1. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Released Items NSF REC #0337595 Knowledge of Mathematics for Teaching Algebra Project (2004 – 2007) © 2005 Division of Science and Mathematics Education Michigan State University

2. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Sample Item 1: Identifying an Exponential Function (School Knowledge) Which of the following situations can be modeled using an exponential function? i. The height h of a ball t seconds after it is thrown into the air. ii. The population P of a community after t years with an increase of n people annually. iii. The value V of a car after t years if it depreciates d% per year. A. i only B. ii only C. iii only D. i and ii only E. ii and iii only

3. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Results: Identifying an Exponential Function Number of casesDifficulty Pre-service teachers1380.297 In-service teachers2870.324 Total4310.313

4. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Sample Item 2: Properties of Number Systems (Advanced Mathematical Knowledge) For which of the following sets S is the following statement true? For all a and b in S, if ab = 0, then either a = 0 or b = 0. i. the set of real numbers i. the set of complex numbers iii. the set of integers mod 6 iv. the set of integers mod 5 v. the set of 2x2 matrices with real number entries A. i onlyD. i, ii, iii and iv only B. i and ii onlyE. i, ii, iii, iv, and v C. i, ii and iv only

5. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Results: Properties of Number Systems Number of casesDifficulty Pre-service teachers 860.151 In-service teachers 960.177 Total1860.161

6. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Sample Item 3: Identifying Student’s Error in Solving a Linear Equation (Teaching Knowledge) A student solved the equation 3(n - 7) = 4 - n and obtained the solution n = 2.75. What might the student have done wrong?

7. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Results: Identifying Student’s Error Number of casesDifficulty Pre-service teachers970.760 In-service teachers140.946 Total1150.787

8. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Sample Item 4: Interpreting and Relating Equivalent Expressions (Teaching Knowledge) Hot tubs and swimming pools are sometimes surrounded by borders of tiles. The drawing at the right shows a square hot tub with sides of length s feet. This tub is surrounded by a border of 1 foot by 1 foot square tiles. s How many 1-foot square tiles will be needed for the border of this pool? a. Paul wrote the following expression: 2s + 2(s+2) Explain how Paul might have come up with his expression. b. Bill found the following expression: (s+2) 2 - s 2 Explain how Bill might have found his expression. c. How would you convince the students in your class that the two expressions above are equivalent?

9. © 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595 Released Items Number of casesDifficulty Pre-service teachers970.459 In-service teachers140.757 Total1150.499 Results: Equivalent Expressions