DATE: Algebra 2. Unit 1, Lesson 2: n th roots and when are n th roots real or not real?
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1 Algebra 2 DATE: Unit 1, Lesson 2: n th roots and when are n th roots real or not real? Objectives - Students are able to evaluate perfect n th roots. - Students are able to estimate non-perfect n th roots and plot them on a number line. - Students are able to simplify n th roots (with numerical radicands) - Students are able to simplify n th roots (with variable expression radicands). - Students are able to determine if an n th root is real or not, based on the index and the sign of the radicand - Students are able to solve basic power equations by taking an n th root. Materials and Handouts - Answer transparency for hw #1-1 - Lesson 2 keynote presentation - Classwork: Practice with n th roots and Power Equations worksheet - Homework: More n th roots and Power Equations Homework #1-2 More n th roots and Power Equations Time Activity 20 min Homework Check / Warm-up Activity - Put the answers to hw #1-1 on the overhead. - When finished checking, they should begin the warm-up (copies will be distributed). - As they are working, circulate to check the homework. - Discuss warm-up problems. 0 min Discussion: What is an n th root? - Show slides that ask students to determine the side lengths for a perfect square and cube. - Show slide with a square with non-perfect area, along with a number line. Students should work in groups to estimate the side length and plot it on the number line. Repeat with a non-perfect cube. Clarify how to position the side length on the number line by thinking of proximity to the relative areas/volumes. - Generalizes what an n th root is algebraically. - Explain simplification by finding prime factorization, with numerical radicands. Variable expressions radicands will be modeled on the worksheet. - Odd index will always yield a real number, whereas an even index will only do so if the radicand is positive. - Various root expressions; students must determine if they are plottable or not. Cross off the ones that are not possible to plot on the real number line. Restate that a root is only real when the radicand is positive. - Discuss number of solutions. Review the idea of including ± when the index is even and you are solving an equation. - Discuss the difference between exact and approximate solutions. - Summarize number of solutions - Show that irrational expressions like root 2 can be drawn on a number line by rotating the diagonal of a square down onto the number line. This is a way to help see that irrational numbers are still real numbers.focus on the difference between not real and not rational as students often say things like you can t take the square root of 2. 2 min Individual Work - Hand out the Practice with n th Roots and Beginning Power Equations sheet. Review on overhead. min Closure Students: o Write down notes template in their logs/planners o Write down practice sheet and answer key in their logs/planners o Write down homework in their logs/planners
2 DATE: Classification Warm-Up Directions: Use the Venn Diagram to determine if each statement is true or false. If it is false, explain why. Venn Diagram #1: Overlapping Circles Omegas 1) It is possible to be both Alpha and Omega at the same time. 2) All Omegas are Alphas. Alphas B C ) If you re not an Alpha, you must be an Omega. A D ) Some Alphas are not Omegas. Venn Diagram #2: Subsets Clydes 1) All Clydes are Inkies. Blinkies 2) All Inkies are Clydes. Inkies A ) No Blinkies are Inkies. B C D ) If you re not a Blinky, you re a Clyde.
3 DATE: Algebra 2 Section: Name: Unit 1, Lesson 2: Lesson Notes Understanding Roots 121 ft 2 12 ft Estimating Roots 68 ft ft Evaluating Roots Definition of n th root:
4 Simplifying Roots To simplify an n th root, remove as many factors as possible from inside the root. Find the prime factorization and remove groups of factors based on the index (the number in front of the root sign). Understanding Roots Odd Index: Even Index: Cross off the roots that CAN T be plotted on the number line (i.e., that are not real)
5 Solving Power Equations How many solutions? What kind of solutions? Summary Even exponent Odd exponent
6 DATE: Algebra 2 Section: Name: Unit 1, Lesson 2: Classwork Part 1: Evaluating Roots Find the exact value of each root. Practice with n th Roots Part II: Estimating Roots Estimate the side length of each shape, and plot it on the number line. ft ft Estimate the value of each root (to one decimal place), and plot it on the number line
7 Part III: Simplifying Roots Numerical Expressions Simplify each root completely Part IV: Simplifying Roots Variable Expressions Simplifying a root with a variable expression is done exactly the same way. Read the example carefully. Example Problem Work Explanation Simplify 2x y 6 z 2 = x x 1 y y z 2 = 2 x y y x 1 z 2 ( ) = 2xy 2 xz 2 1) Write the prime factorization of the coefficient. 2) Split the variables into groups based on the index. ) Remove groups of factors from inside the root, and simplify what remains. Simplify each root completely. 12a b 8 c 0x 12 y 2 z 7
8 Part V: Understanding Roots Cross off all of the roots that are not real. Circle all of the roots that are irrational Part VI: Kinds of Solutions Solve the equation x 2 = 60. First, write the exact solution. Then, write the approximate solution. Part VII: Solving Power Equations (READ THE DIRECTIONS!) Solve each equation, giving exact solutions only. Simplify your answers. Make sure to include ± where necessary. If there are no real solutions, write NRS. x = 1 x 2 = 7 x 2 = 2 x = 16 x = 8 x = 2
9 DATE: Algebra 2 Section: Name: Homework #1-2 More n th Roots and Power Equations Practice Part 1: Evaluating Roots Find the exact value of each root , Part II: Estimating Roots Estimate the value of each root (to one decimal place), and plot it on the number line Part III: Simplifying Roots Simplify each root completely x y 7 6a b 6 c 8
10 Part 1V: Simplifying each root completely a b c 8 Part V: Cross off all of the roots that are not real. Circle all of the roots that are irrational Part VI: Write the exact and the approximate solutions of each equation. Include ± if needed. x 2 = 97 x = 70 Part VII: Solving Power Equations Solve each equation, giving exact solutions only. Simplify your answers. Make sure to include ± where necessary. If there are no real solutions, write NRS. x 2 = 98 x 2 = 98 x = 6 x = 6 x = 80 x = 80 x = 2 x = 2
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