AlgebraII

Algebra II Lesson Plans for Block ScheduleAligned to NCSCOS – 2003

Day 1

Essential Questions: How do I define and compute with numbers? How do I evaluate and simplify algebraic expressions?

Objective(s): 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

“SAP”: Students will work in collaborative pairs to complete the Scientific Calculator Exercises Worksheet.

Lesson Anatomy: 1. Go over classroom rules and expectations.2. Assign books as students complete an information sheet.3. Review basic concepts and properties of real numbers. Use examples 4 and 5 from section 1-1 and examples 1, 2, 4 and 5 from section 1-2 (TE page 7 and pages12-14) as you re-familiarize students with Algebra I concepts that they should have previously learned. Use a graphic organizer to show how the sets of numbers are related.

REAL

RATIONAL IRRATIONAL

INTEGERS

WHOLE NUMBERS

COUNTING NUMBERS

Define the absolute value of a number in terms of its distance from zero on the number line. Monitor students’ progress by using the Check Understanding problems 4-6 (TE pg. 7-8).Review using the order of operations to evaluate and simplify algebraic expressions. Monitor students’ progress by using the Check Understanding problems 1, 2, 4, and5 (TE pgs 12 and 14).4. Demonstrate buttons for basic calculator functions.5. Collaborative Pairs: Scientific Calculator Exercises Worksheet6. Group synthesis on what was discovered about the calculator during this activity.

Summarizing Activity: Ticket out the door: Students complete the Order of Operations activity.

Homework: Prentice Hall, Algebra 2 Text Section 1-1 pages 8-9 (2-8 even, 42-52, 83-87) and Section 1-2 pages 15-16 (2, 8, 12, 14, 18, 28, 32, 63, 64)

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Scientific Calculator Exercises

Try these: Answer:

1. 45 + 285 3 140

2. (45 + 285) 3 110

3. 18 + 6 3 15 – 7

4. –5(42) + 120 -90

5. sin 0

6. sin 45 .7071067812

7. (13.9)3 2685.619

8. ___1__ 25 1.82 3 27

9. e 2.718281828

10. 5√24 24.49489743

11. 3√120 + 5 9.932424149

12. (123)1/4 + ln 1 6.447419591

13. 5√-243 -3

14. log 156 2.193124598

15. 142 – 252 -429

16. 10! 3628800

17. The area of a triangle with base = 7.84 and height = 2.4 9.408

18. The area of a circle with diameter = 15 176.7145868

19. (7.23 x 10-6)(5.87 x 104) .424401

20. (8.4 x 10 12 )(7.28 x 10 -3 ) 2352000 (2.6 x 104)

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Insert parentheses to make each statement true.

1. 6 – 32 X 5 – 3 = -422. 6 – 32 X 5 – 3 = -183. 6 – 32 X 5 – 3 = -364. 6 – 32 X 5 – 3 = -6


1. 6 – 32 X 5 – 3 = -422. 6 – 32 X 5 – 3 = -183. 6 – 32 X 5 – 3 = -364. 6 – 32 X 5 – 3 = -6


1. 6 – 32 X 5 – 3 = -422. 6 – 32 X 5 – 3 = -183. 6 – 32 X 5 – 3 = -364. 6 – 32 X 5 – 3 = -6

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Day 2

Essential Questions: How do I solve equations? How do I solve problems by writing equations?


“SAP”: Students will work in cooperative pairs to complete From Words to Symbols.

Lesson Anatomy: 1. Teacher led discussion of troublesome homework problems.2. Teacher led review of solving equations using the properties of equality. Encourage students to check their answers.

a. 7 – 4x = -5b. 5(3 – 2z) = 28 + 3zc. 10 – (3m + 4) = 5m + 14d. 3n – 4 = 15 (1/3n + 2/5)e. 1/4x – 2/3 = 5/6 + xf. -6(w + 1) = 6 5

3. Teacher led review of solving literal equations and finding restrictions using the following examples:Solve for x.

a. 3x – 4y = 8b. d = 2x/a + bc. ax + bx – 15 = 0

4. Discuss key words (sum, total, plus, increased by, more than, difference, exceeds, less than, less, minus, decreased by, subtracted from, product multiplied by, times, twice, doubled, tripled, quotient, divided by) that indicate operations used in writing algebraic expressions. Explain to students that to represent a word expression by algebraic symbols, first choose a variable to represent the unknowns. Then, identify the key word(s) that indicate the operation and use algebraic symbols to represent the word expression. Use the following examples: Write an algebraic expression for each word expression.

a. $75 less than the cost of a television.b. Seven more than 3 times the number of oranges.c. The sum of the square of a number and two.

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Have students work in pairs to complete the Algebraic Expression Worksheet: From Words to Symbols (5-7 minutes). Review answers.

5. Teacher led discussion of writing equations to model and solve problems using examples 5 and 6 (TE, p.20). Have students work with the same partner on Check Understanding Problems 5 and 6 (TE, p.20)

Summarizing Activity: Have students use an algebraic expression to perform the following trick.a. Think of a number.b. Triple the number.c. Add the original number.d. Subtract 4.e. Divide by 4.f. Add 1.g. Subtract the original number.The answer should always be 0. Have students use the language of algebra to explain how the trick works.

Homework: Prentice Hall Algebra 2 Text Section 1-3 pages 21-22 (2-16 even, 17-28, 31, 32, 48)

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From Words to Symbols

Write an algebraic expression for each word expression.

1. Sixteen less than the cost, k, of a shirt. _____________________

2. Three times the sum of 15 and the cost, j. _____________________

3. Twice the cost, x, of a jar increased by 7. _____________________

4. The quotient of 75 points and the number of baskets, b. _____________________

5. Twenty more than triple the number of votes, v. _____________________

6. Half the number of cars, c, decreased by 10. _____________________

7. The square of a number, n, divided by 3. _____________________

Write an equation for each verbal sentence.

1. A number decreased by 7 is –5. _____________________

2. The product of the square of a number and 5 is 20. _____________________

3. Three decreased by twice a number is 12. _____________________

4. The cube of the sum of a number and 10 is 7. _____________________

5. Six times the sum of a number and its square is 0. ____________________

6. The quotient of a number and 8 is -14. _____________________

7. Twelve less than half the sum of a number and 8 is 54. _____________________

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Day 3

Essential Question: How do I solve and graph linear inequalities?


“SAP”: Students work in collaborative pairs to complete the Graphing on a Number Line activity.

Lesson Anatomy: 1. Teacher led discussion of troublesome homework problems.2. How can we make a visual picture of Exercise 16 from the homework?

Graph on the calculator a y = 5x + 8 – 12x –16 + 15x Use a user friendly window of X-min = -5

X-max = 4.4X-scl = 1Y-min = -10Y-max = 10Y-scl = 1X-res = 1

Relate the solution of the equation to the x-intercept of the graph.Test this by solving 4 – x = x algebraically and graphically. 5 53. What does F = 4(S – 65) + 10, S > 65 mean to you?

(Active discussion that leads to understanding that the variables we use in algebra are really used to “stand for” something and algebra makes more sense to us when we know what the variables stand for. This actually represents the formula used to levy fines in Vermont when drivers go over the posted 65 mph speed limit. What did F stand for? What did S stand for? Which was the independent variable: Which was the dependent variable: What did the condition S > 65 mean?)

4. So what do you think F = 10(S-55) + 4, S > 55 means? This has more meaning to you now doesn’t it? This is the formula used to levy fines in Connecticut when drivers go over the posted 55 mph limit.

5. How can we use the table of values on the calculator to evaluate these and to look at the differences in the fines given in both states? Show how we could also get the same information using the value key on the calculator.

6. Review solving linear inequalities. When do you reverse the sign of the inequality and why? Given two inequalities like –5 < -4 and 8 > 5, is each inequality still true if: 1. You add 2 to each side? 2. You add –2 to each side? 3. You multiply each side by 2? 4. You multiply each side by –2?

7. Demonstrate how to solve an inequality using the calculator.

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8. Demonstrate how to solve compound inequalities like 1.) -3 < 2x + 1 < 5 and 2.) -7 < 2 – 3x < 5.

9. Students complete the Graphing on a Number Line in collaborative pairs. Each partner takes turns solving the inequalities by hand and by calculator.

Summarizing Activity: 1. Ticket out the Door: If I were caught speeding at 82 mph, what would be the fine in Vermont and in Connecticut? Which state is more expensive in which to be caught speeding and what in the formulas seems to cause that to be the case?2. Tell your partner when you need to remember to reverse the sign of an inequality.

Homework: Prentice Hall Algebra 2 Text Section 1-4 page 29-30 (1-13, 41-42, 44, 46)

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Graphing on a Number Line

Inequality Graph

1. 3x + 5 > 9

2. -2x + 6 < 10

3. 5x + 4 < 2x + 3

4. x – 12 > 3x – 15

5. -4x + 2.5 > 15 – 7x

6. 3(2x + 1) < -3(x - 4)

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7. 10 > 4 – 2x > -2

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Day 4

Essential Question: How do I solve absolute value equations and inequalities?

Objective(s): 2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

“SAP”: Students will work in collaborative pairs to complete the Number Tile Puzzle.

Lesson Anatomy: 1. Teacher led discussion of troublesome homework problems.2. Quiz on solving equations and inequalities.3. Solve the following for x: x = 8. Why does this equation have two solutions? Could an absolute value equation have more than two answers? less than two answers? no answer?4. Teach the algebraic method of solving absolute value equations using the number line and difference from 0 to show why these have two answers when solved algebraically. What would an absolute value equation look like that only had one answer? What would the absolute value equation look like that had no answer? Use Guided Notes on Solving Absolute Value Equations.5. Use examples 1, 2 and 3 found in the TE pages 33-34 for guided independent practice.6. Collaborative pairs: Number Tile Puzzle. Review Answers7. Guided notes on solving Absolute Value inequalities. Use a distance from zero model to set up the inequalities. For instance, for (< type), like explain that 2x -3 represents a number that is less than 7 units of distance from 0. That would mean it is a number between -7 and 7. So set up and solve the compound inequality -7 < 2x – 3 < 7. The solution would be -2 < x < 5. Take this opportunity to discuss set notation and show how this same solution could be written on the EOC as {x x , -2 < x < 5 }Then for (> type), like , explain that 2x – 3 represents a number that is more than 7 units of distance from 0. That means it could be a number that is less than -7 or could be a number that is more than +7. So set up and solve the two possibilities as 2x -3 < -7 or 2x – 3 > 7. The solution would be x < -2 or x > 5. Take this opportunity to talk about set notation and how this solution could be written on the EOC as {x x , x < -2

x > 5 }8. Use examples 4 and 5 found in the TE page 35 for guided independent practice.

Summarizing Activity: 3-2-1. Name 3 things you must remember to do when solving an absolute value equation or inequality. Name 2 words used when writing a compound statement. Give 1 example of an absolute value equation or inequality that has no solution.

Homework: Absolute Value Equation and Inequality Worksheet

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Number Tile Puzzle

Arrange the digits 0-9 into the empty boxes so that each of the five equations is mathematically correct. Each of the digits may only be used once.

1. │2X - │ = 11 ; X = -2,

2. │4X - 12│ = ; X = 1,

3. │12 - X │ = 2 ; X = 1 , 1

4. │3X - │ = 12 ; X = 6, -

5. │10 - X │ = ; X = 7, 3

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Number Tile Puzzle (Answers)

Arrange the digits 0-9 into the empty boxes so that each of the five equations is mathematically correct. Each of the digits may only be used once.

1. │2X - 7 │ = 11 ; X = -2, 9

2. │4X - 12│ = 8 ; X = 1, 5

3. │12 - X │ = 2 ; X = 1 0 , 1 4

4. │3X - 6 │ = 12 ; X = 6, - 2

5. │10 - X │ = 3 ; X = 7, 1 3

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Guided Notes on Solving Absolute Value Equations and Inequalities

I. Solve the following absolute value equations:

1. │2x – 5 │= 7 2. 4│x + 1 │= 8 3. │7 – x │= -5

II. Solve the following absolute value inequalities (< type):

1. 3x + 1 < 1 2. 6x + 5 < 3 3. 5x – 2 < 3

III. Solve the following absolute value inequalities (> type):

1. 2x + 2 > 4 2. 5x – 1 > 9 3. 7 – 3x > 10

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Absolute Value Equation/Inequality Worksheet

I. Solve each equation. Check your solution.

1. 2x – 1 = 5 2. 4 x = 20 3. 2 3x + 7 + 7 = 15

II. Solve each inequality. Check your solution graphically.

1. 3x + 5 < 4 2. 3 2x – 4 < 20 3. 2 x + 4 > 10

4. -2 – 5x > 8 5. 2 5x - 8 < 22 6. -2 x – 1 + 4 < -8

7. –3 x + 1 > -6 8. x – 4 - 1 > 8 9. 2 10 – 2x < 2

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Day 5

Essential Question: How do I solve linear and absolute value equations and inequalities?

Objective(s): 2.08 Use equations and inequalities with absolute value to model and solve problems: justify results, a) Solve using tables, graphs, and algebraic properties.

“SAP”: Students will work with a partner to play MATHO.

Lesson Anatomy: 1. Quiz Discussion.2. Have index cards prepared so that every student in the class will get to choose a card. On half of the index cards have an absolute value equation or absolute value inequality. On the other half have answers to the equations/inequalities. Students draw a card and find their match. Then they change seats to sit near the person who held their matching card. The partners should compare their answers to homework and come to a consensus on the correct solution.3. Teacher led discussion of troublesome homework problems.4. Using the same partners, students will play MATHO. This is designed much like the game Bingo. Each pair selects 25 answers to fill in their game card. Students use the third space under the letter T for their free space. The teacher randomly selects problems for the students to solve. Each partner pair works the problems until they have MATHO. Once MATHO is called, the teacher checks the answers for accuracy. The winners of the game receive a bonus point added to their test.

Summarizing Activity: Four Corners Activity: Go to the corner of the room that represents your favorite activity this week A. Working with a partner

B. Working with the calculatorC. Number Tile PuzzleD. MATHO

Ask students to voluntarily explain their choice.

Homework: Prentice Hall Algebra 2 Text Section 1-5 pages 36-37 (4, 5, 12, 13, 16, 20, 23, 26, 34, 35, 41)

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MATHO

Problems Answers Number

1. 5x – 2 < 13 1. x < 3 182. –5x + 12 = -8 2. x = 4 153. –3 4x + 2 = 17 3. 224. 12 – 3x = 36 4. x = -8 or x = 16 15. x + 2 > 5 5. x < -7 or x > 3 206. 6x – 4 > 10 – 8x 6. x > 1 77. 1/3 x + 1 < 2 7. –7 < x < 5 218. 3 4x - 12 = 60 8. x = -6 or x = 6 199. 3x – 8 > 7 9. x > 5 2310. x > 1 10. x > 1 or x < -1 2611. –15x = 66 – 4x 11. x = -6 1712. 2x + 6 < 10 12. –8 < x < 2 1413. -3x = 81 13. x = 27 or x = -27 814. 5 – 3(x + 2) > 7 – x 14. x < -4 1315. x + 3 < 1 15. –4 < x < -2 316. 5x - 16 > 4 16. x < -4 or x > 4 2917. 4x + 3 < -7 17. x < -5/2 3018. 9x + 6 = 3x + 30 18. x = -3 or x = 4 2519. 6(1 – 2x) = -34 + 8x 19. x = 2 2820. 8x < 16 20. –2 < x < 2 1621. 2x – 1 > 5 21. x < -2 or x > 3 2722. 1/3 x + 3 = 0 22. x = -9 223. 12.4 x – 2 = 18.2x + 9.6 23. x = –2 1224. 2x + 4 - 3 = 6x + 1 24. x = 0 525. 7x > -42 25. all real numbers 926. –3x > 27 26. x < -9 2427. 8(x – 4) = -56 27. x = -3 428. x + 1 = 7 28. x = -8 or x = 6 1029. –3x > 6 29. x < -2 1130. 2x – 5 < 3 30. 1 < x < 4 6

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Day 6

Essential Question: How do I prepare for the test on linear and absolute value equations and inequalities?

Objective(s): 1.03, 2.08a

“SAP”: Students will use white board activity to review for the Unit Test.

Lesson Anatomy: 1. Teacher-led discussion of most difficult homework problems.2. Each student should receive Checkpoint Quizzes 1 and 2 from Prentice Hall Chapter 1 Support File. Go through each of the problems. Give students time to work out each problem on their white board. Ask students to hold their answers up so that you can assess their understanding.

Summarizing Activity: Number off 1 or 2. Number 1’s: Create an equation or inequality at least as challenging as the ones discussed in class today and write your solution to it on a different piece of paper. Pass the problem to a Number 2 for them to solve. Check their solution against your own. If both answers agree, staple the papers together and leave them with the teacher for a bonus point for each person. The bonus point will be added to tomorrow’s test grade.

Homework: Study for the test.

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Day 7

Essential Questions: Am I ready to show what I’ve learned on the Unit Test today?How do I graph a line using slope-intercept?

Objective(s): 1.03, 2.08a

“SAP”: Students will complete the graphing calculator activity.

Lesson Anatomy: 1. Begin with a 5-minute review for the test. 2. Unit Test. Your test should be designed so that it takes the slowest working student no more than 50 minutes to complete.3. Use the Algebra Review: The Coordinate Plane (TE page 54) to re-introduce students to graphing. Discuss how to use the slope and y-intercept when graphing a linear equation written in its slope-intercept form using the following examples:

a. 3x – 4y = 2b. 2x = 10 – 5yc. 2x – y = 0

4. In collaborative pairs, students complete the graphing calculator activity. Review answers.

Summarizing Activity: Sentence Stem: One thing I had difficulty with in today’s lesson was …

Homework: Prentice Hall Algebra 2 Text Section 2-2 page 68 (42-53)

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Graphing Calculator Activity on Linear Equations

Use a graphing calculator to graph each equation. Describe the viewing window that you used to view a complete graph for each equation.

1. y – x = 14 2. y = 5x – 35 3. 100x + y = 5

X-min = ____ X-min = ____ X-min = ____ X-max = ____ X-max = ____ X-max = ____ X-scl = ____ X-scl = ____ X-scl = ____ Y-min = ____ Y-min = ____ Y-min = ____ Y-max = ____ Y-max = ____ Y-max = ____ Y-scl = ____ Y-scl = ____ Y-scl = ____ X-res = ____ X-res = ____ X-res = ____

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Day 8

Essential Question: How do I graph relations and identify functions? How do I look at a graph and name the domain and range? Objective(s): (1998 Curriculum)3.01 Describe graphically, algebraically and verbally real-world phenomena as functions; identify the independent and dependent variables; 3.02 Translate among graphic, algebraic and verbal representations of relations.

“SAP”: Students will complete the Domain/Range Worksheet in collaborative pairs. Students also will complete the Graphing Calculator Domain and Range activity in cooperative pairs.

Lesson Anatomy: 1. Test Discussion2. Teacher led discussion of troublesome homework problems.3. Discuss the vocabulary, which is highlighted in yellow in Section 2-1 (TE pages 55-58).4. Using the students in the class and the sports they play, create a concept map that is a function. Alter it so that it is not a function. See examples. Name the independent and dependent variables.5. Give students 5-7 minutes to complete the Domain/Range Activity. Discuss with class.6. Teacher led discussion on function notation. Function notation is used to represent relations which are functions. Some letters commonly used to represent functions are f, g, h, F, G, and H. Have students practice evaluating functions written in function notation. Use Check Understanding problem #6a-c (TE page 58) for independent guided practice.7. Teacher demonstrates through several graphs how to identify the domain and range of a function or relation by looking at a graph. See (TE page 60) problems 38 and 39 for independent guided practice. 8.. In cooperative groups of three, students complete the graphing calculator domain and range activity. Review answers.Summarizing Activity: Put students in groups of 4. Give each group an overhead transparency and overhead pen. Have each group create their own concept map. It may or may not be a function. It can be on any relationship they choose as long as it is “clean.” When all groups are ready, have the groups present their concept maps to the class. When presenting they need to tell whether their creation is or is not a function and which variable is independent and which variable is dependent ORTicket out the door: Graph a function that has a domain of -1≤ x ≤ 5 and a range of -3 ≤ y ≤ 2. Be sure that your function passes the vertical line test.

Homework: Prentice Hall Algebra 2 Text Section 2-1 pages 59-61 (8, 12-15, 16-26, 30, 36, 37, 43-45, 62)

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IS A FUNCTION…

Susan Basketball

Devin Football

Tiffany Cheerleading

Michael Soccer

Amy

IS NOT A FUNCTION…

Sarah Basketball

Jason Wrestling

Eugene Football

Brett Baseball

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Domain/Range

Find the domain and range of each relation. Is the relation a function? Why or why not?

1. 1 1 2. {(-1,1), (2,4), (-1,7), (3,4)}

2 2 Domain: _________________

3 Range: _________________

Function? ____________________Domain: ____________________

Range: ____________________

Function? ____________________

3. y = 12 – x 4. x = -3

Domain: ____________________ Domain: __________________

Range: ____________________ Range: __________________

Function? ____________________ Function? __________________

5. Bob 6. y = 4Beth

Dave Diane Domain: __________________

JohnMarcia Range: __________________

ChuckFunction? __________________

Domain: ____________________

Range: ____________________

Function? ____________________

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Graphing Calculator Domain/Range Activity

Graph each of the following on your graphing calculator. State the domain and range of each relation.

1. y = x2 – 4 Domain: _______________ Range: ______________

2. y = √(4-x) Domain: _______________ Range: ______________

3. y = │x + 3│ - 2 Domain: _______________ Range: ______________

4. y = x + 1 Domain: _______________ Range: ______________x – 3

5. y = x3 – 2x + 3 Domain: _______________ Range: ______________

6. y = -x2 + 2x – 15 Domain: _______________ Range: ______________

7. y = -3 + √(x + 2) Domain: _______________ Range: ______________

8. y = - ½ x – 4 Domain: _______________ Range: ______________

9. x = 7 Domain: _______________ Range: ______________

10. y = 3√x Domain: _______________ Range: ______________

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Day 9

Essential Question: How do I graph a line using x and y-intercepts? How do I graph horizontal and vertical lines?

Objective(s): This is a preview lesson for the following objective:2.04 Create and use best-fit mathematical models of linear functions to solve problems involving sets of data b) Check the model for goodness-of-fit and use it to draw conclusions or make predictions.

“SAP”: Students will practice graphing linear equations using the white boards.

Lesson Anatomy: 1. Teacher led discussion of troublesome homework problems.2. Given the formula F = (9/5)C + 32 which shows the conversion between Fahrenheit and Celsius temperature, which would be the independent variable? the dependent variable? Complete the following table of corresponding temperatures.

Celsius Fahrenheit -40 ? -30 -22 ? -4 -10 ? 0 32 10 ? 20 ? 30 ? 40 ?

3. Teacher led discussion on graphing linear equations using x and y-intercepts, points at which the graph intersects each axis. Use examples 1 and 2 TE page 63, and have students graph Check Understanding problem 1 (a-c) using x and y intercepts

4. Discuss the equations of horizontal and vertical lines and review slope. Graph each of the following:

a. 3y = 5b. x – 4 = 0

5. In collaborative pairs, students practice graphing linear equations on the white boards using the problems on the Graphing Linear Equations (White Board Practice) page.

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Summarizing Activity: Have students on their calculator demonstrate their ability to draw a line that rises from left to right, one that falls from left to right, and one that is horizontal. Talk about the slope of these lines as you are checking their graphs. Finally ask the students to draw a line that is vertical. Hopefully they will tell you that they cannot do this because it is undefined for slope and is not a function. It can’t be graphed on the TI-83 since it is a function graphing calculator.

Homework: Prentice Hall Algebra 2 Text Section 2-2 page 68 (42-53)

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Collaborative Pairs: Graphing Linear Equations (White Board Practice)

Graph using the slope and y-intercept.

1. y = 3x – 5 2. x = 4y – 5 3. 4x – 3y = 9

Graph using x and y intercepts.

1. 5x + 3y = 15 2. 5 + 2y = 3x 3. y = 4x – 1

Graph each equation using the method of your choice.

1. x = 5 2. 2x – 5y = 10 3. ½ x + y = 3

4. y = -3 5. 3x = 2y 6. 2x = 5 – 3y

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Day 10

Essential Question: How do I graph two variable inequalities?

Objective(s): 2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

“SAP”: Students will work in collaborative pairs to check guided practice problems.

Lesson Anatomy:1. Teacher led discussion of troublesome homework problems.2. Teacher led discussion of graphing two-variable inequalities. A dashed boundary line indicates that the line is not part of the solution. A solid boundary line indicated that the line is part of the solution. For an inequality with the symbols < or <, shade above or to the right of the line. For an inequality with the symbols > or >, shade below or to the left of the line. Use Example 1 (TE page 100). Have students complete Check Understanding problem #1a-c for student independent guided practice. If needed, here are some additional examples:a) x + 3y < 2b) 4x – y < 1c) ½ x > y – 3 d) 3x > 5

3. Show students how to use the graphing calculator to graph linear inequalities. Press y= and left arrow to the graph style icon (left of y1.) Press enter until you obtain the correct shading. Upper triangle icon graphs greater than and lower triangle icon graphs less than.4. Pairs checking. Assign to students problems 1-9 on page 102 and problems 38-40 on page 103 of the Text. Each student does their own work. Have students circle even numbered items. When they get to a number that is circled, they should STOP! They cannot go on unless the previous answers agree with their partner’s answers. If they agree, they should continue working. If they do not agree they should justify their answers until they agree. Collect for a class work grade.5. Discuss with students how to write and use inequalities that can help solve application problems. For a student example use Example 2 (TE pg 100) and Check Understanding 2a-b for guided independent practice.6. Use Example 4 (TE page 12) to show students how to write inequalities for graphs. Use Check Understanding 4b for guided independent practice.

Summarizing Activity: Ticket out the Door: Graph 2x – 3y < 9. Is it a function? Why or why not?

Homework: Prentice Hall Algebra 2 Text Section 2-7 pages 102-103 (2, 8, 10, 20, 21, 26-28, 38-40)

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Day 11

Essential Questions: How do I model real-world data using scatter plots? How do I make predictions from linear models?

Objective(s): 2.04 Create and use best-fit models of linear functions to solve problems involving sets of data. b) Check the model for goodness-of-fit and use it to draw conclusions and make predictions.

“SAP”: Students will work in collaborative pairs to analyze data sets.

Lesson Anatomy: 1. Teacher-led discussion of troublesome homework problems.2. Review for quiz on sections 2-1, 2-3, 2-7 on tomorrow using the following problems from the Prentice Hall Text: pg. 70 (91-93), p. 76 (61-63), pg. 77 (69-70), pg. 823 (9, 12, 29, 30, 40, 59, 63, 65)3. Talk about scatter plots created from functions and learn to look at them to decide if there is no correlation in the data, a positive correlation, a strong positive correlation, a negative correlation, or a weak negative correlation.4. Demonstrate how to enter data into the calculator and to analyze the trend line using the following data sets:

a. Trash Productionb. Summer Games Olympic Gold Medal Track Recordsc. Untitled Set (Can students figure out what data this represents?)d. U.S. Postal Rates

Pass out a copy of the directions for data analysis. Copy this on colored paper to make it easier for students to find when they need it.With each set of data analyzed, answer the following questions:

a) How would you describe the correlation?b) What is the correlation coefficient?c) What is the equation of the trend line?d) Using the prediction equation, what would you predict the dependent value would be

for a teacher chosen independent value? For example, what would you expect the gold medal record time was in 2000 based on this data?

e) What is your interpretation of the slope of the trend line?

Summarizing Activity: Sentence Stem - One thing that I am confused about is…

Homework: Prentice Hall Algebra 2 Text: Technology: Finding the Line of Best Fit page 85; Study for Quiz

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Data Analysis Calculator Directions

To enter data:Stat, 1Put data into L1 and L2.

To see scatter plot:Y=, up arrow to Plot1, ENTER, Zoom, 9

To find the correlation coefficient ( r ):Cut the diagnostic on in the catalog and r shows up as you calculate the regression.

To write equation of the trend line (prediction equation):Stat, right arrow to CALC, 4, enter

To graph the trend line:Y=, Vars, 5, right arrow to EQ, 1

To predict from data:2nd, Trace, 1, put in the value for x, Enter(If you get ERR: Invalid, then go to the Window and change the x max or x min so that the value of x lines between xmin and xmax.)

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Linear Data Sets

U.S. Trash Production (millions of tons)

Year Trash1960 881965 1031970 1221975 1281980 1521985 1641990 196

Summer Games Olympic Gold Medal Track Record

Year Time (in seconds)1980 10.25 (Allan Wells, Britain)1984 9.99 (Carl Lewis, US)1988 9.92 (Carl Lewis, US)1992 9.96 (Linford Christie, Britain)1996 9.84 (Donovan Bailey, Canada)

X 32 50 59 77 86 98.6 212 23 5 -4 -31Y 0 10 15 25 30 37 100 -5 -15 -20 -35

United States Postal Rates

Year Stamp Cost1958 .041963 .051968 .061971 .081974 .101975 .131978 .151981 .181985 .221988 .251991 .291995 .321999 .332001 .342002 .37

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Day 12

Essential Question: How do we analyze linear data using the TI-83?

Objective(s): 2.04 Create and use best-fit models of linear functions to solve problems involving sets of data. b) Check the model for goodness-of-fit and use it to draw conclusions and make predictions.

“SAP”: Students will work in collaborative pairs to complete the calculator lab on Linear Data Analysis.

Lesson Anatomy:1. Quiz on Sections 2-1, 2-2, and 2-7.2.. EOC Practice Problem. Correct Answer: D

The table below shows the number of doctors in Bingham City from 1960 to 1986.

Year 1980 1967 1970 1975 1982 1985 1986

Number of Doctors

2,937 3,511 3,754 4,173 4,741 5,019 5,102

If the linear regression model is fit to this data, which statement would best describe the model (let x = 0 for 1960)?

A The equation y = 1.01x – 3,500 is the line of best fit for this data, showing that the number of new doctors in Bingham City has increased by 1% each year.

B The equation y = 82x + 2,937 is the line of best fit for this data, showing that approximately 82 new doctors came to Bingham City over the 28-year period.

C The equation y = 83x + 2,929 is the line of best fit for this data, showing that the number

of new doctors in Bingham City has increased by 83% over the 26-year period.

D The equation y = 83x + 2,929 is the line of best fit for this data, showing that the numberof doctors in Bingham City increased, on average, by 83 each year.

3. Teacher-led discussion of troublesome homework problems.4. Go over steps used when analyzing data on the TI-83 calculator using Practice Problem #13 (TE pg 81)

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5. With a partner and data analysis instruction sheet, work independently through all of the given data sets. Record the answer to the questions in the space provided. Turn papers in to be graded.

Summarizing Activity: none

Homework: Prentice Hall Algebra 2 Text Section 2-4 page 81-83 (4-7, 12, 20, 21)

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Calculator Lab (Linear Data Analysis)

Year Nolan Ryan’s Career Strikeouts

1966 61968 1391969 2311970 3561971 4931972 8221973 12051974 15721975 17581976 20851977 24261978 26861979 29091980 31091981 32491982 34941983 36771984 38741985 40831986 42771987 45471988 47751989 50761990 53881991 5511

a) How would you describe the correlation? __________________________________

b) What is the correlation coefficient? __________________________________

c) What is the equation of the best-fit line? __________________________________

d) Using the prediction equation, what would you predict for the number of career strikeouts he would have had in 1992? ______________________________________________

e) What is your interpretation of the slope of the best-fit line? ______________________________________________________________________________________

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Cancer Deaths

The following data represents the number of cancer deaths per 100,000 people compared to the miles from where they live to the Columbia River in Washington State. A hydroelectric plant had been built by the river and industrial waste was being discarded into the river. The EPS required a massive cleanup of the river after this was discovered through a study of this large number of cancer deaths.

Miles to River Cancer Deaths per 100,0009.5 1479.4 1308.6 13010.7 11410.4 1388.2 1620.4 2085.6 1783.7 210

a) How would you describe the correlation? __________________________________

b) What is the correlation coefficient? __________________________________

c) What is the equation of the best-fit line? __________________________________

d) Using the prediction equation, what would you predict for the number of cancer deaths for people living at a radius of 7 miles from the river? ____________________________

e) What is your interpretation of the slope of the best-fit line? ___________________________________________________________________________________________

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Country Cigarette Consumption Per adult per year

Coronary Heart Disease Deaths Per 100,000 people

United States 3900 257Canada 3350 212New Zealand 3220 212Great Britain 2790 194Ireland 2770 187W. Germany 1890 150Netherlands 1810 125Belgium 1700 118Italy 1510 114Sweden 1270 137

a) What do you think is the independent variable for this data? ________________________

b) Is this a function? Why or why not? ________________________________________________________________________________________________________________________

c) How would you describe the correlation? __________________________________________

d) What is the correlation coefficient? ______________________________________________

e) What is the equation of the best-fit line? __________________________________________

f) Using the prediction equation, what would you predict for the number of heart disease deaths per 100,000 people for adults smoking 3000 cigarettes per year? ______________________________________________________________________________________________________

g) What is your interpretation of the slope of the best fit line? ______________________________________________________________________________________________________

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U.S. Cable Subscribers (rounded to the nearest million)

a) What do you think is the independent variable for this data? ________________________

b) Is this a function? Why or why not? ________________________________________________________________________________________________________________________

c) How would you describe the correlation? __________________________________________

d) What is the correlation coefficient? ______________________________________________

e) What is the equation of the best-fit line? __________________________________________

f) Using the prediction equation, what would you predict for the number of cable subscribers we had in the U.S. in 1990? ______________________________________________________

g) What is your interpretation of the slope of the best fit line? ______________________________________________________________________________________________________

Year Number of Subscribers1975 101976 111977 121978 131979 151980 181981 221982 251983 291984 331985 351986 391987 411988 441989 47

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Day 13

Essential Question: How do I find the composite of two functions?

Objective(s): 2.01 Use the composition and inverse of functions to model and solve problems; justify results.

“SAP”: Students will work in collaborative pairs to name the domain and range of graphs.

Lesson Anatomy: 1. Quiz Discussion2.. Teacher led discussion of troublesome homework problems.3. EOC Practice Problem (below) Correct Answer: C

The table shows the relationship between calories and fat grams contained in orders of fried chicken from various restaurants.

Calories 305 410 320 500 510 440Fat Grams 28 34 28 41 42 38

Assuming the data can be described by a linear model, how many fat grams would be expected to be contained in a 275-calorie order of fried chicken?

A 28 gramsB 27 gramsC 25 gramsD 22 grams4. Students should complete the skills check in Section 7-6 (TE pg 392) in the text. Review answers.5. On overhead, draw several types of graphs such as discrete points, lines, parabolas, and absolute value function graphs. Have collaborative pairs name the domain and range.

6. Pose the following problem:

Temperature is measured in different units in different countries. An American scientist and a German scientist are working on incubating bacterium in their respective countries. They are sharing their findings with each other through their respective countries. They are sharing their findings with each other through the Internet. The last message from the German scientist says that her bacterium died at a temperature of 312˚K, from which she discovered was not warm enough. The American scientist’s temperature for incubation is 98.2˚F. Should the American scientist be worried?

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Ask students to think about how they could figure this out. Soon someone will say that they need to know the conversion formulas. At that time give them the formulas: K(x) = C + 273 for converting Celsius to Kelvin temperature and C(x) = 5/9 (F – 32) for converting Fahrenheit to Celsius temperature. After students in their collaborative pairs have arrived at their conclusion and have defended it, discuss strategies for arriving at the conclusion and the function notation used. Ask them what would have made this process easier? Someone will say that it would have been easier if there had been a direct formula to convert Celsius to Kelvin or vice versa. Use composition of function to create that new formula and then test it out on the problem.

7. Use the following examples to demonstrate the different notation forms.

If f(x) = 3x – 4 and g(x) = 5 – x, find each of the following:a) f(g(x)) b) g(f(x))

If f(x) = x2 and g(x) = 2x + 7, find each of the following:a) (g ◦ f)(x) b) (f ◦ g)(x) c) (g ◦ g)(x) d) (f ◦ f)(x)

If f(x) = x2 – 2x + 3, g(x) = x + 9 and h(x) = 4x – 1, find each of the following:a) f(g(x)) b) g(h(x)) c) (f ◦ g)(x) d) (g ◦ f)(x) e) (h ◦ g)(x) f) f(h(x))g) f(g(3)) h) g(f(-3)) i) (f ◦ h)(1/4) j) (h ◦ g)(10) k) (h ◦ h)(-1) l) h(f(1/2))

8. Show students, on the TI83, how to do composition of functions when putting in a specific number for x. Use Example 3 (TE pg 393). Have students complete Check Understanding problems 3a and 3b.

Summarizing Activity: Ticket out the Door: Write an explanation of how to find f ◦ g and g ◦ f for any given function f and g.

Homework: Function Composition Worksheet

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Function Composition Worksheet

I. If f(x) = x – 4 and g(x) = 2x + 3, find each of the following.1. f(g(x))= __________ 2. g(f(x))= __________ 3. f(g(3))= __________

4. g(f(3))= __________ 5. (f ◦ f)(x)= __________ 6. (g ◦ g)(x)= __________

II. If f(x) = -2x 2 – 3 and g(x) = 2 – x, find each of the following. 1. (g ◦ f)(x)= __________ 2. (f ◦ g)(x)= __________ 3. (g ◦ g)(-2) = __________

4. (f ◦ g) (-2)= __________ 5. (g ◦ f)(-2)= __________ 6. (f ◦ f)(x)= ____________

III. a. Find g(f(x)) and f(g(x)) b. Tell whether (g ◦ f)(x) = (f ◦ g)(x) or (f ◦ g)(x) ≠ (g ◦ f) (x). 1. f(x) = x and g(x) = -x 2. f(x) = 3x and g(x) = x/3 3. f(x) = x – 5 and g(x) = 5 – x

4. f(x) = x + 3 and g(x) = x – 3 5. f(x) = x and g(x) = -x/2 6. f(x) = x and g(x) = 1/x

42


Day 14

Essential Question: How do I find the inverse of a relation?

Objective(s): 2.01 Use the composition and inverse of functions to model and solve problems; justify results.

“SAP”: Students will work in collaborative pairs to complete the Inverse Worksheet.

Lesson Anatomy: 1. Have partners check and come to consensus on homework solutions.2. Teacher-led discussion of troublesome homework problems.3. On the overhead write the following problems and ask collaborative pairs to find the solutions using a TI83:

g(x) = 2x2 – 3x – 6 and h(x) = -x 2 + 5x + 6 x – 3

a) g(-3)b) h(-3)c) g(3)d) h(3)e) g(h(-5))f) h(g(4))g) (g ◦ f)(-1)h) (f ◦ g)(-1)4. Demonstrate how to find the inverse of a relation using Examples 1, 2 and 3 (TE pgs. 401-402) Stress that it is a two-step process:

a. Interchange x and y.b. Solve for y. The y relation is f -1.

Be sure to make the graphical connection between a function and its inverse being symmetric across the y = x line (because of interchanging x and y).5. Collaborative Pairs. Complete the Inverse Worksheet. Check the class results.

Summarizing Activity: Ask students to explain how they can find an equation that models the inverse of a relation or a function.

Homework: Composition of Functions and Inverses Worksheet

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Partners: Find the inverse (f -1 or f -1(x)) for each of the following:Show your work in the space to the right of the problem.

1. f(x) = {(2, 4), (4, 8), (8, 16)}

f -1(x) = __________________________________________

2. f(x) = 4x – 1

f -1(x) = __________________________________________

3. f(x) = 5x – 2

f -1(x) = __________________________________________

4. g(x) = -3x + 3

g -1(x) = __________________________________________

5. h(x) = x2 - 1

h -1(x) = __________________________________________

44

Composition of Functions and Inverses Worksheet

I. Given f(x) = │2x + 3│and g (x) = x 2 – 7 and h(x) = 3x + 2, find the following:

1. f(-6) = ____________ 2. g (-6) = ____________ 3. f(g(-6)) = ____________

4. g(f(-6)) = __________ 5. g(h(-1)) = __________ 6. h(g(-1)) = ____________

7. f(g(-2)) = __________ 8. g(f(-2)) = ___________ 9. g(h(-2)) = ____________

II. Still using the given functions, show your work for composing the requested functions in the space provided.

10. f(g(x)) 11. g(f(x)) 12. h(g(x)

13. g(h(x)) 14. (f ◦ g)(x) 15. (g ◦ h)(x)

16. (h ◦ g)(x) 17. (g ◦ g)(x) 18. ( h ◦ f)(x)

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III. Find the inverse of each of the following:

1. f(x) = {(-1, 2), (-2, -4), (-3, -6)}

f -1(x) = __________________________________________

2. f(x) = -x + 2

f -1(x) = __________________________________________

3. f(x) = -5x – 3

f -1(x) = __________________________________________

4. g(x) = x + 3 5

g -1(x) = __________________________________________

5. h(x) = x2 + 3

h -1(x) = __________________________________________

6. f(x) = ½ x – 7

f -1(x) = __________________________________________

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Day 15

Essential Question: How do I prepare for the test on relations and functions?

Objective(s): 1.05, 2.01, 2.04b, 2.08a and b,

“SAP”: Students will play “tic-tac-toe” to review for the test.

Lesson Anatomy: 1. Partners compare homework answers and try to come to consensus on the correct solution.2. Teacher-led discussion of troublesome homework problems.3. Have students work on the EOC Practice Problems (below). Review answers.

EOC Practice Problem #1: Answer: CThe distance required for a car to stop is directly proportional to the square of its velocity. If a car can stop in 112.5 meters at 15 kilometers per hour, how many meters are needed to stop at 25 kilometers per hour?

A 250.75B 298.00C 312.50D 337.50

EOC Practice Problem #2: Answer: AThomas rented a van for $75 a day plus $0.25 for each mile that he would go over 3,000 miles. How can Thomas represent the cost, C, of renting the van for d days and driving m miles(m > 3,000)?

A C = 75d + 0.25(m – 3,000)B C = 75d + 0.25(m + 3,000)C C = 75d + 25(m – 3,000)D C = 75d + 25(m + 3,000)

EOC Practice Problem #3: Answer: CIf f(x) = x2 – x and g(x) = x – 1, what is f(g(x))?

A x2 – x – 1 B x2 – x – 2C x2 – 3x + 2D x2 – 3x + 1

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EOC Practice Problem #4: Answer: AWhich of the following is the inverse of f(x) = 2x – 3 ?

5A f -1 (x) = 5x + 3

2

B f -1 (x) = -2x + 3 5

C f -1 (x) = 2y – 3 5

D f -1 (x) = -5y – 3 2

4. Students number off 1 and 2 for two teams. Arrange the desks so that the teams are facing each other. Draw a tic-tac-toe game board on the overhead or on the board. Number the squares from 1 to 9. Assign each square a problem from the Chapter Review sections of Chapter 2 (TE pgs. 107-109) and Chapter 7 (TE page 417). Do not reveal the problem until students pick the square. This game is played much like Hollywood Squares. One student from team number 1 (the X team) picks a numbered square. All students must work independently to do the problem because if the student’s answer is incorrect, the other team will have the opportunity to “steal” the square. Some problems may require you to go back and forth several times from team to team until someone answers correctly. Once a student has answered correctly, ask if anyone has questions about the problem before continuing the game. Assign the next problem to the team that did not answer correctly. You can play several rounds of this game. At the end of each game, all students on the winning team receive a bonus point on the test tomorrow.

Summarizing Activity: Number 1’s: Create a problem at least as challenging as the ones discussed in class today and write your solution to it on a different piece of paper. Pass the problem to a Number 2 for them to solve. Check their solution against your own. If both answers agree, staple the papers together and leave with the teacher for a bonus point for each person on tomorrow’s test.

Homework: Study for test.

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Day 16

Essential Questions: Am I ready to show what I’ve learned on the Unit test today? How do I solve a system of equations by graphing?

Objective(s): 2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

“SAP”: Students will complete the Analyzing Graphs Calculator Activity

Lesson Anatomy: 1. Begin with a 5-minute review for the test.2. Unit Test. 3. When all tests are collected, have students work in collaborative pairs on the Analyzing Graphs Calculator Activity. This activity is found on page 116 of the text. Tell students that lines that coincide lie on top of each other. Review Answers.

Summarizing Activity:Ticket out the Door. Give an example of a system of equations with no solution.

Homework: none

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Analyzing Graphs

I. Use a graphing calculator to graph each pair of equations and for each pair, answer the following questions:

1) Do the graphs have any points in common; if so, how many?2) Compare the slopes of the graphs. What is the relationship between the slopes and the number of common points?

a. y = x + 5 b. y = 3x + 2 c. y = -4x – 2 y = -2x + 5 y = 3x – 1 y = 8x + 4

-2

1) 1) 1)

2) 2) 2)

II. Complete the table for the graphs of two linear equations.

Description of Lines

How many Points of Intersection?

Equal slopes?(yes/no)

Same y-intercepts?(yes/no)

Intersecting

Parallel

Coinciding

50


Day 17

Essential Question: How do I use rules of exponents to simplify polynomial expressions?


“SAP”: Students will participate in the Matho Game.

Lesson Anatomy: 1. Test Discussion 2. Teach the rules for exponents, including negative exponents. Be sure that students know that

a simplified expression has no negative exponents. What does a negative exponent and 0 as an exponent mean? Demonstrate why in the following manner:

33 2732 93 030 13-1 1/33-2 1/93-3 1/27

Emphasize that expressions like 3-3 mean the reciprocal of 33. Demonstrate examples of all rules with all types of integer exponents in multiplication and division of monomials.3. White board practice for students on multiplication and division of monomials.4. In collaborative pairs students will play the Matho Game.

Summarizing Activity: Number off 1 and 2.

1 2 What is always the result when you raise a negative monomialbase to an even power? Why?

2 1 What is always the result when you raise a negative monomialbase to an odd power? Why?

Homework: Prentice Hall Algebra 2 Text Skills Handbook: Operations with Exponents page 852 all

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Matho Game on Monomial Multiplication & Division with Negative Exponents

Never leave a negative exponent in your final answer. Problems Answers Problems Answers 1. (2a3)(-4a-5) 9 16. (-3a-2b)2 25

2. -16a -6 18 17. (2a4)(4a-3) 17 -2a-4

3. (4a-3)-2 14 18. (a 4 )(a -1 ) 2 a-7

4. (2a3)-2 24 19. (6a -5 )(2a -4 ) 13 2a-2

5. (-4a-3)2 16 20. -12a -4 19 3a-1

6. (-4a5)-2 1 21. (-2a-3)2 8

7 . 3a -4 2 30 22. (-4ab4)-2 11 a5

8. (-2a-2)4 20 23. (2a2)-5 26

9. (-3a-3)-3 5 24. a -2 -2 4 b-4

10. (3a5)-2 23 25. ( ¼ a-2b3)2 29

11. (-4a-7)2 6 26. 24a -9 7 2a-2

12. a 3 15 27. (3a-2b)-3(2a)2 21 a-3

13. (a-3)-1 28 28. 36a -8 12 9a8

14. (2a-3)(a5)(-2a3) 10 29. ( ½ a-2b3)-2 27

15. (3a2b)-3 22 30. (2a-5)5 3

52


Day 18 Essential Questions: How do I add and subtract polynomial expressions?How do we multiply all types of polynomials?

Objective(s): 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems

“SAP”: Polynomial Races

Lesson Anatomy: 1. How do we add?;

a) 2/5 + 2/5b) 1/3 + 4/3c) 2π + 3πd) 4x + 5xe) 2 + 5f) 4 + 5

What pattern do you see? So how would we need to add?;

a) 2x2 + 2x2

b) 1x2 + 2x2

c) 2x3 + 2x3

d) 4x5 + 5x5

e) 2x2 + 5x2

f) 4x3 + 5x3

4. What would be different about subtraction of the above?5. Demonstrate with red and blue manipulative models.6. Show examples involving addition/subtraction of polynomial expressions. 7. Polynomial Races #1 and #2:Students should move desks side by side with partner and grouped into 4 partner rows to a team. Hand face down the Polynomial Race #1 to the first partner group in each team. On the Go! Command, the first group should complete the directions for Partner Group 1, write down their answer and then pass the paper over their head to Partner Group 2. They do their work and pass the paper on to the next group. This continues until the last group in the team gets the paper. When they have completed writing down their answer, they should raise their hands. The teacher should be watching for the order in which the hands are raised. The first group to have the correct answer earns a bonus point for each member of the team. If a team gets an incorrect answer, tell them to group together to try to correct their answer until a winner is declared. Do again with Polynomial Race #2 .8. Begin reviewing multiplication of polynomials involving binomials and trinomials.

54

Summarizing Activity: Ticket out the door:Find the perimeter and area of the rectangle.

(2x + 3y) m

(7x – 2y) m

Homework: Addition and Subtraction of Polynomials Worksheet

55

Polynomial Race #1

Start with: 3x2 – 2x – 3

Partner Group 1:Add -6x2 – 4x + 3 to the start polynomial. Answer:___________________________

Partner Group 2:Add -6x – 4 to the last answer. Answer:___________________________

Partner Group 3:Add 4x2 – 2x + 3 to the last answer. Answer:___________________________

Partner Group 4:Add -1x2 – 3x – 6 to the last answer. Answer:___________________________

_________________________________________________________________________

Polynomial Race #2

Start with: 2x – 3

Partner Group 1:Add -3x – 1 to the start polynomial. Answer:___________________________

Partner Group 2:Subtract -3x – 1 from the last answer. Answer:___________________________

Partner Group 3:Add 6x2 + 3x – 5 to the last answer. Answer:___________________________

Partner Group 4:Subtract -1x2 – 3x – 6 from the last answer. Answer:___________________________

56

Addition and Subtraction of Polynomials Worksheet

I. Perform the indicated operation of addition or subtraction:

1. (4x2 – 6x + 1) + (-7x2 + 16x – 4) 2. 3x – (6 – 2x)

3. (y4 + 3y2 – 6y + 3) – (5y3 + 8y2 – 6y) 4. (3c – 2c2 + 7) + (-2c2 – 11)

5. (4x2 – 6x + 1) – (-7x2 + 16x – 4) 6. 3x + (6 – 2x)

7. (y4 + 3y2 – 6y + 3) + (5y3 + 8y2 – 6y) 8. (3c – 2c2 + 7) – (-2c2 – 11)

9. (2x + 3) + (5x – 9) – (3x – 8) 10. 4.5rt2 + rt2

57

Algebra II Lesson Plans for Block Schedule

Aligned to NCSCOS – 2003

Day 19

Essential Question:? How do we multiply all types of polynomials?


“SAP”: Students will complete the Magic Square Activity in collaborative pairs.

Lesson Anatomy: 1. Collaborative Pairs check and reach consensus on correct solutions for homework problems. Student-led discussion of troublesome problems.2. Collaborative Pairs: Magic Square Activity 3. Teacher-led discussion of problems fitting the following types of multiplication of polynomials:

(Monomial)(Polynomial)(Binomial)(Polynomial)(Trinomial)(Polynomial)(Binomial)(Binomial)(Binomial)(Binomial)2

4. White Board Practice5. Have students complete the following class work assignment: Skills Handbook Factoring and Operations with Polynomials page 853 (1-6). Collect and grade.

Summarizing Activity: Ticket out the Door:x 7a – 1 y x2a –3y

Homework: Multiplying Polynomial Worksheet

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Polynomial Magic Squares

1. The following is a polynomial magic square. Each of the rows, columns, and diagonals has the same sum. Check this out by adding each row, each column, and each diagonal. Write down the sum you get in each calculation. If your work is correct, each of these sums should be the same.

4A + 2B – 2C -3A + 7B + 3C 2A + 6B – 4C

-A + 9B – 3C A + 5B – C 3A + B + C

4B + 2C 5A + 3B – 5C -2A + 8B

2. Now create a new polynomial magic square by adding –A + 2B – 2C to each of the cells above and putting the sum in the corresponding cell below. Then add each row, each column, and each diagonal. Write down the sum you get in each calculation. If your work is correct, each of these sums should be the same.

59

Multiplication of Polynomials

1. (3x – 2)(7x + 1) 2. 2x2y(8x3y – 6x2 +2xy2 -7) 3. (2x – 6y)2

4. (x-2)(2x+3)(x+5) 5. (3x-2)(x²-6x+1) 6. (3x+8)(3x-8)

7. (x+7)³ 8. (2-3y)(8+y) 9. (5x+2)(x²-6x-5)

10. (x²-y)(x²+y) 11. (x-1)(2x+30)(2x-3) 12. (x+3)( x²+3x+9)

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Day 20

Essential Question: How do I expand a binomial using Pascal’s Triangle and the Binomial Theorem?


“SAP”: Binomial Magic Square; Collaborative Pairs work to expand a binomial.

Lesson Anatomy: 1. Opening Warm-up: Binomial Magic Square2. Collaborative Pairs check and reach consensus on correct solutions for homework problems. Student-led discussion of troublesome problems.3. Give students the following problem: (x + 5)4

Let them multiply it out using foil and the distributive property. Then let them know that there is an easier method that they can use to expand a binomial. Use Pascal’s Triangle which is found on page 347 of the text. Do Examples 1 and 2 (TE, pg 348). For student independent practice use Check Understanding problems 1 and 2 (TE, pg 348).4. Be sure to do some problems like asking for the third term of (2x-5)3. This is a typical EOC question.5. Collaborative Pairs: Section 6-8, page 349-350 (6, 10, 48, 54) Review answers.

Summarizing Activity: Ticket out the Door.

EOC Practice Problem: Answer: D

Expand: (x + y)4

A x4 + y4

B x4 + 4xy + y4

C x4 + 4x3y + 4x2y2 + 4xy3 + y4

D x4 + 4x3y + 6x2y2 + 4xy3 + y4

Homework: Expansion of (Binomial)power Worksheet; Study for Quiz

61

Binomial Magic Square

1. The following is a binomial magic square. Each of the rows, columns, and diagonals has the same sum. Check this out by adding each row, each column, and each diagonal. Write down the sum you get in each calculation. If your work is correct, each of these sums should be the same.

A – 2B -6A + 3B -A + 2B

-4A + 5B -2A + B -3B

-3A 2A – B -5A + 4B

2. Now create a new polynomial magic square by multiplying 4A – 4B by each of the cells above and putting the product in the corresponding cell below. Then add each row, each column, and each diagonal. Write down the sum you get in each calculation. If your work is correct, each of these sums should be the same.

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Expansion of (Binomial) power Worksheet

Use Pascal’s Triangle to expand each binomial.

1. (x – 3)5 2. (2x + 3)4

3. (3x – 4)3 4. (2x + y)5

5. What is the fourth term in the expansion of (3x – 2y)7?

6. What is the third term in the expansion of (7 + y2)4?

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Day 21

Essential Questions: How do I divide a polynomial by a monomial? How do I divide a polynomial by a binomial?


“SAP”: Student will participate in White Board Practice and will complete the Binomial Magic Square.

Lesson Anatomy: 1. Collaborative Pairs: Partners compare homework answers and come to a consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Quiz on Addition, Subtraction, and Multiplication of Polynomials4. Teacher demonstrates the process of dividing a polynomial by a monomial and a polynomial by a binomial. Do the division of a polynomial by a binomial first through long division and then show the shortcut of synthetic division for students to see the comparison. Use textbook Examples 1 and 3 on pages 315-316. Use other examples to show different situations that can occur during the synthetic division process. Be sure to demonstrate the process of dividing a polynomial by a binomial that has a leading coefficient other than 1.5. Cooperative Pairs: Section 6-3 page 319 (37, 38, 39)

Summarizing Activity: Numberered Heads Activity- #1 tell #2 what you would block off on synthetic division if the binomial divisor is x + 3. #2 tell #1 what you would block off on synthetic division if the binomial divisor is 3x + 2. Together talk about how the answer row is affected when a fraction is blocked off.

Homework: Division of Polynomials Worksheet

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Division of Polynomials Worksheet

I. Dividing a monomial by a monomial: (Never leave a negative exponent in the final answer.)

1. -24x 2 y = ________ 2. -8c -3 = ________ 3. 5x 2 y 3 = ________ -3xy 2c4 -5y-3

4. -49c -4 b 2 = ________ 5. -56abc 0 = ________ 6. (-2xy -1 ) 3 = ________ 7c-2b-2 8abc 8x-2y

7. (-2x -2 y) 4 = ________ 8. -7a 3 b 2 c 5 d 0 = ________ 9. (-1/2x -2 ) = ________ 81 -63a3b4c3 x3

II. Dividing a polynomial by a monomial:

1. 8c 2 d – 12d 2 = ________ 2. -15x 2 – 5 = ________ 3. 18d 3 + 12d 2 = ________ -4c -5 6d

4. -10a 3 b 2 – 5a 2 b 3 = _____________ 5. 36a 4 b 2 – 18a 2 b 2 = _______________ -5a2b2 -18a2b2

6. -6a 2 b – 12ab 2 = ______________ 7. -5y 5 + 15y – 25 = _______________-2a2b -5y

III. Dividing a polynomial by a binomial.

1. 4x 3 + 3x 2 – 2x + 1 2. x 2 + 6x + 8 x – 1 x + 2

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3. b 2 + 5b + 7 4. x 2 – 9x + 7 b + 3 x – 2

5. 3x 2 – 13x – 10 6. y 3 – y 2 – 7y – 2 x – 5 y + 2

7. 6x 2 + 5x – 15 8. 3n 2 – 8n + 4 x + 3 n – 2

9. 6x 3 + 5x 2 + 9 10. 8x 3 – 1 2x +3 2x -1

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Day 22

Essential Question: Do I know how to operate with all types of polynomial expressions?

Objective(s): 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions to solve problems.

“SAP”: Students will work in collaborative pairs to complete the warm-up activity.

Lesson Anatomy: 1. Quiz Discussion2. Collaborative Pairs: Partners compare homework answers and come to consensus on correct solutions.3. Teacher led discussion of troublesome homework problems.4. Collaborative Pairs: Multiple Choice Warm-up Problems

(Do and Check) 5. Collaborative Pairs work on the Polynomial Test Review Worksheet. (Do and Check)


Homework: Study for Test on Operations with Polynomials

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Multiple Choice Warm-up Problems

1. Simplify fully: (2x3y-2)2(xy2)-3

A. 4x3y10 B. 4 C. 4x D. 4x 3 x3y10 x3y10 y10

2. Simplify fully: 2xy -2 4xy2

A. 2 B. ½ C. 1 D. y 4 2y4 2

3. Simplify fully: -5(-2x2y-3)2

A. -10x4y6 B. 10x 4 C. -20x 4 D. 20x 4 y6 y6 y5

4. Simplify fully: 4x -2x y3z y-3z2

A. -8x 2 B. -8x 2 y 3 C. -8x D. -8x 2 z3 z2 y9z2 z2

5. Simplify fully: 4x -3 -2

y3z2

A. 16x 6 B. x 6 y 6 z 4 C. 16x 5 D. -8x 6 y3z4 16 yz4 y6z4

6. Simplify fully: (4x)(y3)(5x-3)(2y4)=

A. 11x6y12 B. 40x6y12 C. 20xy4 D. 40y 7 x2

7. Simplify fully: (3a2b)3 (2a4b5)3 =

A. 216a18b18 B. 6a10b18 C. 54a12b11 D. 216a12b11

8. Simplify fully: (-2c-2d3)4 =

A. 16c8d12 B. 16d 12 C. -8c2d7 D. -8d 12 c8 c8

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Polynomial Test Review

Simplify the following fully. Leave no negative exponent in your final answer.1. (-5a3b)(-3ab-2) 2. (5ab)(-4b2)3 3. (-2rs2t3)2

4. -8d 4 h 5 5. -15x -4 y 5 z -6 6. (-4a3b2)-2

(2d4h6)3 5xy-2z3

7. 2(-3y-2)2 8. 6a-2(2a3 + 6a2 – 2a) 9. (2x – 3)(5x – 1)

10. (6a – 2)2 11. (2x + 3)(5x2 + 3x – 2) 12. -3xy(6xy – 1)

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13. (6a – 2)3 14. What is the 3rd term in the expansion of (2x – 5y)4?

15. 12x 3 – 2x 2 – 6x 16. y 3 – 64 2x y – 4

17. 18d 3 – 21d 2 + 9d – 5 3d – 2

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Day 23

Essential Question: Am I ready to show what I have learned about operating with polynomials for the test today?



Lesson Anatomy: 1. Collaborative Pairs: White Board Practice on operations with polynomials as a warm-up for the test.4. Unit Test


Homework: none

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Day 24

Essential Question: How do we factor polynomials by the Greatest Common Factor, Difference of Two Squares, and Quadratic Trinomial Methods?



Lesson Anatomy: 1. Test Discussion2. Collaborative Pairs: Make a list of as many math operations as you can that are inverses of each other.3. Compile a complete class list on the overhead. Did anyone name multiplication as the inverse of factoring? If not, add it to the list.4. Teacher demonstrates through numerous examples the greatest common factor

method, difference of two squares method, and the quadratic trinomial method. Emphasize through each example how the factoring method is an inverse of a multiplication method we just worked with in the last unit.

5. White Board Practice

Summarizing Activity:Ticket out the Door: Complete the following chart representing the signs used when factoring a quadratic trinomial:Signs of Trinomial Signs of Factors + +_____ ( )( )____ –___+_____ ( )( )____ +___ – _____ ( )( )____ – ___– _____ ( )( )

Homework: Factoring Practice Worksheet 1

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Opening/Starting Activator:1. Test Discussion2. Collaborative Pairs: Make a list of as many math operations as you can that are inverses of each other.3. Compile a complete class list on the overhead. Did anyone name multiplication as the inverse of factoring? If not, add it to the list.

Factoring Practice Worksheet 1

I . Factor by the Greatest Common Factor Method:

1. 3x2 + 12y2 = ____________________ 2. 18x2 – 12x = ______________________

3. x2 + 7x = _______________________ 4. 3x2 – 21x3 = ______________________

5. b3 + b2 + b = ____________________ 6. a2b + ab3 + ab _____________________

7. 12x2y + 2y = ____________________ 8. 3a2b2 – 9ab2 = ____________________

9. -12x2 – 6x = _____________________ 10. 60m3n + 48m2n = _________________

11. 2x4 + 6x3 – 10x2 = ______________ 12. -21v3w2 + 14v2w5 = _______________

II. Factor by the Difference of Two Squares Method:

1. a2 – 9 = ________________________ 2. x2 – 49 = ________________________

3. 4x2 – 9y2 = _____________________ 4. x2 – 36y2 = _______________________

5. 1 – 25y2 = ______________________ 6. 16a2 – 9b2 = ______________________

7. 49 – a2b2 = _____________________ 8. 9x2- 16y2 = _______________________

9. y2 – 1 = ________________________ 10. x2 – 64 = _________________________

11. 100r2 – 36 = ____________________ 12. 144 – 9x2 = _______________________

III. Factor by the Quadratic Trinomial Method:

1. 2y2 + 7y + 3 = ____________________ 2. 3n2 – 4n + 1 = ____________________

3. 5x2 – 2x – 7 = ____________________ 4. 2x2 + 9x – 5 = _____________________

5. 3a2 + 2a – 1 = ____________________ 6. 8x2 – 14x + 3 = ____________________

7. 3x2 + 20x – 7 = ___________________ 8. 6x2 – 25x + 14 = ___________________

9. 3y2 – 2y – 5 = ____________________ 10. 6t2 – 11t + 5 = _____________________

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Day 25

Essential Question: How do we factor polynomials by the Sum or Difference of Two Cubes Method?


“SAP”: In collaborative pairs, students will complete the What’s Missing? Activity

Lesson Anatomy: 1. Partners compare homework answers and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Collaborative Pairs: The What’s Missing? Activity4. Teacher demonstrates the sum or difference of two cubes method through several

examples. 5. Collaborative Pairs: Worksheet on Factoring by the Sum or Difference of two Cubes

Summarizing Activity:Ticket out the Door: Explain to your partner how you are going to remember the sign arrangement in the sum and difference of two cubes factoring pattern.

Homework: Factoring Practice Worksheet 2; Study for Quiz on Factoring by the GCF, Difference of two squares, and Quadratic Trinomial Methods

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What’s Missing?

Each problem below has three parts, two of which are missing. Fill in the missing parts of each.

A _____B________ ______C_____Example: (x + 5)(x + 6) = x2 + 6x + 5x + 30 = x2 + 11x +30

A B C1. (x + 1)(x+ 2) = __________________ = _______________

2. _________________ = x2 + 2x + 3x + 6 = ________________

3. _________________ = ___________________ = x2+ 11x + 18

4. (x + 5) ( x + 5) = __________________ = ________________

5. __________________= x2 + 2x + 4x + 8 = ________________

6. __________________ = ___________________ = x2 + 10x + 21

7. (x + 3)(2x + 4) = ____________________ = ________________

8. _________________ = x2 + 3x + 5x + 15 = __________________

9. __________________ = ___________________ = 2x2 + 5x +3

10. _________________ = 2x2 + 3x + 10x + 15 = _________________

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Factoring by the Sum or Difference of two Cubes Factoring Method

1. x3 – 64 = ____________________________________________________________

2. 64y3 + z3 = __________________________________________________________

3. c3 + 8 = _____________________________________________________________

4. 8x3 + 27y3 = _________________________________________________________

5. m3 – 125 = ___________________________________________________________

6. 27x3 + 64 = __________________________________________________________

7. x3 – y6 = ____________________________________________________________

8. 8x9 - 1 = _____________________________________________________________

9. x9 – y3z3 = __________________________________________________________

10. x3 + 1 = _____________________________________________________________

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Factoring Practice Worksheet 2

I. Factor by the GCF method:

1. 6x3 – 3x2 = _______________________ 2. x5 – x3 + x2 = _____________________

3. 9a2b + 6a2b = _____________________ 4. 12x3y + 2y = ______________________

5. 8x2 – 4x = ________________________ 6. b3 + b2 = _________________________

7. -3a2b2 – 9ab2 = ____________________ 8. -6x2 – 4x +2 =_____________________

II. Factor by the Difference of two Squares Method:

1. y4 – 36 = _________________________ 2. x4 – 1 = __________________________

3. 36x2 – 1 = ________________________ 4. 1 – 16y2 = ________________________

5. 9a2 – 25b2 = ______________________ 6. 4a2b2 – 25 = _____________________

7. 1 – 100z2 = ______________________ 8. 64x2 – 25 = ______________________

III. Factor by the Quadratic Trinomial Method:

1. x2 – 5x + 6 = _____________________ 2. 3x2 – 5x + 2 = ____________________

3. x2 – 5x – 6 = _____________________ 4. 5x2 – 2x – 7 = _____________________

5. x2 + 5x – 6 = _____________________ 6. 2y2 + 7y + 3 = _____________________

7. x2 + 5x + 6 = _____________________ 8. 6t2 – 11t + 5 = _____________________

IV. Factor by the Sum or Difference of two Cubes Method:

1. x3 – 1 = __________________________ 2. x3 + y3 = _________________________

3. x3 + 8 = _________________________ 4. y3 – 125 = ________________________

5. x3 – 27 = ________________________ 6. x9 – 27 = _________________________

7. x6 – y3 = ________________________ 8. x6 + 64 = _________________________

9. 64 – x3 = ________________________ 10. 8 – y6 = __________________________

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Day 26

Essential Question: How do we use a combination of our factoring methods to factor polynomials completely over the real numbers?


“SAP”: .Students will use the White Boards to practice problems on factoring completely.

Lesson Anatomy: 1. Partners compare homework answers and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Quiz on GCF, Difference of Two Squares, and Quadratic Trinomial Factoring Methods4. Remember the factoring trees you did in middle school? How do you break a number like

56 down into prime factors using the factor tree? What we are going to do is to follow this model to break a polynomial down into its prime factors? When all factors of a polynomial cannot be factored by any of our factoring methods, they are called prime factors. When we factor a polynomial into its prime factors, it is factored completely.

5. Teacher uses textbook examples to demonstrate how to factor a polynomial completely. At the beginning factor using the factor tree model. Work into a step by step procedure that would use less paper after several examples.

6. Use the following graphic organizer to help students organizer their thinking:

1. GCF 1. GCF 1. GCF2. Diff. 2 squares 2. Quadratic Trinomial 2. Factoring by grouping3. Sum 2 cubes4. Diff. 2 cubes

7. White Board Practice

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Factoring Completely

For Binomials

For Trinomials

For 4 term polynomials

Summarizing Activity: Collaborative Pairs: Pass out the homework graphic organizer and the Factoring Completely Worksheet 1. Put the number of the problem under the appropriate method in the organizer that represents the FIRST method that would be used to factor the problem completely.

Homework: Factor completely all the problems on Factoring Completely Worksheet 1

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Graphic Organizer

1.

1. _____________________ 1. ____________________ 1. ____________________

2. _____________________ 2. ____________________ 2. ____________________

3. _____________________ 3. ____________________ 3. ____________________

4. _____________________ 4. ____________________ 4. ____________________

5. _____________________ 5. ____________________ 5. ____________________

1. ______________________ 1. _____________________ 1. ____________________

2. _____________________ 2. ____________________ 2. ____________________

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FactoringCompletely

Prime GCFMethod

Diff. of 2 squaresMethod

Quadratic TrinomialMethod

Sum of 2 CubesMethod

Difference of 2 Cubes Method

3._____________________ 3. ____________________ 3. ____________________

4._____________________ 4. ____________________ 4. ____________________

5._____________________ 5. ____________________ 5. ____________________

Factoring Completely Worksheet 1

1. Categorize the problem into the graphic organizer by the FIRST factoring method that would be used to factor the problem completely. 2.Then actually take each of these problems and use all methods necessary to factor each of the problems completely.

1. 15a2b – 10ab2 2. 2y2 – 242 3. 16r2 – 49

4. c3 – 49c 5. 4x6 – 4x2 6. 8m3 – 1

7. b4 – 81 8. x2 – 3x- 10 9. r3 + 3r2 – 54r

10. 4a2 + a – 3 11 2t3 + 32t2 + 128t 12. x6 + 8y3

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13. 6n2 – 11n – 2 14. 81x4 – 16 15. 4x2y2 + 1


Day 27

Essential Questions: How do we use a combination of our factoring methods to factor polynomials completely over the real numbers? How do we use the factor by grouping method to factor polynomials of 4 or more terms?


“SAP”: In collaborative pairs students will complete the Factoring by Grouping Worksheet. They will also begin their homework worksheet with their partner.

Lesson Anatomy: 1. Quiz Discussion2. Partners compare homework and come to consensus on correct solutions.3. Teacher-directed discussion of troublesome homework problems.4. Teacher demonstrates factoring by grouping using the top three examples of the Guided

Notes. 5. Collaborative Pairs: Complete together the remaining six problems at the bottom of the

page.6. Group Review of the answers and correct methods of factoring the problems completely.

Summarizing Activity: Collaborative Pairs: Begin working together on Factoring Completely Worksheet

Homework: Finish Factoring Completely Worksheet 2; Study for Quiz on Factoring Completely

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Factoring by Grouping Guided Notes

Examples:1. m3 + 5m2 – 4m – 20 2. x3 – 21 – 3x2 + 7x 3. n2g + n2t – 36g – 36t

Partner Problems:1. 2p2 + 2a + 4ap + p 2. m2r2 – 9s2 – 9r2 + m2s2 3. 4ac + 2bc – 2ad –bd

4. 3x3 + 2x2 – 12x – 8 5. 2ax – 6bx + ay – 3by 6. 3x3 + 3x2 – 27x – 27

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Factoring Completely Worksheet 2

Factor completely:1. 48x3 – 48 2. 4r2 – 4r – 48 3. n2 –18n – 40

4. 5r2 –13r + 6 5. 12a2 – 27b2 6. x4 – 7x2 – 60

7. 5x3 + 10x2 – 20x – 40 8. 64x6 + 8 9. 2ax2 – 2ax –12a

10. x3 – 4x 11. 25x2 – 20x + 4 12. z6 – z2

13. r3 – s6 14. x4 + 2x2 – x – 2 15. 49y4 – 144z2

84

16. 64x3 – 27 17. -8x3 – 1 18. 4x2(x2–1) + 5x(x2–1) + (x2–1)


Day 28

Essential Question: How do we use a combination of our factoring methods to factor polynomials completely over the real numbers?


“SAP”: Basketball Shootout

Lesson Anatomy: 1. Challenge Problem: x 7 + x 6 + x 5 + x 4 + x 3 + x 2

x6 + x5 + x4 + x3 + x2 + x2. Partners compare and come to consensus on homework problems.3. Teacher-directed discussion of troublesome homework problems.4. Quiz on Factoring Completely ( 10 problems)5. “Basketball Shootout Game” for Review and Extra Practice for Test ( This test will be a cumulative test on Units 2 and 3. Therefore, problems from review should include exercises from both units.Game Problems:

p. 853, Ex. 1-18p. 324, Ex. 12-14p. 260, Ex. 51-65


Homework: Study for Test on Operations with Polynomials and Factoring Completely over the real numbers.

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“Basketball Shootout” Review Game

Directions: 1. Divide the class into three teams. Each student in the team should pick a partner to work

with during the game. They may only talk to their partner while working on a problem, not with other people on their team.

2. The game leader reads a problem or writes it on the board or the overhead, if needed, and allows enough time for the all the partner groups to agree on their answer.

3. The game leader randomly calls on a partner group in Team #1 to give their answer.4. If the answer is correct, Team #1 receives one point and a chance for bonus points by

shooting a ball at a trash can placed on a desk against a wall. Masking tape is used to mark a “2 point line” and a “3 point line” on the floor in front of the basket. One try is allowed from the “2 point line” and two tries are allowed from the “3 point line”. If the shot is made from the “2 point line”, the team gets a total of 2 points, one for getting the question right and one for the shot. If the shot is made from the “3 point line”, the team gets a total of 3 points, one for getting the question right and two for the shot. No more than 3 points can be earned on each question.

5. If the answer is incorrect, the same question is asked of a randomly chosen partner group in Team #2, without allowing any extra time to work. If Team #2 gets it right, then they take the shots and earn the points for their team. If they miss the question, then it passes to Team #3 until finally a team is able to answer the question correctly.

6. A new question will be started with the team coming after the one that received the last points, to keep the questions rotating fairly.

7. The team members with the highest number of total points at the end of the game wins a prize such as candy or bonus points on a quiz or test.

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Day 29

Essential Question: Am I ready to show on the test today all that I have learned about operating with polynomials and factoring polynomials completely over the real numbers?


“SAP”: none

Lesson Anatomy: 1. Quiz Discussion2. Answer questions that might have come up from studying for the test.3. Unit Test #3


Homework: none

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Day 30

Essential Questions: How do I graph the rational function? How do I simplify rational expressions through factoring?

Objective(s): 2.05 Use rational equations to solve problems. b) Interpret the constants and coefficients in the context of the problem. c) Identify the asymptotes and intercepts graphically and algebraically.1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

“SAP”: Students will work in collaborative pairs on the Making the Message Activity.

Lesson Anatomy: 1. Test Discussion

2. Graph on your calculator. How would you describe this graph?

3. Teach the definition of a rational function and identify the vertical and horizontal asymptotes in the graph. Students sometimes call this a “boomerang” graph.

4. Cooperative Pairs: Graph , , , and explain the similarities

and differences. As a group, generalize how you could look at each equation and recognize how the graph would appear on the calculator. 5. Teach students how to find the vertical asymptote by deciding when the denominator of the fractional term would equal 0. Connect this to the domain of the function. Remind students of the Zero Product Property so that when they will be able to handle finding the vertical asymptote

of a degree 2 denominator. An example of this would be . When setting the

denominator equal to 0, they will need to remember how to solve (x – 2)(x + 3) =0 by recognizing from the Zero Product Property that either x – 2 = 0 or x + 3 = 0. So the vertical asymptotes are at x = 2 and x = -3. This would also mean that the domain of this function would {x x x 2, -3}.6. Teach students how to find the horizontal asymptote by comparing the degree of the numerator to the degree of the denominator. Consider only rational functions of lower over higher degree, or same degree in numerator and denominator.7. White Board Practice: Finding Vertical asymptotes page 495 (10, 12 and 18)

88

Finding Horizontal asymptotes page 496 (19-24)8. Show how to sketch graphs of rational functions using the vertical and horizontal asymptotes and by plotting a point to the left and to the right of the vertical asymptote. 8. Demonstrate through several examples how to reduce rational expressions through factoring and show students when and how to state restrictions on the variable. Connect this to synthetic division when appropriate. Use Example 1 and Check Understanding 1a-c (TE, pg. 499).

Summarizing Activity:

Cooperative Pairs: Find the vertical and horizontal asymptote of and predict the graph

you will see on the calculator. Check it out.

Homework: Graphing and Reducing Algebraic Fractions Worksheet

89

Graphing and Reducing Algebraic Fractions Worksheet

I. For each of the following name the vertical asymptote(s) and the horizontal asymptotes(s):

1. va=_______ 2. va=_______ 3. va=_______

ha=_______ ha=_______ ha=_______

4. va=_______ 5. va=_______ 6. va=_______

ha=_______ ha=_______ ha=_______

7. va=_______8. va=_______ 9. va=_______

ha=_______ ha=_______ ha=_______

10. va=_______ 11. va=_______ 12. va=_______

ha=_______ ha=_______ ha=_______

II. Use the vertical and horizontal asymptote information to help you sketch the following rational function graphs. Make a Tchart with a point to the left and right of the vertical asymptote to you help sketch the graph without your calculator.

1. 2. 3.

90

III. Reduce the following rational expressions by factoring:

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

IV. Divide the following using synthetic division.

1. 2. 3.

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Day 31

Essential Question: How do I add and subtract rational expressions?

Objective(s): 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions)to solve problems.

“SAP”: Cooperative Pairs: Make the Message Activity

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Pairs complete the Make the Message Activity.4. Review how to find the Least Common Multiple. Connect that to add or subtract rational expressions with different denominators. You must first write all the expressions with a common denominator (LCM). Use Example 2 and Check Understanding problem 2a, 2b (TE, pg. 505). 5. Demonstrate, using Examples 3-4, as well as Check Understanding problems 3-4, how to add and subtract rational expressions. Make sure to connect the steps in the algebraic procedure to the steps used to add or subtract arithmetic fractions.5. White Board Practice: page 507, 4-20 even

Summarizing Activity:Cooperative Pairs: Christine says that the sum of and is because the numerators are

the same. Andre’ says, “No way! It doesn’t work that way!” Christine asks, “Why not?” What does Andre’ tell her? Convince your partner of the answer.

Homework: Prentice Hall Algebra 2 Text Section 9-5 page 507 (5-21 odd)

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Make the Message

Simplify each expression.Record the letter corresponding to the correct answer in the space provided.

A = x – 1 E = x + 5 H = x – 7 I = x + 1

T = x + 2 S = x – 3 Y = x + 7

1. __________ 6. __________

2. __________ 7. __________

3. __________ 8. __________

4. __________ 9. __________

5. __________ 10. __________

Resources for Algebra **B-72** Public Schools of North Carolina

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Day 32

Essential Question: How do I add and subtract rational expressions? (second day)


“SAP”: Students will use white boards to practice adding and subtracting rational expressions.

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. White Board Practice on more problems involving addition and subtraction of rational expressions using examples from the Reteach 9-5 worksheet.4. Collaborative Pairs work on Practice Worksheet 9-5. Do and check.

Summarizing Activity:Ticket Out the Door:How are the problems 1/4 + 2/3 and 1/x + 2/y alike in the way we arrive at the answers? Describe the similarities as you show how to work each problem correctly.

Homework: Prentice Hall Algebra 2 Text Section 9-5 page 507, 4-20 (even)

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Day 33

Essential Questions: How do I multiply and divide rational expressions? How do I simplify complex fractions?


“SAP”: Students will use white boards to practice multiplying and dividing rational expressions.

Lesson Anatomy: 1. p.496 Basketball Application Problem #422. Partners compare homework solutions and come to consensus on correct solutions.3. Teacher-directed discussion of troublesome homework problems. When discussing the 2nd page of the homework, name the restriction on the domain of each function.4. EOC Practice Problem: Answer: D

Which of the following is a horizontal asymptote of ?

A x = -4B y = 4C x = 1D y = 05. Teacher demonstrates through Examples 1, 3 and 4 of the text, Section 9-4 pages 499-501, the procedure of multiplying and dividing rational expressions. Connect this to the way we multiply and divide arithmetic fractions.6. Teacher demonstrates through selected examples on the Simplifying Complex Fractions Worksheet the procedure of simplifying complex fractions. Teach students to do this by multiplying each term in the numerator and denominator of the complex fraction by the common denominator of all the individual fractional terms. This is a usually one-step process to getting the answer and is much easier for students than to simplify the operations in the numerator and simplify the operations in the denominator and then dividing. Also because of using the method of multiplying by the LCD in the numerator and denominator (a fancy name for 1), this topic can be taught before adding and subtracting rational expressions.7. White Board Practice: Practice and Problem Solving Problems Section 9-4 (39-41).

Summarizing Activity:Ticket out the Door. Explain the procedural difference between multiplying and dividing rational expressions.

Homework: Complete the Simplifying Complex Fractions Worksheet and Prentice Hall Algebra 2 Text Section 9-4 page 501 (1-18)

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Simplifying Complex Fractions

Simplify fully:

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

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Day 34

Essential Question: How do we add, subtract, multiply, and divide rational expressions? How do we simplify complex fractions?How do we solve rational equations?

Objective(s): 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions to solve problems. 2.05 Use rational equation to model and solve problems.

“SAP”: Cooperative Pairs will complete the Rationals in Review Worksheet

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Cooperative Pairs: (Do and check) “Rationals in Review”4. Group Discussion of rational application problems: p. 496, Ex 42; p. 509, Ex. 52 and 54; 5. Teacher demonstrates through several problems the procedure of solving rational equations

by multiplying each term in the equation by the common denominator. Check for restrictions on the variable.

6. Cooperative Pairs: p.514, Exercises 1-5. Do and discuss.7. Group Discussion: p. 515, Ex 10-21

Summarizing Activity: Ticket out the Door:

State what you would multiply each side of by in order to solve the

equation. What value(s) of x cannot be a solution?

Homework: p. 530, Chapter Test, Ex. 10-32 (even); Study for Quiz on Operations with Rational Expressions and finding asymptotes of rational functions

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“Rationals in Review”

I. Name the vertical and horizontal asymptote(s) for the following:

1. y = 2. f(x) = 3. f(x) = 4. f(x) =

va=___________ va=__________ va=___________ va= ___________

ha=___________ ha=__________ ha=___________ ha=___________

I. Name the vertical and horizontal asymptotes and sketch the graph of y = .

va = __________ ha = __________

II. Simplify fully:

1. 2.

3. 4.

99

5. 6.

7. 8.

9. 10.

100


Day 35

Essential Question: How do we solve problems involving direct, inverse, joint, or combined variation?

Objective(s): 1.05 Model and solve problems using direct, inverse, combined and joint variation.

“SAP”: Guided Note-taking and work practice exercises in cooperative pairs

Lesson Anatomy: 1. Quiz on Operations with Rational Expressions2. Teacher-led discussion of homework problems. Check each rational equation done

algebraically by graphing it on the TI-83 to find the zeros. Identify the asymptotes and intercepts.

3. Sometimes when relating two variables they vary directly or they vary inversely. To understand direct variation I want you all to put your arms up about shoulder level and pretend that you are holding a weight lifting bar. As your arms push the bar up, the weight goes up. To understand inverse variation I want you to think of lifting an object with a lever. As your arm goes down, the object goes up. Direct variation is represented

as a linear equation y= kx and inverse variation is represented as a rational equation y=

where k is the constant of variation or constant of proportionality.4. Discuss the vocabulary highlighted in yellow in Section 2-3 (TE page72-74). Work through examples 1-4 (TE pages 72-74). Have students complete the Check for Understanding problems 1-4 (TE pages 72-74) with a partner. Review answers.5. Guided Notes on Linear and Inverse Variation6. Teacher demonstrates through several examples in text that joint variation is an extension

of direct variation and that combined variation is a combination of direct and inverse variation. (Use Examples 4 and 5 (TE, pg. 480) and Check Understanding 4 and 5 (TE, pg. 480).

7. Cooperative Pairs: p. 481, Ex. 13-27 (odd)8. Group Discussion of p. 482, Ex. 52

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Summarizing Activity: Explain to your partner what equation you will use when you know y varies directly as x.

Explain to your partner what equation you will use when you know y varies inversely as x. AND

Ticket out the Door: Cooperative Pairs: (Do and discuss)

1. Write the equation for the following variation. The weight of an object varies inversely as the square of the distance of the object from the center of the earth.2. Solve: The time required for a team of road workers to dig a ditch is directly

proportional to the number of people digging and inversely proportional to the length of the ditch. It takes 10 workers 5 hours to dig a 100 yard ditch. How many hours would 20 workers take to dig a 1700 yard ditch?Homework: Algebra 2 Prentice Hall text, p. 481-2, Ex. 14-26, 36, 49-51

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1 2

2 1

Guided Notes on Direct and Inverse Variation

I. Direct Variation: y varies directly as x or y is directly proportional to x (y = kx) where k is the constant of proportionality or constant of variation As x gets larger, y gets larger or as x gets smaller, y gets smaller.A line with a y-intercept of 0 is a direct variation.Real world examples of direct variation include fruit sold by the pound, distance traveled by a car over time, characters printed from a computer per second, circumference of a circle varies directly as the diameter, and wages varying directly to hours worked. Can you think of others? __________________________________________

If you buy three pounds of grapes at $2.99 per pound, how much would you pay for the grapes. What are the two variables and what is the constant of variation? __________________________________________________________________________How can you write this as a linear equation?_______________________________________

Data that represents direct variation:

What is the constant of proportionality? ___________

Example 1: If y varies directly as x and s=10 when y=9, then what is y when x=4?

Method 1: y = kx y = Method 2: (You could also use proportions.)

9 = k(10) y =

y = 3.6 10y = 36

y = 3.6Example 2: When a bicycle is pedaled in a certain gear, it travels 16 meters for every 3 pedal revolutions. How many revolutions would be needed to travel 600 meters?

Method 1: d = kr d = Method 2:

16 = k(3) 600 = 16x = 1800

1800 = 16r x = 112.5

112.5 = r

x y-1 3 0 0-2 6 4 -12

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Problem 1: A refund r you get varies directly as the number of cans you recycle. If you get a $3.75 refund for 75 cans, how much should you receive for 500 cans?

II. Inverse Variation: y varies inversely as x or y is inversely proportional to x

( y = or xy = k ) where k is the constant of proportionality or constant of variation

As x gets larger, y gets smaller or as x gets smaller, y gets larger.A boomerang graph (rational function) with a vertical asymptote at x=0 and a horizontal asymptote at y=0 is an inverse variation.Boyles’ Law of Gases is a real world example of inverse variation. Likewise, for a trip to Myrtle Beach, the greater your car speed, the less time it would take you to get there. If a rectangle has an area of 15 square units, then as the length increases the width decreases.

Data that represents inverse variation:

What is the constant of variation? _________

Example1: If y varies inversely as x and x=3 when y=9, then what is x when y=27?

y = y =

9 = 27 =

27 = k x = 1

Example 2: If y varies inversely as the square of x and y = 20 when x =4, find y when x =5.

y = y =

20 = y =

320 = k y = 12.8

Problem 1: Find x when y = 3, if y varies inversely as x and x = 4 when y = 16.

Problem 2: The amount of resistance in an electrical circuit required to produce a given amount of power varies inversely with the square of the current. If a current of .8amps requires a resistance of 50 ohms, what resistance will be required by a current of .5 amps?

x y3 42 6 9

10

1 12

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Day 36

Essential Question: How do I prepare for the test on Rational Expressions?

Objective(s): 1.03, 1.05, 2.05

“SAP”: Students will work in cooperative groups on the Applications of Variation Worksheet.

Lesson Anatomy: 1. Quiz Discussion2. Partners compare homework solutions and come to consensus on correct solutions.3. Teacher-led discussion of troublesome homework problems.4. Cooperative Pairs: a) At a water pollution control facility one pipe can fill a tank in 4 hours. Another pipe can fill the tank in 6 hours. How many hours will it take to fill the tank if both pipes are working together?b) Gasoline Mileage Problem on page 489 # 47c) EOC Practice Problem: Answer: B

Solve:

A {-3}B {-5}C {2}D {5}

5. Cooperative Groups: Applications of Variation Worksheet (Do and discuss)

Summarizing Activity:Ticket out the Door: Cooperative Pairs: (Do and discuss)

1. Write the equation for the following variation.The weight of an object varies inversely as the square of the distance of the object from the center of the earth.2. Solve: The time required for a team of road workers to dig a ditch is directlyproportional to the number of people digging and inversely proportional to the length of the ditch. It takes 10 workers 5 hours to dig a 100 yard ditch. How many hours would 20workers take to dig a 1700 yard ditch?

Homework: Study for Unit test on Rational Functions, Operations with Rational Expressions, Solving Rational Equations, and Variation

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Applications of Variation

1. The distance in meters that a body falls from rest varies as the square of the number of seconds it has fallen. If a body falls 78.08 meters in 4 seconds, how far will it fall in 7 seconds?

2. A bicycle wheel with a 26-inch diameter takes 10 revolutions to go a certain distance. How many revolutions will a wheel with a 20-inch diameter make in covering the same distance?

3. The pressure of wind on a sail varies jointly as the area of the sail and the square of the velocity of the wind. When the velocity of the wind is 20 miles per hour, the pressure on 2 square feet of sail is 4 pounds. Find the velocity of the wind when the pressure on 9 square feet of sail is 32 pounds.

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Day 37

Essential Questions: Am I ready to show what I have learned about rational functions on the Test today?

Objective(s): 1.03, 1.05, 2.05

“SAP”: White Board Practice for Warmup to Test

Lesson Anatomy: 1. White Board Practice to Review for Test. (15 minutes)2. Unit Test


Homework: none

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Day 38

Essential Questions: How can I solve systems of equations by graphing?


“SAP”: Graphing using the TI-83

Lesson Anatomy: 1. Test Discussion2. Pose the following problem: “Farmer McDonald raises ducks and cows. The animals have a total of 9 heads and 20 feet. How many cows and how many ducks does Farmer McDonald have?” As students raise their hands with a solution, check out their answer, and ask them to explain how they got it. (Most methods will be guess and check.)3. Re-pose the problem with 10 heads and 24 feet. Then ask about 8 heads and 18 feet. Then

change it to 9 heads and 50 feet. We will revisit this problem later in the class.4. Here is another problem for you:

“I’m thinking of two numbers. When you add them the answer is 7. What are the numbers? Generate a list on the overhead of all of the suggestions. Who is right? Let me give you another clue. When you subtract them the answer is 3.” When I gave you the second clue, you were automatically able to tell which numbers I was thinking of.That’s what happens when you have two unknowns (variables). If there are two variables and you have two clues (equations) you can solve the problem.Write this problem algebraically as x + y = 7 and x – y = 3. We are going to learn four algebraic ways of solving this system. Can you remember some of the ways you solved it in Algebra I? Hopefully students will come up with the graphing method, substitution method, and elimination method. The teacher can add matrices as the fourth method that will be studied in the next unit.

5. Teacher demonstrates how to solve the system by all three methods.6. Revisit the ducks and cows problem. Let the students write the problem

algebraically as a system and solve it by the three methods. Use the TI-83 for the graphing method. 7. Focus on solving by the graphing method using Examples 1 and Check Understanding problems on p. 117. Talk about the user-friendly graphing window and how to adjust it easily when needed.

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Summarizing Activity: Ticket out the door: Explain how you will solve: x + y = 6 3x – 4y = 4 by the graphing method.

Homework: Prentice Hall Algebra 2 Text: p.118-9, Ex. 1-9, 25, 27, 28

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Day 39

Essential Questions: How do we solve systems of equations using the substitution method and the elimination method?


“SAP”:

Lesson Anatomy: 1. Collaborative Pairs: Check homework answers and come to consensus on the correct

solutions.2. Student-led discussion of troublesome problems.3. Using examples 1-5 on p. 123-125, the teacher demonstrates the substitution and

eliminations methods. Stress how to decide when one method is easier to use than the other.

4. Talk about the three ways two lines can be drawn in a coordinate plane. Relate the three possibilities to the number of solutions for the system. How will we know algebraically that a system has no solution or has infinite solutions? Show examples of both situations.

5. Collaborative Pairs: Solve the following three problems by your choice of methods: a) 3x – 2y = 10 b) 4x – 5y = 60 c) 4x – 2y = 5

y = 6 – 4x 8x + 15y = -80 2x = y – 1

Summarizing Activity: Solve the system 5x – y = 2 x + y = 4 by all three methods.Show a sketch of your graph from the calculator and show your algebraic work for the substitution method and the elimination method.

Homework: Prentice Hall Algebra 2 Text, p. 126, Ex. 1-9, 18-23, 31-35

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Day 40

Essential Questions: How do we solve a system of equations by the graphing method?How do we solve a system of inequalities by graphing?


“SAP”: Collaborative pairs work on Cablevision Subscription Problem and Function Notation and Composition of Function/Graphs of Inequalities Review Worksheet

Lesson Anatomy: 1. Collaborative Pairs: Check homework answers and come to consensus on the correct

solutions.2. Student-led discussion of troublesome problems.3. Collaborative Pairs: System of Equations Activity (Cablevision Subscription Problem).

Do and discuss.4. Teacher demonstrates several examples of graphing systems of inequalities by hand using

Example 1 on p. 130 and the Check for Understanding Problems at the top of p. 131. Show how to solve the systems using the shade function under y= on the TI 83.

Summarizing Activity: Collaborative Pairs:Function Notation and Composition of Function/Graphs of Inequalities Review Worksheet

Homework: Prentice Hall Algebra 2 text, p.132, Ex. 4-15Pass out graph paper on which to sketch the graphs of the solutions.Study for Quiz on Solving Systems by graphing, substitution, and elimination methods

111

Systems of Equations

The area cable company has two options for new subscribers. They can choose between Option 1 which offers a $39.99 installation charge and a monthly fee of $27.95 or Option 2 which offers a $60.00 installation charge and a monthly fee of $19.99. How would a new subscriber decide which was the best deal for them?

(a) What linear function could be used to represent Option 1?

_____________________________________________________________________(b) What linear function could be used to represent Option 2?

______________________________________________________________________ (c) What would be a good “user-friendly” viewing window to see this graph?

X-min= _______X-max=_______X-scl = _______Y-min= _______Y-max=_______Y-scl = _______X-res = _______

(d) Complete the chart: Accumulated Costs

Month0

Month1

Month2

Month3

Month4

Month 5

Month6

Month7

Month8

Month9

Month10

Option 1

Option 2

(e) List the different ways that the decision could be made.

(f) What is the best deal for the new subscriber? Explain why.

112

Function Notation and Composition of Functions

Given f(x)=x2 - 16x + 35 and g(x)= -2x + 15

1. Find f(-3) ___________________

2. Find g(-1) ___________________

3. Find f(g(-1)) _________________

4. Find g(f(-3)) __________________

5. Find f(g(14)) __________________

6. Find g(f(14)) _________________

Graphing Functional Inequalities

1. y x2 - 16x + 35 What is an appropriate “user-friendly” window for this graph?

X min= _______ y min= _________X max= _______ y max= _________X scl = ________ y scl = _________

Sketch the graph:

2. y -2x + 15 What is an appropriate “user-friendly” window for this graph?

X min= _______ y min= _________X max= _______ y max= _________X scl = ________ y scl = _________

Sketch the graph:

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Day 41

Essential Question: How do I solve problems by using systems of equations?


“SAP”: Students will work in cooperative groups on the “A Problem To Solve” activity.

Lesson Anatomy: 1. Quiz on Solving Systems by graphing, substitution, and elimination methods2. Collaborative pairs: Check homework answers and come to consensus on the correct solutions.3. Student-led discussion of troublesome problems.4. For the “A Problem to Solve” activity, students must be broken into groups of 3 or 4. Groups should be selected randomly. Each group will pick a folded/stapled slip of paper out of a bag. Instruct them not to open it until you tell them to. Because there are only two problems, two groups will have to solve the same problem. Make sure that the groups that have the same problem do not work together. Have students complete the “A Problem to Solve” activity. The members of each group should staple their work and answers together and turn it in as they leave the room. Most groups will take 15-20 minutes to solve the problem. 5. Use the Guided Notes to help students with the problem solving process.


Homework: Systems of Linear Equations Word Problem Worksheet

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Guided Notes on Using Linear Systems to Solve Word Problems

Systems of equations can be used to solve real-world problems involving two variables. To use linear system to solve work problems involving two variables:

a) Use two different variables to represent the unknowns.b) Organize the given information in a table, if possible.c) Identify the conditions of the problem. Write an equation to represent each condition.d) Solve the system.e) Check the solution in the original conditions of the problem.

I. Complete each table. Then write a system of equations that could be used to solve each problem.

1. Jill owns a fruit stand and plans to make baskets of oranges and apples to sell. Each basket will have 24 pieces of fruit and will sell for $7.50. Jill wants to get $0.25 for each orange, $0.20 for each apple, and $1.95 for the basket. How many oranges will there be in each basket?

Item Quantity Value

Apples x ?

Oranges y ?

2. Justin’s piggy bank has 30 coins in dimes and nickels. Their combined value is $2.10. How many coins of each type are there?

Item Number Value

Dimes x ?

Nickels y ?

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II. Use a system of linear equations to solve each problem.

3. The drama club at South High School sells hot chocolate and coffee at the school’s football games to make money for their upcoming field trip. At one game they sold $200 worth of hot drinks. They need to report how many of each type of drink they sold for their club records. Michelle knows that they used 295 cups that night. If hot chocolate sells for $0.75 and coffee sells for $0.50, how many of each type of hot drinks did they sell?

4. The sum of two numbers is 52. Twice the first number is 17 more than the second number. Find the numbers.

5. The perimeter of a rectangular swimming pool is 84 meters. The width of the pool is ¾ the length. What are the length and width of the pool?

116

Systems of Linear Equations Word Problem Worksheet

Use a system of equations to solve the following:

1. The number of boys in a math class is one less than three times the number of girls. There are 39 students in the class. How many of each are there in the class?

2. The length of a rectangular garden is 3 meters less than twice the width. Three times the width of the garden equals two times the length. Find the width and the length of the garden.

3. Twice one number is 28 more than a second number. When 23 is added to the second number, the result is 12 less than 3 times the first number. What are the numbers?

4. Help Mark solve the following homework problem in Geometry: Two angles are complementary and one angle is 36˚ more than twice the other. Find the measure of each angle.

5. The sum of 4 times Macy’s age and 3 times Kevin’s age is 47. Kevin is one year less than twice as old as Macy. Find each of their ages.

6. The sum of two numbers is 48 and their difference is 80. Find each of the numbers.

7. On Friday Kelly cashed her pay check for $880 at her bank. The teller gave her only $20-bills and $10-bills. There were 8 fewer $10-bills than $20-bills. How many $20-bills did Kelly receive?

8. Time Warner offers the following cable packages: basic cable television and one movie channel for $39 a month or basic cable and two movie channels for $45.50 a month. What is their monthly charge for basic cable? (Assume that each movie channel has the same monthly charge.)

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Day 42

Essential Question: Am I ready for the Midterm Exam?

Objective(s): Review of all objectives taught on Days 1-41.

“SAP”: Basketball Shootout

Lesson Anatomy: 1. Quiz Discussion2. Homework check and discussion3. Use resources from DPI, GCS, Prentice Hall to pull problems that would prepare student for the Midterm Exam. Present these problems to students to work in the Basketball Shootout Game.

“Basketball Shootout” Review GameDirections:

1. Divide the class into three teams. Each student in the team should pick a partner to work with during the game. They may only talk to their partner while working on a problem, not with other people on their team.2. The game leader reads a problem or writes it on the board or the overhead, if needed, and allows enough time for the all the partner groups to agree on their answer.3. The game leader randomly calls on a partner group in Team #1 to give their answer.4. If the answer is correct, Team #1 receives one point and a chance for bonus points by






Homework: Study for Midterm Exam (Lessons 1-41 tested)

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Day 43

Essential Question: Am I ready to show on the Midterm Exam what I have learned this semester?

Objective(s): Review of all objectives taught on Days 1-41.

“SAP”: none

Lesson Anatomy: Mid-Term Exam


Homework: none

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Day 44

Essential Question: How do we use linear programming with systems of three or more inequalities to solve problems?


“SAP”: Students will work on the Investigation with teacher guidance.

Lesson Anatomy: 1. Mid-term Exam Discussion2. Use the Investigation on page 135 of the Text to introduce students to the concept of finding a minimum value. (See investigation page following.)3. Teach the linear programming method using Examples 1 and 2, and Check for Understanding Problems 1 and 2 on pages 136-137 of the Text. Refer to the shaded region of the system of inequalities as the feasible region and inequalities as constraints. Summarizing Activity: Sentence Stem. What I am confused about is . . .

Homework: Prentice Hall Algebra 2 Text Section 3-4 page 138 (1, 2, 5, 5, 6, 9, 10)

121

Investigation: Finding a Minimum Value

Suppose you want to buy some tapes and CDs. You can afford as many as 10 tapes or 7 CDs. You want at least 4 CDs and at least 10 hours of recorded music. Each tape holds about 45 minutes of music, and each CD holds about an hour.

1. Write a system of inequalities to model the problem. Let x represent the number of tapes purchased and y represent the number of CDs purchased.

2. Graph your system of inequalities.

3. Does each ordered pair satisfy the system you have graphed?a. (4, 7) b. (12, 7) c. (7, 6) d. (9, 4) e. (10, 4)

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Day 45

Essential Question: How do we use linear programming with systems of three or more inequalities to solve problems?


“SAP”: Students will work in cooperative groups to solve linear programming problems.

Lesson Anatomy: 1. Teacher led discussion of troublesome homework problems.2. In collaborative pairs to complete the following class work assignment: Practice and Problem Solving TE, page 139 (14, 16).3. Cooperative Group Activity:

Solve the problem: In a double-elimination tournament, a team is out if it loses two games. The team left is the champion. If there are 30 teams playing in the tournament, how many games will need to be played to determine a champion? Explain your answer. Encourage groups to be creative when explaining their answer. For example, groups can act out the wins and losses in the tournament.4. Collaborative Pairs work on Reteach Worksheet 3-4 and Practice Worksheet 3-4 (odds)

Summarizing Activity:EOC Practice Problem (below) Answer: C

Students plan to spend 300 hours preparing and 250 hours packaging popcorn for sale. The student council has $160 for supplies. The table below gives data on the time and money required to purchase materials and produce the finished product.

Plain Popcorn Caramel PopcornPreparation Time (per lb) 0.1 hr 0.8 hrPackaging Time (per lb) 0.2 hr 0.45 hrMaterial Costs (per lb) $0.10 $0.40

The students want to set up a linear program to assist them in this project. Given that x = pounds of plain popcorn and y = pounds of caramel popcorn, what should be the constraints of this situation?

A 0.1x + 0.8y < 1600.2x + 0.45y < 2500.1x + 0.4y < 300x > 0 and y > 0

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B 0.1x + 0.2y < 3000.8x + 0.45y < 2500.1x + 0.4y < 160x > 0 and y > 0

C 0.1x + 0.8y < 3000.2x + 0.45y < 2500.1x + 0.4y < 160x > 0 and y > 0

D 0.1x + 0.8y > 3000.2x + 0.45y > 2500.1x + 0.4y > 160

Homework: Prentice Hall Algebra 2 Practice Workbook (Practice 3-4, evens)

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Day 46

Essential Question: How do I prepare for the test on Linear Systems?


“SAP”: Students will work in collaborative pairs to complete the EOC Practice Problem.

Lesson Anatomy: 1. Collaborative pairs: Check homework answers and come to consensus on the correct solutions.2. Student-led discussion of troublesome problems.3. Collaborative Pairs: EOC Practice Problem (below): Answer: D

Specialty Furniture can produce a maximum of 400 tables and chairs each week. At least 25 tables and 100 chairs must be produced each week. The profit on each table is $350 and the profit on each table is $75. What is the maximum profit that can be generated weekly?

A $16,250B $36,875C $98,750D $112,500

4. Test Review Assignment (from text): pages 157 – 159 #10, 18, 19, 20, 22, 23, 29, 30. Review Answers.

Summarizing Activity: Tell your partner one particular concept you plan to study more before the test tomorrow.

Homework: Study for the test.

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Day 47

Essential Questions: Am I prepared to show what I have learned in this Unit on the test today?


“SAP”: Students work in collaborative pairs to complete Check for Understanding after test.

Lesson Anatomy: 1. 5-minute review before testing.2. Unit test on solving systems of equations (by graphing, elimination and substitution), solving systems of inequalities by graphing, and linear programming .3. After students turn in their test, have them complete the Check Skills You’ll Need on page 164 of the text. Review answers.


Homework: none

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Day 48

Essential Question: How do I add, subtract, and multiply matrices?

Objective(s): 1.04 Operate with matrices to model and solve problems

“SAP”: Students will work in collaborative pairs on the Matrix Operations Worksheet.

Lesson Anatomy: 1. Test Discussion2. Discuss vocabulary of matrices from section 4-1. Collaborative pairs: Do Practice 4-1 in the Prentice Hall Algebra 2 Practice Workbook. Check answers.3. Discuss vocabulary of matrices in sections 4-2 and 4-3 of the text.4. Do the examples from both sections 4-2 and 4-3. In collaborative pairs students should do the Check for Understanding problems from both sections. Review answers.5. Collaborative pairs: Have students do the Matrix Operations Worksheet. Review answers.

Summarizing Activity: EOC Practice Problem (below): Answer: DNagel’s Bagel Shop makes a monthly report to summarize the cost of making a single bagel of each type and the price at which it is sold. Matrix C represents cost, and matrix P represents selling price.

Plain Blueberry Wheat Onion Plain Blueberry Wheat OnionC = [0.12 0.17 0.13 0.15] P=[0.45 0.50 0.50 0.50]

Which matrix represents the profit on a single bagel of each type?

Plain Blueberry Wheat OnionA [0.57 0.67 0.63 0.65]

Plain Blueberry Wheat OnionB [0.33 0.33 0.35 0.37]

Plain Blueberry Wheat OnionC [0.33 0.33 0.33 0.33]

Plain Blueberry Wheat OnionD [0.33 0.33 0.37 0.35]

Homework: Prentice Hall Algebra 2 Practice Workbook Practice 4-2 – 4-3, odds

127

Matrix Operations

Directions: Find the value of each variable.

1.

2.

3.

4.

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Day 49

Essential Question: How do I use the TI-83 to operate with matrices?

Objective(s): 1.04 Operate with matrices to model and solve problems.

“SAP”: Students will complete the Matrix Operations on the TI- 83 Worksheet and the Football Statistics Exercise in collaborative pairs.

Lesson Anatomy: 1. Collaborative pairs: Check homework answers and come to consensus on the correct solutions.2. Teacher demonstrates correct solution on troublesome homework problems. Using some of the most missed homework problems, show how to enter these into the matrices function of the calculator and operate with them to get the answer. Show students how to use the calculator to find the determinant of a matrix.3. In collaborative pairs do the Matrices Operations Worksheet on the TI-83. Talk about why some of the answers are undefined.4. Collaborative pairs: Football Statistics Exercise

Summarizing Activity: On small cards put matrix notation like A2 and B4 to match to the arrangement of desks in rows and columns in the classroom. Have students individually draw a card and rearrange themselves in the classroom to match the card they drew. Match new collaborative pairs by who they end up sitting beside. Remind them to remember this new seat for when they come to class tomorrow. Homework: Prentice Hall Algebra 2 Text Section 4-2 pages 174-175 (2-24 even), Section 4-3 pages 182-183 (10-18 even, 22-28 all, 38,40)

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Matrices

Matrix Operations:

Let A= and B= and C=

Find: 1. A +B = ___________ 7. C2= __________

2. A - B= ___________ 8. det [A]= _______

3. A x B= ___________ 9. det [B]= _______

4. A x C= ___________ 10. det [C]= _______

5. 3B= ____________ 11. A-1= _______

6. B2= _____________

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Football Statistics

The tables below represent statistics on several NFL quarterbacks for 1998 and 1999 seasons. These statistics were taken from:http://sportsillustrated.cnn.com/football/nfl/rosters/Quarterbacks.html.The idea for this problem comes for page 13 of Contemporary Mathematics in Context, by Coxhead et al., 1998.

1998 StatisticsPlayer Attempts Completions Touchdowns InterceptionsTroy Aikman 315 187 12 5Tony Banks 408 241 7 14Jeff Blake 93 51 3 3Steve Beuerlein 343 216 17 12

1999 StatisticsPlayer Attempts Completions Touchdowns InterceptionsTroy Aikman 442 263 17 12Tony Banks 320 169 17 8Jeff Blake 389 215 16 12Steve Beuerlein 571 343 36 15

Directions: Input the 1998 statistics as Matrix A and the 1999 statistics as Matrix B.

1. What would be the meaning of B – A?

2. What would be the meaning of B + A?

3. What would be the meaning of ½ A?

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http://sportsillustrated.cnn.com/football/nfl/rosters/Quarterbacks.html


Day 50

Essential Questions: How do we use determinants to find the area of a triangle?How do we use matrices to solve any system of equations?

Objective(s): 1.04 Operate with matrices to model and solve problems; 2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

“SAP”: Collaborative Pairs Activities

Lesson Anatomy: 1. Collaborative Pairs:

Solve these matrix equations:

1. X + = -2

2. – X =

3. 2X – 3 =

2. Teacher demonstrates how to find the area of a triangle using determinants and the formula below where the x and y variables represent the vertices of the triangle.

Area = Note: The sign is chosen so that a positive area is the result.

3. Gypsy Moth Problem:Gypsy moths are native to France but in 1869 some were accidentally brought to

the United States. Since that time the gypsy moth population has increased dramatically and these insects have done great damage to the forests, particularly in the northeastern U.S.

The state was going to crop dust the area with an insecticide that would kill the gypsy moths. The infested region was roughly triangular. From the northern vertex of the region, the distance to one of the other vertices is 20 miles south and 28 miles east. The distance to the third vertex is 25 miles south and 20 miles east.

Find the area of the infested region so that the pilot will know how much insecticide to load on the plane.

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4. Collaborative Pairs: EOC Practice Problem: (Answer is B)

The student store decided to sell school spirit pennants. The art teacher laid out a design of the triangular pennant on a coordinate grid. The vertices of the pennant ended on (6,6), (6,16), and (24,11). To find the area of a triangle in a coordinate system, the determinant of a 3x3 matrix can be used with the following formula:

Area = Note: The sign is chosen so that a positive area is the result.

What is the area of the material needed to make a school pennant?A. 50 square units B. 90 square units C. 120 square units D. 180 square units

5. Teacher demonstrates how to solve a system of equations using matrices. Emphasize the set up of the matrices and why you will solve with A-1 B.6. Collaborative Pairs: Solving Systems using Matrices Worksheet

Summarizing Activity: Ticket out the door: Write an explanation to how you would solve the system

x – y + z = 32x – z = 12y – x + 1 = 0

Homework: Study for Quiz on Matrices

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Solving Systems of Equations on the TI-83

1. 3x – 4y = 7 ANS: __________ 9. a + b – 2c = 4 ANS: __________ 4x + 6y = 15 2a + b + 2c = 0 a – 3b – 4c = -2

2. 2x + y = 0 ANS: __________ 10. –a + b – c – d = 1 ANS: __________ 5x + 3y = 2 2a + 2b + c + d = 3

-3a + b – 2c – 2d = 0 5a + 3b – 4c + d = -3

3. 5x – 2y = -10 ANS: __________ 3x + 6y = 66

6. x = -4y ANS: __________3x + 2y = 20

5. 3x + y = 1 ANS: __________x – y = 3

6. 2x – y – 1 = 0 ANS: __________ 3x – 2y + 9 = 0

7. x + y – 3z = 8 ANS: __________ 2x – 3y + z = -6 3x + 4y – 2z = 20

8. a + 2b + c = 0 ANS: __________ 2a + 5b + 4c = -1 a – b – 9c = -5

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Day 51

Essential Question: How can I use matrices to organize data and solve real-world problems?

Objective(s): 1.04 Operate with matrices to model and solve problems.2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

“SAP”: Students will play the Looping Game using Chapter 4 vocabulary words.

Lesson Anatomy: 1. Quiz on Matrices2. How does Integrade use matrices to calculate your grade?

Set up grade of an imaginary student based on the weights used by the teacher in calculating grades. For example:

[83 77 92 89] *

where a student’s homework average is 83; their quiz average is 77; their first test grade is 92; their second test grade is 89 and the homework average weights at 20%, the quiz

average at 20%, the first test at 30% and the second test at 30%.3. Play the Looping Game using Chapter 4 vocabulary found on page 225 of the text. Depending on the size of your class, some students may not get a card or some students may get more than one card.Directions: Create a set of cards which have a vocabulary word on it and a definition, but, the definition is for a word on another card. All the words are defined but the definition is on another student’s card. The first person reads their definition (NOT THE WORD) and the person with the word being defined says, “I have . . .” and then reads their definition. Continue on and it will loop back around to the first person.

This activity is taken from the Professional Development Institute, Inc. Copyright 2001

4. Collaborative Pairs: Use Checkpoint Quiz 1 (all) & Quiz 2(5-10) from the Prentice Hall Chapter 4 Resource book to review for the quiz on tomorrow.5. Guided Practice Matrix Application Problems

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Summarizing Activity: Collaborative Pairs:

Solve the following problem:In a three team track meet, the following chart shows the number of first, second, and third place finishes per team.

1st 2nd 3rd Blandon 4 10 6Walnut Cove 7 6 9Heritage 8 3 4

If five points are awarded for first, 3 for second, and 1 for third, find the total points awarded for each school to decide the winning team.

Homework: none

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Guided Practice Matrix Application Problems

1. You and your friend invested in three stocks and followed their value for one year. You each bought 1000 shares of stock, but you bought 450 shares of stock A, 300 shares of stock B, and 250 shares of stock C. Your friend bought 275 shares of both stocks A and B, and 450 shares of stock C. You and your friend both paid $12.25 for each share of stock A, $9.75 for each share of stock B, and $30.50 for each share of stock C. Find the amount of money you and your friend each spent for the stock purchases.

2. A shoe manufacturer compiled data on the performance of their largest outlet mall stores. Sales (in thousands of pairs) at Outlet A totaled 72 running shoes, 38 basketball shoes, and 51 walking shoes. Sales at Outlet B were 12 running shoes, 8 basketball shoes and 15 walking shoes. At Outlet C, sales were 25 running shoes, 21 basketball shoes and 38 walking shoes. The total sales figure for these shoes at Outlet A was $7996.55 while Outlet B’s sales figure was $1711.85 and Outlet C’s figure was $4116.05. What was the selling price for each type of shoe?

3. The manager of the Snack Shack, tracks the time it takes to make and serve hamburger and chicken sandwiches. It takes 5 minutes to prepare a hamburger and 2 additional minutes to assemble it with cheese, lettuce, tomato, and ketchup. It takes 7 minutes to prepare a chicken sandwich and 1 additional minute to assemble it with the “fixins”. How many sandwiches can be prepared and served by an employee if 42 minutes is spent on preparation and 15 minutes is spent on serving?

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Day 52

Essential Question: How can I use matrices to organize data and solve real-world problems? How can I prepare for the test on Matrices?

Objective(s): 1.04 Operate with matrices to model and solve problems;2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

“SAP”: Students will complete Pairs Checking and “The Envelope Activity” in cooperative groups.

Lesson Anatomy: 1. Quiz Discussion2. Review for test using the Chapter 4 Review in the text on pages 225-227 (11-27, 36-46) Pairs Checking: Have students circle the even numbered problems. Instruct them to do their own work, however, when they get to a number that is circled they should stop and check their answers with their partner. They can not go on unless their previous answers agree with their partner’s answers. If they agree, they can continue working. If they do not agree, they must justify their answers until they can agree on the same answer3. Cooperative Problem Solving: “The Envelope Activity”

Directions: Assign students to groups of 4 and rearrange the room for students to sit with their groups. Make 8-9 copies of the Matrix Application Problems and cut the problems apart and put them into labeled envelopes. Every group will work at their own pace in solving the problems as the teacher walks around the groups, checking their solutions and giving advice as a last resort. Every group member has to assume on of these four roles: facilitator, encourager, checker, or recorder. The facilitator is the “take charge guy” who pushes to get the job done. The encourager makes sure every one in the group participates and stays on task. The checker approves the solution before the teacher is called to the group. The recorder explains the solution to the teacher and shows the answer on their TI-83. Only when the teacher has approved the solution does the group get to turn in their envelope and get the next problem. With every new problem the students in each group has to change to a different role. Add one bonus point to each group member’s Matrices Test Grade for each problem correctly solved by the group.

Summarizing Activity: Complete the group synthesis on the bottom of the recording sheet before turning it in to the teacher.

Homework: EOC Practice Problems and Study for Unit Test on MatricesAnswers: 1. D

2. D3. B

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Name(s) _________________________ _________________________ _________________________ _________________________

Group Recording Sheet for “The Envelope Activity”

Point Scored

1. “Bookstore Problem” _______________________2. “The Travel Agent” _______________________3. “Shipping Business” _______________________4. “Flags Problem” _______________________5. “Candy Sale” _______________________

Total Points for the Group = ________

Group Summarizing Activity

a) Which of the four roles of recorder, encourager, checker, or facilitator was the hardest in your opinion? _____________________________

b) Why was it the hardest? __________________________________________________________________________________________________________________________________________________________

c) Did you like this activity? ____________________________________________________

d) What could be done to improve it? ____________________________________________

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Matrix Application Problems

1. The Campus Bookstore’s inventory of books consists of the following quantities:Hardcover: textbooks---5280; fiction---1680; nonfiction---2320; reference---1890Paperback: textbooks---1940; fiction---2810; nonfiction---1490; reference---2070

The College Bookstore’s inventory of books consists of the following quantities:Hardcover: textbooks---6340; fiction---2220; nonfiction---1790; reference---1980Paperback: textbooks---2050; fiction---3100; nonfiction---1720; reference---2710

Use matrix algebra to determine the total inventory of a new company formed by the merger of the College Bookstore and the Campus Bookstore.

2. A travel agent offers three different travel packages to Williamsburg, Virginia. Package I consists of 4 nights at a hotel, 3 passes to local attractions, and 5 meals. Package II consists of 3 nights at a hotel, 4 passes, and 7 meals. Package III consists of 5 nights at a hotel, 4 passes, and no meals. The agent can book a hotel room for $90 per night, get passes for $28 and provide meal vouchers at a local restaurant for $15 per meal. She wants to run an ad featuring the least expensive package. Which plan should she advertise?

3. Janice, Nancy, and Davonna work after school and weekends for a local shipping business. They get paid a different rate for afternoon, evenings, and weekends. The number of hours they worked during one week is given in the following matrix:

Afternoons Evenings WeekendsJanice 5 2 3Nancy 1 2 6Davonna 2 2 3

If Janice had worked twice the number of hours for the week, her salary would have been $98. If Nancy had worked 2 more hours in the evening, her salary would have been $62. If Davonna had worked 1 more hour on the weekend, her salary would have been $43. Find the rate of pay for each of the times of day worked by the girls.

4. Suppose you and your friend Candace are responsible for making flags for the flag team at Algebra High. There are two types of flags, large ones with 4 orange strips and 3 black strips, and small flags with 2 orange strips and 1 black strip. You plan to make 15 large flags and 10 small ones. Candace will make 5 large and 8 small flags. Use matrices to determine the number of orange and black strips that you and Candace each need.5. The math club holds a fall and spring candy sale for a fundraiser.Last year the club sold in the fall: Last year the club sold in the spring:40 Trail Treats 75 Trail Treats100 Carob Chews 108 Carob Chews0 Fruit Clusters 80 Fruit Clusters40 Nut Bars 65 Nut BarsEach individual bar of candy sold at the following prices:$1.00 Trail Treats$1.00 Carob Chews$0.50 Fruit Clusters$1.50 Nut Bars

(a) What is the total revenue in the fall?(b) What is the total revenue in the spring?(c) What is the total revenue for the year?(d) If the club made a 40% profit in the fall and a 50% profit in the spring, what is the yearly

profit for the group?

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EOC Practice Problems

1. John and his father each own a boat for bass fishing. The motor on one boat takes regular gasoline, and the other needs premium. John records the following information in his expense book:

Regular Premium Cost

3 gal 2 gal $7.351 gal 4 gal $7.75

What is the cost of one gallon of premium gasoline?

A $1.29B $1.39C $1.49D $1.59

2. For a campaign, a company gave away 5,000 toys to children. Toys x and y cost the company $1.29 and $0.98, respectively. The company spent a total of $5,613. How many of toy x did the company give away?

A 229B 2,000C 2,200D 2,300

3. Two pickup trucks have capacities of ¼ ton and ½ ton. They made a total of 18 round trips to haul 7 ½ tons of crushed rock to a job site. Which matrix equation could be used to determine how many round trips each truck made?

A B

C D

Algebra II Lesson Plans for Block Schedule

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Day 53

Essential Question: Am I ready to show on the test today that I know how to operate with matrices?

Objective(s): 1.04 Operate with matrices to model and solve problems.2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

“SAP”: Collaborative Pairs Warm Up Exercises

Lesson Anatomy:1. EOC Homework Practice Discussion2. Collaborative Pairs Warm Up Exercises: Prentice Hall Algebra 2 Text, p. 228, Ex. 4-6,

16,18, 27, 28, 303. Unit Test on Matrices


Homework: none

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Day 54

Essential Questions: How do I simplify radicals fully? How do I add and subtract radicals?

Objective(s): 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems; 1.02 Define and compute with complex numbers.

“SAP”: Students will use White Boards to practice simplifying, adding and subtracting radicals.

Lesson Anatomy: 1. Test Discussion2. Review the Real Number System.3. Since every real number is either rational or irrational, how do you recognize the difference? Have students name different types of numbers and classify them as either rational or irrational. Emphasize that some radicals are rational and some are irrational and explain why.4. Teacher demonstrates through several examples how to simplify square roots, cubed roots, and some fourth roots. Focus only on the principal root. Use Examples 2 and 3 (TE, pgs. 364-365)5. Cooperative Pairs: Check Understanding problems 2a-c and 3a-c (TE, pgs. 364-365)6. Teacher demonstrates how to add and subtract radicals and then simplify the answer fully. Use Example 3 and Check Understanding problem 3 (TE, pg. 375). See also the additional examples 1-3 (TE, pg 375). 7. White Board Practice

Summarizing Activity: Ticket out the Door: Answer these and explain the pattern you see.

1/3 + 4/3 =1x + 4x =1x2 + 4x2 =1 + 4 = 1 + 4 =

Homework: Prentice Hall Algebra 2 Text: Section 7-1 page 366 (39 – 51), Section 7-3 page 376 (1-12)

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Day 55

Essential Question: How do I multiply radical expressions?


“SAP”: Students will play MATHO to review simplifying radical expressions.

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solution.2. Teacher demonstrates correct solution on troublesome homework problems.3. Matho Game on Simplifying Radical Expressions4. Teacher demonstrates the following examples to show the different situations that can occur with multiplication of radical expressions:

* *

2 * 66 * 73(2 + 2 )(2 + )(3 - )(3 - )(3 + )(1 + )(2 - )

(1 + 2 )(1 + )2

(3 - 2 )2

5. White Board Practice: Practice and Problem Solving Exercises page 372 (37-45) and page 377 (13-21 odd)

Summarizing Activity:Ticket out the Door: Find the perimeter and area of the given rectangle.

2 +

5 -

Homework: Addition/Subtraction/Multiplication of Radicals Worksheet

144

Matho Game on Simplifying Radicals

Problem Answer Problem Answer1. 9 2. 15

3. 25 4. 18

5. 10 6. 4

7. 26 8. 19

9. 23 10. 21

11. 7 12. 14

13. 20 14. 22

15. 1 16. 11

17. 29 18. 5

19. 28 20. 12

21. 27 22. 3

23. 16 24. 8

25. 17 26. 2

27. 24 28. 6

29. 13 30. 30

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Addition, Subtraction & Multiplication of Radicals

I. Addition/Subtraction of Radicals: Simplify all answers fully.1. 2 + 3 2. 7 - 2 3. 4 - 5

4. + 5. + 6. 6 + 3

7. 14 - 3 8. + 9. 8 - 3

10. + 11. - 12. +

13. 3 - 2 14. + + 2 15. + 2 - 5

II. Multiplication of Radicals: Simplify all answers fully.1. * 2. * 3. *

4. * 5. * 6. 3 * 2

7. -6 * -2 8. - * 2 9. 6 * 2

10. ( + 2) 11. 3 ( - 2) 12. 4 (3 - 1)

13. (2 + )(2 - ) 14. (3 - )(4 + ) 15. (2 + )2

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Day 56

Essential Question: How do I divide radicals?


“SAP”: Students will compete in the “Radical Race” for review.

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solutions.2. Teacher demonstrates correct solution on troublesome homework problems.3. Radical Races 4. Teacher demonstrates through the following examples the different situation that can occur in dividing radical expressions:

1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

12. 13. 14.

5. White Board Practice: Practice and Problem Solving Exercises page 371 (23-33 odd) and page376 (23-26).

Summarizing Activity:Cooperative Pairs: 1. Find the reciprocal of 2 + .

2. Find the product of 2 + and its conjugate.

Homework: Division of Radicals Worksheet

148

Radical Race #1

Start with:

Partner Group 1:Add 2 to the starting radical. Answer:___________________

Partner Group 2:Subtract 5 from the last answer. Answer:___________________

Partner Group 3:Add 7 to the last answer. Answer:___________________

Partner Group 4:Subtract from the last answer. Answer:___________________

__________________________________________________________________

Radical Race #2

Start with: 2

Partner Group 1:Add 6 to the starting radical. Answer:___________________



Partner Group 4:Add 3 to the last answer. Answer:___________________

149

Division of Radicals Worksheet

1. 2. 3.

4. 5. 6.

7. 10 8. 9.

10. 11. 12.

13. 14. 15.

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Day 57

Essential Question: What do fractional exponents mean?


“SAP”: Students will use white boards to practice simplifying rational exponents.

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solution.2. Teacher demonstrates correct solution on troublesome homework problems.3. Cooperative Pairs: Do and check Practice 7-3(odds) of the Prentice Hall Algebra 2 Workbook. 4. Teacher demonstrates the connection between rational exponents and radical expressions through the definition on p.380. Use Example 2, 4 and 5 (TE, pgs. 380-382) and Check Understanding problems 2, 4 and 5 (TE, pgs. 380-382).5. White Board Practice


Ticket out the Door: Explain how you would find the value of

without your calculator.

Homework: Prentice Hall Algebra 2 Text Section 7-4 page 382 (1-37) and study for Quiz on Operations with Radical Expressions

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Day 58

Essential Question: How do I solve simple radical equations?

Objective(s): 2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties. “SAP”: Students will work in cooperative pairs to solve equations involving radicals.

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solution.2. Teacher demonstrates correct solution on troublesome homework problems.3. Quiz on Operations with Radical Expressions4. Teacher demonstrates how to solve radical equations using the properties of equality through the following examples:

a) = 4b) = c) = 2d) = 2e) y – 4 – = 0f) - 2 = yg) + 4 = 0h) 2 = 10

5. Solve the same problems graphically on the TI-83 and compare answers.

Summarizing Activity: Cooperative Pairs: (Do algebraically and check graphically)

a) = b) = 3c) = x – 1

How should we have recognized algebraically that c) had no solution in the real numbersystem? How does this show up graphically?

Homework: Prentice Hall Algebra 2 Workbook Practice 7-5 (5, 10, 14, 17, 19-21, 25, 30, 34, 36, 37, 39, 42)

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Day 59

Essential Questions: How do I solve equations involving rational exponents? How do I solve problems involving radical expressions?

Objective(s): 2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

“SAP”: Students will work in cooperative pairs to solve problems involving radical expressions.

Lesson Anatomy: 1. Quiz Discussion2. Partners check the algebraic homework solution by graphing them on the TI-83.3. Teacher demonstrates correct algebraic and graphical solution on troublesome homework problems.4. Use Example 2 and Checking Understanding problem 2 (TE, pg 386) to demonstrate how to solve radical equations with rational exponents.5. Cooperative Pairs: Do and check p. 388 9-126. Who is the best weight-lifter in this class? Let class make 3-4 suggestions. The best weightlifter is not always the person who can lift the largest amount of weights. What else might matter in deciding who is the best? Decide who is the best weightlifter through the following formula.

O’Carroll’s Formula for handicapping weight lifters:

W =

Where W = handicapped weight lifted w = weight lifted in kilograms b = your weight in kilograms

one pound = .45 kilograms

Example: Larry weighs 215 pounds (96.75 kg). He can lift 305 pounds (1.37.25 kg).His handicapped weight lifted is 34.724.

Derek weighs 160 pounds (72 kg). He can lift 275 pounds (123.75kg). His handicappedWeight lifted is 36.137.

Derek is the better weight lifter.

153

7. After an accident, police investigators use the formula s = 2 to estimate the speed (s) of a car in miles per hour. The variable L represents the length in feet of the tire skid marks on the pavement. On one occasion, an accident scene investigation team measured skid marks 120 feet long. How fast was the car traveling?

8. If a standard baseball diamond has sides 90 feet long, how far does the catcher have to throw the ball to reach to second base?

9. As part of a tryout for a girls’ softball team, each player must hit a series of balls in a batting cage. The coach determines the velocity of each hit with a speed gun. She uses the formula

d = v to estimate the distance the ball would have traveled if the ball had been hit in an

open field. In the formula, v represents the velocity (in meters per second) of the baseball, and h is the height (n meters) above the ground from which the ball is hit. If one of he girls hits the ball at a speed of 45 m/s from a height of 0.8 meters above the ground, what distance would this ball have traveled?

10. Cooperative Pairs: “How do they fit?” Puzzle (from NCDPI Resources for Algebra)Be sure to have partners cut out the nine puzzle pieces first thing. Then ask them to arrange the puzzle pieces into a 3X3 square so that a problem and its corresponding answer are on adjacent edges of the squares. Problems or answers that do not have a match must be on an outer edge. Have partners glue or tape their finished product to a piece of paper before turning it in.

Summarizing Activity: EOC Practice Problem: Answer: D

What should be the first step in solving the equation = 4?

A Square both sides.B Divide both sides by 2.C Raise both sides to the 1/3 power.D Cube both sides.

Homework: Prentice Hall Algebra 2 Text Chapter 7 Review page 416-417 (11-53 odd, 54, 56) to do as they Study for the Test tomorrow.

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Day 60

Essential Question: Am I ready to show what I have learned about radicals on the Unit Test today?

Objective(s): 2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

“SAP”: Students will complete review problems in cooperative pairs.

Lesson Anatomy:1. Partners compare homework solutions and come to consensus on correct solution.2. Teacher demonstrates correct solution on troublesome homework problems.3. Answer any questions that came up as students were studying for the test.4. Cooperative Pairs:

a) Find the reciprocal of 4 - b) What is the product of 1 + and its conjugate?c) Find the area and perimeter of each figure. Figures are not necessarily drawn to scale.

#1) 2

3

#2) a triangle with legs of 2 , 2 , and 3 and a height of 5. Unit Test on Radical Expressions and Equations.


Homework: none

156


Day 61

Essential Questions: What is an imaginary number? How do I operate with i numbers?

Objective(s): 1.02 Define and compute with complex numbers.

“SAP”: Students will use white boards to practice operating with complex numbers. Students will play MATHO to practice operations with complex numbers.

Lesson Anatomy: 1. Test Discussion2. Introduce the Complex Number System and give examples of each set using the diagram in the text on page 271.3. How do we solve x2 + 1 = 0? Imaginary numbers were created by the French mathematician Rene’ Descartes because he wanted to be able to have an answer for equations like this. We need to learn to operate with imaginary numbers for when we are solving quadratic equations in this unit.4. Explain that every complex number can be written in the form a + bi.5. Demonstrate through examples found in the text on pages 270-273 how to simplify, add, subtract and multiply imaginary numbers. (Exclude example 3 in the text)When the need arises as you are multiplying i numbers, demonstrate the powers of i and how to simplify those powers. Show i as a cyclic group of four through the following graphic organizer. Start at i and move in a counter-clockwise rotation.

i

-1 1

-i

6. Demonstrate through the following examples how to divide complex numbers:

a) b) c) d) e)

7. White Board Practice8. MATHO Game on Operations with Complex Numbers

Summarizing Activity: Ticket out the door:

What would be the simplified form of 3 i102? How did you figure out your answer?

Homework: Operations with Imaginary Numbers Worksheet

157

Operations with Imaginary Numbers Worksheet

I. Express each complex number in the form a + b i . 1. 4i + 6 2. -3i – 2 3. -6 4. 5i

5. 4i 6. 0 7. 7 + 8. -

Simplify.9. (9 + 2i) + (6 + i) 10. (4 + 2i) + (6 + 7i) 11. (-7 – 5i) + (-8 + 2i)

12. (-3 + 2i) + (5 + 6i) 13. (3 + ) + (2 + ) 14. (3 + ) + (5 + )

15. (9 + ) + (16 + ) 16. (9 + 5i) – (10 + 3i) 17. (-7i – 5) – (4 – i)

18. (8 + 6i) – (4i + 5) 19. (9 – 2i) – (-3i + 7) 20. (2i – 14) – (6 – 4i)

II. Simplify.1. = __________ 2. = __________ 3. = __________

4. = __________ 5. 3 = __________ 6. -10 = __________

7. 7 = _________ 8. - = __________ 9. = __________ 10. 3 = __________

III. Simplify.1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

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Matho Game on Operations with Complex NumbersSimplify all problems fully.

Problems Answers Problems Answers1. 9 2. 12

3. 18 4. 19

5. 27 6. 5

7. 23 8. 15

9. 8 10. 6

11. 28 12. 1

13. 7 14. 25

15. 17 16. 30

17. 24 18. 20

19. 16 20. 26

21. 22 22. 2

23. 14 24. 29

25. 4 26. Solve: 3

27. 21 28. 13

29. 10 30. 11

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Day 62

Essential Question: What is the connection between the graph of the quadratic function and the solution of the related quadratic equation?

Objective(s): 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. b) interpret the constants and coefficients in the context of the problem. “SAP”: Students will complete a Paideia Seminar on the Quadratic Function.

Lesson Anatomy: 1. Partners compare homework answers and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Paideia Seminar: Introduction to Quadratic Function graphing (Alternative: Students complete the lab in cooperative pairs and then have a class discussion of the questions with group cooperation in completing the graphic organizer.)

Summarizing Activity: Ticket out the Door: (Show work!) 1. EOC Practice Problem: Answer: B

Simplify:

A

B

C

D

2. Show that and are inverses of each other.

Homework: Prentice Hall Algebra 2 text page 274, Ex. 29-40

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Seminar Plan

Graphing Quadratic Functions

1. Arrange desks into a rectangle or circle.2. Have partners complete worksheets 1-3 on graphing quadratic function through exploration on the TI-833. Lead seminar discussion questions 4. Homework: Prentice Hall Algebra 2 Practice Workbook; Practice 5-5 (first column, 1-40 solve by factoring only)

Seminar Discussion Questions

1. The first sentence reads……. Why does it say in y= ax2 +bx + c that a ≠ 0? For the graphs 1-8, what were the values of a, b, and c?2. What did you find similar about the graphs in Part I? What were the differences?3. How does a in y= ax2 + bx + c affect the graph?4. In part II, what does it mean when it reads y= ax2 + bx + c, where b=0?5. How does the value of c affect the graph? What else could we call the value of c, i.e. what would c be called on the graph?6. In part III, what were your descriptions of the graphs and how did you decide?7. In part IV, how did the b value seem to affect the graph?8. What were your predictions in part V and why?9. What were your solutions to the equation in Part VI? How did those solutions relate to the graph?10. What would be real-world examples of parabolas?11. Discuss Part VII. What is the y intercept? Why does it turn down?12. What do you know, without graphing on the calculator, about the graph of y= 2x2 + 4x + 2?

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Name(s) ___________________________________________________

Exploring the graphs of Quadratic FunctionsOf the form y= ax2 + bx + c

Part I:The graph of a quadratic function of the form y= ax2 + bx + c, where a ≠ 0, is called a parabola. In this part you will investigate how the value of a affects the shape and/or position of the parabola. Start graphing with a user friendly window. Graph the following on your TI-83, and sketch what you see in the space provided.

1. y= 3x2 2. y= -3x2 3. y= 2x2 4. y= -2x2

5. y= 1/2x2 6. y= -1/2x2 7. y= 1/4x2 8. y= -1/4x2

What is the same about all the graphs?

What is different about all the graphs?

How does a affect the graph?

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Part II:Next consider the equation y= ax2 + bx + c, where b=0. Graph each of the following on your TI-83, and then sketch what you see in the space provided.

1. y= 2x2 + 3 2. y= 2x2 – 3 3. y= 2x2 – 4 4. y= 2x2 + 4

How does the value of c affect the graph?

____________________________________________________________________________________________

____________________________________________________________________________________________

Part III:

Do not use a calculator on this section. Predict how each equation will look from what you learned in Part I and Part II. Describe or sketch your prediction.1. y= x2 + 7

2. y= -x2 + 7

3. y= 5x2 + 3

4. y= 1/3x2 -5

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Part IV:Now consider the equation y= ax2 + bx + c. Graph each of the following on your calculator and then sketch what you see.1. y= x2 + 4x + 4 2. y= x2 -4x + 4 3. y= x2 – 3x -10 4. y=x2 + 3x - 10

How does the value of b seem to affect the graph?

____________________________________________________________________________________________

____________________________________________________________________________________________

Part V:Do not use your calculator on this section. Use what you have learned from the previous sections to predict the answers to the following questions about y = 4x2 + 4x + 1.

Which way does it turn? _______________________________________________________________________

What is the y-intercept? ________________________________________________________________________

Is it average width, narrow, or wide? ____________________________________________________________

Is the lowest point on the graph (called the vertex) to the left or right of the y-axis? ________________________

Check the graph on the calculator to see if your predictions were correct.

Part VI:Solve the following quadratic equations algebraically by factoring and using the zero-product property.1. x2 + 4x + 4 = 0 2. 0 = x2 – 4x + 4 3. 0 = x2 – 3x – 10 4. x2 + 3x – 10 = 0

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Compare your answers in this part to the functions you graphed in Part IV. What is the relationship?

__________________________________________________________________________________________

__________________________________________________________________________________________

Part VII:

A golf ball hit into the air forms a parabola. If y = 32x – 5x2 models the height of the golf ball in meters for the x seconds it is in the air, graph this on your calculator. What window did you use to see the complete parabola?Xmin ______________ Xmax ______________ Ymin _________________ Ymax _______________

Find the maximum height reached by the golf ball. ____________________

How many seconds was it in the air when it reached that maximum height (vertex)? ____________________

What is the y-intercept? ____________________________________________________________________

What is/are the x-intercept(s)? _______________________________________________________________

What do/does the x-intercept(s) represent? _____________________________________________________

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Graphing

Quadratic

Equations

Of the Form

y=ax2 + bx + c

167

What does “a” tell us?

What does “b” tell us?

What does “c” tell us?


Day 63

Essential Questions:How do I identify quadratic functions and graphs? How do I find the zeros of a function? How do I find and interpret the maximum and minimum values and the intercepts of a quadratic function? How do I model data with quadratic functions?

Objective(s): 1.02 Define and compute with complex numbers; 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. b) interpret the constants and coefficients in the context of the problem; 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

“SAP”: Students will complete the Quadratic Graphing Exercise and the Modeling Data with Quadratic Functions Worksheet in collaborative pairs.

Lesson Anatomy: 1. Teacher-led homework discussion.2. Discuss the vocabulary, which is highlighted in yellow in Section 5-1 (TE, pgs. 234-236). Show students how to tell the difference between a linear and quadratic function. Use Example 1 (TE, pg 234) and Check Understanding problem 1a-c (TE, pg 235).3. Demonstrate how to identify the vertex, the axis of symmetry and the zeros of a parabola. Use Example 2 (TE, pg 235).4. Discuss the following vocabulary (some of this will be review): quartic, cubic term, quadratic term, linear term, constant term, vertex, zeros, x-intercept, y-intercept, turning points, minimum value, maximum value as related to quadratic and cubic and quartic functions.5. Name the zeros of the function f(x) = by interpreting the graph on the calculator. Discuss the relationships in the vocabulary of x-intercepts, zeros, roots, and solutions.6. Cooperative Pairs: Quadratic Graphing Exercise

7. Many applications require knowledge of quadratic functions. Have students find a quadratic model for the following data and answer the questions:

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A small manufacturing firm collected the data below on advertising expenditures (in thousands of dollars) and total revenue (in thousands of dollars).

Advertising Total Revenue20 $610122 $622225 $635025 $637827 $645328 $642329 $636031 $6231

a) Describe the viewing window that you used to view a complete graph.Xmin = ______Xmax = ______Xscl = _______Ymin = ______Ymax = ______Yscl = _______

b) What is the independent variable? dependent variable?c) Find the quadratic function of best-fit.d) Determine the maximum revenue for this firm.e) Predict the revenue if the firm spends 33,500 on advertising.

5. Collaborative Pairs: Modeling Data with Quadratic Functions (Do and check)

Summarizing Activity: Ticket out the Door (with partner):Compare and Contrast the following two functions:f(x) = and g(x) = List as many similarities and differences as you can in 5 minutes.

Homework: Prentice Hall Algebra 2 Text Section 5-1 page 237 (1-12, 16-22) *need graphing calculator; Study for Quiz on Operations with Imaginary Numbers

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Quadratic Graphing Exercise

y =

Find the following:

a) vertex __________________________________________________________

b) y- intercept ______________________________________________________

c) __________________________________________________

d) __________________________________________________

e) What is the minimum value of the function? ____________________________

f) What is the value of y when x= .5? ____________________________________

g) What is the value of x when y = -1? ___________________________________

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Modeling Data with Quadratic Functions Worksheet

Fuel Consumption

Year Average FuelConsumption

1980 71.91981 71.01982 70.11983 69.91984 68.71985 69.31986 71.41987 70.61988 71.91989 72.71990 72.01991 70.71992 73.91993 75.1

a) What is the equation the quadratic function of best-fit? ________________________

b) Determine the year in which average fuel consumption was lowest. ______________

c) Predict the average fuel consumption for 1996. ________________

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A farmer collected data which shows crop yields for various amounts of fertilizer.

Plot Fertilizer(Pounds/100 ft2)

Yield(Bushels)

1 0 42 0 63 5 104 5 75 10 126 10 107 15 158 15 179 20 1810 20 2111 25 2012 25 2113 30 2114 30 2215 35 2116 35 2017 40 1918 40 19

a) What do you think is the independent variable for this data? _____________________

b) What is the equation the quadratic function of best-fit? _________________________

c) What is the maximum amount of fertilizer to use? _____________________________ What is the crop yield for the maximum amount of fertilizer? ___________________

d) Predict the crop yield if 42 pound of fertilizer is used. __________________________

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Day 64

Essential Questions: How do we solve quadratic equations by factoring?

Objective(s):1.02 Define and compute with complex numbers;2.04(a) Solve (quadratic equations) using tables, graphs and algebraic properties.

“SAP”: Basketball Shootout Game

Lesson Anatomy: 1. Cooperative Pairs: (Do and check)

Simplify fully:1. 2. 3. 4.

5.

6.

7.

8.

9. 10i 10. -10

2. Quiz on Operations with Imaginary Numbers3. Teacher-led discussion of troublesome homework problems.4. Teacher demonstrates through several examples the procedure for solving a quadratic or

cubic equation by factoring. Emphasize the Zero Product Property.5. White Board Practice6. Basketball Shootout Game on Solving Quadratic Equations by Factoring using the problems from the text on pages 266-267, Ex. 1-10, 36-39, 41-42, 46-53Summarizing Activity: Ticket Out the Door: Number 1’s tell Number 2’s the three important steps for solving a quadratic equation by factoring. Number 2’s write them down to turn in. Put both names on the paper.

Homework: Practice 5-5 Workbook, Ex. 1-41 (odd)

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Basketball Shootout” Review Game

Directions: 8. Divide the class into three teams. Each student in the team should pick a partner to work

with during the game. They may only talk to their partner while working on a problem, not with other people on their team.

9. The game leader reads a problem or writes it on the board or the overhead, if needed, and allows enough time for the all the partner groups to agree on their answer.

10. The game leader randomly calls on a partner group in Team #1 to give their answer.11. If the answer is correct, Team #1 receives one point and a chance for bonus points by





174


Day 65

Essential Questions: How do I write a polynomial equation given its solutions? How do I solve quadratic inequalities by factoring and by graphing?

Objective(s): 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. b) interpret the constants and coefficients in the context of the problem.

“SAP”: Students will work in collaborative pairs to complete the Partner Practice Worksheet.

Lesson Anatomy: 1. Quiz Discussion2. Partners compare answers to homework and come to consensus on correct solutions.3. Teacher-led discussion of troublesome homework problems.4. If you know the solutions of a polynomial function are -2 and 2, what could be the polynomial equation?5. Cooperative Pairs: What equation could have a solution set of {-1/2, 0, 5}?6. Cooperative Pairs: The solutions of a polynomial equation are given. Find the polynomial equation.

a) -3, 0, 2 b) ½, 37. The following problem and student solution was on an exam paper in Pre-Calculus. The teacher marked the problem entirely wrong. Explain to me why the teacher gave no credit for the student solution. Then show me the correct solution you would have written.

Solve for x: (2x – 1)(x + 2) = 3---------------------------------

Student Solution:2x – 1 = 3 or x + 2 = 3 2x = 4 x = 1 x = 2

8. What is the difference between y < x2 + 4x + 4 and x2 + 4x + 4 < 0?9. Teacher demonstrates the procedure for solving quadratic inequalities by factoring and sign-line test.10. Partner Practice Worksheet. Do and check.11. Using the same problems as the Partner Practice Worksheet demonstrate how to graph the functions on the calculator and interpret the solution set for the equations and inequalities from their graphs.(<0: name values of x for the points on the graph lying below the x axis, >0: name values of x for the points on the graph lying above the x axis)

175

Summarizing Activity: Cooperative Pairs: Teacher graphs y = x3 – 4x on the overhead calculator. Ask the students to use the graph to name the solution of x3 – 4x < 0.

Homework: Quadratic Inequality Worksheet.

176

Partner Practice Worksheet

I. Solve the following quadratic equation:

II. Solve the following quadratic inequalities:1. 2.

3. 6x + 18x 4. x – 7x

5.

177

Quadratic Inequality Worksheet

Solve each inequality by factoring and the sign-line test:1. 2. 3.

4. 5. 6.

7. 8. 9.

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Day 66

Essential Question: How do I solve quadratic equations by completing the square?


“SAP”: Students will work in cooperative pairs to complete the square.

Lesson Anatomy: 1. EOC Practice Problem. Answer: C

What are the approximate solutions to the following equation?

A {-0.19, -2.14}B {-0.14, -1.19}C {0.14, 1.19}D {0.19,2.14}

2. Ask students the following questions:1. Why are these solutions approximate?2. Does anyone know if or how we could find the exact solutions? Some students mayremember the quadratic formula.

3. Demonstrate through several examples the process of completing the square to solve a quadratic equation.4. Cooperative Pairs: Section 5-7 Check for Understanding pages 279-280 3a, 3b, 5a. (Do and check)5. Have students solve Example 4 on page 279. Then ask them to try to solve the same equation with the graphing calculator. Have them explain their results.6. Cooperative Pairs: Section 5-7 Check for Understanding pages 279-280 4c and5b


1 2

Explain how you would find the value of c that would be used to complete the square when solving the equation .

Homework: Prentice Hall Algebra 2 Text Section 5-7 page 281 (7-17 odd, 23-27 odd)

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Day 67

Essential Question: How do I solve quadratic equations by the Quadratic Formula?


“SAP”: Students will use White Boards to practice the quadratic formula.

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Tell students that an alternative method of solving quadratic equations that do not factor is the Quadratic Formula. Present the formula and show how to use it using one example from the homework. Derive the formula from , by completing the square, for honors classes.4. Teach students how to remember the Quadratic Formula by putting it in song to the tune of Frere Jacques. Have students practice singing it several times.5. Demonstrate how to use the Quadratic Formula using several more examples from the homework. Be sure to show at least one that has rational solutions, one that has irrational solutions, and one that has complex solutions.6. Show all three methods of factoring, completing the square, and the Quadratic Formula using the equation .7. White Board Practice: Check Understanding problems 1 and 2 (TE, pg. 286) and Additional Examples 1 and 2 (TE, pg. 287)

Summarizing Activity: EOC Practice Problem Answer: A

Solve :

ABCD

Homework: Prentice Hall Algebra 2 Text Section 5-8 page 289 (1-21 odds)

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Day 68

Essential Questions: What does the discriminant in the Quadratic Formula tell us about the nature of the roots? How do we use polynomial equations to solve problems?


“SAP”: Students will complete the Quadratic Graphing Exercise 2 in cooperative pairs.

Lesson Anatomy: 1. Partners compare homework answers and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Cooperative Pairs: Quadratic Graphing Exercise 24. Discuss the discriminant in the Quadratic Formula and what it tell about the kind of answers that the Quadratic Equation has.5. Graphic Organizer:

Solving Quadratic Equations

by Factoring by Completing the Square by Quadratic formula

Only gives solutions if they Always gives solutions Always gives solutions are rational easiest to use

Discriminant

b2 – 4ac = 0 b2 – 4ac > 0 b2 – 4ac < 0 1 real, rational root 2 real roots 2 imaginary roots

If perfect square If non-perfect square

2 rational real roots 2 real, irrational roots

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6. The longest home run recorded in major league baseball history was hit by Babe Ruth in an exhibition game between the Boston Red Sox and the NY Giants in 1919. The path of the ball is described by the equation . What was the greatest height that the ball reached? How far from home plate did the ball land? What did x represent and what unit of measure would be appropriate for it? What did y represent and what unit of measure would be appropriate for it? What was the independent variable and what was the dependent variable?7. Write the function that represents the area of a barn that has a length that is 5 yards less than twice the width. Find the area of the barn when the width is 10.15 yards. Find the width when the area is 102 square yards.8. The length of the base of a triangle is 10cm less than 4 times the height. Find the length of the base and the height of the triangle if its area is 25 square centimeters.

Summarizing Activity: Cooperative Pairs: Find the width of a rectangle if its length is three centimeters more than its width and its area is 108 square centimeters.

Homework: Prentice Hall Algebra 2 Practice Workbook and Text: Practice 5-8 (1-15 odd), Section 5-7 page 282-283 (39, 50), Section 5-8 page 290-291 (53, 56, 67, 68), Study for Quiz on solving quadratic equations by graphing, factoring, completing the square and the Quadratic Formula

182

Quadratic Graphing Exercise 2

Draw a sketch of what you see:

Find the following:

a) vertex _________________________________________________________

b) y-intercept _____________________________________________________

c) x-intercept(s) or zeros ____________________________________________

d) ≤ 0 ________________________________________________

e) > 0 ________________________________________________

f) What is the maximum value of the functions? _________________________

g) What is the value of y when x = 1.65? _______________________________

h) What is the value(s) of x when y = 1? _______________________________

i) Algebraically find the exact solutions of = 0 using the QuadraticFormula. Check to see if the solutions match to the approximate x-intercepts foundthrough graphing.

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Day 69

Essential Questions: How do I solve radical equations by factoring and graphing? How do I solve equations with rational exponents?

Objective(s): 2.07 Use equations with radical expressions to model and solve problems; justifyresults. a) Solve using tables, graphs, and algebraic properties.

“SAP”: Students will work in cooperative pairs to solve radical and rational exponent equations.

Lesson Anatomy: 1. Partners compare homework solutions and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Quiz on solving quadratic equations by graphing, factoring, completing the square and the Quadratic Formula.4. Cooperative Pairs: How would you solve ?5. Let students present the solution methods they used. Hopefully some group will figure out how to solve it algebraically and some will show it graphically. If not, demonstrate both methods. Since radical equations were first presented in a prior Unit this will be an extension of what was learned. Use the following examples to demonstrate both the graphing method and the algebraic method several more times. Be sure to compare the solutions both ways.

a) b) c)

6. Cooperative Pairs: Solve 7. Demonstrate through the following examples the algebraic and graphical procedures for solving equations with rational exponents:

a)

b)

c)

d)

8. Cooperative Pairs: Solve

184

Summarizing Activity: Ticket out the Door: Explain the steps you would take to solve

Homework: Radical Equations Worksheet

185

Radical Equations Worksheet

Solve each equation algebraically. Check each equation graphically.1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

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Day 70

Essential Questions: How do I solve problems involving polynomial functions by graphing? How do I model data using polynomial functions?

Objective(s): 2.04 Create and use best-fit models to solve problems.2.06 Use cubic equations to model and solve problems. a) Solve using graphs, b) Interpret constants and coefficients.

“SAP”: Students will work in groups of 3-4 on the Polynomial Problem Solving Activity.

Lesson Anatomy: 1. Quiz Discussion.2. Partners compare homework solutions.3. Teacher-led discussion of troublesome homework problems.4. EOC Practice Problem: Answer: A

Solve:

A {4}B {1/4, 1}C {-1, 4}D no solution

5. Cooperative Pairs: Section 6-1 Investigation on page 300 of the text. Review answers.6. Discuss the following vocabulary: degree of a polynomial (relate degree to the number of zeros of a polynomial) and end behavior. Revisit key vocabulary, such as monomial, binomial, trinomial, standard form and standard form. Tell students that the end behavior of a graph describes the far left and the far right portions of the graph. The graphs of polynomial functions show four types of end behavior; up and up, down and down, down and up, and up and down. You can determine, by inspection the end behavior of the graph of a polynomial function in standard form. Show students several examples.7. Cooperative Pairs: End Behavior Extension: page 306 1-10 (Do and Check)8. Tell students that they have already used lines and parabolas to model data. Sometimes you can fit data more closely by using a polynomial model of degree three or greater. Use Example 2 on page 302 of the text. Do Check for Understanding #2.

187

9. Cooperative pairs: EOC Practice Problem Answer: CWhich cubic polynomial best describes the data in the table below?

x -3 -2 -1 0 1 2 3 4y -12 0 0 -6 -12 -12 0 30

AB CD

Ask students why D could automatically be excluded from the list of answers.

10. Use the following problems as Guided Practice:

a) If describes the newspaper circulation (in millions) in the United States for 1920-98 (x = 20 for 1920). Identify the periods of increasing or decreasing circulation. When did newspaper circulation peak? When will the circulation be approximately 45 million?

b) If models U.S. exports and models U.S. imports for the period 1970-1998 (x = 0 for 1970), find the years when U.S. trade was balanced

. Graphically identify and algebraically define the U.S. trade surplus/deficit according to the functions given.

11. Cooperative Groups: Polynomial Problem Solving Worksheet


Homework: Finish the Polynomial Problem Solving Worksheet

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Polynomial Problem Solving Activity Worksheet

1. From a hand-stand position, a diver completes a dive from a 10 meter platform. The diver’s height h, in meters, at any time t, in seconds, is given by :

a) What would be a good “user-friendly” viewing window to see this graph?Xmin = ________Xmax = ________Xscl = _________Ymin = ________Ymax = ________Yscl = _________Xres = _________

b) At exactly one-second, what is the height of the diver? ____________________________

c) After how many seconds is the diver halfway between the top of the platform and the surface of the water? ___________________________________

d) How long did the dive last? _____________________

2. Graph


b) What is the value of y when x = -3? _______________

c) What is (are) the x-intercepts of the graph? _____________________________________

d) What is the maximum value of the function? ____________________________________

e) At what value(s) of x is y =4? ____________________

f) If the function were changed to , what is (are) the zeros of the function?

189

3. The height, in feet, of a ball being tossed upwards from a person’s hand is modeled by the function , where t represents time in seconds.


b) How high was the ball one second after it was released? __________________________

c) After how many seconds did the ball hit the ground? _____________________________

d) What was the maximum height reached by the ball? ______________________________

e) When was the ball 15 feet above the ground? ___________________________________

4. The following data represents the number of larceny thefts (in thousands) in the United States for the years 1983-1993. (Let x = 1 for 1983)

Year Larceny Thefts

1983 67131984 65921985 69261986 72571987 75001988 77061989 78721990 79461991 81421992 79151993 7821

a) What is the independent variable? ____________________ What is the dependent variable? ______________________

b) Find the cubic function of best fit. ____________________________________________

c) Predict the number of larceny thefts in 1994. __________________________________

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Day 71

Essential Question: How do I analyze the factored form of a polynomial? How do I write a polynomial function from its zeros? Am I ready for the unit test on Quadratics?


“SAP”: Students will play “4 Corners” as a review for the test.

Lesson Anatomy: 1. Teacher-led discussion of troublesome homework problems.

2. Solve =0 by graphing y = on the TI-83. Remind students how to interprete the graph to solve inequalities where y < 0 and y > 0

3. Cooperative Pairs:Using the TI-83 to solve Cubic and Quartic Equations and Inequalities

4. Demonstrate by several examples how to use the zeros of a polynomial to write the polynomial in factored form. Discuss what happens to the graph of a polynomial with repeated zeros. Review how to write the standard form of a polynomial given its roots. Include examples of polynomials that have 2 irrational roots and 2 imaginary roots.

5. Review for Unit Test: Four Corners Activity

Four Corners: Have students complete the EOC Practice Problem Worksheet independently. Each corner of the room should be labeled A, B, C, or D. Before reviewing each answer, ask students to go to the appropriate corner of the room that is the answer to the problem. (You may need to ask a few students to purposely go to a wrong corner of the room to deter some students from following others.)

Answers: 1. C 2. C 3. A 4. B 5. B 6. C 7. C 8. A 9. B

Summarizing Activity: Create two problems you think I will have on your Test tomorrow. Solve them on another piece of paper and trade problems with your partner. Then check your answer against the answers your partner had. Discuss any differences that occur until you are convinced both of you could solve all four problems correctly

Homework: Study for Unit Test

191

Using the TI-83 to solve Cubic and Quartic Equations and Inequalities

I. 1. Solve x3 – 2x2 – 5x + 6 = 0 by graphing. _______________ Sketch the graph:

1. Solve: x3 – 2x2 – 5x + 6 0_________________________________________________

2. Solve: x3 – 2x2 – 5x + 6 0_________________________________________________

4. Approximate the turning points of the function. ___________________________________________________

II. 1. Solve: x4 – 9x2 + 4x + 1 = 0 by graphing. ________________Sketch the graph:

2. Solve: x4 – 9x2 + 4x + 1 0_________________________________________________

3. Solve: x4 – 9x2 + 4x + 1 0___________________________________________________

4. Approximate the turning points of the function.____________________________________________

192

EOC Practice Problem Worksheet

1. What are the approximate zeros of the function ?

A {3, -1}B {4, 0}C {-1.2, 0.3, 2.9}D {-1.1, 0.2, 3.0}

2. What is the approximate positive zero of ?

A 2.01B 1.97C 1.89D 1.75

3. A company’s total revenue R (in millions of dollars) is related to its expenses by the equation , where x is the amount of expenses (in tens of thousands of dollars). What values of x will produce zero revenue?

A x = 0, x = 1, x = 3B x = 1, x = 3, x = 4C x = 1, x = 3D x = 0, x = -1, x = -3

4. Which quadratic equation has roots and ?

ABCD

5. The height, h(t), in feet, of an object shot from a cannon with initial velocity of 20 feet per second can be modeled by the equation , where t is the time, in seconds, after the cannon is fired. What is the maximum altitude that the object reaches?

A 13.5 feetB 12.25 feetC 10.25 feetD 1.5 feet

193

6. A company found that its monthly profit, P, is given by where x is the selling price for each unit of product. Which of the following is the best estimate of the maximum price that the company can charge without losing money?

A $300.24B $210.00C $10.58D $6.00

7. Solve:

A {-3, 4}B {3, 5}C {4, 5}D {4, 6}

8. Solve:

A

B

C

D

9. A model rocket is fired upward at an initial velocity of 240 ft/s. The height h(t) of the rocket is a function of time t in seconds and is given by the formula . How long will it take the rocket to hit the ground after takeoff?

A 16 secondsB 15 secondsC 7.5 secondsD 4 seconds

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Day 72

Essential Question: Am I ready to show what I have learned about polynomial functions on the test today?

Objective(s): 1.02, 1.03, 2.02, 2.04, 2.06, 2.07

“SAP”: Students will do and check the Solving Equations and Inequalities Calculator Lab.

Lesson Anatomy: 1. Do and check the Solving Equations and Inequalities Calculator Lab.2. Unit Test


Homework: none

195

Solving Equations and Inequalities Calculator Lab

1. a) -x2 + 4 = 0 __________________ 2. a) 2x2 – 3x – 7 = 0 ___________________

b) -x2 + 4 < 0 __________________ b) 2x2 – 3x – 7 < 0 ___________________

c) -x2 + 4 > 0 __________________ c) 2x2 – 3x – 7 > 0 ___________________

d) -x2 + 4 0 _________________ d) 2x2 – 3x – 7 0 __________________

e) -x2 + 4 0 _________________ e) 2x2 – 3x – 7 0 ___________________

3. a) 2x __________________ 4. a) = 9 _______________________

b) 2x __________________ b) < 9 _______________________

c) 2x __________________ c) > 9 _______________________

d) 2x __________________ d) 9 _______________________

e) 2x __________________ e) 9 _______________________

5. a) -2x2 + 13x – 20 = 0 ____________ 6. a) x3 – 3x2 –x + 3 = 0 _________________ b) -2x2 + 13x – 20 < 0 ____________ a) x3 – 3x2 –x + 3 < 0 _________________

c) -2x2 + 13x – 20 > 0 ____________ a) x3 – 3x2 –x + 3 > 0 _________________

d) -2x2 + 13x – 20 0 ____________ a) x3 – 3x2 –x + 3 0 _________________

e) -2x2 + 13x – 20 0 ____________ a) x3 – 3x2 –x + 3 0 _________________

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Day 73

Essential Question: How do I write equations of and graph circles and ellipses?

Objective(s): 2.09 Use the equations of parabolas and circles to model and solve problems. a) Solve using tables, graphs, and algebraic properties. b) Interpret the constants and coefficients in the context of the problem.

“SAP”: Students will work in cooperative pairs on the Circles and Ellipses Practice Worksheet.

Lesson Anatomy: 1. Test Discussion2. Collect Graphing Functions Exploration Worksheet to be graded for accuracy.3. What is a circle? What is an ellipse? Are these functions? What are their equations? How can you tell from their equations if they are functions or just relations? Show how a circle and an ellipse are formed from intersecting a cone and a plane to explain why these are referred to as conic sections.4. Demonstrate how to find the center and the radius of a circle and draw the graph from the equation. Also reverse the process and write the equation if given the center and the radius information or the information is shown in graph form. Use Examples 1, 2, 4 and 5 (TE, pgs. 550-552). Have students practice leaving answers in factored form and in expanded form.5. Cooperative Pairs: Practice and Problem Solving problems p. 552-553 (3, 15, 17, 23, 63) - do and check.6. Demonstrate how to find the center, horizontal and vertical vertices and draw the graph of an

ellipse from both the forms of and

7. Cooperative Pairs: Begin the Circles and Ellipses Practice Worksheet.

Summarizing Activity: Ticket out the Door:Draw the graph of x2 + y2 = 9. What is the domain and range? Function or relation?Draw the graph of 2x2 + 3y2 = 6. What is the domain and range? Function or relation?

Homework: Complete the Circles and Ellipses Practice Worksheet

197

Circles and Ellipses Practice Worksheet

I. Given the equation of the circle, find the center and radius. Draw a sketch from that information.

1. (x – 3)2 + (y + 7)2 = 19 2. x2 + y2 – 6x + 4y – 12 = 0 3. 4x2 + 4y2 = 36

II. Given the center and the radius of the circle, write the equation of the circle in its expanded form.

1. center (0, 0); radius 4 2. center (-3, 7); radius 2 3. center (-2, -4); radius 7

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III. Given the equation of the ellipse, answer the requested information

1. a) center _____________________________

b) horizontal vertices ___________________Sketch:

c) vertical vertices _____________________

d) major axis _________________________

2. a) center _____________________________



d) major axis _________________________

3. 9x2 + 4y2 – 18x + 16y = 11 a) center _____________________________



d) major axis _________________________

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Day 74

Essential Question: How do I graph hyperbolas and non-functional parabolas from their equations?


“SAP”: Students will practice graphing hyperbolas and non-functional parabolas on white boards.

Lesson Anatomy: 1. Return and discuss graded Graphing Functions Exploration Worksheet.2. Partners compare homework solutions and come to consensus on correct solutions.3. Teacher-led discussion of troublesome homework problems.4. Cooperative Pairs: EOC Practice Problem: Answer: D

Which of the following points is in the interior of the graph of the relation ?

A (3, 5)B (2, 4)C (0, -5)D (-2, -3)

5. Demonstrate that the equation of a non-functional parabola is x = ay2 + by + c and explain how the values of a, b and c affect the graph.6. White Board Practice: Have students tell the direction of the opening and explain how the values of a, b and c affect the graphs on problems 46-49, page 547. 7. Demonstrate how to graph hyperbolas from the equation forms of

and and

Use Examples 1, 2 (TE, pgs. 564-565) and x2 – 4y2 – 2x – 8y = 7. Also, show how these two conics are formed from intersecting the cone(s) with a plane.8. White Board Practice9. Cooperative Pairs: Begin Hyperbola Practice Worksheet

Summarizing Activity:What’s My Conic?

Homework: Complete the Hyperbola Practice Worksheet

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What’s My Conic?

1. ______________________

2. ______________________

3. ______________________

4. ______________________

5. ______________________

6. ______________________

7. (x – 2)2 + (y + 1)2 = 1 ______________________

8. x2 – 8x – y + 19 = 0 ______________________

9. x2 – 8x = y2 + 19 ______________________

10. x2 – 8x + 4y2 + 19 = 0 ______________________

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Hyperbola Practice Worksheet

1.

a) center __________________________________ Sketch:

b) turns in which direction? ___________________

c) distance from center to vertex points? _________

d) slope of diagonal asymptotes ________________

2. 25x2 – 16y2 = 400

a) center __________________________________ Sketch:




3. x2 – 4y2 – 2x + 16y – 19 = 0

a) center __________________________________ Sketch:




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Day 75

Essential Question: Do I know how to recognize conic sections from their graphs?


“SAP”: Students will complete the Conic Sections Activity Worksheet in collaborative pairs.

Lesson Anatomy: 1. Partners compare homework answers and come to consensus on the correct solutions.2. Teacher-led discussion of troublesome homework problems.3. Cooperative Pairs: (Do and discuss) Conic Sections Activity4. Cooperative Pairs:

Solve: 2x – 3y = -14 x + y = 5

What are the different methods of solution that you used?

How would we solve this system x2 + 4y2 = 25 2y – x = 1 ?

5. Use the system above and the following two examples to show students how to solve quadratic systems both algebraically and graphically.

x2 + y2 = 9 x2 + 3y2 = 125x – y = 0 x2 – y2 = 9

Discuss how many different solutions there could be if you intersect a line with a line, a circle with a line, or a circle with a hyperbola.

Summarizing Activity: EOC Practice Problem: Answer: C

1. Which curve opens to the left?

A y2 = 8x + 24

B y =

C y2 = -8x – 24

D y =

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EOC Practice Problem: Answer: B

2. Solve: y = 3x2 + 3y = 5 – 5x

A

B

C

D

Homework: Study for Partner Quiz on Conic Sections

204

Conic Section Activity Worksheet

1. How does the graph of (x – 2)2 + (y + 1)2 = 36 compare to the graph of x2 + y2 = 36?

2. Compare the similarities and differences of and .

3. How does the graph of compare to the graph of ?

4. What are the similarities and differences of and ?

5. Graph x2 + y2 = 25.a) What will happen to the graph if we replace x with x – 2?_________________________________________________________________________b) What will happen to the graph if we replace x with x + 2?_________________________________________________________________________c) What will happen to the graph if we replace y with y – 1?_________________________________________________________________________d) What will happen to the graph if we replace 25 with 36?_________________________________________________________________________

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Day 76

Essential Questions: How do I write and graph exponential functions of the form f(x)= a(b)x

and f(x) = a(1+r)x? How do I write and interpret an equation of a curve(exponential) which models a set of data? How do I use exponential functions to model and solve problems? How do I create and use best-fit models of exponential functions to solve problems involving sets of data?

Objective(s): 2.03 Use exponential functions to model and solve problems.2.04 Create and use best-fit mathematical models of exponential functions to solve problems involving sets of data.

“SAP”: Students, in collaborative pairs, will complete the Investigating Exponential Functions Worksheet.

Lesson Anatomy: 1. Partner Quiz on Conics 2. Consider the experiment that begins with 500 bacteria. The bacteria doubles every hour. Complete the chart that shows the relationship between the number of bacteria over time.

3. Graph the data on the calculator and look at the scatterplot. This curve is called an exponential function. Its equation is always in the form y = a(b)x. Analyze the data using exponential regression and write the equation of the best-fit line. What is the independent variable? What is the dependent variable? What is the domain?4. Use the prediction equation to tell how many bacteria would be in the sample after 11.5 hours. When were there 2000 bacteria in the sample?5. Collaborative Pairs complete the Investigating Exponential Functions Worksheet. Have group discussion about what the “a” value and the “b” value tell us about the graph of an exponential function? Complete Exponential Function Graphic Organizer.6. Relate the equation of best fit of the bacteria to the formula y = a(1+r)x. 7. Demonstrate through Examples 1-3 on Exponential Problem Solving Worksheet the use of the formulas y=a(b)x, y=a(1+r)x, and A=Pert.

x= time in hours y=number of bacteria0 50012345

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Summarizing Activity: Ticket out of Door:The price of a hamburger and fries on the menu of Rockin’ Burger is $3.99 but the corporate management team adjusts the prices on its menu each year on January 1. If the annual rate of increase is 3%, what would you expect to pay for that same burger and fries in 5 years?

Homework: Complete Exponential Problem Solving Worksheet Problems 4-10

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Investigating Exponential Functions

I. On the same coordinate axes, sketch the graphs of the following functions.

a) y = 10x b) y = 3 x c) y = 4 x d) y = (1/2) x e) y = (1/3) x f) y = 1 x

1. Do the graphs have any points in common? If so, name them. _____________________

2. Look at graphs a, b, and c. How are they alike? _________________________________How are they different? ______________________________________________________

3. Look at graphs d and e. How are they alike? ___________________________________How are they different? ______________________________________________________

4. What is what is different about graph f? _______________________________________

5. Will any of graphs a-f intersect the x axis? Why or why not? ______________________

II. Use your TI-83 to compare and contrast each of the following groups of equations:

1. y = 2(3) x, y =2(1) x, y = 2(1/3) x

2. y = 3(2) x, y =5(2) x, y = 2(1/2) x

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“a” means . . . “b” means . . .

General Form of

Exponential Functions

y = a * bx, b>0

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Exponential Problem Solving

1. The cost of attending college has been rising at an average increase of 7% for the past ten years. How much will a college that now costs $10,368 for room, board, tuition and fees cost two years from now?

2. My Toyota Avalon was purchased for $27,500 in 1997. It has been decreasing in value at an average rate of 15% annually since then. What is its approximate value now?

3. What would be the value of $20,000 invested at 5.75% APR for 40 years if it wasa) compounded annually b) compounded quarterly c) compounded monthly d)compounded daily e) compounded continuously?

4. The island of Manhattan was sold for $24 in 1626. How much would this amount have grown to in 2003 if it had been invested at 6% per year compounded quarterly?

5. In a certain city, the value of a home is increasing at a rate of 9% annually. Find the value of a $100,000 home in four years. In how many years will the value of the house be about double what it is now?

6. You have a choice either to be given $1,000 now to invest in an account earning 5.1% APR, compounded daily, for 30 years, or be given $2,000 in fifteen years to invest in a similar account for 15 years? If you were not allowed to touch either account until its time lapsed, which investment would you choose?

7. Bacteria in a culture grows exponentially. If the original number of bacteria in the culture was 20,000 and the bacteria was growing at a 9% growth rate per hour, how many would be in the culture after seven hours? How long would it take the bacteria to triple in size?

8. Jerry has joined the Peace Corps and is studying Malagasy at the language school. Because he entered the school late, he was able to learn only 100 key phrases before the month long Winter Break. Jerry returns to his hometown for the holiday and does not practice. According to language experts, without continual practice or immersion, a person will forget 0.5% daily of any new language they are studying. How many of the key phrases does Jerry remember the day before he returns to the language school in January? Back at school, it takes Jerry just a short time to recall forgotten phrases.

9. If A= 800(1.03)8 is a formula used to find the value of farmland that has been increasing yearly over time, interpret the meaning of a) 800 b) 1.03 c) 8 d)A.

10. The half life of radium (Ra226) is 1620 years. If y = y0e-0.00043t is the formula that represents the decay of radium, what would be the mass remaining from a 20 gram sample that was sealed in a box for 5000 years? How long would it take the sample to deteriorate to one-fourth of its original mass? When would the sample be gone completely?

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Day 77

Essential Question: How do I use exponential functions to model and solve problems? How do I write and evaluate logarithmic expressions?

Objective(s): 1.01 Simplify and perform operations with rational exponents and logarithms(common and natural) to solve problems.

“SAP”: Students will play MATHO to practice evaluating logarithmic expressions.

Lesson Anatomy: 1. Quiz Discussion 2. Partners compare homework solutions and come to consensus on correct solutions.3. Teacher-led discussion of troublesome homework problems.4. Cooperative Pairs: Solve each of the following.(a) Suppose you invested $25,000 for 30 years at 4.25% APR. After 30 years how much would be in the account if it were a) compounded annually, b) compounded quarterly c) compounded monthly d) compounded daily e) compounded continuously?(b) Growth of a human embryo is approximately exponential, increasing at about 28.5% each day. A model for the weight of a human embryo is W = .0125 * 1.285t, where W is the weight in milligrams and t is the age of the embryo in days. What would be the approximate weight of the embryo in 90 days? After how many days would the embryo weigh one pound?(c) A general rule-of-thumb for used car dealers is that the trade-in value of a car decreases by 30% each year. If your car is worth $3200 now, what would be its trade-in value in 3 years? What was your car worth 2 years ago?5. Teacher demonstrates the inverse relationship between the exponential function and the logarithmic function. Practice through several examples changing from exponential form to logarithmic form and vice versa. Include natural logarithms in these examples.6. MATHO Game on evaluating logarithms (without the calculator)

Summarizing Activity: Ticket out the Door:

What I learned about logarithms . . . What I still have questions about . . .

Homework: Logarithm Worksheet

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Matho Game on Evaluating Logarithms

Problems Answers Problems Answers1. 11 2. 12

3. 13 4. 1

5. Solve: 10 6. Solve: 20

7. 21 8. 18

9. 2 10. Solve: 7

11. 8 12. Solve: 16

13. 29 14. 5

15. Solve: 15 16. Solve: 27

17. 17 18. 24

19. 14 20. Solve: 9

21. 26 22. 3

23. 30 24. 19

25. Solve: 22 26. 4

27. Solve: 6 28. Solve: 23

29. Solve: 28 30. 25

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Logarithm Worksheet

I. Express the following in exponential form:1. __________ 2. __________ 3. __________

4. __________ 5. __________ 6. ___________

7. __________ 8. ____________ 9. ____________

II. Express the following in logarithmic form.

1. 34 = 81 ___________ 2. 27 = 128 ___________ 3. = ____________

4. ___________ 5. ___________ 6. __________

7. __________ 8. __________ 9. _____________

III. Evaluate the following:

1. __________ 2. = ____________ 3. ___________

4. __________ 5. ____________ 6. __________

7. _________ 8. _________ 9. _________

10. ________ 11. ____________ 12. ___________

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Day 78

Essential Questions: How do I create and use best-fit models of exponential functions? How do I use the properties of logarithms?

Objective(s): 2.04 Create and use best-fit mathematical models of exponential functions to solve problems involving sets of data.1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

“SAP”: Students will graph exponential data to find equations of best-fit.

Lesson Anatomy: 1. Partners compare homework answers and come to consensus on correct solutions.2. Teacher-led discussion of troublesome homework.3. Divide the class into three groups and assign one of the Exponential Data Sets to each group. Have them analyze their data and find the equation of the best fit line. Then they need to make a prediction based on their data. Each group will present their findings to the class.4. Teacher presents the change of base formula and uses several of the previously worked homework problems to show that it gives us the correct answer using the calculator.5. White Board Practice

a) d) g)

b) log 39 e) h) 2x = 5c) ln 10 f)

6. Teacher presents the laws of logarithms and uses numbers to prove that they work to allow us to combine logarithm terms. Use Section 8-4 Practice and Problem Solving problems on page 449 (11-18).7. Group Discussion: Practice and Problem Solving page 450 (58-65)

Summarizing Activity: Ticket out the Door: Use the laws of logarithms to combine the following expression into one term:

Homework: Exponential and Logarithmic Calculations Worksheet

215

Exponential Data Sets

I. Hurricane Fran hit North Carolina on the evening of September 5, 1996 leaving over one million homes and businesses without power. This information was in The News & Observer of Raleigh on September 14, 1996.

Day after storm hit Date Customers without power1 Sept. 6 1,159,0002 Sept. 7 804,0003 Sept. 8 515,0004 Sept. 9 340,5005 Sept. 10 195,2006 Sept. 11 136,3007 Sept. 12 77,0008 Sept. 13 37,600

II. The following table indicates the cost of a 30-second television commercial during the Super Bowl game from 1977 through 2003.

Year Cost (in thousands of dollars)1977 1251978 1851979 1801980 2341981 2751982 3451983 4001984 3751985 5251986 5501987 5901988 6501989 6891990 7001991 8151992 8401993 8501994 10001995 12001996 12001997 12001998 13001999 20002000 23002001 21002002 2100

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III. The following represents the temperature of coffee cooling on the table in a 70˚ Farenheit room.

Time (in minutes) F˚ Above Room Temperature0 1355 10010 7415 5520 4125 3030 2235 1740 1245 950 755 560 4

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Exponential and Logarithmic Calculations Worksheet

I. Solve for the unknown value with your TI-83. Round decimals to the nearest hundredth.

1. ln 5 = _______________ 2. _______________ 3. ____________4. log 32 = _____________ 5. ________________ 6. ____________7. ______________ 8. ________________ 9. ___________10. _____________ 11. _______________ 12. 3___________

13. ____________ 14. _________________ 15. ___________

16. ____________ 17. _________________ 18. ___________

II. Suppose you had $100,000 to invest for five years at an APR of 3.25%. Calculate the amount of money you would have after 5 years if the money were compounded under these different options.a) compounded annually b) compounded quarterly c) compounded monthly

d) compounded daily e) compounded continuously

III. Use the properties of logarithms to combine these log terms together.1. log 2 + log 3 2. 3.

4. log 14 – log 7 5. 2 log 32 + log 2 6. log 24 – 3 log 23

7. ln 25 + ln 3 8. 9. 2 log 32 + 4 log 2

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Day 79

Essential Question: How do I solve exponential and logarithmic equations? Objective(s): 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.2.01 Use the inverse of functions to model and solve problems.

“SAP”: Students will work in cooperative pairs to analyze graphs of exponential functions.

Lesson Anatomy: 1. EOC Practice Problems: Answers: C, D, A2. Collect homework to be graded for accuracy.3. Cooperative Pairs:

a) For f(x) = 5(b)x and b>1 as b increases, how does the graph of f(x) change?b) For f(x) = 5(b)x and 0< b < 1 as b approaches zero, how does the graph of f(x) change?c) For f(x) = a * 1.9x and a ≥ 1 as a increases, how does the graph of f(x) change?

4. Teacher demonstrates the procedure for solving exponential equations through the following examples:

a) d) b) e) c) f) Solve each problem algebraically and then check the solution graphically.

5. White Board Practice: Check Understanding problems 1a-c (TE, pg. 453) and Check Understanding problem 3 (TE, pg. 454)6. Teacher demonstrates the procedure for solving logarithmic equations through the following examples:

a) d) b) e)

c) f)

Solve each problem algebraically and then check the solution graphically.7. White Board Practice: Check Understanding problems 6 and 7 (TE, pgs. 455-456)

Summarizing Activity: Ticket out the door:

Explain how you would solve . At each step explain the rule, property or reasoning used.

Homework: Prentice Hall Algebra 2 Text Section 8-5 pages 456-457 (1-47) odd

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EOC Practice Problems

1. Gas prices at a local gas station for the year 1999-2000 are as follows:Month Price Month Price

January 1.27 July 1.30February 1.36 August 1.43March 1.40 September 1.52April 1.52 October 1.58May 1.38 November 1.65June 1.34 December 1.69

Which type of function best models this data?

A exponentialB logarithmicC polynomialD rational

2. In 1984, the population of Greensboro, N.C. was 197,910. According to the U.S. Census Bureau, Greensboro has been growing at the rate of 6.9% annually since 1984. What equation models the population of Greensboro t years after 1984?

ABCD

3. Which equation best fits the data in the given table?Number ofHalf-Lives

RemainingAmount of Substance

(in grams)0 4,0001 2,0002 1,0003 5004 2505 1256 62.5

A

B

C

D

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Day 80

Essential Question: How do I prepare for the test on Exponential Functions and Logarithms?

Objective(s): 2.03, 2.04, 1.01, 2.01

“SAP”: Students will work in cooperative pairs on the EOC test review.

Lesson Anatomy: 1. Return and discuss graded homework paper.2. Partners compare homework solutions and come to consensus on correct solutions. Check solutions graphically.3. Teacher-led discussion of troublesome homework problems.4. Cooperative Pairs: EOC Practice Problems (Part 2): Answers: B, A, A, B, A, C, D, C

(Do and discuss for test review)5. Cooperative Pairs work on test review assignment:

Chapter 8 Review from Text: pages 469-471 (7, 9, 15, 33-39 odd, 44, 45, 53-63 odd)Do and discuss.

Summarizing Activity: Tell your partner one particular concept you plan to study more before the test tomorrow.

Homework: Study for Unit Test on Exponential Functions and Logarithms

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EOC Practice Problems (Part 2)

1. The equation models United States copper production in pounds from 1987-1992. Which statement best interprets the coefficient and base of this equation?

A The copper production in 1987 was 523,430 pounds, and it had been increasing at arate of 1.93% per year during that period.

B The copper production in 1987 was 523,430 pounds, and it had been increasing at a rate of 19.3% per year during that period.

C The copper production increased by a factor of 523,430x1.193 pounds per year duringthat period.

D The copper production at the beginning of 1987 was at 1.193 pounds, and it had beenincreasing by a factor of 523,430 pounds per year during that period.

2. Which of the following is the logarithmic form of the equation ?

A

B

C

D

3. If , what is m?

A

B

C

D

4. An $18,000 car depreciates at a rate of 16% per year. How old will the car be when it is worth $12,000?

A 0.2 yearsB 2.3 yearsC 2.6 yearsD 3 years

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5. The resting heart rate (h), in beats per minute, for a mammal is related to its mass (m) in kilograms by the equation . What is the approximate resting heart rate, in beats per minute, of a polar bear weighing 326 kilograms?

A 57B 67C 82D 92

6. Alan deposited $300 in an account that pays 6% interest compounded continuously. Approximately how long will it take for Alan’s money to triple? (Use the formula A = Pert where A is the accumulated amount, P is the initial amount, r is the annual rate of interest, and t is the elapsed time in years.)

A 7.95 yearsB 11.55 yearsC 18.31 yearsD 23.10 years

7. The table shows the growth of a certain bacteria.

If N represents the number of cells at time t, which equation best models this set of data?

ABCD

8. Which function models the population of Ethiopia from 1940 to 2000 (let x = 0 in 1940)?Year Population of

Ethiopia(in millions)

1940 161950 201960 251970 311980 391990 502000 64

A BC D

Time in Hours, t 0 1 2 3 4 5Number of Cells, N 50 71 100 141 200 283

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Day 81

Essential Question: Am I prepared to show what I have learned on the test today about Exponential Functions and Logarithms?

Objective(s): 2.03, 2.04, 1.01, 2.01

“SAP”: Students will complete the Graphing Functions Exploration Activity.

Lesson Anatomy: 1. 5 minute review for Unit Test. 2. Unit Test3. After completing the Test, begin the Graphing Functions Exploration Worksheet

Summarizing Activity: The Turtle Race

Homework: Finish the Graphing Functions Exploration Worksheet

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Graphing Functions Exploration Worksheet

I. Consider the following functions: y = x + 1 and y = x – 1. Without graphing, what do you know about the graphs of these two functions? Think of all the similarities and differences that you can. _______________________________________________________________________________________________________________________________________________________________________________________________________________________________

II. If we were to add these two functions it would create a new relationship coming fromy = (x + 1) + (x – 1). What would be the simplified equation of this new relationship? ______________ Would this be a function? __________ Before graphing, predict the general shape of the graph of the new relationship. __________________________________________Now graph on the calculator and draw a sketch of what you see.

What is the domain? _______________________

What is the range? _________________________

III. If you were to subtract these two functions it would create a new relationship coming from y= (x + 1) – (x – 1). What would be the simplified equation of this new relationship?_______________ Would this be a function? ___________ Before graphing, predict the general shape of the graph of the new relationship.___________________________________Now graph on the calculator and draw a sketch of what you see.

What is the domain? ______________________

What is the range? ________________________

IV. If you were to multiply these two functions it would create a new relationship coming from y= (x + 1) (x – 1). What would be the simplified equation of this new relationship?_______________ Would this be a function? ___________ Before graphing, predict the general shape of the graph of the new relationship.___________________________________Now graph on the calculator and draw a sketch of what you see.

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What is the range? ________________________

V. If you were to divide these two functions it would create a new relationship coming fromy= (x + 1) ÷ (x – 1). What would be the simplified equation of this new relationship?_______________ Would this be a function? ___________ Before graphing, predict the general shape of the graph of the new relationship.___________________________________Now graph on the calculator and draw a sketch of what you see.


What is the range? ________________________

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The Turtle Race

Turtle #1: Ploddie

Y = 1.5x

Turtle #2: Quaddie

Y = .07x2

Turtle #3: Exppie

Y = .000286(2)x

What is an appropriate “user-friendly” window for these graphs?

X min = ______ Y min = ______X max = ______ Y max = ______X scl = _______ Y scl = _______

Which turtle will reach the finish line at y = 30 first? __________________

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Days 81-90

Discuss Unit Test and Graphing Function Exploration Worksheet (Day 80). These remaining days are allocated as needed for PSAT Testing, Assemblies, Teacher absence, EOC Review Days, EOC Practice Test and EOC Exam Days.

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AlgebraII

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