UNIT ONE RADICALS 15 HOURS MATH 521B

Transcription

1 UNIT ONE RADICALS 15 HOURS MATH 521B Revised Nov 9, 00 19

2 SCO: By the end of grade 11 students will be expected to: A demonstrate an understanding of the role of irrational numbers in applications Elaboration - Instructional Strategies/Suggestions What is a radical? Invite a short discussion by student groups on the term radical. Ideas they come up with could be listed on the board. These might include: < a radical is a compact way of writing an irrational number similar to scientific notation being a compact and convenient way of writing very small or very large numbers. < parts of a radical are! radical symbol! radicand! index (degree of the root) A4 approximate square roots Index º radical symbol 9 4» radicand B9 use the calculator correctly and efficiently for various computations Estimation of Radicals Encourage student groups to use guess and check to estimate the values of various degree roots and check their accuracy using a scientific calculator. Simplifying Radicals a) entire to mixed Challenge students to reduce the size of the radicand as much as possible and still have an equivalent radical expression. The idea is to find two factors of the radicand, one of which is the highest possible perfect square (cube, etc.). Ex. 48 = 16 = 4 48 = 8 6 = 2 6 For even degree roots of variable radicands the concept of principal root must be considered. Example: the square root of 16 can be either ± 4 The symbol means principal square root (or positive root) ˆ 16 = 4 and not! 4 20

3 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources What is a radical? Journal Write a short explanation of the advantages of writing numbers in radical form. Estimation of Radicals Estimation/Technology Use guess and check to approximate, to the nearest hundredth, the value of the following. Then use a calculator to check the accuracy of your guess. a) 40 b) 9 4 c) 20 5 d) 200 Technology Use decimal approximations to arrange the following from smallest to largest. 15, 2 2, 5, 4, 50, 2. Simplifying Radicals a) entire to mixed Write each of the following in simplest form. 1) 128 2) 68x ) 128 4) ) 16x What is a radical? Problem Solving Proof by contradiction p.9 #2 Estimation of Radicals Math Power 10 p.14 # -17 odd Algebra, Structure and Method Book 2 p.268 # -8 Math Power 10 p.15 # 1- Simplifying Radicals a) entire to mixed Math Power 10 p.19 # 1-15 odd p.20 # 59 Algebra, Structure and Method Book 2 p.268 # 19,20,1,9-42 Applications Math Power 10 p.20 # 56,57 21

4 SCO: By the end of grade 11 students will be expected to: A8 demonstrate an understanding of and apply properties to operations involving square roots Elaboration - Instructional Strategies/Suggestions Simplifying Radicals (cont d) Only when the radicand is a variable, must we consider the concept of a principal root. For example, if the question is: Simplify: then we must ensure that the answer is positive. We do this by using absolute value symbols. x 2 = x b) mixed to entire Challenge students to write a mixed radical as an entire radical. Note to Teachers: As a rationale, look at arranging in order from smallest to largest: 2 x 2, 2,, 5 2 Operations with Radicals a) add and subtract Challenge students to attempt problems like Have the 8 students use a calculator to get the approximate values of and 2, then add these values. Now invite students to do the problem without evaluating the square roots first. Students might try adding the radicands to get 10. Through discussion and discovery the rule below should be developed: Adding or subtracting radical expressions is equivalent to combining variable expressions with like terms. Example: 2x + x = x + x + x + x + x = 5x = =

5 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources b) mixed to entire Write each of the following as entire radicals 1) 4 2) 2 4 ) 2 b) mixed to entire Math Power 10 p.19 # 25-0,1- Without using a calculator, arrange the following from smallest to largest: a) 5, 2 11, 4, b) 6 2, 2 10,, 7 2 Operations with Radicals a) add or subtract Simplify: 1) ) ) Operations with Radicals a) add or subtract Math Power 10 p.2 # p.24 # 67 Algebra, Structure & Method Book 2 p.272 # 1-12 Journal Explain the rules regarding adding or subtracting radical expressions. 2

6 SCO: By the end of grade 11 students will be expected to: A8 demonstrate an understanding of and apply properties to operations involving square roots Elaboration - Instructional Strategies/Suggestions Operations with Radicals b) Multiply and divide Challenge student groups to carry out investigations about multiplying and dividing radical expressions like: or 5 Students could use a calculator to evaluate the individual radical expressions before doing the multiplying or dividing. They could develop notions about the rules for solving these types of problems if the multiplying or dividing is done first. 2 8 = 16 = = 4 = 2 The students should appreciate the reversibility of the operations and when it is advantageous to do a particular operation first. Note to Teachers; Any radical expressions can be multiplied if changed to exponential form. (See p.11) 24

7 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Operations with Radicals b) Multiply and divide Simplify: 1) 2 5 2) ) 4 ( ) 4) ( 2 + 5)( 7 6 2) 5) 2( ) Determine the area of this rectangle. 4 2 Operations with Radicals b) Multiply and divide Choose sparingly from: Math Power 10 p.19 #4-45 p.2 # p.24 # 65 Applications Math Power 10 p.26-7 # 1-4 green p.8 # 9 Algebra, Structure & Method Book 2 p.268 # 1-18,25-28 p.272 # 19-24,1,2 p.276 # 1-6, , Find the distance from A to B in simplest radical form

8 SCO: By the end of grade 11 students will be expected to: Elaboration - Instructional Strategies/Suggestions Rationalizing denominators Initiate a discussion on what rational and non-rational denominators would look like. Hopefully students could predict that a non-rational denominator is in an irrational form (for our purposes that means a radical). Challenge student groups to develop a method for making the denominator rational. A8 demonstrate an understanding of and apply properties to operations involving square roots For a rational expression with a monomial radical denominator, multiply by the radical part of the denominator over itself. B2 develop algorithms and perform operations on irrational numbers Ex = = = For a rational expression with a binomial radical denominator, multiply by the conjugate of the denominator over itself. Ex = = 4 For rational expressions with higher degree monomial radical denominators, multiply by the correct form of (1). Ex = =

9 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Rationalizing denominators Rationalize the following denominators: 2 5 1) ) ) 4) 2 2 If a rectangle has an area of 6 cm 2 and a width of 5 2 cm, what is the length in simplest radical form? Rationalizing denominators Choose sparingly from: Math Power 10 p.20 # p.2 # 4-52 p.25 # Applications Math Power 10 p.24 # Algebra, Structure & Method Book 2 p # 9-12,22,29, 0,2 p.272 # 1-18 Calculate the perimeter of each figure in simplest radical form. Then write the ratio of the perimeter of the larger figure to that of the smaller figure in simplest radical form (remember to rationalize the denominator). Journal Write a short series of instructions to explain how radical denominators are rationalized. 27

10 SCO: By the end of grade 11 students will be expected to: B9 use the calculator correctly and efficiently for various computations Elaboration - Instructional Strategies/Suggestions Rational exponents Invite students to discuss and give examples of expressions with rational exponents. Challenge students to write the following in order from smallest to largest. 27 1/, 27 1, 27 4/, 27 1/2, 27 2, 27 0, 27 2/, 27!1/ Because all have the same base, they should be able to order them simply by looking at the relative sizes of the exponents. If, however, the bases were not all the same they would have to be able to evaluate each expression. On the TI-8: Students should also understand that raising a base to a power of ½ is equivalent to taking the square root of the base. Allow student groups to try various operations on their calculator to evaluate 27 1/2. Hopefully 12 / they will on their own find that 27 = 27. If they look at 27 1/ on their calculator hopefully it can be equated with cube rooting. Looking at 27 2/ and using power law, this can be thought of as (27 1/ ) 2 or 27 m 2 e j a n n m n m = a = ( a) ˆ the operations done to the 27 are cube rooting and squaring. The order that these are done in are reversible. In general, for rational exponents: a) the denominator of the exponent is the index of the radical b) the numerator of the exponent is the power to which the answer from part (a) is raised Students should see the connection between radical expressions and rational exponents. Most radicals represent irrational numbers but many times it is more convenient and compact to do operations like +,!, and with the irrational numbers in radical form. Extension: Perhaps a mention of the historical use of logarithm tables could occur here. For example, a problem like 7 /5 is easily solved today with a scientific calculator. Not too many years ago this type of problem would be solved using logarithms. A logarithm is a specialized form of an exponent. 28

11 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Rational exponents Discussion Arrange the following in order from smallest to largest: 27 1/, 27 1, 27 4/, 27 1/2, 27 2, 27 0, 27 2/ Write in simplest radical form (simplify further, if possible): Rational exponents Math Power 10 p.7 Algebra, Structure & Method Book 2 p.457 # 1-8 p.458 # 1-19 odd p.458 # 29-5 odd a) 8 4/ e) 6 8 b) 4!/2 f) Problem solving Guess and Check p.11 #,5,6,9 c) 625 1/4 g) d) 9 5/2 Write in exponential form and simplify if possible: 6 a) x x x 4 6 b) x x x Express in simplest radical form (Hint: it may be necessary to temporarily convert to exponential form)

12 SCO: By the end of grade 11 students will be expected to: C12 solve linear, simple radical, exponential and absolute value equations or linear inequalities Elaboration - Instructional Strategies/Suggestions Radical equations (5.7) Invite students to discuss strategies for solving equations. Hopefully they will see that the same method is always used, ie. whatever operations have been performed on the variable must be undone. For the problem: 2 x = 0 the steps are: 1) Isolate the radical term 2 x + 1 = 2 x + 1 = 1 2) Square both sides x + 1 = 1 ) Solve for the variable x = 2 4) Verify by substituting into the original equation. Here we see the answer does not verify; it is an extraneous root. The left and right sides are not equivalent. 5) State the conclusion There are no real roots for this problem. Note to the Teacher: Steps 4 and 5 can be eliminated for odd-degree radical equations. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 0

13 Radical equations (5.7) Guess and Check Solve and verify: 1) x = Radical equations Math Power 11 p.2 # 2-6 Algebra, Structure & Method Book 2 p.280 # ) x 6= 0 ) x + 1 = 5 4) 2 x = 4 5) x + 2 = 0 Solve algebraically, verify your answer: 1) x 1 = 20 2) 4 x = 7 ) x = 4 4) x + 1 = 2 4 5) x 2 = 2 Written Assignment Create a problem where there is a radical equation to solve. Research Search various fields of study (ex. Physics, Chemistry, Economics, Business, etc.) to find two formulas containing a radical. Give a short description of their use in those fields. Research Write a short paper on the life and contributions to mathematics of Julius Wilhelm Richard Dedekind. 1

14 SCO: By the end of grade 11 students will be expected to: C12 solve linear, simple radical, exponential and absolute value equations or linear inequalities Elaboration - Instructional Strategies/Suggestions Equations with Rational Exponents Invite students to discuss possible methods of solving problems like: x 2/ = 16 One possible method might be by guess and check. Perhaps someone might say; let s read the operations done to the x and undo them. The operations are: 1) cube rooting 2) squaring We must square root and cube to solve for x. If we employ power law and raise both sides to the reciprocal power that is in the problem... (x 2/ ) /2 = 16 /2 x = e16j = 64 Exponential Equations The more difficult problems require logarithms but the simpler ones can be done fairly easily. Allow students to try to solve problems like x!2 = 81 by any method; perhaps guess and check. Some students might notice that the bases can be made the same thus allowing them to be disregarded. Ex. x!2 = 81 x!2 = 4 The only way the left side equals the right side is if the exponents are equivalent. Looking at only the exponents: x! 2 = 4 x = 6 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 2

15 Equations with Rational Exponents Group Activity Solve each of the following: 1) (2 + x) ½ = Equations with Rational Exponents Algebra, Structure & Method Book 2 p.457 # 2-0 p.458 # ) (x! 1) 2/ = 4 ) (5 + y)!1/ = ½ Exponential Equations Solve each of the following: 1) 2 x!1 = 2 Exponential Equations Algebra, Structure & Method Book 2 p.461 # 9-12 oral p.462 # ) 4 1!2x = 128 Application A bacteria culture starts with 2,000 bacteria. After 5 hours the estimated number of bacteria is 64,000. What is the time required for the population to double for this culture? Solution N(t) = N(0) 2 t/d 64,000 = 2, /d 2 = 2 5/d 2 5 = 2 5/d 5 = 5/d d = 1 hour Project Look for at least two examples where formulas have the variable in the exponent.