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MATH ALGEBRA II 9-12 Performance Objective Task Analysis Benchmarks/Assessment

*1. Students solve equations and

inequalities involving absolute value.

Students: 1. Solve equations involving absolute

value with both equalities and inequalities.

• Solve for x.

| x - 5| = 4 3|x| + 7 = 14

-5|x| > 20 2| x + 1|< 6 Express the solution using interval notation. ( FW) |2x - 3| > 4 Sketch the interval in the real number line that is the solution for: ( FW) |x - 3| < 5 2

*2. Students solve systems of linear

equations and inequalities (in two or three variables) simultaneously, by substitution, graphically, or with matrices.

Students: 1. Demonstrate methods of solving

systems of linear equations (inequalities).

• Substitution • Linear combination • Graphically • Using matrices

• Solve for x and y algebraically for each system and verify the solutions graphically or with matrices.

a) y < 5x - 4 2x + 3y > 27 b) -x + y = 3 2x - y = -5 c) 2x + z = 7 x + y + z = 0 2x + 3y - 2z = -8

*Power Standard FW = California Mathematics Framework

H:\DATA\WORD\MATH\S&B\ALG2.DOC2/3/04

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Solve the system of linear equations: ( FW)

x + 2y = 0 x + z = -1 y - z = 2

Draw the region in the plane that is the solution set for the inequality. ( FW) (x - 1) (x + 2y) > 0

*3. Students are adept at operations on

polynomials, including long division.

Students: 1. Use operations with polynomials. • Addition • Subtraction • Multiplication • Division

• Simplify the following completely:

(3x2 + 7x - 12) + (5x2 - 8x + 4) ( 5x2 + 7x + 2) - ( 4x2 - 13x - 9) (2x - 7)(3x2 + 7x -4) x3 + 1 x + 1 Divide x4 - 3 x2 + 3x by x2 + 2. Use this to write x4 - 3x2 + 3x in the x2 + 2 form: polynomial + linear polynomial ( FW) x2 + 2



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*4. Students factor polynomials

representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

Students: 1. Factor polynomials. • Difference of squares • Perfect square trinomials • Sum of two cubes • Difference of two cubes

• x2 - 25 • x2 + 2xy + y2 • x3 + y3 • 8x3 - 27y3

Simplify: ( FW) x3 - y3

x2 - y2

Express without any square roots in the numerator: ( FW)

*5. Students demonstrate knowledge of

how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.

Students: 1. Plot complex numbers as points in a

plane.

• Graph each complex number in the complex plane.

a) 3 + 2i b) - 5i c) 4.5

Give the absolute value of the complex number: | 5 + 12i |

x + y x - y2

imaginary real



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Locate all complex solutions to z2 + 4 in the complex plane. ( FW)

*6. Students add, subtract, multiply, and

divide complex numbers.

Students: 1. Use operations with complex numbers. • Add • Subtract • Multiply • Divide

• Simplify the following completely: (4-5i) + (7 - i) • ( 4 - 2i) - (9 + 5i) • (4 + 3i)(4 - 3i) • 7 - 2i 2 + 1i

Write in the form a + bi, where i is a square root of -1: ( FW) (3 - 2i)2 2 + i

*7. Students add, subtract, multiply,

divide, reduce and evaluate rational expressions with monomial and polynomial denominators, and simplify complicated fractions including fractions with negative exponents in the denominator.

Students: 1. Simplify and evaluate rational

expressions.

• Simplify the following completely: 6 - 3 2x + 4 + 3x - 7 4b 5b - 1, 5x - 3 5x - 3 5a4b3 x 20a2b3 , x + 3 ÷ x - 2 4a-5 75ab2 x -2 x + 6

Simplify: (x2 - x)2 ( FW) x(x - 1)-2(x2 + 3x - 4)



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*8. Students solve and graph quadratic

equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.

Students: 1. Demonstrate methods of solving

quadratic equations. • Factoring • Completing the square

• Solve by factoring y =2x2 - 5x -3 • Solve by completing the square y = 2x2 + 5x + 1

• Quadratic formula

• Solve by using the quadratic formula. y = x2 + 3x - 3, y = x2 + x + 10

• Word problem • What is the maximum area of a pillow that has a perimeter of 36 inches?

In the figure shown below, the area

between the two squares is 11 square inches. The sum of the perimeters of the two squares is 44 inches. Find the length of a side of the larger square. (ICAS) ( FW) Find all solutions to the equation. ( FW) x2 + 5x + 8 = 0



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*9. Students demonstrate and explain

the effect changing a coefficient has on the graph of quadratic functions. That is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b)2 + c.

Students: 1. Determine how the graph of a

quadratic function changes as a, b, and c vary in an equation.

• Find the vertex of y = 10( x - 2)2 + 4

Graph y = -2 (x - 3)2 + 4 The function f(x) = (x - b)2 + c is graphed below. Use this information to identify the constants b and c. ( FW)

*10. Students graph quadratic functions

and determine the maxima, minima, and zeros of the function.

Students: 1. Graph quadratic functions and

determine their maxima and minima and zeros.

• For the function h(x) = 3x2 + 6x - 24

a) Find the vertex b) Find the line of symmetry c) Fine the maxima/minima point d) the zeros

Find a quadratic function of x that has zeroes at x = -1 and x = 2. Find a cubic equation of x that has zeroes at x = -1 and x = 2 and nowhere else. (ICAS) ( FW)

(2,1)

-1 4 3 2 1

y = f (x)

-1 -2

6 5 4 3

1 2

x



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Graph the function f(x) = 2(x + 3)2 -4 and determine the minimum value for the function. ( FW) Find the vertex for the graph of f(x) = 3x2 - 12x + 4. ( FW)

*11. Students prove simple laws of

logarithms.

Students: *1. Students understand the inverse

relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.

• Solve each equation. l0g5 x = 3 x = log452 1200(e4x) = 8866.87

2. Students judge the validity of an argument based on whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.

• Determine if the following work is

correct or not and state why.

logx + log 3 = log 12 log(x + 3) = l0g12 x + 3 = 12 x = 12 Solve for x and explain each step: ( FW) log3(x + 1) - log3x = 1

log 7 = logbx b



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*12. Students know the laws of exponents,

understand exponential functions, and use these functions in problems involving exponential growth and decay.

Students: 1. Use the laws of exponents and

exponential functions.

• Sharon is investing $1000 in a

certificate of deposit that pays 6% interest each year. If she keeps the money in the account, how much will she have after 8 years?

You have 10 grams of radioactive

isotope gold-202. After 1 minute there are only 8 grams left. How long will it take until there are 5 grams left? The number of bacteria in a colony was growing exponentially. At 1 p.m. yesterday the number of bacteria was 100 and at 3 p.m. yesterday it was 4000. How many bacteria were there in the colony at 6 p.m. yesterday? (TIMSS) ( FW) Scientists have observed that living matter contains, in addition to carbon, C12, a fixed percentage of a radioactive isotope of carbon, C14. When the living material dies, the amount of C12 present remains constant, but the amount of C14



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decreases exponentially with a half life of 5,550 years. In 1965, the charcoal from cooking pits found at a site in Newfoundland used by Vikings was analyzed and the percentage of C14 remaining was found to be 88.6%. What was the approximate date of this Viking settlement? (ICAS) ( FW)

13. Students use the definition of

logarithms and the product formula for logs to translate between logarithms in any bases.

Students: 1. Translate between logarithms in any

base.

• Express each of the following quantities in terms of base ten logarithms.

log25 log510 ln3 logb2 log2b

Calculate the following logarithms to three decimal places. log514 log31/2

Simplify to find exact numerical values for: b3log

b2-log

b5

log (b2) b



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14. Students understand and use the

properties of logarithms to simplify logarithmic numeric expressions and identify their approximate values.

Students: 1. Simplify logarithmic expressions and

approximate their values.

• Give log 2 ≈ 0.30 and log3 ≈ 0.48, find the approximate value of the following:

log6 log8 log(0.75) log2/3

Solve the following equations: 3x+2 = 81 2logx = log144 Find the largest integer that is less than: ( FW) log10(1,256) log10(.029) Write as a single logarithm: ( FW) log37 log35

*15. Students determine if a specific

algebraic statement involving rational expressions, radical expressions, logarithmic or exponential functions, is sometimes true, always true, or never true.

Students: 1. Determine if expressions are true.

• Determine whether each statement is true for:

a) all real numbers. b) some real numbers. c) no real numbers

1/x + 1/2 = x + 2 , log (2x) = log(x2)

2x



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a + 3 = a = 3 5x+2 = 25(5x)

Is the following true for all real numbers x, for some real numbers, or for no real numbers x? ( FW)

16. Students demonstrate and explain how

the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.

Students: 1. Graph conic sections and determine

how the graphs are affected when coefficients are changed.

• Graph the following equation. x2 + y2 = 25

Write and equation for the circle in standard form. Then state the center and radius. x2 - 6x + y2 = 0

Graph the ellipse and give the coordinate of its foci,. Find its eccentricity, and express it as a decimal rounded to the nearest hundredth. 25x2 + 9y2 = 225

(1 - x2)2 1 - x 1 + x =



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Graph the hyperbola and give the coordinates of its foci and the equations for its asymptotes.

(y + 2)2 - (x - 3)2 = 1 4 9

If xy = 1 and x is greater than 0, which of the following statements is true? ( FW)

a) When x is greater than 1, y is negative.

b) When x is greater than 1, y is greater than 1.

c) When x is less than 1, y is less than 1.

d) As x increases, y increases. e) As x increases, y decreases

(TIMSS) ( FW)

17. Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method of completing the square to put the equation into standard form and can recognize whether its graph is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.

Students: 1. Use quadratic equation to recognize its

graph.

• Complete the square for the following

equations. Then state if the conic is a circle, an ellipse, a parabola, or a hyperbola. Then give the center and graph the conic.



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x2 - y - 8x + 16 = 2 3x2 + 4y2 - 6x + 16y = -7 16x2 - 9y2 -32x + 90y = 353 x2 + 2x + y2 - 6y + 11 =9 Does the origin lie inside of, outside of, or on the geometric figure whose equation is x2 + y2 - 10x + 10y - 1 = 0? Explain your reasoning. (ICAS) ( FW)

Write the conic section whose equation is given by 4x2 - 8x-y2 + 4y = 4 in standard form to determine whether it is a parabola, hyperbola, or ellipse. ( FW)

*18. Students use fundamental counting

principles to compute combinations and permutations.

Students: 1. Use counting principle.

• During your turn in a dice game, you

rolled two dice. a) How many ways can you get a

sum of 5? b) How many ways can you get a

sum of 10? c) How many ways can you get a

sum of 5 and 10?



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At Romano’s Pizzeria, you can order pizza with thick or thin crust and with any combination of the five toppings they offer.

a) How many types of crust are

there? b) How many choices of toppings

are there? c) How many different types of

one-topping pizzas can be ordered?

An examination consists of 13 questions. A student must answer only one of the first two questions and only nine of the remaining ones. How many choices of questions does the student have? (TIMSS, adapted) ( FW)

*19. Students use combinations and

permutations to compute probabilities.

Students: 1. Compute probabilities.

• Suppose you draw 2 cards at random

from a deck of 52 playing cards. What is the probability that both will be black?

Evaluate the following expressions: 8! 8P6 8C6 7!/3!



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Six people with different last names line up randomly. What is the probability they are lined up in alphabetical order?

A lottery will be held to determine which three members of a club will attend the state convention. This club has 12 members, 5 of whom are women. What is the probability that none of the representatives of the club will be women?

*20. Students know the Binomial

Theorem and use it to expand binomial expressions which are raised to positive integer powers.

Students: 1. Use the Binomial Theorem to expand

binomial expressions which are raised to positive powers.

• Write the first three terms of (a + b)9

Write an informal description of the following words: a) term b) power c) coefficient d) exponent

Answer the following question for the binomial power: (x + 4)4. a) How many terms are in its

expansion? b) Write the coefficients of each

term using combination notation. c) Write the complete expansion in

simplest form.



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Determine the middle term in the binomial expansion of (ICAS) ( FW)

21. Students apply the method of

mathematical induction to prove general statements about the positive integers.

Students: 1. Use mathematical induction to prove

statements about the positive integers.

• Use mathematical induction to prove

each formula is valid for all positive integral values of n. 1 + 3 + 5 + 7 +....+ (2n -1) = n2 2 +4 + 6 + ... + 2n = n(n + 1) 1 + 2 + 4 + ... + 2n-1 = 2n - 1

Use mathematical induction to show that n(n + 1) 1 + 2 + 3 + 4 + ... + n = 2 . ( FW)

22. Students find the general term and

the sums of arithmetic series and both finite and infinite geometric series.

Students: 1. Fine the general term and the sums of

arithmetic series .

• Put each of the following series or

partial sums in sigma notation and find the sum. 3 + 8 + 13 + 18 + 23 + .... + 108 2 + 6 + 18 + 54 + .... the first 15 terms. 54 + 36 + 24 + 16 + ...

x( - 2x

) 10



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Find the sum of the following infinite series: ( FW) 35

+ 925

+ 27125`

+ 81625

+ ...

*23. Students derive the summation

formulas for arithmetic series and both finite and infinite geometric series.

Students: 1. Derive the summation formulas for

arithmetic series.

• Show that the nth term of an arithmetic

sequence with first term a1 and common difference d is given by

an = a1 + (n - 1) d. • Given that s6 = 3 + 15 + 75 + 375 +1875 + 9375 s6 = 3 + 3⋅51 + 3⋅52+ 3⋅54+ 3⋅54 + 3⋅55

5⋅s6 = 3⋅51+3⋅52+3⋅53 +3⋅54+3⋅55 + 56 s6(1-6) = 3 - 3⋅56

s6 = 3 - 3⋅ 56 1 -5

then derive sn using the same logical steps.

24. Students solve problems involving

functional concepts such as composition, inverse, and arithmetic operations on functions.

Students: 1. Use arithmetic inverse and

composition concepts on functions.

• Given that f(x) = 2x + 6 and g(x) = 3x -

2; find the following: f + g(x) f - g(x) f x g(x) f ÷ g(x) f g(-3)



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Find f -1(x) when f(x) = x2 - 6. Then compute f-1(-2)

Which of the following functions are their own inverse functions? Use at least two different methods to answer this, and explain your methods. (ICAS) ( FW)

f(x) = 2x

g(x) = x3 + 4

h(x) = 22

+−

lnlnxx

Sketch a graph of a function g that satisfies the following conditions: g doesn’t have an inverse function, g(x) <x for all x, and g(2) >0. ( FW)

25. Students use properties from number

systems to justify steps in combining and simplifying functions.

Students: 1. Use properties from number systems to

justify steps in combining and simplifying functions.

• Identify the field axiom illustrated below.

f(x) + g(x) = g(x) + f(x) f(x) +{g(x) + h(x)} = { f(x) + g(x) + h(x) f(x) { g(x) + h (x) } = f(x)h(x) + g(x)h(x) Let b(x) = x + 3 and c(x) = 4x, show that b(-2) + c(-2) = (b+c)(-2)

x3 + 1 x3 – 1

j(x) = 3



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