MATHEMATICS 20-1 Radical Expressions and Equations High School collaborative venture with M. E LaZerte, McNally, Queen Elizabeth, Ross Sheppard, Strathcona and Victoria M. E. LaZerte: Teena Woudstra Queen Elizabeth: David Underwood Ross Sheppard: Dean Walls Strathcona: Christian Digout Victoria: Steven Dyck McNally: Neil Peterson Facilitator: John Scammell (Consulting Services) Editor: Jim Reed (Contracted) 2010 - 2011 Mathematics 20-1 Radical Expressions and Equations Page 2 of 53 TABLE OF CONTENTS STAGE 1 DESIRED RESULTS PAGE Big Idea 4 Enduring Understandings 4 Essential Questions 4 Knowledge 5 Skills 6 STAGE 2 ASSESSMENT EVIDENCE Transfer Task Captain Red Ickle’s Booty Teacher Notes for Transfer Task Transfer Task Rubric Possible Solution 7 9 15 16 STAGE 3 LEARNING PLANS Lesson #1 Reviewing Radicals 17 Lesson #2 Adding and Subtracting 23 Lesson #3 Multiplying 27 Lesson #4 Division and Rationalizing 34 Lesson #5 Domain of Radicals 40 Lesson #6 Solve Radical Equations 45 Appendix – Worksheets/Keys Mathematics 20-1 49 Radical Expressions and Equations Page 3 of 53 Mathematics 20-1 Radical Expressions and Equations STAGE 1 Desired Results Big Idea: Radical numbers allow us to use exact values in real life situations like measurement, distance and surveying. Radical number operations are the fundamental skills useful in other mathematical topics like trigonometry and coordinate geometry. Implementation note: Post the BIG IDEA in a prominent place in your classroom and refer to it often. Enduring Understandings: Students will understand … That operations performed on radicals are similar to other number systems and algebraic operations. Radicals with even indices are limited to non-negative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. There are conventions for simplifying answers after performing radical operations. Essential Questions: Where are radicals used in real life? When is it appropriate to round or approximate values? When are exact values necessary? What does the square root of a negative number represent? How are operations performed on radicals similar or different to those performed on other number families? Implementation note: Ask students to consider one of the essential questions every lesson or two. Has their thinking changed or evolved? Mathematics 20-1 Radical Expressions and Equations Page 4 of 53 Knowledge: Enduring Understanding List enduring understandings (the fewer the better) Students will understand… There are conventions for simplifying answers after performing radical operations. 8888 I*AN = when to simplify a radical what like terms are when to perform the operations +, - , x, ÷ radicals can be expressed in different forms a radical yields a numerical approximation Students will know … AN.2, AN.3 AN.3 AN.2 the radicand must be a non-negative number when the index is even the radicand can be a positive or negative number when the index is odd Students will know … Solving radical equations can yield extraneous roots. Students will understand… Students will know … Radicals with even indices are limited to non-negative radicands, while odd indices have no restrictions on the radicands. Students will understand… Description of Knowledge The paraphrased outcome that the group is targeting *AN.2, AN.3 That operations performed on radicals are similar to other number systems and algebraic operations. Students will understand… Specific Outcomes List the reference # from the Alberta Program of Studies to check each root for validity why some roots are extraneous Students will know … when to rationalize denominators when to simplify radicals Algebra and Number Mathematics 20-1 Radical Expressions and Equations Page 5 of 53 Skills: Enduring Understanding List enduring understandings (the fewer the better) Students will understand… Specific Outcomes List the reference # from the Alberta Program of Studies Description of Skills The paraphrased outcome that the group is targeting Students will be able to… *AN.2, AN.3 That operations performed on radicals are similar to other number systems and algebraic operations. Students will understand… AN.3 There are conventions for simplifying answers after performing radical operations. *AN = Algebra and Number Mathematics 20-1 AN.2 determine the domain of radical expressions and equations Students will be able to… Solving radical equations can yield extraneous roots. Students will understand… Students will be able to… AN.2, AN.3 Radicals with even indices are limited to non-negative radicands, while odd indices have no restrictions on the radicands. Students will understand… add, subtract, multiply and divide radicals simplify entire and mixed radicals compare and order radical expressions with numerical radicands in a given set express an entire radical with a numerical radicand as a mixed radical express a mixed radical with a numerical radicand as an entire radical identify extraneous roots through verification solve radical equations algebraically Students will be able to… rationalize monomial or binomial denominators Implementation note: Teachers need to continually ask themselves, if their students are acquiring the knowledge and skills needed for the unit. Radical Expressions and Equations Page 6 of 53 STAGE 2 1 Assessment Evidence Desired Results Desired Results Captain Red Ickle’s Booty Teacher Notes There is one transfer task to evaluate student understanding of the concepts relating to radical expressions, operations and equations. A photocopy-ready version of the transfer task is included in this section. Implementation note: Students must be given the transfer task & rubric at the beginning of the unit. They need to know how they will be assessed and what they are working toward. Each student will: Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands. Solve problems that involve radical equations (limited to square roots). Map Master map student template Master map no solutions Mater map with student solutions Files were added to the EPSB Understanding by Design share site Mathematics 20-1 Radical Expressions and Equations Page 7 of 53 Teacher Notes for Captain Red Ickle’s Booty Transfer Task Glossary conjugates – Two binomial factors whose product is the difference of two squares [Math 20-1 (McGraw-Hill Ryerson: page 587)] entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)] equation - A statement of equality between two expressions expression – A general term that ultimately represents a number. An expression can consist of numbers, variables and operations on these. extraneous root – A number obtained in the process of solving an equation that does not satisfy the equation like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill Ryerson: page 273)] mixed radicals - A radical with a coefficient other than 1 operation – Associates two or more members of a set with one of the members of the set. The basic operations in mathematics are addition, subtraction, multiplication, division and exponentiation. radical – n x - Includes radical sign, radical index and radicand. radical sign – n radical index – n radicand – x radical equation – An equation with a variable within a radicand rationalize – A procedure for converting to a rational number without changing the value of the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)] Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. Implementation note: Teachers need to consider what performances and products will reveal evidence of understanding? What other evidence will be collected to reflect the desired results? Mathematics 20-1 Radical Expressions and Equations Page 8 of 53 Captain Red Ickle’s Booty - Student Assessment Task After years of piracy on the high seas, Captain Red Ickle has finally been captured. Alone he waits in a cold dark cell; waiting for the trial that will sentence him to life in the dungeon, or to his death. The excitement of his capture and the anticipation of his judgement day have created quite a stir among the citizens. You feel anxious to know the outcome, but feel unsure of which you hope for. Piracy is unforgiveable, but you imagine yourself sailing port to port on an endless adventure and feel a strange sense of sympathy. You decide, against your better judgment, to sneak down to the holding cell to meet the Captain. But what would a good person like you want to learn from a thieving pirate? You arrive at the prison shortly after sunset. A small bribe is all is takes to get past the night guard and suddenly you are standing in front of the Captain’s cell. He walks towards the bars to size you up. His mouth curls up into a sly grin, as if he recognizes you. “Aye Peter, ye’ve come fer me spoils, have ye?” He has obviously mistaken you for someone else, but you feel that now is not the time to correct him. You nod slowly. “Yarr,it pains me to give it up after all these years, but I fear me ship be approaching her final port. When ye find me treasure chest, don’t share the booty with anybody.” Reaching into his boot, he draws a small roll of parchment. You know at once what this is: a map. The Captain says no more as he returns to the dark corner of his cell. You leave feeling confused. Who is Peter? Where does this map lead? And what is this treasure? Destiny has spoken. You know what must be done. The following morning you set out with nothing but two weeks rations and your map. You find a merchant ship willing to take you to your destination. The captain of the merchant ship lets out a deep chuckle as he tells you: “Foolish lad, if death is what you seek, I’ll take you there – for a fee of course. But I warn you that no one comes back from Skwaroot Island!” As the merchant ship nears the island, you are forced to jump ship and swim ashore. Walking up the beach, dripping wet, you wonder if this was such a good idea. “No time for doubt now,” you tell yourself, “I’ve got a treasure to find.” Captain Red Ickle’s Booty Part I 1. Looking more closely at the map, you notice that Captain Ickle has designated a path along which you will find the key to his treasure chest. You can see from his legend that you must pass by five landmarks, in the order specified. A note at the bottom tells you the correct path should have a distance of 12 10 + 6 5 + 5 2 (assume that you always walk directly from one landmark to the next). Noticing the scale on the map, you see that each square of the map’s grid has side length of 2 km. Along which route will you find the key? Indicate your chosen path and show the work supporting your choice. 2. Wherever there is a fork in the road, you will have at least two options. If you always went right, how long would the path be to the treasure? If you always went left, how long would the path be to the treasure? Which route is longer and by how much? Show all your work and express your answer in simplest form. 3. At long last, you have arrived at the treasure. Judging by the Sun, you estimate the time spent searching to be 5 5 hours. Given that the path length was 12 10 + 6 5 + 5 2 km, was your average walking speed? total distance average speed = . total time 4. Along your way, you noticed a beautiful lake in the shape of a parallelogram and were impressed by its size. Calculate the approximate area of this lake. (Area = base x height). Captain Red Ickle’s Booty Part II Think of a situation that would require you to hide something. Write a brief explanation of at least one paragraph of why you’ve hidden this thing and why it must now be found. Create a map to guide someone to your secret hiding spot. Your map must include the following: A title A start point and a finish point At least 3 different routes; each route must have at least 4 segments A legend which indicates: a scale for the map in which each square has an irrational side length, a desired path to the hiding spot, and a clue related to the distance of the path You will be asked to submit two copies of your map: an original map and a solution map. Your project must also include: Questions about: o Total distance along a route o The difference between the distances of two routes o The average speed of travel (given the time) Detailed solutions to each question Radicals are not limited to numerical radicands. You will therefore be asked to find the distances for each segment for any scale. Do this by using x km as the side length of your grid’s squares. Calculations showing the distance of each segment and total distance of each route if each grid square had a side length of x km. Captain Red Ickle’s Booty ● Key on Path: ■ ▲ Distance of correct path: 9 10 + 10 5 + 10 2 Scale: each grid square 2 km ´ 2 km Captain Red Ickle’s Booty Glossary conjugates – Two binomial factors whose product is the difference of two squares [Math 20-1 (McGraw-Hill Ryerson: page 587)] entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)] equation - A statement of equality between two expressions expression – A general term that ultimately represents a number. An expression can consist of numbers, variables and operations on these extraneous root – A number obtained in the process of solving an equation that does not satisfy the equation like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill Ryerson: page 273)] mixed radicals - A radical with a coefficient other than 1 operation – Associates two or more members of a set with one of the members of the set. The basic operations in mathematics are addition, subtraction, multiplication, division and exponentiation. radical – n x - Includes radical sign, radical index and radicand. radical sign – n radical index – n radicand – x radical equation – An equation with a variable within a radicand rationalize – A procedure for converting to a rational number without changing the value of the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)] Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. Assessment Mathematics 20-1 Radical Expressions and Equations Captain Red Ickle’s Booty Rubric COMPONENT Description of Requirements Assessment Part II – Design Your Own Secret Map Required Components Questions Clarity of Work Calculations Visual/Artistic Presentation & Creativity The map has a title to indicate the name of the island The map has a start point and a finishing point There are at least 3 routes with at least 4 line segments on each route The map’s legend is clearly indicated and includes: a scale, a desired path, and a clue related to the distance of desired path Project includes a description of at least one paragraph explaining the context of the map The project includes an appropriate question about total distance The project includes an appropriate question about a difference between two distances The project includes an appropriate question involving distance, speed and time Questions include a solution key showing detailed calculations Start point, finishing point, and landmarks are clearly indicated Routes on the solution map are drawn using a straight edge Routes on the solution map have distances clearly indicated for each segment Descriptive paragraph is written using proper spelling and grammar Calculations are included for each question Calculations are done correctly Calculations show simplification of entire radicals to mixed radicals in simplest form Calculations demonstrate an understanding of like radicals Calculations demonstrate an understanding of simplifying numerical and variable radicands The limits of the map are clearly defined The map has a grid overlaid (should students create their own canvas) The map is visually appealing The project showcases the student’s creative talents IN 1 2 3 4 IN 1 2 3 4 IN 1 2 3 4 IN 1 2 3 4 IN 1 2 3 4 Assessment: IN – Inadequate 1 – Limited 2 – Adequate 3 – Proficient 4 – Excellent Possible Solution to Captain Red Ickle’s Booty 4 10 8 5 4 5 5 10 5 10 10 5 10 2 ● Key on Path: ■ ▲ Distance of correct path: 3 26 3 10 6 5 12 10 + 6 5 +10 2 Scale: each grid square 2 km ´ 2 km Mathematics 20-1 Radical Expressions and Equations Page 16 of 53 STAGE 3 Learning Plans Lesson 1 Reviewing Radicals STAGE 1 BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement, distance and surveying. Radical number operations are the fundamental skills useful in other mathematical topics like trigonometry and coordinate geometry. ENDURING UNDERSTANDINGS: ESSENTIAL QUESTIONS: Students will understand … That operations performed on radicals are similar to other number systems and algebraic operations. Radicals with even indices are limited to nonnegative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. There are conventions for simplifying answers after performing radical operations. Where are radicals used in real life? When is it appropriate to round or approximate values? When are exact values necessary? What does the square root of a negative number represent? How are operations performed on radicals similar or different to those performed on other number families? KNOWLEDGE: SKILLS: Students will know … Students will be able to … radicals can be expressed in different forms a radical yields a numerical approximation compare and order radical expressions with numerical radicands in a given set express an entire radical with a numerical radicand as a mixed radical express a mixed radical with a numerical radicand as an entire radical Implementation note: Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete. Mathematics 20-1 Radical Expressions and Equations Page 17 of 53 Lesson Summary Achievement Indicators: 2.1, 2.2, 2.3 2.7, 2.9 and 3.5 will be embedded in the lesson Students will estimate radicals and convert between entire and mixed radicals Lesson Plan Consider the following structure: Hook Show students a Number 9 clock. With the class, discuss the different operations and symbols that are used to build the clock. See if they can come up with other possibilities. 9 An example is: 9 + 9 - = 5 9 Get students to build their own # 4 clock. Students can work in groups. http://www.flickr.com/photos/ohrphan/1373936341/ Lesson Goal To activate prior knowledge of what they know about comparing and ordering radicals and converting between mixed and entire radicals. Mathematics 20-1 Radical Expressions and Equations Page 18 of 53 Lesson Complete the following activity with students: Create a Pythagorean spiral. http://www.geom.uiuc.edu/~demo5337/Group3/spiral.gif Have students create the first four triangles of the Pythagorean spiral. Encourage a discussion about the need to represent distance as exact values. Mathematics 20-1 Radical Expressions and Equations Page 19 of 53 The following interactive shows the solution: http://members.shaw.ca/jreed/math20-1/pythagoreanSpiral.htm Instructions: Things You'll Need: paper pencil, pen, or other writing utensil protractor or other object that has a 90° angle http://www.ehow.com/how_4621697_spiral-pythagorean-theorem.html After students have completed the first 4 triangles give them a printout of Figure 1. Directions: Cut out the triangles from the printout. Make a number line from 1 to 10 using the referent of unit 1 from the first triangle of Figure 1. Using the length of number line: Mathematics 20-1 2 (a from Figure 1) put the following lengths on the 2, 2 2, 3 2, ... Radical Expressions and Equations Page 20 of 53 Using the length of 3 (b from Figure 1) put the following lengths on the number line: 3, 2 3, 3 3, ... Continue the process with triangles c through m. Generate discussions on equivalent lengths expressed in various forms. ie: mixed, entire 4 = 2, 2 2 = 8 The following interactive may be useful to reinforce some of these equivalent lengths. http://members.shaw.ca/jreed/math20-1/radical.htm Going Beyond Resources Supporting Assessment Mathematics 20-1 Radical Expressions and Equations Page 21 of 53 Glossary equation - A statement of equality between two expressions expression – A general term that ultimately represents a number. An expression can consist of numbers, variables and operations on these. operation – Associates two or more members of a set with one of the members of the set. The basic operations in mathematics are addition, subtraction, multiplication, division and exponentiation. radical – n x - Includes radical sign, radical index and radicand. radical sign – n radical index – n radicand – x radical equation – An equation with a variable within a radicand Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. Other Mathematics 20-1 Radical Expressions and Equations Page 22 of 53 Lesson 2 Adding and Subtracting STAGE 1 BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement, distance and surveying. Radical number operations are the fundamental skills useful in other mathematical topics like trigonometry and coordinate geometry. ENDURING UNDERSTANDINGS: ESSENTIAL QUESTIONS: Students will understand … That operations performed on radicals are similar to other number systems and algebraic operations. Radicals with even indices are limited to nonnegative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. There are conventions for simplifying answers after performing radical operations. Where are radicals used in real life? When is it appropriate to round or approximate values? When are exact values necessary? What does the square root of a negative number represent? How are operations performed on radicals similar or different to those performed on other number families? KNOWLEDGE: SKILLS: Students will know … Students will be able to … when to simplify a radical what like terms are when to perform the operations + and the radicand must be a non-negative number when the index is even the radicand can be a positive or negative number when the index is odd check each root for validity when to simplify radicals add and subtract radicals simplify entire and mixed radicals Lesson Summary Achievement Indicators: 2.4 2.7, 2.9 and 3.5 will be embedded in the lesson Mathematics 20-1 Radical Expressions and Equations Page 23 of 53 Lesson Plan Hook Give students a couple of familiar questions and have them add and subtract and note how and why they do this. In particular, make sure students know what like terms are. 1 1 + 2 3 2x + 5x2 – 3x2 + y Lesson Goal Students learn to efficiently add and subtract expressions with radicals. Activate Prior Knowledge Have students simplify several radicals: 8, 96, 3 363 . Lesson Have students explore how to add and subtract radicals. Have them use a calculator to find the decimal approximation of each of the following. Ask students to compare/discover equivalent expressions. a. f. 2- 2 b. 2 2 c. g. 0 2 h. 3- 3 d. 2 3 e. i. 0 3 j. 3+ 2 3- 2 Have students come up with a conclusion and what “like terms” are for radicals and compare this to like terms for adding algebraic expressions. Have students explore how to add and subtract radicals. Have them use a calculator to find the decimal approximation of each of the following. a. e. b. 3 2- 2 3 f. 4 5 2+ 2 2- 2 c. 2 3 2 3 g. 0 2 d. 6 2 h. 3 2 Have students come up with a conclusion and an explanation of what “like terms” are for radicals and compare this to like terms for adding algebraic expressions. Mathematics 20-1 Radical Expressions and Equations Page 24 of 53 Is each of the following pairs of expressions ‘like’ radicals? a. 2 and 8 b. 3 and 5 b. 3 and 6 Have a further discussion with students about “like terms” and how the radicals must be simplified to know if you have like terms. Students should also note that the indices have to be the same. Alternate Lesson: Use activity attached to review concepts of like terms. Use similar activity to expand to the operation of addition and subtraction of radicals. Going Beyond Resources Math 20-1 (McGraw-Hill Ryerson: sec 5.1) Supporting Assessment Glossary entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)] equation - A statement of equality between two expressions expression – A general term that ultimately represents a number. An expression can consist of numbers, variables and operations on these. like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill Ryerson: page 273)] mixed radicals - A radical with a coefficient other than 1 Mathematics 20-1 Radical Expressions and Equations Page 25 of 53 operation – Associates two or more members of a set with one of the members of the set. The basic operations in mathematics are addition, subtraction, multiplication, division and exponentiation. radical – n x - Includes radical sign, radical index and radicand. radical sign – n radical index – n radicand – x Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. Other Mathematics 20-1 Radical Expressions and Equations Page 26 of 53 Lesson 3 Multiplying STAGE 1 BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement, distance and surveying. Radical number operations are the fundamental skills useful in other mathematical topics like trigonometry and coordinate geometry. ENDURING UNDERSTANDINGS: ESSENTIAL QUESTIONS: Students will understand … That operations performed on radicals are similar to other number systems and algebraic operations. Radicals with even indices are limited to nonnegative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. There are conventions for simplifying answers after performing radical operations. Where are radicals used in real life? When is it appropriate to round or approximate values? When are exact values necessary? What does the square root of a negative number represent? How are operations performed on radicals similar or different to those performed on other number families? KNOWLEDGE: SKILLS: Students will know … Students will be able to … when to simplify a radical what like terms are when to perform the operations +, - , x, ÷ when to simplify radicals add, subtract, multiply and divide radicals simplify entire and mixed radicals Lesson Summary Achievement Indicators: 2.4 2.7, 2.9 and 3.5 will be embedded in the lesson Mathematics 20-1 Radical Expressions and Equations Page 27 of 53 Lesson Plan Hook Review perimeter and then discuss how to find the area of the polygon in order to determine what students instinctively know about multiplying radicals. In the diagram, AB, BC, CD, DE, EF, FG, GH, and HK all have length 4, and all angles are right angles, with the exception of the angles at D and F. Determine the perimeter of ABDFHK. Solution: P = 4(4) + 4(4√2) = 16 + 16√2 Mathematics 20-1 Radical Expressions and Equations Page 28 of 53 Lesson Goal Multiplying monomial x monomial Multiplying monomial x binomial Multiplying binomial x binomial Activate Prior Knowledge Already activated in Hook. Determine the area of the rectangle ABHK. Solution: A = 4(4+ 4√2 + 4) = 4(8+ 4√2) = 32 + 16√2 Lesson In groups of 4, have the students complete the Multiplying Polynomial and Radical Practice Worksheet. Have the students make comparisons between the multiplication operations of polynomials and radicals. Each group should record their solutions on chart paper. Record the polynomials on one piece of paper and radicals on a second piece. Once all the groups are finished they can hang them on the walls of the classroom. As a large group you can discuss the similarities between the multiplication operations of both polynomials and radicals. Finally as a class you can write the rules for multiplying radicals. Going Beyond Mathematics 20-1 Radical Expressions and Equations Page 29 of 53 Resources Math 20-1 (McGraw-Hill Ryerson: sec 5.2) Supporting Lesson 3 Worksheet: “Multiplying Polynomial and Radical Practice Worksheet” Multiplying Polynomial and Radical Practice Worksheet copy was added to Appendix Mathematics 20-1 Radical Expressions and Equations Page 30 of 53 Multiplying Polynomial and Radical Practice A. Multiply the following Polynomials: ( ) 1. 2x 3 4x 2 ( ) 2. -3x 3 4x 3 ( 3. 2x 3 x + 5x 2 ) ( 4. 4xy - xy - x 2 ( )( ( )( ) ) 5. x + 5 x + 4 ) 6. 2x - 7 3x + 6 B. Multiply the following Radicals: 1. ( 5)( 6) 2. (2 3 ) (-3 5 ) 3. (3 6 ) ( 2 + 5 3 ) 4. -8 Recall 18 = 9 2 =3 2 ( 6 ) (1+ 4 10 ) 5. ( )( 6. (5 + 6 ) ( 2 - 6 ) 3 +4 3+ 3 ) How are multiplying radicals similar to those on polynomials? How are multiplying radicals different from those on polynomials? Answers: A. 1. 8x 5 2. -12x 6 3. 2x 4 +10x 5 4. -4x 2 y 2 - 4x 3 y 5. x 2 + 9x + 20 6. 6x 2 - 9x - 42 B. 1. 30 2. -6 15 3. 6 6 + 45 2 4. -8 6 - 64 15 5. 7 3 + 15 6. -3 6 + 4 How are multiplying radicals similar to those on polynomials? Multiplying performed on radicals and on polynomials both follow the same principles of distribution. How are operations on radicals different from those on polynomials? There are conventions for simplifying answers after performing radical operations. Radicals with even indices are limited to non-negative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. Mathematics 20-1 Radical Expressions and Equations Page 32 of 53 Assessment Glossary conjugates – Two binomial factors whose product is the difference of two squares [Math 20-1 (McGraw-Hill Ryerson: page 587)] entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)] extraneous root – A number obtained in the process of solving an equation that does not satisfy the equation like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill Ryerson: page 273)] mixed radicals - A radical with a coefficient other than 1 operation – Associates two or more members of a set with one of the members of the set. The basic operations in mathematics are addition, subtraction, multiplication, division and exponentiation. radical – n x - Includes radical sign, radical index and radicand. radical sign – n radical index – n radicand – x rationalize – A procedure for converting to a rational number without changing the value of the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)] Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. Other Mathematics 20-1 Radical Expressions and Equations Page 33 of 53 Lesson 4 Division and Rationalizing STAGE 1 BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement, distance and surveying. Radical number operations are the fundamental skills useful in other mathematical topics like trigonometry and coordinate geometry. ENDURING UNDERSTANDINGS: ESSENTIAL QUESTIONS: Students will understand … That operations performed on radicals are similar to other number systems and algebraic operations. Radicals with even indices are limited to nonnegative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. There are conventions for simplifying answers after performing radical operations. Where are radicals used in real life? When is it appropriate to round or approximate values? When are exact values necessary? What does the square root of a negative number represent? How are operations performed on radicals similar or different to those performed on other number families? KNOWLEDGE: SKILLS: Students will know … Students will be able to … when to simplify a radical. what like terms are when to perform the operations +, - , x, ÷ the radicand must be a non-negative number when the index is even the radicand can be a positive or negative number when the index is odd when to rationalize denominators when to simplify radicals add, subtract, multiply and divide radicals simplify entire and mixed radicals determine the domain of radical expressions and equations identify extraneous roots through verification solve radical equations algebraically rationalize monomial or binomial denominators Lesson Summary Achievement Indicators: 2.4, 2.5, 2.6 2.7, 2.9 and 3.5 will be embedded in the lesson Mathematics 20-1 Radical Expressions and Equations Page 34 of 53 Lesson Plan Hook 1. Sometimes we need to perform operation on or compare fractions. Have students try: 1 9 + 6 27 9 1 to . 27 3 1 1 + 6 3 Then find an equivalent fraction with the lowest common denominator by multiplying 1 2 the by before adding. 3 2 1 1 2 1 2 3 1 + · ® + ® ® 6 3 2 6 6 6 2 Discuss how it is advantageous to reduce 2. This leads to the convention of rationalizing the denominator. Have students try: 1 1 + 2 2 Discuss the similarity with the above example. Lesson Goal Students will be able to divide expressions with radicals and leave all responses with rational denominators. Activate Prior Knowledge Review the concept of difference of squares, distributive property, simplifying fractions Mathematics 20-1 Radical Expressions and Equations Page 35 of 53 Lesson Exploration 1: 8 , 2 8 , 2 4, 2 2 , 2 16 ,2 2 Discuss with students that there are multiple ways to simplify the expression can divide first multiply 8 by 2 8 = 2 2 4 , simplify to 2 2 8 . They 2 or they may see right away that they can 2 . This gives students 6 different representations of the same value. 2 Exploration 2: 6 ,2 2 6 2 6 , and 3 8 8 In the form, 2 6 , students cannot divide 6 and 8 evenly, so they will have to try 8 different things: change 8 to 2 2 and then simplify change 6 to 2 3 and divide Students will need to make some connection on how they want to do this. Discuss the method(s) students prefer. Exploration 3: 7 2 Students are faced with the dilemma that they cannot divide the two numbers evenly. Discuss rationalization of denominators. Exploration 4: 10 20 Students will simplify this differently and you can discuss which method is most efficient. At this point you can stress to students what simplest form is, but they can construct their own way to get there. Mathematics 20-1 Radical Expressions and Equations Page 36 of 53 Exploration 5: 5 3 and 3 6 5 2 12 Tell students their answer should be in simplest form. The second example will give students a couple of different options and you can discuss the pros and cons of each. Exploration 6: 22 15 363 At this point we can move onto a harder example where we want them to simplify first. If a student chooses to rationalize the denominator first they will get, 22 5445 . 363 Many students will have difficulty reducing 5445 . Use this opportunity to stress simplifying first when possible to do so. 22 15 22 15 22 15 2 15 æ 3 æ 2 45 2 9 · 5 6 5 = = = = = = 2 5 compared to æ æ= 3 3 3 363 3 ·121 11 3 3 æ 3æ 22 15 363 = 22 15 3 * 121 = 22 15 =2 5 11 3 Exploration 7: Binomial numerators and denominators: Expand the lessons and rules learned to include expressions with binomials in numerators and/or denominators. 1+ 2 2 Have students simplify and then compare their responses with their peers. As a group, discuss the correct answer and the process to achieve this. Exploration 8: Conjugates Have the students expand and simplify. (1+ 2 ) (1- 2 ), (1+ 2 ) (1+ 2 ), (1- 2 ) (1- 2 ) Discuss the similarity and differences in the 3 products. Introduce the term conjugates. Have the students expand and simplify. (3 3 + x ) (3 3 - x ) , (3 3 + x ) (3 3 + x ) , (3 3 - x ) (3 3 - x ) Mathematics 20-1 Radical Expressions and Equations Page 37 of 53 Have students expand, simplify and then compare responses with their peers. (3 7 + 4 x ) (3 7 - 4 x ) and 2 3 +2 2 Going Beyond Investigate the concept of rationalizing a numerator or rationalize an expression with an index of 3 or 4. Students would most likely need direction, but if they have some exposure to factoring sum and difference of cubes, you can relate rationalizing cube roots with rationalizing binomial square roots with difference of squares. Resources Math 20-1 (McGraw-Hill Ryerson: sec 5.2) Supporting Assessment Glossary conjugates – Two binomial factors whose product is the difference of two squares [Math 20-1 (McGraw-Hill Ryerson: page 587) rationalize denominator – create an equivalent expression that does not have a radical in the denominator. Other Like Terms Worksheet copy was added to Appendix Mathematics 20-1 Radical Expressions and Equations Page 38 of 53 Like Terms Examine all the different terms below. Place each into the box with the matching like term at the bottom of this page. Note: Like terms for radicals can only be determined when the radicals are reduced to simplest form, like terms have the same index and radicand, much like polynomials where like terms have the same variables with the same exponents. 75 50 3 80 100 -3 2 8x2 72 3x2 -8 18 5a 32 5 3 48 3 20 8 27 8x -2x2 2a 16 16 -a 3 8ax2 -x 8 128 You must state why each group is a like term. 3 3 5 x ax 2 a 2 x2 constant 2 Mathematics 20-1 Radical Expressions and Equations 128 Page 39 of 53 Lesson 5 Domain of Radicals STAGE 1 BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement, distance and surveying. Radical number operations are the fundamental skills useful in other mathematical topics like trigonometry and coordinate geometry. ENDURING UNDERSTANDINGS: ESSENTIAL QUESTIONS: Students will understand … That operations performed on radicals are similar to other number systems and algebraic operations. Radicals with even indices are limited to nonnegative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. There are conventions for simplifying answers after performing radical operations. Where are radicals used in real life? When is it appropriate to round or approximate values? When are exact values necessary? What does the square root of a negative number represent? How are operations performed on radicals similar or different to those performed on other number families? KNOWLEDGE: SKILLS: Students will know … Students will be able to … the radicand must be a non-negative number when the index is even the radicand can be a positive or negative number when the index is odd determine the domain of radical expressions and equations Lesson Summary Achievement Indicators: 2.8, 3.1 Mathematics 20-1 Radical Expressions and Equations Page 40 of 53 Lesson Plan Hook Have students find 3 values that can be substituted for x in each of the following x,- x , and - x and evaluate the 3 numbers to the nearest hundredth. Students will discuss in small groups the smallest and largest possible number for each of the above expressions and answer the question: Is there a number that can be substituted into all three expressions that would create a valid response? The teacher can then show the graphs of each expression and have students identify which graph is for which expression based on the domains and make sure that students understand that the value of the radicand needs to be greater or equal to zero. Lesson Goal Understand the restriction to the domain of a radical function/expression. Activate Prior Knowledge Review the concept of domain and range as discussed in 10C and 20-1 (quadratics). Lesson Teacher Note: A file was created using GeoGebra to illustrate the transformation of radicals. L5 BasicRadicalTransformations.ggb file was added to the EPSB Understanding by Design share site To use this file you will need to download GeoGebra. At the time this document was prepared the program could be downloaded for free from: http://www.geogebra.org/cms/en/download. Mathematics 20-1 Radical Expressions and Equations Page 41 of 53 Exploration 1: Continue the discussion (with a software graphing tool) of what happens to the domain when the expression is x + 2 . Students could explore this with their own graphing calculator or as a class on the board with students observing the graph moves 2 units to the left compared to the graph of f ( x ) = x . Students can then make the connection that + 2 changes the domain of the radical expression. You can further explore what happens with x - 2 and 2- x. From here we want students to have a good grasp that the domain of a radical expression is the values of the variable that will make the radicand greater or equal to zero. Exploration 2: Solve the radicand to determine the domain without the graph. 2 - 3x x 2 +1 x2 x 2 + 5x + 6 At this point, we want to stress that the radicand cannot be negative for square roots, so we are always solving the radicand to be greater or equal to zero no matter what the expression looks like inside the radical. Students can also look at the graphs for further verification and to discuss the domain restrictions. Exploration 3: A square root has an index of 2. Have students investigate higher order indices with the following: 3 x 3 x +3 4 x 5 x 6 x 7 x Have students graph these expressions and find their domains. The following questions can be investigated. 1. Are there any restrictions on cube roots? 2. Are there any restrictions on 4th roots? 3. Are there any restrictions on 5th roots? 4. What rule in general relates to restrictions involving radicals of higher indices? With a graphing calculator, students can graph these quickly and should see the pattern of even and odd indices. Mathematics 20-1 Radical Expressions and Equations Page 42 of 53 Going Beyond Look at restrictions if there is a radical in the denominator of a fraction such as: 1 x You can also investigate imaginary numbers as ways to deal with domain restrictions that may occur when performing calculations with radicals. Resources Math 20-1 (McGraw-Hill Ryerson: sec 5.1) Supporting Assessment Glossary conjugates – Two binomial factors whose product is the difference of two squares [Math 20-1 (McGraw-Hill Ryerson: page 587)] entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)] equation - A statement of equality between two expressions expression – A general term that ultimately represents a number. An expression can consist of numbers, variables and operations on these. extraneous root – A number obtained in the process of solving an equation that does not satisfy the equation like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill Ryerson: page 273)] mixed radicals - A radical with a coefficient other than 1 Mathematics 20-1 Radical Expressions and Equations Page 43 of 53 operation – Associates two or more members of a set with one of the members of the set. The basic operations in mathematics are addition, subtraction, multiplication, division and exponentiation. radical – n x - Includes radical sign, radical index and radicand. radical sign – n radical index – n radicand – x radical equation – An equation with a variable within a radicand rationalize – A procedure for converting to a rational number without changing the value of the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)] Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. Other Mathematics 20-1 Radical Expressions and Equations Page 44 of 53 Lesson 6 Solve Radical Equations STAGE 1 BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement, distance and surveying. Radical number operations are the fundamental skills useful in other mathematical topics like trigonometry and coordinate geometry. ENDURING UNDERSTANDINGS: ESSENTIAL QUESTIONS: Students will understand … That operations performed on radicals are similar to other number systems and algebraic operations. Radicals with even indices are limited to nonnegative radicands, while odd indices have no restrictions on the radicands. Solving radical equations can yield extraneous roots. There are conventions for simplifying answers after performing radical operations. Where are radicals used in real life? When is it appropriate to round or approximate values? When are exact values necessary? What does the square root of a negative number represent? How are operations performed on radicals similar or different to those performed on other number families? KNOWLEDGE: SKILLS: Students will know … Students will be able to … when to simplify a radical. what like terms are when to perform the operations +, - , x, ÷ the radicand must be a non-negative number when the index is even the radicand can be a positive or negative number when the index is odd check each root for validity. why some roots are extraneous when to rationalize denominators when to simplify radicals add, subtract, multiply and divide radicals simplify entire and mixed radicals determine the domain of radical expressions and equations identify extraneous roots through verification solve radical equations algebraically rationalize monomial or binomial denominators Lesson Summary Achievement Indicators: 3.2, 3.3, 3.4 2.7, 2.9 and 3.5 will be embedded in the lesson Mathematics 20-1 Radical Expressions and Equations Page 45 of 53 Lesson Plan Hook: A cable television company is laying cable in an area with underground utilities. Two subdivisions are located on opposite sides of Willow Creek, which is 500 m wide. The company has to connect points P and Q with cable, where Q is on the north bank, 1200 metres east of R. It costs $40/m to lay cable underground and $80/m to lay cable underwater. What is the least expensive way to lay the cable? 1200 R S Q x 500 P Lesson Exploration 1: Reactivate prior learning of solving linear equations: -3 = x + 2 x = -5 The equation should still be true, if we square both sides. (-3)2 = (x + 2) 2 square both sides 9 = x2 + 4x + 4 solve resulting equation 0 = x2 + 4x - 5 ( x + 5) ( x - 1) = 0 x = -5 and x = 1 () x = 1 is an extraneous root, it does not verify, because -3 ¹ 1 + 2 . x = -5 When would it be useful to square both sides? Mathematics 20-1 Radical Expressions and Equations Page 46 of 53 Exploration 2: Start with a couple of straightforward examples with students in small groups: x = 3 x =4 x = -4 With the first example, it may also be useful to note that x = 4 and x - 4 = 0 are the same equation. This change may be necessary for more complicated equations when we have to isolate the radical. Use the last example to emphasize the need of verifying. It may also be helpful to remind students that the square root of a number is always positive. There will be no solution for x = -4. Consider using this opportunity to remind students that x2 = 16 could have x = -4 as a solution. At this point you can also use the graphing calculator as a tool to show the solution is where the lines intersect on the graph. If you used the graphs with domain and range, students will already be familiar with the shape of a radical graph. Exploration 3: At this point students may have been able to work out a solution by guessing and checking. We will investigate equations where guessing and checking is an inefficient way to find the solution. It may be a good idea to make sure that all students have an understanding of the process of squaring each side. x -1 = x - 7 Exploration 4: Move onto more complicated example(s): x + 2x + 1 = 2x -1; x - 4x + 8 = 2x + 8 ; 7 + 3x = 5x + 4 + 5 At this point if students struggle you may want to have a class discussion about what processes occur in every question: Isolate radical. If there are two radicals, isolate one of them. Square both sides. Remind students about the distributive property (ex. ( x + 2) 2 ¹ x 2 + 22 ). Have students investigate this and see that they do not find the solution if they do not square the whole side. o You may need to review some basic equation solving rules or have students come up with their own and put on chart paper. If there were originally two radicals, there should now be only one. Isolate this radical and square both sides. Solve the resulting equation Verify the solution in the ORIGINAL equation. Mathematics 20-1 Radical Expressions and Equations Page 47 of 53 o Again, you could have students verify in the original and in an intermediate and check the results by graphing and see which one is correct. Going Beyond Resources Math 20-1 (McGraw-Hill Ryerson: sec 5.3) Supporting Assessment Glossary radical – n x - Includes radical sign, radical index and radicand. radical sign – n radical index – n radicand – x operation – Associates two or more members of a set with one of the members of the set. The basic operations in mathematics are addition, subtraction, multiplication, division and exponentiation. expression – A general term that ultimately represents a number. An expression can consist of numbers, variables and operations on these. equation - A statement of equality between two expressions Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. Other Mathematics 20-1 Radical Expressions and Equations Page 48 of 53 Appendix Appendix 1: Multiplying Polynomial and Radical Practice Worksheet and Key Appendix 2: Like Terms Worksheet and Key Mathematics 20-1 Radical Expressions and Equations Page 49 of 53 Appendix 1: Multiplying Polynomial and Radical Practice A. Multiply the following Polynomials: ( ) 1. 2x 3 4x 2 ( ) 2. -3x 3 4x 3 ( 3. 2x 3 x + 5x 2 ) ( 4. 4xy - xy - x 2 ( )( ( )( ) ) 5. x + 5 x + 4 ) 6. 2x - 7 3x + 6 B. Multiply the following Radicals: 1. ( 5)( 6) 2. (2 3 ) (-3 5 ) 3. (3 6 ) ( 2 + 5 3 ) 4. -8 Recall 18 = 9 2 =3 2 ( 6 ) (1+ 4 10 ) 5. ( )( 6. (5 + 6 ) ( 2 - 6 ) 3 +4 3+ 3 ) How are multiplying radicals similar to those on polynomials? How are multiplying radicals different from those on polynomials? Answers: A. 1. 8x 5 2. -12x 6 3. 2x 4 +10x 5 4. -4x 2 y 2 - 4x 3 y 5. x 2 + 9x + 20 6. 6x 2 - 9x - 42 B. 1. 30 2. -6 15 3. 6 6 + 45 2 4. -8 6 - 64 15 5. 7 3 + 15 6. -3 6 + 4 Appendix 2: Like Terms Examine all the different terms below. Place each into the box with the matching like term at the bottom of this page. Note: Like terms for radicals can only be determined when the radicals are reduced to simplest form, like terms have the same index and radicand, much like polynomials where like terms have the same variables with the same exponents. 75 50 3 80 100 -3 2 8x2 72 3x2 -8 18 5a 32 5 3 48 3 20 8 27 8x -2x2 2a 16 16 3 8 3 5 x ax 2 2 a 2 x2 constant 3 128 8ax2 -x You must state why each group is a like term. -a 128 Answers 3 3 48 , a 2 50 , 18 , 27 32 , 72 , 5a , -a , 2a 8 , 128 3 5 2 x2 4 3 2 , 3 128 , - 3 2 8x 2 , 3x 2 , -2x 2 ax 2 x Constant 8ax 2 8x , - x 75 , 80 , Mathematics 20-1 20 100 , 5 , 8 , 16 Radical Expressions and Equations Page 53 of 53