example, the three arrangements for a length of 3 are shown
below. You can do the worksheet now.
With three 1s
With a 2 and a 1
1
2
With a 1 and a 2
Problems 4-5. For Problem 4, you are climbing stairs. You can
climb either one at a time or two at a time. The left figure below
shows a set of three stairs. The middle figure shows climbing the
stairs, 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—first
two steps at a time and then one at a time. You are to draw all
combinations for 3, 4, and 5 stairs.
For Problem 5, the instructions explain the bee problem. The bee can
enter 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 think
of a good one, let me know at joancotter@alabacus.com. See some
problems at http://www.rightstartgeometry.com.
G: © Joan A. Cotter 2008
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
______________
1. Continue the
Fibonacci sequence.
1
1
2
3
2
1
© Joan A. Cotter 2005
What kind of numbers did you write? ________________________
How many arrangements could you make if the wall were 6 units wide? _______
2. A brick wall is 2 units high. The bricks are used two ways as shown.
Find all the different arrangements to make a wall 3-5 units wide. Record
the number of arrangements on the lines at the left.
Date ____________________________
Name ___________________________________
Worksheet 120-1, Fibonacci Sequence
© Joan A. Cotter 2005
What kind of numbers did you write? ________________________
2
1
3. Sketch all the possibilities for making lengths 1-6 using 1s and 2s.
Write the number of arrangements at the left.
1
2
In how many different ways could you make 7? ______
Date ____________________________
Name ___________________________________
Worksheet 120-2, Fibonacci Sequence
4. You are climbing stairs either one at a time
Date ____________________________
or two at a time. Draw all the ways you can
climb 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 1
or 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.
1
3
2
5
4
7
6
1
To reach cell #1. __________________________________________________________________
To reach cell #2. __________________________________________________________________
2 12
To reach cell #3. __________________________________________________________________
23
To reach cell #4. __________________________________________________________________
To reach cell #5. __________________________________________________________________
__________________________________________________________________
How many ways can the bee enter cell #7? _________
© Joan A. Cotter 2005
Worksheet 120-3, Fibonacci Sequence
Name ___________________________________
168
Volume of Cubes
Lesson 140
GOALS
MATERIALS
ACTIVITIES
1. To learn the terms volume, cubic centimeter, and surface area
2. To calculate volumes and surface areas of cubes
Worksheet 140
Geometry panels and rubber bands
Centimeter cubes
Ruler
Volume. Volume is the amount of space taken up by a solid. To
measure this space you need a unit that takes up three-dimensional
space. Units of volume are called cubic units because they are
usually cubes.
Recall that you can measure a line segment with centimeters (cm)
and area with square centimeters (cm2). You will measure the
volume of a solid with cubic centimeters (cm3). A cubic centimeter is a
cube with edges 1 cm long. See these units of measurements below.
1 cm
2
1 cm
3
1 cm
Construct a cube with the geometry panels.
Volume of cubes. Use the centimeter cubes to construct a cube
that measures 2 cm on a side. (If you don’t have these cubes, look at
the figures on the worksheet.) How many centimeter cubes do you
need? The number of cubic centimeters you need to fill the solid is
the volume. What is the cube’s volume? The answers are at the
bottom of the page.
Cubing a number. With the cube you made, did you notice that
the total number of cubes is 2 × 2 × 2? You can also write this
expression with exponents as 23. Read it as “two cubed.”
Surface area of a cube. Surface area is the area of all the surfaces
of a solid. For polyhedra, the surface area is the sum of the area of
all the polygons. The symbol for surface area is “S.”
Worksheet. For the first table, you will be considering
cubes of various dimensions. The second table pertains
to a cube made with the panels, but measured with
different units. Measure the edges of a panel polygon as
shown in the figure on the right. For the inch, use the
nearest whole inch. [Answers: 8, 8 cm3]
10 cm
The crystalline structure of
common salt. The larger
chloride ions form a cubic
shape. The smaller sodium ions
fill in the gaps between them.
G: © Joan A. Cotter 2008
a
2
2
9 cm
Area of a Face
Volume of the Cube
Surface Area
Length of a Side
Area of a Face
Volume of the Cube
a
© Joan A. Cotter 2006
*5. Which is more, 1 cm or 1 cm 3? Explain your answer. _____________________________________________________________
4. Imagine a cube that is 1 m on an edge. How many panel cubes would fit in the 1-meter cube? ___________
3. How many centimeter cubes will fit in the cube made from the panels? ___________
Inch
Meter
Millimeter
Centimeter
Unit
Surface Area
Date ____________________________
2. Construct a cube from the panels. Measure a side using the units given and complete the table.
4 cm
2 cm
1 cm
Length of a Side
1. Complete the table for various cubes.
Name ___________________________________
Worksheet 140, Volume of Cubes