Math Makes Sense 9 PDF Textbook

PEARSON Wk ÊÊ M É Math Lorraine Baron Trevor Brown Garry Davis Sharon Jeroski Susan Ludwig Sandra Glanville Maurer Kanwal Neel Robert Sidley Shannon Sookochoff David Sufrin...

PEARSON Wk ÊÊ M É Math Lorraine Baron Trevor Brown Garry Davis Sharon Jeroski Susan Ludwig Sandra Glanville Maurer Kanwal Neel Robert Sidley Shannon Sookochoff David Sufrin David Van Bergeyk Jerrold Wiebe PEARSON Publisher The information and activijtiies presented in this Mike Czukar book have been carefully édited and reviewed. However, the publisher sh;il I not be liable for any Research and Communications Manager damages resulting, in who e or in part, from the Barbara Vogt reader's use of this materU I. Publishing Team Brand names that appear iin photographs of Enid Haley Claire Burnett products in this textbook are intended to provide Lesley Haynes Marina Djokic students with a sense of tf e real-world loana Gagea Ellen Davidson applications of mathemati :s and are in no way Lynne Gulliver Jane Schell intended to endorse specil ic products. Bronwyn Enright Karen Alley Alison Dale David Liu The publisher wishes to th ankthe staff and Judy Wilson students of St. John's High School, Winnipeg and Anderson Collegiate and Vocational Institute, Photo Research Whitby, for their assistance; with photography. Lisa Brant Statistics Canada informati on is used with the Design and Art Direction permission of Statistics Capada. Users are Word & Image Design Studio Inc. forbidden to copy the dat and redisseminate them in an original or moc ified form, for Composition commercial purposes, with out permission from Lapiz Digital Services, India Statistics Canada. Informat on on the availability of the wide range of data froiijn Statistics Canada can Copyright © 2009 Pearson Education Canada, be obtained from Statistic: Canada's Regional a division of Pearson Canada Inc. All Rights Reserved. Offices or the Statistics Cariiada Web site. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission, write to the Permissions Department, ISBN-13 978-0-321-49558-7 ISBN-10 0-321-49558-6 Printed and bound in the United States. 3 4 5 CC5 13 12 11 10 09 PEARSON Consultants, Advisers, and Reviewers w Series Consultants Trevor Brown Maggie Martin Connell Mignonne Wood Craig Featherstone I m John A. Van de Walle I Assessment Consultant Sharon Jeroski Aboriginal Content Consultant Sonya Ellison, Frontier School Division, MB Advisers and Reviewers Pearson Education thanks its advisers and reviewers, who helped shape the vision for Pearson Math Makes Sense through discussions and reviews of prototype materials and manuscript. Thanks to all our Previewers and the Test and Try schools, acknowledged on our Math Makes Sense Web site. A special thank you to the staff and grade 9 students at Holy Cross High School in Saskatoon and St. Michaels University School in Victoria for their recommendations for improvements to this resource prior to publication. Alberta Bob Berglind Lauri Goudreault Calgary Board of Education Holy Family Catholic Regional District #37 Jacquie Bouck Kyle Honish Lloydminster Public School Division 99 Calgary Board of Education Theresa S.Chalifoux Mary-Elizabeth Kaiser Edmonton Catholic Schools Calgary Board of Education Carolyne Chipiuk Ted Mclnnis Edmonton Public School Board Rocky View Schools Lissa D'Amour Leslie McRae Calgary Board of Education Medicine Hat School District 76 Ken Pistotnik Margo Fosti Calgary Board of Education Edmonton Catholic Schools Florence Glanfield Delcy Rolheiser University of Alberta Edmonton Public School Board III Trevor Rosenfeldt Andrew Hirst Edmonton Public School Board River EastTranscona Sqhool Division Catherine A. Savin Christine Ottawa Calgary Board of Education Mathematics Consultant, Winnipeg Andrea Suley Gretha Pallen Calgary Board of Education Formerly Manitoba Education Jeffrey Tang Calgary R.C.S.S.D. 1 Doug Roach Fort La Bosse School Division Lisa Walpole Suzanne Sullivan Calgary Separate School Division Brandon School Divisioh Sandra Smith Winnipeg British Columbia Lorraine Baron Saskatchewan School District 23 (Central Okanagan) Michelle Dament Whitney Christy Prairie Spirit School Div sion School District 28 (Quesnel) Edward Doolittle Dean Coder First Nations University of Canada School District 73 (Kamloops/Thompson) Lisa Eberharter Carol M.Funk Greater Saskatoon Catholic Schools School District 68 (Nanaimo-Ladysmith) Andrea Gareau Saskatchewan Rivers Sciool Division Daniel Kamin School District 39 (Vancouver) Shelda Hanlan Stroh Andrew Shobridge Greater Saskatoon Catholic Schools School District 44 (North Vancouver) Kevin Harbidge Chris Van Bergeyk Greater Saskatoon Catholic Schools School District 23 (Central Okanagan) Lisa Hodson Denise Davis Greater Saskatoon Catholic Schools School District 41 (Burnaby) Barbara Holzer Matthew Wang Prairie South School Division #210 School District 37 (Delta) Faron Hrynewich Prairie Spirit School Dist ict Mark Jensen Manitoba North East School Division Lori Bjorklund Lisa Kobelsky Mountain View School Division Greater Saskatoon Catholic Schools Lori Edwards TanisWood Huber Regina Public Schools Assiniboia School Division Jason Hawkins River EastTranscona School Division iv  Project: Making Squares into Cubes p | Square Roots and Surface Area Launch 4 1.1 Square Roots of Perfect Squares 6 1.2 Square Roots of Non-Perfect Squares 14 Mid-Unit Review 21 Start Where You Are: How Can I Begin? 22 Game: Making a Larger Square f r o m Two Smaller Squares 24 1.3 Surface Areas of Objects Made from Right Rectangular Prisms 25 1.4 Surface Areas of Other Composite Objects 33 Study Guide 44 Review 45 Practice Test 48 Unit Problem: Design a Play Structure 49 Powers and Exponent Laws Launch 50 2.1 What Is a Power? 52 2.2 Powers of Ten and the Zero Exponent 58 2.3 Order of Operations w i t h Powers 63 Mid-Unit Review 69 Start Where You Are: What Strategy Could 1 Try? 70 Game: Operation Target Practice 72 2.4 Exponent Laws 1 73 2.5 Exponent Laws II 79 Study Guide 86 Review 87 Practice Test 90 Unit Problem: How Thick Is a Pile of Paper? 91 UNIT 2 Rational Numbers Launch 92 3.1 What Is a Rational Number? 94 Start Where You Are: How Can I Learn from Others? 104 3.2 Adding Rational Numbers 106 3.3 Subtracting Rational Numbers 114 Mid-Unit Review 121 Game: Closest to Zero 122 3.4 Multiplying Rational Numbers 123 3.5 Dividing Rational Numbers 130 3.6 Order of Operations w i t h Rational Numbers 137 Study Guide 143 Review 144 Practice Test 146 147 Unit Problem: Investigating Temperature Data 148 Cumulative Review Units 1-3 Linear Relations Launch 150 Start Where You Are: How Can I Explain My Thinking? 152 4.1 Writing Equations to Describe Patterns 154 Technology: Tables of Values and Graphing 163 4.2 Linear Relations 164 4.3 Another Form of the Equation for a Linear Relation 174 Mid-Unit Review 181 Game: What's My Point? 182 4.4 Matching Equations and Graphs 183 4.5 Using Graphs t o Estimate Values 191 Technology: Interpolating and Extrapolating 199 Study Guide 200 Review 201 Practice Test 204 205 Unit Problem: Predicting Music Trends Project: Number Systems 206 C Polynomials J Launch 208 5.1 Modelling Polynomials 210 5.2 Like Terms and Unlike Terms 217 5.3 Adding Polynomials 225 5.4 Subtracting Polynomials 231 Mid-Unit Review 237 Start Where You Are: How Can I Summarize What I Have Learned? 238 Game: Investigating Polynomials that Generate Prime Numbers 240 5.5 Multiplying and Dividing a Polynomial by a Constant 241 5.6 Multiplying and Dividing a Polynomial by a Monomial 249 Study Guide 258 Review 259 Practice Test 262 Unit Problem: Algebra Patterns on a 100-Chart 263 Linear Equations and Inequalities Launch 264 6.1 Solving Equations by Using Inverse Operations 266 6.2 Solving Equations by Using Balance Strategies 275 Start Where You Are: How Can I Use My Problem-Solving Skills? 284 Mid-Unit Review 286 Game: Equation Persuasion 287 6.3 Introduction to Linear Inequalities 288 6.4 Solving Linear Inequalities by Using Addition and Subtraction 294 6.5 Solving Linear Inequalities by Using Multiplication and Division 300 Study Guide 307 Review 308 Practice Test 310 311 Unit Problem: Raising Money for the Pep Club 312 Cumulative Review Units 1 - 6 vii  Similarity and Transformations Launch 314 Start Where You Are: What Should I Recall? 316 7.1 Scale Diagrams and Enlargements 318 7.2 Scale Diagrams and Reductions 325 Technology: Drawing Scale Diagrams 332 7.3 Similar Polygons 334 7.4 Similar Triangles 343 Mid-Unit Review 352 7.5 Reflections and Line Symmetry 353 Game: Make Your Own Kaleidoscope 360 7.6 Rotations and Rotational Symmetry 361 7.7 Identifying Types of Symmetry on the Cartesian Plane 368 Study Guide 376 Review 377 Practice Test 380 Unit Problem: Designing a Flag 381 Launch 382 8.1 Properties of Tangents to a Circle 384 8.2 Properties of Chords in a Circle 392 Technology: Verifying the Tangent and Chord Properties 400 Game: Seven Counters 402 Mid-Unit Review 403 8.3 Properties of Angles in a Circle 404 Technology: Verifying the Angle Properties 413 Start Where You Are: How Do I Best Learn Math? 415 Study Guide 417 Review 418 Practice Test 420 Unit Problem: Circle Designs 421 viii (J Probability and Statistics Launch 422 9.1 Probability in Society 424 Game: Cube Master 430 9.2 Potential Problems w i t h Collecting Data 431 9.3 Using Samples and Populations to Collect Data 437 Technology: Using Census at School 442 Mid-Unit Review 444 9.4 Selecting a Sample 445 Technology: Using Spreadsheets and Graphs to Display Data 450 Start Where You Are: How Can I Assess My Work? 452 9.5 Designing a Project Plan 454 Study Guide 457 Review 458 Practice Test 460 Unit Problem: What Can You Discover about the World around You? 461 Project: Constructing a Math Quilt 462 Cumulative Review Units 1 - 9 464 Answers 468 Illustrated Glossary 541 Index 549 Acknowledgments 553  Welcome to Pearson Math Makes Sense 9 M a t h helps y o u understand your world. This b o o k w i l l help y o u improve your problem- solving skills and show y o u h o w y o u can use your m a t h now, and i n your future career. The opening pages o f e a c h u n i t are designed to help y o u prepare for success. Square R( Surface A you'll Learn t a|*tne the sauare roots of , W h y It's : Important i\ & i e squar^oots of Heat-workf measure tUtace expressed as fraction « We use th® i c o b of «hoe H 4 b m when we w o i M A formulas suth a& w * 3-D objects to setve problem Pythagoiesn T h e o r y. the amount of paper n e ^ f l to wrap a g i f t the number of cans of paint needed to paint a room; and the amount of siding needW to covw a building Find out W h a t You'll Learn and W h y It's I m p o r t a n t. Check the list of Key Words.  Square Roots of Non-Porfect Squares UAfcr i» lc*iiia§ WrtWIiOT»»»™ 1(6 Start W h e r e You Are illustrates strategies you may use to show your best performance. Study Guide UH* Otagnms Study Guide for as crrfatçnitent ot nitm An enlargement hu » scale » summarizes key ideas Similar Pshrtoa Similar poiyjjjiQs are related I a icdactum. When two poivgom are s«dfer: Measure ihe photo What arc the from the unit. » their corresponding angles an: «jtiafe AA * «ift £.& « F t -* _ and ~ is the xfih. factor. i) W'hai is tlit scalefactor for wis 1 reduction» » Astandatd-siK ool ette is about 144 cm I S The Practice and lifts 8f1R«ttefrV long. What is thlength of j modri ot ( hn,^ A shape has Une symmetry when a It* h the «aiefactorfrom j divides the shape into two congnwotfwrts so thai oik pan. is the image of the ofcr part ^7"~" after a reflection in the tine of symraarf, I. [J 4. iiere he^ih!»ofscal theeramp amofararop. The dîagits Mea60° about its centre. The number ot tint» the dupe arincite with itsdl' LC lesigibofthe ramp ] 1 provides additional support. à the order of rotation. «# of "»*** ™«*»r " VutâfcxKi 376 < xiii  Raising Money for the Pep CltA i i ! 1. The Pep Oub caw i»iy i w uniform» fioin 2 differed! suppliers Company A charge* plus $22 per uniform. Cwrapiuiy « charges SAW, j%lus $28 pes œnifor». S. a) Write a fioipviriiwi for she perimeter ot this shape. Sinipiily she pohnomij!. a) Define a vsii«l>le> the" write ao equation that c^n be used todeieronne the number of uniforms that wilt rewftwiCTpuicosts at both K 'V>K'c the equation Wirify rhi' solution. tl WHieh company jliouki îhe 1'cfi Club choose? Juwity your reammendation. 8. a' SVctth the rixtargïe aod (abet it vrith its dimensions, Hww » solving a linear inequality iikesoi'itig a Knear equation» bj What i» the a«s >rf the reciiJnglc? How i» il di lièrent? h»iw. on Monday nsorninj;. j How muchfendngis needed to enckw IT. A wheatfieldis 10 iSt> m wide. The ares j the g,irik-n? How d» vin know?. decin»is that have stfuarc a) Use the exponentfciwsto determine the length of Ihe field. } 7, Use arty tflrah^y you wish w estimate the 1t. WriBr ciu'h ptudutt as ,t pou-er, tlteti craitu le. OiXiKl B) What u the perimeter ni the field? Did you use any exjumetn lavrs to KM) ; vainc of eadi rooi. cak'tiljte tâe perimeter? titpiain. 106 X 6 X (, X i ! tijl nJf « VUS d) «M -.';i ?)(-3)- Build a cube f r o m 1-cm grid card stock. Each face o f your cube should be the square you drew i n Part 2. Use tape to assemble your cube. > Draw a net of your cube. > Calculate the surface area of your cube. Show your calculations. > Describe how you w o u l d calculate the volume o f your cube. >- Is the volume of your cube less than or greater than the volume o f a cube w i t h edge length 6 cm? Explain. Take It Further V Draw squares w i t h these areas: 5 cm 2 ,20 cm 2 ,45 cm 2 ,80 cm 2 , and 125 cm 2 H o w are these squares related? What makes these squares a "family"? >- What other families o f squares could you draw? Draw 3 squares f r o m that family. Describe the family o f squares. Determine the side length o f each square. Making Squares into Cubes  p r a s Square Roots an Surface Area Which geometric objects can you name? How could you det >rmine their surface aieas' What You'll Learn Delfermine the square roots of Why It's fracti Dns and decimals that are perfect squares. Important Approximate the square roots of Real-world treasures arp often fractions aril decimals that are expressed as fractions or decimal non-perfect squares. We use the square roots of ' » Determine the surface areas of composite when we work with formulas si 3-D objects to solve problems. Pythagorean Theorem. An understanding of surface area to solve practical problems such as calculating: the amount of paper needed to wrap a gift; the number of cans of paint needed to paint a room; and the amount of siding needed to cover a building 4  A children's playground is a square w i t h area 400 m 2. What is the side length of the square? H o w much fencing is needed to go around the playground? Investigate Each square below has been divided 10 into 100 equal parts. I n each diagram, what is the area of one small square? 1 For the shaded square on the left: > What is its area? > - Write this area as a product. >- H o w can you use a square root to relate the side length and area? For the shaded square on the right: >- What is its area? > Write this area as a product o f fractions. >- H o w can you use a square root to relate the side length and area? 6 UNIT 1: Square Roots and Surface Area For the area o f each square i n the table: > Write the area as a product. > - Write the side length as a square root. Area aslft Product Side Length as a Square Root 49 = JIL - 100 ~ 04 = 64 _ 100 121 = 121 - 100 144 = 144 _ 100 Compare your results w i t h those o f your classmates. H o w can y o u use the square roots o f whole numbers to determine the square roots o f fractions? Suppose each fraction i n the table is w r i t t e n as a decimal. H o w can y o u use the square roots o f whole numbers to determine the square roots o f decimals? Connect To determine the area o f a square, we m u l t i p l y the side length by itself. That is, we square the side length. Area = I j j 15 15 _ 10 x 10 _ 225 _ 100 225 The area is — square units. 15.. units 1.1 Square Roots of Perfect Squares To determine the side length of a square, we calculate 169 Area: — square units the square root of its area. Side length = = 13x13 V10 10 _ 13 ~ 10 13 The side length is — units Squaring and taking the square root are opposite, or inverse, operations. The side length of a square is the square root of its area.. [225 15 j fl69 13 Thatls = and = ' \lm m ^m w We can rewrite these equations using decimals: 1.5 and 1.3 are terminating decimals. \ f l 2 5 = 1.5 and VÎL69 = 1.3 The square roots of some fractions are repeating decimals. To determine the side length of the shaded square, take the square root of 9* 1 unit 3 3 To find the square root of g, _ 1 ~ 3 - unit \ I look for a number that when = 0.333 333 333... multiplied by itself gives Area: „y square units = 0.3 When the area of a square is - square units, its side length is or 0.3 of a unit. A fraction i n simplest f o r m is a perfect square i f it can be written as a product of two equal fractions. When a decimal can be written as a fraction that is a perfect square, then the decimal is also a perfect square. The square root is a terminating or repeating decimal. 8 UNIT 1: Square Roots and Surface Area  Determining a Perfect Square Given its Square Root Calculate the number whose square root is: a) ! b) 1.8 A Solution 3 3 a) Visualize - as the side length of a square. - units The area o f the square is: ( - I = - x 3.. -units 9_ O 64 3 9 So, - is a square root of —. o 64 — — square units b4 b) Visualize 1.8 as the side length of a square. The area of the square is: 1.82 = 1.8 X 1.8 = 3.24 - - 1. 8 units So, 1.8 is a square root of 3.24. Identifying Fractions that Are Perfect Squares Is each fraction a perfect square? Explain your reasoning, a) 18 "f M Solution 18 Simplify the fraction first. Divide the numerator and denominator by 2. _8_ _ 4 _ 18 9 Since 4 = 2 X 2 and 9 = 3 X 3, we can write: — = —x — 9 3 3 Since | can be written as a product of two equal fractions, it is a perfect square. So, ^ is also a perfect square. 1.1 Square Roots of Perfect Squares 9  The fraction is i n simplest form. So, look for a fraction that when multiplied by itself gives —. The numerator can be written as 16 = 4 X 4, but the denominator cannot be written as a product of equal factors. So, -^r is not a perfect square. 2 9 The fraction is i n simplest form. So, look for a fraction that when multiplied by itself gives The denominator can be written as 9 = 3 X 3, but the numerator cannot be written as a product of equal factors. So, ^ is not a perfect square. m r n m a. — g Decimals «hat Arc Perfect Spares Is each decimal a perfect square? Explain your reasoning. a) 6.25 b) 0.627 Solutions Method 1 Method 2 a) Write 6.25 as a fraction. Use a calculator. Use the square root function. a) V625 = 2.5 Simplify the fraction. Divide the numerator The square root is a terminating decimal, and denominator by 25. so 6.25 is a perfect square. 25 5 U s, 5 — can be written as - x -. So, — , or 6.25 is a perfect square, b) Write 0.627 as a fraction. b) Vo.627 = 0.791 833 316 The square root appears to be a decimal that neither terminates nor repeats, so This fraction is i n simplest form. 0.627 is not a perfect square. To be sure, Neither 627 nor 1000 can be written as a write the decimal as a fraction, then product of equal factors, so 0.627 is not a determine i f the fraction is a perfect perfect square. square, as shown i n Method 1. 10 UNIT 1: Square Roots and Surface Area  1. H o w can y o u tell i f a decimal is a perfect square? 2. H o w can y o u tell i f a fraction is a perfect square? Practice Check 7. Use your answers to questions 4 and 6. Determine the value o f each 3. Use each diagram to determine the value o f square root. the square root. a). /169 /400 yjie b) V 196 a) VÔ25 1 unit. 1256 d) /225 C) V 361 Y 289 b >^ 1 unit e) V Ï 4 4 f) Vo.0225 g) V0.0121 h) V3.24. /16 C) ^25 i) Vo.0324 j) Vo.0169 1 unit Apply 8. W h i c h decimals and fractions are perfect 4. a) List all the whole numbers f r o m squares? Explain your reasoning. 1 to 100 that are perfect squares, a) 0.12 b) 0.81 c) 0.25 b) Write a square root o f each number y o u »£ d) 1.69 f) - ' 81 listed i n part a. 81 9) 49 i) 0.081 5. Use your answers to question 4. 25 j) k) 2.5 > 50 Determine the value o f each square root. 10 a) VÔ36 b) VÔI49 9. Calculate the number whose square root is: c) V 0 8 1 d) V 0 J 6 a) 0.3 b) 0.12 e) ^ f) c) 1.9 d) 3.1 f) 5 e ) ! 6 h) 2 9) h) 6. a) List all the whole numbers f r o m 101 10. Determine the value o f each square root, to 400 that are perfect squares. a) V l 2. 2 5 b) V30.25 b) Write a square root o f each number y o u listed i n part a. c) V20.25 d) V56.25 1.1 Square Roots of Perfect Squares 11 11. a) Write each decimal as a fraction. b) Sketch the number line i n part a. Write W h i c h fractions are perfect squares? 3 different decimals, then use the letters i) 36.0 ii) 3.6 iii) 0.36 G, H , and J to represent their square iv) 0.036 v) 0.0036 vi) 0.000 36 roots. Place each letter o n the number b) To check your answers to part a, use a line. Justify its placement. calculator to determine a square root o f each decimal. 14. A square has area 5.76 cm 2. c) W h a t patterns do y o u see i n your answers a) W h a t is the side length o f the square? to parts a and b? b) W h a t is the perimeter o f the square? d) W h e n can y o u use the square roots o f H o w do y o u know? perfect squares to determine the square 15. A square piece o f land has an area not less roots o f decimals? than 6.25 k m 2 and not greater than 10.24 km 2. a) W h a t is the least possible side length o f 12. a) Use the fact that V9 = 3 to write the the square? value o f each square root. b) W h a t is the greatest possible side length i) V9Ô000 ii) V9ÔÔ o f the square? iii) V a 0 9 iv) V0.0009 c) A surveyor determined that the side b) Use the fact that V25 = 5 to write the length is 2.8 k m. W h a t is the area o f value o f each square root. the square? i) VoÔÔ25 ii) VÔ25 iii) V2500 iv) V250 000 c) Use the patterns i n parts a and b. Choose a whole number whose square root y o u know. Use that number and its square root to write 3 decimals and their square roots. H o w do y o u k n o w the square roots are correct? a) W h i c h letter o n the number line below corresponds to each square root? Justify your answers. i) V I T 2 5 ii) iii) V l 6. 8 1 16. A student said that V0.04 = 0.02. Is the student correct? iv) ^ v) vi) I f your answer is yes, h o w could y o u check that the square root is correct? F B A C E D I f your answer is no, what is the correct I -I h 1 — H 1 H- 0 1 2 3 4 5 6 square root? Justify your answer. 12 UNIT 1: Square Roots and Surface Area 17. Look at the perfect squares you wrote for 19. A student has a rectangular piece of paper questions 4 and 6. 7.2 cm by 1.8 cm. She cuts the paper into The numbers 36, 64, and 100 are related: parts that can be rearranged and taped to 2 2 2 36 + 64 = 100, or 6 + 8 = 10 f o r m a square. These numbers f o r m a a) What is the side length of the square? Pythagorean triple. b) What are the fewest cuts the student a) W h y do you t h i n k this name is could have made? Justify your answer. appropriate? b) H o w many other Pythagorean triples can you find? List each triple. Take It Further 18. Are there any perfect squares between 0.64 and 0.81? Justify your answer. Explain the term perfect square. List some whole numbers, fractions, and decimals that are perfect squares. Determine a square root of each number. History The Pythagorean Theorem is named for the Greek philosopher, Pythagoras, because he was the first person to record a proof for the theorem, around 540 BCE. However, clay tablets from around 1700 BCE show that the Babylonians knew how to calculate the length of the diagonal of a square. And, around 2000 BCE, it is believed that the Egyptians may have used a knotted rope that formed a triangle with side lengths 3,4, and 5 to help design the pyramids. \ 1.1 Square Roots of Perfect Squares 13  % k/- Square Roots of Non-Perfect Squares i it A ladder is leaning against a wall. For safety, the distance f r o m the base o f a ladder to the wall must be about ^ o f the height up the wall. FOCUS H o w could y o u check i f the ladder is safe? Approximate the square roots of 9 m fil decimals and fractions that are non-perfect squares. 2m Investigate A ladder is 6.1 m long. The distance f r o m the base o f the ladder to the wall is 1.5 m. Estimate h o w far up the wall the ladder w i l l reach. Compare your strategy for estimating the height w i t h that o f another pair o f classmates. D i d y o u use a scale drawing? D i d y o u calculate? W h i c h m e t h o d gives the closer estimate? Connect M a n y fractions and decimals are n o t perfect squares. That is, they cannot be w r i t t e n as a product o f two equal fractions. A fraction or decimal that is n o t a perfect square is called a n o n - p e r f e c t square. Here are two strategies for estimating a square root o f a decimal that is a non-perfect square. 14 UNIT 1: Square Roots and Surface Area > - Using benchmarks, To estimate \FÏ.5, visualize a number line and the closest perfect square on each side of 7.5. Vi = 2 and V9 = 3 7.5 is closer to 9 than to 4, so \ f l. 5 is closer to 3 than to 2. From the diagram, an approximate value for \ f ? 3 is 2.7. We write V7^5 = 2.7 >- Using a calculator = 2.738 612 788 This decimal does not appear to terminate or repeat. There may be many more numbers after the decimal point that cannot be displayed on the calculator. To check, determine: 2.738 612 7882 = 7.500 000 003 Since this number is not equal to 7.5, the square root is an approximation. Example 1 illustrates 4 different strategies for determining the square root of a fraction that is a non-perfect square. MiUJMM Estimating a Square Bool of a Fraction Determine an approximate value of each square root. M Solution a) Use benchmarks. T h i n k about the perfect squares closest to the numerator and g denominator. I n the fraction - , 8 is close to the perfect square 9, and 5 is close to the perfect square 4. 1.2 Square Roots of Non-Perfect Squares 15  b) W r i t e the fraction as a decimal, then t h i n k about benchmarks. 3 W r i t e — as a decimal: 0.3 T h i n k o f the closest perfect squares o n either side o f 0.3. VÔ25 = 0.5 and VÔ36 = 0.6 0.2 0.25 0.3 0.36 0.4 Y Y Y 0.5 ? 0.6 0.3 is approximately halfway between 0.25 and 0.36, so choose 0.55 as a possible estimate for a square root. To check, evaluate: 0.55 2 = 0.3025 0.3025 is close to 0.3, so 0.55 is a reasonable estimate. 3 c) Choose a fraction close to - that is easier to w o r k w i t h. 3 7 0 3 1 - is a little less than VÔ5 = V o 4 9 A n d , V o 4 9 = 0.7 d) Use the square root f u n c t i o n o n a calculator. To the nearest hundredth, =1.78 16 UNIT 1: Square Roots and Surface Area MSBSBBBk Finding a Number with a Square Root between Two Given Numbers Identify a decimal that has a square root between 10 and 11. Check the answer. Solutions Method 1 Method 2 The number w i t h a square root of 10 is: One decimal between 10 and 11 is 10.4. - Use the tiles to make as many different-sized larger squares as you can. Write the area of each square as a product. Record your results in a table. Number of Tiles Area (square units) Side Length (units) Area as a fjroduct 1 1x1 >- Use the cubes to make as many different-sized larger cubes as you can. Write the volume of each cube as a product. Record your results in a table. Number of tubes ju Volume (cubic units) Edge Length (units) Volume as a product 1x1x1 What patterns do you see in the tables? Use the patterns to predict the areas of the next 3 squares and the volumes of the next 3 cubes. How are these areas and volumes the same? How are they different? 52 UNIT 2: Powers and Exponent Laws  Connect When an integer, other than 0, can be written as a product of equal factors, we can write the integer as a power. For example, 5 X 5 X 5 is 5 3. 5 is the base. exponent 3 is the exponent. ( c 3 base 5 3 is the power. power 3 5 is a power of 5. We say: 5 to the 3rd, or 5 cubed >- A power with an integer base and exponent 2 is a square number. When the base is a positive integer, we can illustrate a square number. Here are 3 ways to write 25. Standard form: 25 5 x 5 = 52 = 25 As repeated multiplication: 5 X 5 25 is a square As a power: 5 2 number. > A power with an integer base and exponent 3 is a cube number. When the base is a positive integer, we can illustrate a cube number. 5 5 Here are 3 ways to write 125. 5 x 5 x 5 = 53 Standard form: 125 = 125 As repeated multiplication: 5 X 5 X 5 5 3 125 is a cube As a power: 5 number. Writing Powers Write as a power, a) 3 X 3 X 3 X 3 X 3 X 3 b)7 /I Solution a) 3 X 3 X 3 X 3 X 3 X 3 The base is 3. There are 6 equal factors, so the exponent is 6. So, 3 X 3 X 3 X 3 X 3 X 3 = 36 b) 7 The base is 7. There is only 1 factor, so the exponent is 1. So, 7 = 71 2.1 W h a t Is a Power? 53  Evaluating Powers Write as repeated multiplication and in standard form, a) 3 5 b) 7 4 M Solution a) 3 5 = 3 X 3 X 3 X 3 X 3 As repeated multiplication = 243 Standard form 4 b) 7 = 7 X 7 X 7 X 7 As repeated multiplication = 2401 Standard form Examples 1 and 2 showed powers with positive integer bases. A power can also be negative or have a base that is a negative integer. M M M i M Evaluating Expressions Involving Negative Signs Identify the base of each power, then evaluate the power, a) (—3)4 b) - 3 4 c) —(—34) A Solution a) The base of the power is —3. ( —3)4 = ( - 3 ) X ( - 3 ) X ( - 3 ) X ( - 3 ) As repeated multiplication Apply the rules for multiplying integers: The sign of a product with an even number of negative factors is positive. So, (—3)4 = 81 Standard form b) The base of the power is 3. The exponent applies only to the base 3, and not to the negative sign. - 3 4 = — (34) = - ( 3 X 3 X 3 X 3) = -81 c) From part b, we know that — 3 4 = —81. So, - ( - 3 4 ) = - ( - 8 1 ) — (—81) is the opposite of —81, which is 81. = 81 54 UNIT 2: Powers a n d Exponent Laws We may write the product of integer factors without the multiplication sign. In Example 3a, we may write (—3) X ( — 3) X ( — 3) X (—3) as ( — 3)(—3)(—3)( — 3). A calculator can be used to evaluate a power such as (—7)5 r -n ~ r _ in standard form. - IJ D = i c p m - l O O U I. 1. Can every integer, other than 0, be written as a power? Explain. 2. Why is — 3 4 negative but (—3)4 positive? Give another example like this. 3. Two students compared the calculator key sequences they used to evaluate a power. Why might the sequences be different? Practice Check 7. Write the base of each power, a) 2 7 b) 4 3 4. Write the number of unit squares in each 2 c) 8 d) ( - 1 0 ) 5 large square as a power. e) ( - 6 ) 7 f) - 8 3 a) b) 8. Write the exponent of each power, a) 2 5 b) 6 4 c) 9 1 d) — 3 2 9 e) ( - 2 ) f) (—8) 3 5. Write the number of unit cubes in each 9. Write each power as repeated large cube as a power, multiplication. a) b) c) a) 3 2 b) 10 4 s c) 8 d) (—6) 5 e) — 6 5 f) - 4 2 10. a) Explain how to build models to show the 6. Use grid paper. Draw a picture to represent difference between 32 and 2 3. each square number. b) Why is one number called a square a) 4 2 b) 6 X 6 c) 49 number and the other number called d) 10 2 e) 81 f) 12 X 12 a cube number? 2.1 W h a t Is a Power? 55 11. Use repeated multiplication to show why 6 4 a) Captain George Vancouver was a Dutch 6 is not the same as 4. explorer who named almost 400 Canadian places. To commemorate his 12. Write as a power. 250th birthday in 2007, Canada Post a) 4 X 4 X 4 X 4 created a $1.55 stamp. b) 2 X 2 X 2 i) How many stamps are in a 3 by 3 c) 5 X 5 X 5 X 5 X 5 X 5 block? Write the number of stamps d) 10 X 10 X 10 j as a power. e) ( - 7 9 X - 7 9 ) ii) What is the value of these stamps? f) - ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) b) In July 2007, Canada hosted the FIFA U-20 World Cup Soccer Championships. Apply Canada Post issued a 52- The expression must have at least one power. The base of the power can be a positive or negative integer. > The expression can use any of: addition, subtraction, multiplication, division, and brackets Evaluate the expression. Write and evaluate as many different expressions as you can. Share your expressions with another pair of students. f p a Where does evaluating a power fit in the order of operations? Why do you think this is? 2.3 Order of O p e r a t i o n s w i t h Powers  Connect To avoid getting different answers when we evaluate an expression, we use this order of operations: Evaluate the expression in brackets first. Evaluate the powers. Multiply and divide, in order, from left to right. Add and subtract, in order, from left to right. Adding and Subtracting with Powers Evaluate. a) 3 3 + 2 3 b) 3 - 2 3 c) (3 + 2) 3 A Solution a) Evaluate the powers before adding. b) Evaluate the power, then subtract. 3 3 3 + 2 = (3)(3)(3) + (2)(2)(2) 3 - 23 = 3 - (2)(2)(2) = 27 + 8 = 3 - 8 = 35 = -5 c) Add first, since this operation is within the brackets. Then evaluate the power. (3 + 2 ) 3 = 5 3 = (5)(5)(5) = 125 When we need curved brackets for integers, we use square brackets to show the order of operations. When the numbers are too large to use mental math, we use a calculator. /'/ym Multiplying and Dividing with Powers Evaluate. a) [2 X ( —3) 3 — 6 ] 2 b) (18 2 + 5°) 2 -5- ( —5) 3 A Solution a) Follow the order of operations. Do the operations in brackets first: evaluate the power (—3)3 [2 X ( —3)3 - 6] 2 = [2 X ( - 2 7 ) - 6] 2 Then multiply: 2 X ( - 2 7 ) 2 = [ —54 — 6] Then subtract: —54 — 6 = (—60)2 Then evaluate the power: (—60)2 = 3600 64 UNIT 2: Powers a n d Exponent Laws  b) Use a calculator to evaluate (182 + 5°)2 ( — 5)3. For the first bracket: 182 + l Use mental math when you can: 5° = 1 3c?5. Evaluate 182 + 1 to display 325. Evaluate 3252 to display 105 625. i n 5S25. For the second bracket: ( —5)3 is negative, so simply evaluate 5 3 to display 125 5"3 To evaluate 105 625 + ( — 125), the integers have 155. opposite signs, so the quotient is negative. Evaluate 105 625 -h 125 to display 845. 105655/155 So, (182 + 5°)2 -h (—5)3 = - 8 4 5 8H5. w m m Solving Problems Using Powers Lyn has a square swimming pool, 2 m deep with side length 4 m. The swimming pool is joined to a circular hot tub, 1 m deep with diameter 2 m. Lyn adds 690 g of chlorine to the pool and hot tub each week. This expression represents how much chlorine is present per 1 m 3 of water: 690 2 X 4 + tt X l 3 2 The suggested concentration of chlorine is 20 g/m 3 of water. What is the concentration of chlorine in Lyn's pool and hot tub? Is it close to the suggested concentration? A Solution Use a calculator. Since the denominator has a sum, draw brackets around it. This ensures the entire denominator is divided into the numerator. Key in the expression as it now appears: — — 3 , — 19.634 85 ( 2 X 4 + 7T X 1 j The concentration is about 19.6 g/m 3. This is very close to the suggested concentration. 1. Explain why the answers to 3 3 + 2 3 and (3 + 2) 3 are different. 2. Use the meaning of a power to explain why powers are evaluated before multiplication and division. 2.3 O r d e r of O p e r a t i o n s w i t h Powers 65  Practice Check 8. State which operation you will do first, 3. Evaluate, then evaluate. a) ( 7 ) ( 4 ) - (5) 2 b) 6(2 - 5) 2 a) 3 2 + 1 b) 3 2 - 1 c) ( —3) 2 + ( 4 ) ( 7 ) d) ( - 6 ) + 4° X ( - 2 ) c) (3 + l ) 2 d) (3 - l)2 2 2 e) 10 + [10 - (—2)] f) [18 -f- (—6)] 3 X 2 e) 2 2 + 4 f) 2 2 - 4 g) (2 + 4 ) 2 h) (2 - 4 ) 2 9. Sometimes it is helpful to use an acronym i) 2 - 4 2 j) 2 2 - 4 2.. _ An acronym is as a memory trick. Create an acronym to a word formed help you remember the order of operations, from the first 4. Evaluate. Check using a calculator. Share it with your classmates. ' etters ot er ^ a) 2 3 X 5 b) 2 X 5 2 words. c) (2 X 5 ) 3 d) (2 X 5) 2 3 10. Evaluate. e) ( - 1 0 ) - 5 f) ( - 1 0 ) - 5° a) (3 + 4 ) 2 X (4 - 6 ) 3 g) [ ( - 1 0 ) - 5 ] 3 h) [ ( - 1 0 ) - 5]° b) (8 22 + l) 3 - 3 5 3 c) 4 h- [8(6° - 2 1 )] 5. Evaluate. d) 32 -L. [9 - ( - 3 ) ] 2 a) 2 3 + (—2) 3 b) (2 - 3 ) 3 2 3 2 e) (2 X l ) c) 2 3 - ( —3) 3 d) (2 + 3 ) 3 f) ( l l 3 4- 5 2 ) 0 + (42 - 24) e) 2 3 + ( - 1 ) 3 f) (2 - 2 ) 3 g) 2 3 X (—2) 3 h) (2 X l ) 3 11. Explain why the brackets are not necessary to evaluate this expression. Apply (—4 3 X 10) - (6 - 2) 6. a) Evaluate. Record your work. Evaluate the expression, showing each step. i) 4 2 + 4 3 ii) 5 3 + 5 6 12. Winona is tiling her 3-m by 3-m kitchen b) Evaluate. Record your work. floor. She bought stone tiles at $70/m 2. It i) 6 3 - 6 2 ii) 6 3 - 6 5 costs $60/m 2 to install the tiles. Winona has 7. Identify, then correct, any errors in the a coupon for a 25% discount off the student work below. Explain how you think installation cost. This expression represents the errors occurred. the cost, in dollars, to tile the floor: 70 X 32 + 60 X 32 X 0.75 How much does it cost to tile the floor? I 1 + 7 2 x 2 * (~