RIGHTSTART MATHEMATICS

Transcription

1 Activities for Learning, Inc. RIGHTSTART MATHEMATICS by Joan A Cotter Ph D A HANDS-ON GEOMETRIC APPROACH LESSONS

2 Copyright 2009 by Joan A. Cotter All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without written permission of Activities for Learning. Three-D images are made with Pedagoguery Software, Inc s Poly ( Printed in the United States of America For questions or for more information: info@rightstartmath.com To place an order or for additional supplies: order@rightstartmath.com Activities for Learning, Inc. PO Box 68; 321 Hill Street Hazelton ND or fax ISBN December 201

3 Table of Contents Lesson 1 Lesson 2 Lesson 3 Lesson Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 1 Lesson 15 Lesson 16 Lesson 17 Lesson 18 Lesson 19 Lesson 20 Lesson 21 Lesson 22 Lesson 23 Lesson 2 Lesson 25 Lesson 26 Lesson 27 Lesson 28 Lesson 29 Lesson 30 Lesson 31 Lesson 32 Lesson 33 Lesson 3 Lesson 35 Lesson 36 Lesson 37 Lesson 38 Lesson 39 Lesson 0 Lesson 1 Lesson 2 Lesson 3 Lesson Lesson 5 Lesson 6 Lesson 7 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 and More 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 Parallelograms 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 (First Quarter test) External Angles of a Triangle Angles Formed With Parallel Lines

4 Table of Contents Lesson 8 Lesson 9 Lesson 50 Lesson 51 Lesson 52 Lesson 53 Lesson 5 Lesson 55 Lesson 56 Lesson 57 Lesson 58 Lesson 59 Lesson 60 Lesson 61 Lesson 62 Lesson 63 Lesson 6 Lesson 65 Lesson 66 Lesson 67 Lesson 68 Lesson 69 Lesson 70 Lesson 71 Lesson 72 Lesson 73 Lesson 7 Lesson 75 Lesson 76 Lesson 77 Lesson 78 Lesson 79 Lesson 80 Lesson 81 Lesson 82 Lesson 83 Lesson 8 Lesson 85 Lesson 86 Lesson 87 Lesson 88 Lesson 89 Lesson 90 Lesson 91 Lesson 92 Lesson 93 Lesson 9 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 (Second Quarter test) Finding the Center of Rotation More Double Reflections Angles of Incidence and Reflection Lines of Symmetry

5 Table of Contents Lesson 95 Lesson 96 Lesson 97 Lesson 98 Lesson 99 Lesson 100 Lesson 101 Lesson 102 Lesson 103 Lesson 10 Lesson 105 Lesson 106 Lesson 107 Lesson 108 Lesson 109 Lesson 110 Lesson 111 Lesson 112 Lesson 113 Lesson 11 Lesson 115 Lesson 116 Lesson 117 Lesson 118 Lesson 119 Lesson 120 Lesson 121 Lesson 122 Lesson 123 Lesson 12 Lesson 125 Lesson 126 Lesson 127 Lesson 128 Lesson 129 Lesson 130 Lesson 131 Lesson 132 Lesson 133 Lesson 13 Lesson 135 Lesson 136 Lesson 137 Lesson 138 Lesson 139 Lesson 10 Lesson 11 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 More Golden Goodies 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

6 Table of Contents Lesson 12 Lesson 13 Lesson 1 Lesson 15 Lesson 16 Lesson 17 Lesson 18 Lesson 19 Lesson 150 Lesson 151 Lesson 152 Lesson 153 Lesson 15 Lesson 155 Lesson 156 Lesson 157 Lesson 158 Lesson 159 Lesson 160 Lesson 161 Lesson 162 Lesson 163 Lesson 16 Lesson 165 Volume of Prisms 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 (Final test)

7 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, triangle, 5 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@rightstartmath.com

8 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@RightStartMath.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@rightstartmath.com 8/06

9 Vocabulary First Introduced Lesson 1 Lesson 2 Lesson 3 Lesson Lesson 5 Lesson 6 Lesson 10 Lesson 13 Lesson 1 Lesson 15 Lesson 16 Lesson 17 Lesson 18 Lesson 19 Lesson 21 Lesson 23 Lesson 2 Lesson 25 Lesson 26 Lesson 28 Lesson 30 Lesson 32 Lesson 3 Lesson 36 Lesson 38 Lesson 2 Lesson 3 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 50 Lesson 51 Lesson 52 Lesson 5 Lesson 55 Lesson 56 Lesson 57 Lesson 58 Lesson 59 Lesson 60 Lesson 62 Lesson 6 Lesson 65 Lesson 66 Lesson 67 line segments, parallel lines, intersection 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

10 Vocabulary First Introduced Lesson 68 Lesson 69 Lesson 70 Lesson 71 Lesson 72 Lesson 73 Lesson 7 Lesson 75 Lesson 76 Lesson 80 Lesson 81 Lesson 83 Lesson 86 Lesson 87 Lesson 88 Lesson 93 Lesson 9 Lesson 95 Lesson 97 Lesson 98 Lesson 99 Lesson 100 Lesson 101 Lesson 102 Lesson 103 Lesson 105 Lesson 108 Lesson 109 Lesson 110 Lesson 111 Lesson 112 Lesson 11 Lesson 115 Lesson 116 Lesson 118 Lesson 119 Lesson 120 Lesson 121 Lesson 123 Lesson 129 Lesson 131 Lesson 133 Lesson 135 Lesson 136 Lesson 137 clockwise, counterclockwise 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 Euler path trigonometry, opposite, adjacent, sine, cosine, tangent scientific calculator angle of elevation, stride, clinometer angle of depression sine wave

11 Vocabulary First Introduced Lesson 138 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 1 Lesson 15 Lesson 16 Lesson 19 Lesson 151 Lesson 152 Lesson 153 Lesson 15 Lesson 155 Lesson 157 Lesson 158 Lesson 159 Lesson 160 Lesson 161 Lesson 162 Lesson 163 Lesson 16 Lesson 165 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 quasiregular polyhedron

12 Lesson 2 A sharp pencil, an eraser, and tape are essentials. They will not be listed in future lessons. GOALS MATERIALS ACTIVITIES Drawing Diagonals 1. To review the terms horizontal and vertical 2. To learn the mathematical meaning of diagonal 3. To review the term hexagon. To find the correct edge of the triangle to draw diagonals Worksheet 2, Math Dictionary Drawing board, T-square, triangle Horizontal and vertical. Horizontal refers to the horizon, the intersection between the earth and sky. You can see it if there aren t too many buildings and trees in the way. Vertical refers to straight up 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 vertical line on paper goes from top to bottom, parallel to the sides of the paper. 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, the lines segments are still diagonals. Now diagonal DB is horizontal and diagonal AC is vertical. A A B diagonal diagonals D diagonal B Diagonal lines on a building. D C C In the word diagonal, dia means across and gon means angle. So, a diagonal is a line across angles, that is, a line connecting two vertices. 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 the sides of the hexagon and the diagonals using your tools. The horizontal and vertical lines need only a T-square. The left figure below is a hexagon; the right figure shows the diagonals.

13 Name 1. Use your T-square to draw horizontal lines in the octagon below. Be sure your T-square is snug against the edge of the drawing board. Date 2. Use your T-square and triangle to draw vertical lines in the hexagon. Be sure that your triangle is snug against the T-square. 3. Draw lines parallel to the sides of this equilateral triangle. Use your T-square and triangle to draw the lines. Hold your pencil about 2.5 cm from the tip ( ).. In which figure(s) have you drawn parallel octagon hexagon lines? 5. In which figure(s) have you drawn intersecting triangle lines? Worksheet 1, Getting Started Joan A. Cotter 2009 Name 1. First, trace the dotted lines forming the two hexagons. Use your T-square for drawing all lines. Use your triangle for all lines except horizontal lines. 2. Next, draw all the diagonals in the hexagons, using your drawing tools. There are 3 diagonals at each vertex. Date Include both hexagons: 3. How many diagonals are horizontal? 3 3. How many diagonals are vertical? How many diagonals at each vertex are either horizontal or vertical? 6. How many diagonals at each vertex are not horizontal or vertical? Worksheet 2, Drawing Diagonals Joan A. Cotter 2009

14 12 Lesson 9 Equilateral Triangle into Twelfths and More GOALS 1. To discover how to divide an equilateral triangle into congruent pieces greater than 9 2. To divide an equilateral triangle into twelfths 3. To divide an equilateral triangle into a number greater than 12 MATERIALS Worksheets 9-1, 9-2 Drawing board, T-square, triangle Colored pencil, optional ACTIVITIES Dividing a triangle into twelfths. How would you divide an equilateral triangle into twelfths 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. 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 32 equal parts. Some divisions are shown below. Sixteenths Eighteenths Eighteenths Triangle into 32nds by Joseph Hermodson-Olsen, 1. How could he have done it? The answer is at the bottom of the page. Twenty-fourths Twenty-sevenths Twenty-sevenths Worksheet 9-1. For this worksheet, you will divide the equilateral triangle into twelfths. Work carefully. For Problem 2, figure out how you would divide equilateral triangles into various congruent pieces. Worksheet 9-2. After drawing the equilateral triangle, divide it into congruent triangles. Either copy one of the designs above, or better yet, design your own. You might like to color your design. [Answer: ninths, fourths, fourths, and thirds.] Thirty-seconds

15 1. Draw an equilateral triangle. Divide it into fourths. Then divide each fourth into thirds, as shown. Name Date 2. Fill in the chart. Number of Pieces First Division Second Division [or 6] 3 9 Third Division 2 3 Worksheet 9-1, Equilateral Triangle into Twelfths and More Joan A. Cotter 2009 Name 3. Draw an equilateral triangle. Divide it into more than 12 equal parts. Date. Describe how you did it. [RESULTS WILL VARY.] Worksheet 9-2, Equilateral Triangle into Twelfths and More Joan A. Cotter 2009

16 32 Lesson 27 Comparing Areas of Rectangles GOALS 1. To calculate more areas of rectangles 2. To compare areas of rectangles with constant perimeter MATERIALS Worksheets 27-1, 27-2 Drawing board, T-square, triangle -in-1 ruler ACTIVITIES 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. 3 cm cm 5 cm 2 cm 1 cm 2 cm 3 cm cm 5 cm 1 cm This type of problem is easily solved with a branch of mathematics called calculus. The areas, which you can do in your head using A = wh, are from left to right, 5 cm 2, 8 cm 2, 9 cm 2, 8 cm 2, and 5 cm 2. Graphing the frame problem. It is interesting to graph the results as shown below. Why is the area equal to 0 when the width is equal to 0 or 6? cm 2 The area in Rectangle Areas with Perimeter = 12 cm The width of the rectangle in cm The shape of this graph is called a parabola. You can see the greatest area occurs when the width of the rectangle is to 3. What is the height when the width is 3? The answer is at the bottom of the page. Worksheets. There is a similar problem on Worksheets 27-1 and Draw the rectangles by measuring with your ruler like you did on Worksheet 11. [Answer: 3]

17 Name 1. Find the areas of the small groups of squares. Write the answer in the lower right square. Also write it in corresponding space in the large square. Then fill in the remaining spaces in the large square Date 2. What do you call the large square in Problem 1? Multiplication table 3. If a rectangle is 8 cm wide by 9 cm high, how many square centimeters do you need to cover it? 72 cm 2. If a rectangle is w cm wide and h cm high, how many square centimeters do you need to cover it? w h cm 2 Read Lesson 26 before answering the next question. 5. What is the area of the figure below? cm cm cm or or 16 cm 6 cm A = lg rect sm rect A = A = 1 cm 2 A = A = 1 cm 2 A = A = 1 cm 2 Worksheet 26, Area of a Rectangle Joan A. Cotter 2009 Name Date 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. 1. On each of the five lines below, draw a rectangle with a perimeter of 20 cm. Write the dimensions. 9cm 1cm A = wh A = 1 9 A = 9cm 2 8cm 2 2. Below each rectangle, calculate its area in cm. Which rectangle gives the most area? 5 cm by 5 cm 2 cm A = wh A = 2 8 7cm A = 16 cm 2 3cm A = wh A = 3 7 A = 21 cm 2 6cm cm A = wh A = 6 A = 2 cm 2 5cm 5cm A = wh A = 5 5 A = 25 cm 2 Worksheet 27-1, Comparing Areas of Rectangles Joan A. Cotter 2009

18 Name Date. 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 the graph. Then connect the points in a smooth curve; do this freehand (without any drawing tools). 2 Areas in cm Area of Rectangles with a Perimeter of 20 cm The base of the rectangle in cm parabola 5. What is the name of the shape of the curve? 6. According to the graph, what is the maximum area? 25 cm2 The square has the greatest area. 7. How does the graph compare with the example in the lesson? Worksheet 27-2, Comparing Areas of Rectangles Joan A. Cotter 2011 Name 1. For problems, A-C, crosshatch the top row of squares. Place the crosshatched squares on the right of the new figure. Complete the square with dashed lines. The steps are shown below. Date n n B. C. A. 2. Complete the table. A. B. C. n (n 1) (n + 1) Squares in the Original Fig. 2 = = 15 6 = = 7 9 = 10 8 = 11 9 = n 1 Squares in the New Fig = = = = = = = Write the results (or rule) you found in your own words. When multiplying two numbers that are two numbers apart, square the number between them and subtract one. Apply this result to find the following: = = = = = = 299 = n (n 1) (n + 1) Worksheet 28, Product of a Number and Two More Joan A. Cotter 2009

19 96 Lesson 8 Rotating GOALS 1. To learn the mathematical meaning of rotation 2. To construct rotations at various angles MATERIALS Worksheet 8 Goniometer A set of tangrams Drawing board, T-square, 5 triangle ACTIVITIES 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 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 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.

20 Worksheet 8, 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. Name Date The shows you where to start. Name Date cm Worksheet 83-2, Reflecting For each figure, flip horizontal and flip vertical about the center lines Joan A. Cotter cm 9. What angle of rotation is the same turning something 180 upside down? 10. Is a rotation of 180 the same as reflecting about a no horizontal line? Joan A. Cotter 2010