MATH: A2. ADE Summer Item Writing Institute. Performance-Based Assessment. x f(x) g(x)

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1 Date: 6/11/2013 A-REI.11-2 Task Type: I II III Math Practice: x f(x) g(x) If g(x) = 1 4 (x 2)2 + 1, find all values of x to the nearest tenth where f(x) = g(x). X= Click to enter Another solution X=4.3 Table is interactive where students can input x and f(x) and g(x) appear to the nearest hundredth.

2 Date: 6/11/2013 HS.C.3.1 Task Type: I II III Math Practice: Sort each expression into the appropriate category. Bank of Choices Real Complex i 2 i ( 3) 2 4 (1 + 2i)(1 2i) (1 + 2i) 2 Explain algebraically why is real. Explain algebraically why is complex. Real: i 2, ( 3) 2, (1 + 2i)(1 2i) Complex: i, 4, (1 + 2i) 2 Example for part 2: 3 2 = 3 3 = i 3 i 3 = i 2 9 i 2 9 = ( 1)(3) = 3 1 choice from each category will populate in the explain boxes in the second part of the item after students have sorted in part one.

3 Date: 6/12/2013 HS.C.6.2 Task Type: I II III Math Practice: Explain if each of the points is a solution to the system shown. A. (3, 4) B. (-2, 1) C. (2, 0) A. (3, 4) No does not lie in the shaded area. B. (-2, 1) Yes lies in the shaded area. C. (2, 0) No lies on the dashed line which in not a part of the solution.

4 Date: 6/12/2013 HS.C.6.2 Task Type: I II III Math Practice: A student analyzed the following function. He determined that ( 2,0) (3,0) (5,0) are the only solutions for f(x). Do you agree or disagree. Justify your response. Disagree. Any point on the graph of the function is a solution to the function. The points listed are zeroes of the function or solutions to the specific case of f(x) = 0.

5 Date: 6/12/2013 HS.C.9.2 Task Type: I II III Math Practice: Compare the shape a position of the graphs of f(x) = 2 x to g(x) = 2 x and explain the differences in terms of the algebraic expressions for the functions. The graph for g(x) is translated 3 unit to the right and 4 units up from the graph of f(x). The position is moved but the shape of the graph is unaffected.

6 Date: 6/12/2013 HS.C.16.3 Task Type: I II III Math Practice: Examine the steps a student showed for solving the following rational equation. 4x x x 2 4 = 1 x + 2 Step 1: 4x(x + 2) + 7 = (x 2) Step 2: 4x 2 + 8x + 7 = x + 2 Step 3: 4x 2 + 9x + 5 = 0 Step 4: (4x + 1)(4x + 5) = 0 Step 5: 4x + 1 = 0 4x + 5 = 0 Step 6: x = 1 4 x = 5 4 Is the solution correct? If not identify the error and correct the solution. Error in step 4. The student factored incorrectly. The correct factorization is (x + 1)(4x + 5) = 0. After setting these new factors equal to zero in step 5 the correct solutions would then be x = { 5, 1 }. 4 4

7 Click here to enter item author. Date: 6/12/2013 HS.C.16.3 Task Type: I II III Math Practice: Evaluate the process shown for solving 4x 3 6 = 1. Describe the operation that was performed in each line. 4x 3 6 = 1 Given equation 4x 3 = 5 4x 3 = 25 4x = 28 x = 7 Is this a valid process? Justify your answer. Operations performed. Add the constant 6 to both sides Square both sides Add the constant 3 to both sides Divide both sides by the coefficient 4 Process is valid. (Answer cont.) Substitute solution into original equation. 4(7) 3 6 = = = 1 1 = 1

8 Date: 6/11/2013 A-REI.11-2 Task Type: I II III Math Practice: Given the following equations determine the x-value(s) that results in an equal output for both functions. f(x) = 3x + 7 g(x) = (x 2) 2 3 x = 3 x = 2

9 Date: 6/11/2013 A-REI.11-2 Task Type: I II III Math Practice: Between which two successive integer values of x is the output equal for f(x) and g(x)? f(x) = x + 2 g(x) = ( 1 2 )x Between and -1 and 0

10 Date: 6/11/2013 A-REI.4b-2 Task Type: I II III Math Practice: Which of these equations has only real solutions? 6x 2 = 3x 1 2x 2 3 = 8x 5x = 4x x + 8 = 3x 2 2x 2 3 = 8x

11 Date: 6/11/2013 A.SSE.2-3 Task Type: I II III Math Practice: Fill in the blank to make the statement true. 2(8) x = 2 3x + 1

12 Date: 6/11/2013 A-SSE.2-3 Task Type: I II III Math Practice: Rewrite 64 x using two different bases. Identify the base, 64 x = Identify the base, 64 x = Student inputs 2 of the following: Base 8, 64 x = 8 2x Base 4, 64 x = 4 3x Base 2, 64 x = 2 6x

13 Date: 6/11/2013 A-REI.2 Task Type: I II III Math Practice: Solve: 3x = 3 No Solution

14 Date: 6/11/2013 A-REI.4b-2 Task Type: I II III Math Practice: Which of these equations has no real solutions? 8x 2 5x = 1 4x 2 = 6x 1 2x = 7x x 2 = 2x x = 7x

15 Date: 6/11/2013 A-REI.2 Task Type: I II III Math Practice: Solve: x + 3 = 8 x+2 x 2 x 2 4 x = 1

16 Date: 6/10/2013 N-RN.2 Task Type: I II III Math Practice: Rewrite x 2 in at least three distinct alternate forms. x x x 5 x 5 x6 x 1 1 x6 1 6 x Note: Multiplying by a 1 does not constitute a new form (ex. x 1 6 and 2 2 x 1 6 would be considered the same form).

17 Date: 6/11/2013 HS.N.RN.2 Task Type: I II III Math Practice: x 2 3 can be rewritten as which rational expressions? Select all that apply. 1 x 3 3 x 2 1 x x 2 x 3 1 x x3 1 3 x 2

18 Date: 6/11/2013 N-CN.1 Task Type: I II III Math Practice: Simplify 36 6i

19 Date: 6/11/2013 N-CN.1 Task Type: I II III Math Practice: Drag and drop each expression into the appropriate category. 8i i ( 3i) 2 5 Real Complex Real Complex 8i i ( 3i) 2

20 Date: 6/11/2013 A-REI.4b-2 Task Type: I II III Math Practice: Which expression(s) are equivalent to x 4 16? (x 2 + 4)(x + 2)(x 2) (x + 4)(x + 1)(x + 2)(x 2) (x 2 + 4)(x 2 4) (x + 2)(x 2)(x + 2)(x 2) (x + 2)(x + 2)(x + 2)(x 2) (x 2 + 4)(x 2 4) (x 2 + 4)(x + 2)(x 2)

21 Date: 6/10/2013 N-CN.7 Task Type: I II III Math Practice: Find all solutions and express them in the form a + bi. 2x = 4x 1 ± 1 i 6 2

22 Date: 6/10/2013 A-SSE.2-3 Task Type: I II III Math Practice: Given the equation of a parabola y = x 2 6x + 14, complete the square to express the same parabola in vertex form. A. y = (x 6) 2 22 B. y = (x + 3) 2 5 C. y = (x + 9) 2 14 D. y = (x 3) E. y = (x 3) 2 23 D. y = (x 3) 2 + 5

23 Date: 6/12/2013 HS.C.11.1 Task Type: I II III Math Practice: What is the cosine of the angle in the third quadrant having sine equal to -3/4? Explain how you computed your answer. 7 4 Student s answer may include the unit circle model, or could show a diagram of the 3 rd quadrant, or demonstrate the Pythagorean theorem with x=- 7, y = -3, and r = 4.

24 Date: 6/12/2013 HS.C.11.1 Task Type: I II III Math Practice: Show that sin 2 θ + cos 2 θ = 1 when θ = 2π 3 sin 2 ( 2π 3 )+ cos2 ( 2π 3 ) = 1 ( 3 2 )2 + (- ½) 2 = 1 ¾ + ¼ = 1 4/4 = 1 1=1

25 Date: 6/12/2013 HS.C.8.3 Task Type: I II III Math Practice: Find the quotient and remainder for the rational expression x 3 + 5x 2 10x 6 x Explain how you can use your remainder when checking your answer. X+5 * 15x 31 x 2 +5 When you multiply (x+5) and (x 2 + 5) you get x 3 +5x 2 +5x+25, so you add (-15x 31) and that produces x 3 +5x 2-10x-6.

26 Date: 6/12/2013 HS.C.9.2 Task Type: I II III Math Practice: Is f(x) = x 3 + 2x 2 3x + 5 even, odd, or neither? Explain your answer. Neither. Student may show that f(-x) = x 3 + 2x 2 + 3x + 5, which is not f(x) or f(x). Also may explain that the problem has a mix of even and odd integers, thus is neither even nor odd.

27 Date: 6/12/2013 HS.C.7.1 Task Type: I II III Math Practice: Given the function, g(x) = x 3 8x 2 + 5x + 14 and g(2) =0, find all zeros. Explain how to find them. Using the remainder theorem, 2 is a zero. Now use synthetic or long division to find the remaining polynomial x 2-6x-7, factor (or use the quadratic equation) to fine (x-7)(x+1). By setting each factor equal to 0 and solving, the three zeros include x = {-1, 2, 7}.

28 Date: 6/12/2013 HS.C.7.1 Task Type: I II III Math Practice: Explain how the distinct zeros of the function, g(x) =(x+2)(x+3)(5x-4) are related to the factors. Student s response must relate each of the factors will have to be set equal to zero and when solved, all signs of zeros will be opposite of their factor. (i.e. (x+2) is the factor but the zero is x=-2, (x=3) is a factor, but the zero is x= -3, and (5x-4) is a factor, but zero is x=4/5.

29 Date: 6/12/2013 HS.C.7.1 Task Type: I II III Math Practice: Given f(x0 = (X+5)(X+7) 2, at which zero(s) will the graph of the function cross through the x-axis? Explain your answer. Student s response must indicate the multiplicity (even) of two means x+7=0 (x=-7) is a zero and the graph cross through x at -7 because of even multiplicity. Whereas at x=-5, it has multiplicity of odd (one) so the graph is tangent but reverses direction at x=-5.

30 Date: 6/12/2013 HS.C.12.2 Task Type: I II III Math Practice: The projected population of a school is given by the formula p(t) = 1400(1.05) t. You have been selected by the principal to serve on a committee to plan for future student populations. Explain what the formula p(t) = 1400(1.05) t means to the committee is the initial student population at t=0. The student population is growing. The percent rate of growth is 5%.

31 Date: 6/10/2013 N-CN.2 Task Type: I II III Math Practice: Simplify: (2 5i) 2 a i b i c i 2 d. 4 25i 21 20i

32 Date: 6/12/2013 HS.C.7.1 Task Type: I II III Math Practice: Given f(x) = (x 1) 2 (x + 3) Explain why the number of factors is different from the distinct number of zeroes. Because (x 1) 2 means that there are two factors each being x 1 and (x + 3) is only one, there are three factors; (x 1)(x 1)(x + 3). However there are only two distinct zeroes 1 and -3 which can be found by setting each factor equal to zero and solving.

33 Date: 6/12/2013 HS.C.6.4 Task Type: I II III Math Practice: Part 1: Given the focus (3, -1) and the directrix y = 3, write the equation of the parabola. (x - ) 2 = y Part 2: Identify two points on the parabola and explain how you know these points are on the parabola. Part 1: 4(x 3) 2 = y ( 2) Part 2: Student correctly selects any two points on the parabola. Student explains that the points selected satisfy the equation for the parabola. That is the points can be substituted into the equation for x and y and result in a true statement.

34 Date: 6/12/2013 HS.C.5.4 Task Type: I II III Math Practice: Which equation(s) has extraneous solutions? a. 3x 5 = 5 b. 3x 5 = 5 3 c. 2x d. 2x + 8 = 2 = 2 Justify your response. a. 3x 5 = 5 An algebraic solution to the problem is 10, when this answer is substituted back in to the original equation the resulting equation is not true, 5 5

35 Date: 6/12/2013 HS.C.5.11 Task Type: I II III Math Practice: How many solutions exist for the system? f(x) = x 3 4 g(x) = (x 6) Justify your answer. 2 Solutions exist. If the system is graphed you have an upward opening absolute value with a vertex at (3, -4) and a downward opening parabola with a vertex at (6, 2). This means that the graphs will intersect at two points). OR students solve for the two points algebraically or with a table of values.

36 Date: 6/11/2013 HS.C.5.11 Task Type: I II III Math Practice: Given: f(x) = (x 5)2 + 2 g(x) = 3 5 x 1 How many solutions exist for the system? Justify your answer. 1 Solution. When graphed the 2 graphs intersect at the vertex of the parabola in quadrant 1.

37 Date: 6/11/2013 HS.C.5.11 Task Type: I II III Math Practice: How many solutions exist for the system? y = 2 x y = x 2 1 Justify your answer. 1 Solution. When graphed the parabola insects the exponential in the second quadrant once. When examining the end behaviors, the exponential increases more rapidly than the quadratic so they will not intersect again.

38 Date: 6/11/2013 HS.C.3.1 Task Type: I II III Math Practice: Given the statement: The quotient of two irrational numbers can be rational or irrational. Choose 2 irrational numbers from the list below that result in a rational quotient and explain your reasoning Sample correct answers: Student correctly uses the properties of exponents to rewrite the quotient of radicals as a rational value (ie. Rationalizing the denominator and dividing, Rewriting as a fraction under the radical and reducing and taking the root).

39 Date: 6/11/2013 HS.C.3.2 Task Type: I II III Math Practice: Examine the student s work below for any error. Explain your reasoning. Line 1: 1 x 2/3 x 2 3 Line 2: 1 x 3 Line 3: 1 x x a. Error in line 1, b. Error in line 2, c. Error in line 3, d. No error was made b. Error in line 2 The square root and exponent are incorrect. The 3 correct denominator should be x 2

40 Date: 6/11/2013 HS.C.3.1 Task Type: I II III Math Practice: Given the statement: 6 b = a rational number Determine at least three distinct values of b to make the statement true. Describe the characteristics of all values of b that make the statement true. Sample responses for b include 6, 24, 54 b = 6x 2 where x is any number not equal to zero.

41 Date: 6/11/2013 A.Int.1 Task Type: I II III Math Practice: Find the value of p that makes the equation true x + 9x 2 x 3 = p (x 3) 3 p = 54

42 Date: 6/11/2013 A.Int.1 Task Type: I II III Math Practice: Solve: 2 x 2 8 x = + 3 = 2x 1 Click to add another solution 0, 3, 4

43 Date: 6/12/2013 HS.C.8.2 Task Type: I II III Math Practice: a b = (x y)(x + y) = x 2 y 2 is a useful rule for mental math calculations. Fill in the chart then explain how you determined your x and y values. a b = (x y)(x + y) = x 2 y = ( )( + ) = ( ) 2 ( ) 2 = = ( )( + ) = ( ) 2 ( ) 2 = = ( )( + ) = ( ) 2 ( ) 2 = = (20 1)(20 + 1) = = = (20 2)(20 + 2) = = = (20 3)(20 + 3) = = 391 To find the y-value: y = (b a)/2 To find the x-value: x = a + y Student could also describe this relationship. Answer cont. For example a student might respond that x is halfway between a & b and y is the distance from a & b to the middle number.

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