Flip Your Grade Book With Concept-Based Grading CMC 3 South Presentation, Pomona, California, 2016 March 05

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1 Flip Your Grade Book With Concept-Based Grading CMC South Presentation, Pomona, California, 016 March 05 Phil Alexander Smith, American River College Additional Resources and Information about Concept-Based Grading Read More About It Marzano, R. J., & Heflebower, T. (011). Grades that show what students know. Educational Leadership, 69(), 4-9. Miller, J. (01). A better grading system: Standards-based, student-centered assessment. English Journal, 10(1), Internet Blog Discussions: Want to Try Concept Grading Yourself? If so, I have concept quizzes, tests, and final exams for the following classes and textbooks. I can provide them to you as PDFs or as LaTex files. You are welcome to have them. Just send me an request. PreAlgebra Calculus II Trigonometry PreAlgebra is a three-unit course at my college. I used the following textbook: College of the Redwoods Department of Mathematics. (01). Prealgebra Textbook. Online: Calculus II is a five-unit course at my college. Our department adopted Calculus: Early Transcendentals (014) by Briggs, Cochran, Gillet (nd Edition) published by Pearson. We cover chapters Trigonometry is a three-unit course at my college. I used the following open-source textbook: Smith, P. A. (015). College Trigonometry With Extensive Use of the Tau Transcendental. [Derivative work based upon Stitz, C. & Zeager, J. (01). Precalculus (Corrected rd ed.), Chapters 10 & 11, implemented under the Creative Commons Attribution- NonCommercial-ShareAlike.0 Unported.] (Note: I use rather than as the basis for Concept Grading Description from Syllabus 1

2 introducing radian measure. My materials would need to be translated into if you prefer that approach.) Precalculus Linear Algebra Precalculus is a five-unit course at my college. I used the following open-source textbook: Lippman, D., & Rasmussen, M. (015). Precalculus: An Investigation of Functions. This is an open-source textbook available online. I modified the text for my students. (Note: As for the trigonometry course, I use rather than as the basis for introducing radian measure. My materials would need to be translated into if you prefer that approach.) Linear Algebra is a three-unit course at my college. I m doing concept-based grading with Linear Algebra for the first time in Spring 016, but I d be happy to share my materials during the semester as they get written. Concept-Based Grading Description Excerpted From Syllabus Concepts This course is comprised of about 5 concepts like Convert Between Degrees and Radians,, Verify Trigonometric Identities, and Solve Right Triangles. On quizzes, tests, even one-on-one office visits, I will give you problems to assess how well you understand a particular concept. I use the levels below to score conceptual understanding: 0. No Basis for Assessment. No information or only scant work provided. Usually means missed assessment. 1. Entry Level of Understanding. Demonstrates an initial, partial understanding. Limited answer, may include incorrect work or misunderstandings.. Basic, Incomplete. Demonstrates some understanding of main concepts. Analysis of the problem is evident with some accuracy.. Competent, Adequate. Demonstrates a decent understanding and analysis of the main concepts, but some details or nuance are missing. 4. Proficient, Accomplished, Skillful. Demonstrates a complete and thorough understanding. Shows conceptual analysis and skills independently with high accuracy. 5. Mastered. Demonstrates repeated understanding by achieving Proficient (4.0) skill level two times in a row. Because I will ask questions about each concept on several quizzes and tests, you ll have multiple times to raise scores on each concept. Only the most recent concept scores on quizzes and tests get recorded in my grade book. Older concept scores are simply replaced. Concept level scores tend to go up over time, but it s also possible for them to go down depending on the quality of the response. If you are unhappy with your concept score at any time, you are welcome to visit me during semester office hours to take another assessment ( concept quiz ) to replace your score on that concept. On quizzes and tests, I report concepts scores on a scale of five (i.e., 0.0, 1.0,.0,.0, 4.0, and 5.0). In order to merge overall concept average scores on the 5-level scale with percentage scores on the tests and the final, I convert concept scores to the following percentages: 0.0 = 0%, 1 = 61%, = 71%, = 81%, 4 = 91%, and 5 = 99%. Your average score over all the concepts during the semester is worth 70% of your grade. Concept Grading Description from Syllabus

3 Concept Score Example SQUARING BINOMIALS From Algebra Simplify the following: (x +y) Response A. (x +y) =(x +y)(x +y) =(x)(x)+(x)(y)+(y)(x)+(y)(y) =4x + 1xy +9y Level: 4.0 Response B. (x +y) =(x) + (x)(y)+(y) =4x + 1xy +9y Level: 4.0 Response C. (x +y) =(x) +(x)(y)+(y) =4x +6xy +9y Level:.0 Response D. (x +y) =(x +y)(x +y) =(x)(x)+(y)(y) =4x +9y Level:.0 Response E. (x +y) =4x +9y Level: 1.0 Response G. (x +y) =(x +y)(x +y) Level: 1.0 Concept Grading Description from Syllabus

4 Grade Book Example Grades on course concepts alway reflect students current level of understanding. Students receive concept scores on quizzes, tests, and the final exam. The example below illustrates the procedure for concepts C1, C, and C. The table on the right is the official grade book after each update. On Quiz 1, a student scores.0,.0, 1.0 on concepts 1,, and. On Quiz, a student scores.0,.0,.0 on concepts 1,, and. On Quiz, a student scores 4.0, 4.0,.0 on concepts 1,, and. On Quiz 4, a student scores 4.0,.0,.0 on concepts 1,, and. C1 C C Student C1 C C Student C1 C C Student C1 C C Student 5.0*.0.0 *The student has now scored two fours in a row with Concept 1 so a score of 5.0 is entered for C1. On Quiz 5, the student leaves C1 blank** and scores.0 and 4.0 on concepts and. C1 C C Student 5.0** **The student was excused from attempting Concept 1 on Quiz 5 because mastery was already shown. The score of 5.0 stays as it was for Concept 1. Concepts and get updated. On Test 1, a student scores.0, 4.0, and 4.0 on concepts 1, and. C1 C C Student For tests, problems from all concepts must be attempted, even for concepts with a score of 5.0. The student has now scored two 4s in a row for Concept. So a score of 5.0 is entered for C. On the Final Exam, a student scores 4.0,.0, and 4.0 on concepts 1, and.* C1 C C Student 4.0.0** 5.0*** *The final exam is like the tests in the sense that problems from all concepts must be attempted, even for concepts with a score of 5.0. **With the final exam only, the concept score will be the average of the previous concept score and the final exam concept score when the final exam concept score is lower. ***Three 4s in a row for Concept remains a 5.0. A score of 5.0, followed by a 4.0, remains a 5.0. Concept Grading Description from Syllabus 4

5 Math Quiz 4.5 & 4.6 (cont d) Name: Date: Level: EQUIVALENT FRACTIONS 1. Fill in the missing equivalent fractions: = = = 48 5 = = = MULTIPLYING AND DIVIDING FRACTIONS 1 1. Multiply and simplify as necessary: x Divide and simplify as necessary: 9 4. Divide and simplify as necessary: 50 MULTIPLYING AND DIVIDING MIXED FRACTIONS 1. Divide and simplify as necessary: Multiply and simplify as necessary: ( 6) To make some jewelry, a bar of silver 4 1 pieces were made? inches long was cut into pieces 1 1 inch long. How many ADDING MIXED FRACTIONS 1. Add and simplify as necessary: Add and simplify as necessary: Sarah and Joseph are finishing up a remodel. They have a length of curtain material that measures 1 feet. They cut a length of 6 feet from the material to fit the living room window. What is the length of the remaining piece?

6 Math 401 Quiz Name: Date: Level: INTEGRATION Z BY PARTS Evaluate: ln (x)dx PARTIAL FRACTIONS Evaluate: Z Z 1 1. dx. (x )(x + ) 8x 17 x + x 1 dx

7 Math 401 Quiz **IDENTIFY AND IMPLEMENT APPROPRIATE INTEGRATION TECHNIQUE** Use either basic integration (with algebra/trig/hyp transformation), u-substitution, integration by parts, trig powers, trig substitution, partial fractions to integrate the following: 1. Z sec (x) tan(x) dx. Z ln(e x )dx. Z sin (5x) dx 4. Z xdx p x +4 Page of

8 Math 0 Quiz Name: Date: Level: VERIFYING TRIGONOMETRIC IDENTITIES Verify the trigonometric identity below. Assume that all quantities are defined. tan ( ) + cot ( )=sec( )csc( ) RECOGNIZE AND APPLY FUNDAMENTAL TRIGONOMETRIC IDENTITIES Product-to-Sum Sum-to-Product Identities + cos( ) cos( )= 1 [cos( ) + cos( + )] cos( ) + cos( ) = cos cos + sin( )sin( )= 1 [cos( ) cos( + )] cos( ) cos( )= sin sin ± sin( ) cos( )= 1 [sin( )+sin( + )] sin( ) ± sin( )=sin cos 1. Write as a sum: cos ( ) cos (5 ). Write as a product: cos (5 ) cos (6 ). Write as a sum: sin ( )sin( ) 4. Write as a product: sin ( ) + sin(5 ) 5. If is a Quadrant II angle with cos ( ) = 8 9, and if is a Quadrant IV angle with sin ( )= 17 41, find: (a) sin ( + ) (b) cos ( )

9 Math 0 Quiz GRAPH TRIGONOMETRIC FUNCTIONS Graph the following trigonometric functions for one period. Use a table of quadrantal angles and their respective trigonometric values to help you plot the graph accurately. State the period and the amplitude (if any). If the graph has any asymptotes, clearly indicate those with dotted lines. 1. y =sin(x). y = cos (x). y = tan (x) Page of

10 Math 70 Quiz 1. (cont d) Name: Date: Level: RATE OF CHANGE Find the average rate of change of the function f(x) =4 will include x as well as the parameter h. x over the interval [x, x + h]. Your final answer ABSOLUTE VALUE 1. Solve 1 4x apple15. Solve x > 5. Show that x + 1 < 4 generates the neighborhood ( 5, ). Then give the center and the radius. 4. Write an absolute value inequality that generates the neighborhood (, 10). Find the center and the radius of the neighborhood. LINEAR FUNCTIONS 1. Given the line f(x) = 4x + 10, find an equation for the line passing through (7, ) that: (a) is parallel to f(x) (c) Then graph both lines on the same set of axes. (b) is perpendicular to f(x). Find the value of a for which the graph of the equation y = ax + is perpendicular to the graph of the y 4x = 7.

11 Math 70 Quiz 1. (cont d) TRANSFORMATION OF FUNCTIONS Consider the graph of the sawtooth function f(x) graphed below. Sketch the graph of each function: 1. y = f(x). y = f 1 x. y =f(x) 4. y = 1 f (x) POLYNOMIAL FUNCTIONS 1. What is the maximum number of x-intercepts and turning points for a polynomial of degree 11?. Find the vertical and horizontal intercepts of f(x) = (x + )(x )(x + 4). QUADRATIC FUNCTIONS 1. Find the vertex, the vertical intercept, and the horizontal intercepts of g(x) = x 14x+1.. Write an equation for a quadratic with x- intercepts (1, 0) and (, 0), and y-intercept (0, 4). Page of

12 Math 410 Linear Algebra Quiz 08 11, 1, & 1 Name: Date: Level: ELEMENTARY MATRICES Given the product: List in order the set of elementary row operations that reduce 4 codes like R 1 + R! R to describe each row operation = to Use CALCULATE DETERMINANTS BY THE DEFINITION AND THE APPLICATION OF ROW OPERATIONS Compute the following determinants by inspection using the definition of determinant and by applying row operations strategically. Explain how you arrived at the answer. Don t use a calculator EVALUATE EXPRESSIONS USING DETERMINANT PROPERTIES Let M and N be matrices with det (M) = 10 and det (N) =. Use the properties of the determinants of matrices to find the values of the following. Explain how you arrived at the value. 1. det (M ). det (N 1 ). det (M T N T ) 4. det (M)

13 Math 410 Linear Algebra Quiz 08 11, 1, & 1 PROOF BASICS Prove the Inverse-Product Theorem: If A and B are invertible matrices of the same size, then AB is also invertible and (AB) 1 = B 1 A 1 Proof. (write the 1st step of an if-then proof) Since A and B are both invertible, then we know that matrices A 1 and B 1 exist such that AA 1 = A 1 A = I and BB 1 = B 1 B = I In order to demonstrate that AB is invertible, it su ces to produce a matrix N such that (AB)N = N(AB) =I by the Let N = B 1 A 1. Then (AB)N =(AB)(B 1 A 1 ) = A(BB 1 )A 1 = AIA 1 = AA 1 = I And N(AB) = Substitution = MAPT(c): Associative Property of Multiplication. = Substitution = Definition of Identity Matrix = Substitution Therefore, since (AB)(B 1 A 1 )=I =(B 1 A 1 )(AB), then AB is invertible and (AB) 1 = B 1 A 1 by definition of inverse matrix. Q.E.D. Page of

14 Math 410 Linear Algebra Quiz 08 11, 1, & 1 DEFINITIONS, THEOREMS, & OTHER JUSTIFICATIONS USED IN PROOFS Definition of Transpose of a Matrix. The transpose of an m n matrix A =[a ij ], written, A T,issimply the interchange of the rows and columns of A. In symbols, A T is the n m matrix A T =[a ji ]. Definition of Symmetric Matrix. A square matrix A is symmetric if A T = A. Definition of Skew-Symmetric Matrix. A square matrix C is symmetric if C T = C. Definition of an Identity Matrix. An n n matrix I is identity matrix if it possesses the property AI = IA = A for all matrices A. In form, the identity matrix is a matrix in which the( upper left to lower 1, if i = j right diagonal consists of all ones and all other entries are zero: I n =[a ij ]wherea ij = 0, if i 6= j Definition of a Matrix Raised to a Power. If A is a square matrix, then we define the nonnegative integer powers of A as follows: A n means A multiplied by itself n times. Definition of Matrix Addition. If two matrices A m n =[a ij ] and B p q =[b ij ] have the same number of rows and columns (i.e., m = p and n = q), then the sum of A and B is defined A + B =[a ij + b ij ]. Definition of an Inverse Matrix. The inverse of matrix A, designated A 1, is the matrix that when multiplied by the original matrix returns the identity. That is, AA 1 = A 1 A = I for all A. Transpose Properties Theorem. If the sizes of the matrices are such that the stated operations can be performed, then (a) (A T ) T = A (b) (A + B) T = A T + B T and (A B) T = A T B T. (c) (ka) T = ka T,wherek is any scalar (d) (AB) T = B T A T Inverse-Transpose Theorem. If A is an invertible matrix, then A T is also invertible and (A T ) 1 = (A 1 ) T. Substitution. This justification is used when two expressions are equal, and one expression replaces the other in the proof. Definition of a Square Matrix. A matrix A is square if A has the same number of rows and columns. Matrix Arithmetic Properties Theorem. If the sizes of the matrices A, B, and C are such that the stated operations can be performed and k and l are any two real number constants, then (a) Commutative Property of Addition: A + B = B + A (b) Associative Property of Addition: A +(B + C) =(A + B)+C (c) Associative Property of Multiplication: A(BC) =(AB)C (d) Left Distributive Property: A(B + C) =AB + AC (e) Right Distributive Property: (A + B)C = AC + BC (f) (k + l)a = ka + la (g) k(la) =(kl)a (h) k(ab) =(ka)b = A(kB) (i) k(a + B) =ka + kb Page of