rightstart™ mathematicsrightstartmath.com/wp-content/uploads/2015/11/rightstart-geometric... ·...

bbbbyyyy JJJJooooaaaannnn AAAA CCCCooootttttttteeeerrrr PPPPhhhh DDDD

Activities for Learning, Inc.

RIGHTSTART™ MATHEMATICS

A HANDS-ON

GEOMETRIC APPROACH

LESSONS

Copyright © 2009 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)

Printed in the United States of America

www.RightStartMath.com

For questions or for more information:[email protected]

To place an order or for additional supplies:[email protected]

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

ISBN 978-1-931980-38-8December 2014

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 Twelfths and MoreLesson 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 ParallelogramsLesson 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 Angles (First Quarter test)Lesson 46 External Angles of a TriangleLesson 47 Angles Formed With Parallel Lines

Table of Contents

G: © Activities for Learning, Inc. 2010


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 Reflection (Second Quarter test)Lesson 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 119 More Golden GoodiesLesson 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 Boxes

Table of Contents


Lesson 142 Volume of PrismsLesson 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 (Final test)

Table of Contents

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

8/06


Lesson 1 line segments, parallel lines, intersectionLesson 2 horizontal, vertical, diagonal, hexagonLesson 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 28 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 58 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 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 105 tartan, plaid, warp, weft, woofLesson 108 EscherLesson 109 MondrianLesson 110 fractals and the terms iteration and self-similar, exponentLesson 111 Sierpinski TriangleLesson 112 Koch SnowflakeLesson 114 similar, similar trianglesLesson 115 proportionLesson 116 cross-multiplyingLesson 118 golden rectangle, golden ratio, phi, φ Lesson 119 golden spiral, golden triangleLesson 120 sequence, Fibonacci sequenceLesson 121 Fibonacci spiralLesson 123 generalizeLesson 129 Euler pathLesson 131 trigonometry, opposite, adjacent, sine, cosine, tangentLesson 133 scientific calculatorLesson 135 angle of elevation, stride, clinometerLesson 136 angle of depressionLesson 137 sine wave



Lesson 138 solid, polyhedron, polyhedra, face, edge, vertex, net, dimensionLesson 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 149 Platonic solidsLesson 151 dual polyhedraLesson 152 sphere, great circle, small circleLesson 153 planes of symmetryLesson 154 axes of symmetryLesson 155 reciprocalLesson 157 stella octangula, concave polyhedronLesson 158 truncate, semiregular polyhedra, Archimedean solidsLesson 159Lesson 160Lesson 161Lesson 162Lesson 163 quasiregular polyhedronLesson 164Lesson 165


4


BD

C

A

C

A

D

B

diagonals

diagonal

diagonal

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 2, Math DictionaryDrawing board, T-square, 30-60 triangle

ACTIVITIES Horizontal and vertical. Horizontal refers to the horizon, theintersection between the earth and sky. You can see it if there aren’ttoo many buildings and trees in the way. Vertical refers to straightup and down, like a flagpole.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 verticalline on paper goes from top to bottom, parallel to the sides of thepaper.Diagonals. In common everyday English, the word diagonalusually means at a slant. It often means a road that runs neithernorth and south 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, the lines segments are still diagonals. Now diagonalDB is horizontal and diagonal AC is vertical.

A sharppencil, aneraser, andtape areessentials.They will notbe listed infuturelessons.

In the word diagonal, dia means “across” and gon means “angle.”So, a diagonal is a line across angles, that is, a line connecting twovertices. 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 the sides of the hexagon and the diagonals using your tools.The horizontal and vertical lines need only a T-square. The leftfigure below is a hexagon; the right figure shows the diagonals.

Diagonal lines on a building.

Name ___________________________________

Date ____________________________

© Joan A. Cotter 2009

Worksheet 1, G

etting Started

3. Draw lines parallel to the sides of thisequilateral triangle. Use your T-squareand 30-60 triangle to draw the lines.Hold your pencil about 2.5 cmfrom the tip ( ) .

1. Use your T-square to draw horizontal lines inthe octagon below. Be sure your T-square is snugagainst the edge of the drawing board.

2. Use your T-square and 30-60triangle to draw vertical lines in the hexagon. Be sure thatyour triangle is snug against the T-square.

4. In which figure(s) have you drawn parallel

lines? ________________________________

_____________________________________

5. In which figure(s) have you drawn intersecting

lines? ________________________________

octagonhexagon

triangle

Name ___________________________________

Date ____________________________


1. First, trace the dotted lines forming the two hexagons. Use yourT-square for drawing all lines. Use your 30-60 triangle for all linesexcept horizontal lines.2. Next, draw all the diagonals in the hexagons, using your drawingtools. There are 3 diagonals at each vertex.

Worksheet 2, Drawing Diagonals

3. How many diagonals are horizontal? ________4. How many diagonals are vertical? ________5. How many diagonals at each vertex are either horizontal or vertical? ________6. How many diagonals at each vertex are not horizontal or vertical? ________

33

12

Include both hexagons:

12


Lesson 9 Equilateral Triangle into Twelfths and MoreGOALS 1. To discover how to divide an equilateral triangle into congruent

pieces greater than 92. To divide an equilateral triangle into twelfths3. To divide an equilateral triangle into a number greater than 12

MATERIALS Worksheets 9-1, 9-2Drawing board, T-square, 30-60 triangleColored pencil, optional

ACTIVITIES Dividing a triangle into twelfths. How would you divide anequilateral triangle into twelfths – into twelve congruent parts?Think about it for a while before reading further. Would it work todivide the triangle into thirds and divide each third into fourths?One student even suggested dividing the triangle into tenths andthen dividing each tenth in half. Let’s hope he was joking!If you have thought about it, you probably realize you first dividethe triangle into fourths and then each fourth into thirds.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 9-1. For this worksheet, you will divide the equilateraltriangle into twelfths. Work carefully. For Problem 2,figure out how you would divide equilateral trianglesinto various congruent pieces. Worksheet 9-2. After drawing the equilateraltriangle, divide it into congruent triangles. Eithercopy one of the designs above, or better yet,design your own. You might like to color yourdesign. [Answer: ninths, fourths, fourths, and thirds.]

Sixteenths Eighteenths Eighteenths

Twenty-fourths Twenty-sevenths Twenty-sevenths

Thirty-seconds

Triangle into 432nds byJoseph Hermodson-Olsen, 14.How could he have done it?The answer is at the bottom ofthe page.

Name ___________________________________

Date ____________________________


1. Draw an equilateral triangle. Divide itinto fourths. Then divide each fourth intothirds, as shown.

Worksheet 9-1, Equilateral Triangle into Twelfths and M

ore

8

12

16

18

24

27

48

2

Number of Pieces

342334

FirstDivision

4449494

SecondDivision

ThirdDivision

34

2. Fill in the chart.

6481

4

9

4

9

2[or 6]

Name ___________________________________

Date ____________________________


Worksheet 9-2, Equilateral Triangle into Twelfths and M

ore

3. Draw an equilateral triangle. Divide it intomore than 12 equal parts.

[RESULTS WILL VARY.]

4. Describe how you did it. ____________________________________________________________________________________________________________

32


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

2. To compare areas of rectangles with constant perimeterMATERIALS Worksheets 27-1, 27-2

Drawing 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 frame problem. It is interesting to graph the resultsas shown below. Why is the area equal to 0 when the width is equalto 0 or 6?

The shape of this graph is called a parabola.

You can see the greatest area occurs when the width of the rectangleis to 3. What is the height when the width is 3? The answer is at thebottom 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.

Name ___________________________________

Date ____________________________


Worksheet 26, Area of a Rectangle

1. Find the areas of the small groups of squares. Write the answer inthe lower right square. Also write it in corresponding space in thelarge square. Then fill in the remaining spaces in the large square.

12

2. What do you call the large square in Problem 1?_______________________________________________3. If a rectangle is 8 cm wide by 9 cm high, howmany square centimeters do you need to cover it?_____________________4. If a rectangle is w cm wide and h cm high, howmany square centimeters do you need to cover it?_____________________

5. What is the area of the figure below?

12 cm

4 cm

6 cm

16 cm

1 2 3 4 5 62 4 8 10 123 6 9 12 214 8 16 20 245 10 15 20 306 18 30 427 14 28 42

15

76 14

182835

12 24 3621 49

25

35

Multiplication table

or

or

72 cm2

w ×××× h cm2

4 cm

A ==== lg rect −−−− sm rectA ==== 16 ×××× 10 −−−− 4 ×××× 4A ==== 144 cm2

A ==== 16 ×××× 6 ++++ 12 ×××× 4A ==== 144 cm2

A ==== 12 ×××× 10 ++++ 4 ×××× 6A ==== 144 cm2

Read Lesson 26 before answering the next question.

123

6 14

212425

35

49

12

36

28

7

18

Name ___________________________________

Date ____________________________


If you had 20 cm of expensive trim to decorate the edge of a rectangular bulletin board, what should the dimensions of therectangle be to give you the most area for photos and notes? Follow the steps below for the solution.1. On each of the five lines below, draw a rectangle with a perimeter of 20 cm. Write the dimensions.

2. Below each rectangle, calculate its area in cm . Which rectangle gives the most area?__________________________________

Worksheet 27-1, Com

paring Areas of Rectangles

5 cm by 5 cm2

A ==== whA ==== 1 ×××× 9A ==== 9 cm2

2 cm1 cm

A ==== whA ==== 2 ×××× 8A ==== 16 cm2

A ==== whA ==== 3 ×××× 7A ==== 21 cm2

A ==== whA ==== 4 ×××× 6A ==== 24 cm2

A ==== whA ==== 5 ×××× 5A ==== 25 cm2

3 cm

4 cm5 cm

9cm

8cm

7cm

6cm

5cm

Name ___________________________________

Date ____________________________


0 1 2 3 4 5 6 7 8 9 1002468

1012141618202224262830

The base of the rectangle in cm

Area of Rectangles with a Perimeter of 20 cm

Are

as in

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 rectangle widths: 0, 6, 7, 8, 9, and 10. Plot those areas on thegraph. Then connect the points in a smooth curve; do this freehand (without any drawing tools).

5. What is the name of the shape of the curve? ______________________________________6. According to the graph, what is the maximum area? ___________________________________7. How does the graph compare with the example in the lesson? _________________________________________________________

parabola 25 cm The square has the greatest area.

Worksheet 27-2, Com

paring Areas of Rectangles

2

2

Name ___________________________________

Date ____________________________


1. For problems, A-C, crosshatch the top row of squares. Place thecrosshatched squares on the right of the new figure. Complete thesquare with dashed lines. The steps are shown below.

(n – 1) × (n + 1)Squares in the Original Fig.

n – 1Squares in the New Fig.

2

A.B.

C.

7 × 9 =

8 × ____ =

9 × ____ =

A.

B.

C.

3. Write the results (or rule) you found in your own words.

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

Apply this result to find the following:

4. 19 × 21 _____________________________________

5. 31 × 29 _____________________________________

6. 49 × 51 _____________________________________

7. (n – 1) × (n + 1) _____________________________

2. Complete the table.

6 × 8 =

Worksheet 28, Product of a Num

ber and Two More

When multiplying two numbersthat are two numbers apart,square the number betweenthem and subtract one.

32 −−−− 1 ==== 842 −−−− 1 ==== 1552 −−−− 1 ==== 2472 −−−− 1 ==== 4882 −−−− 1 ==== 6392 −−−− 1 ==== 80

102 −−−− 1 ==== 99

202 −−−− 1 ==== 399302 −−−− 1 ==== 899502 −−−− 1 ==== 2499

==== n2 −−−− 1

2 ×××× 4 ==== 83 ×××× 5 ==== 154 ×××× 6 ==== 24

============

48638099

n

3

4

5

7

8

9

10

n

1011

n

96


12° 90°

180°250°

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

2. To construct rotations at various anglesMATERIALS Worksheet 84

GoniometerA 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 ____________________________


For each figure, flip horizontal and flip vertical about the center lines.

6.

1. 2.

54.

3.

Worksheet 83-2, Reflecting

Nam

e ___________________________________

Date ____________________________

© Joan A

. Cotter 2010

Worksheet 84, Rotating

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 w

ithyour geom

etry tools. Use your ruler only to

measure the line representing the bottom

ofthe ship and the side of the arrow

.

The × shows you w

here to start.

3.0 cm

9. What angle of rotation is

the same turning som

ething

upside down? ______

10. Is a rotation of 180° the

same as reflecting about a

horizontal line? ______

180no

rightstart™ mathematicsrightstartmath.com/wp-content/uploads/2015/11/rightstart-geometric... ·...

Documents