MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 1 of 13 1. What are the three categories of common algebra errors as described by Matz? MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 2 of 13 2. Categorize the following errors according to Matz’s framework. (i) AX BY A B X Y 1 1 3 2 2 6x 2 (ii) x x x x 1 3 6x 2 (x a)(x b) c (iii) (x a) c or (x b) c x c a or x c b (iv) A A A BC B C (v) If x 1 5 , then x 2 and x 4. x4 6 MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning 3. Find the error in the argument below. xx x 11 x 0 (1 x) 1 (x 1) 1 (1 x) 1 (x 1) 1 1 x 1 x 1 1 04 How would you describe this type of error? Handout - Page 3 of 13 MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 4 of 13 4. Analyze the argument below. xx (x) 2 (x) 2 (x 5) 2 5 2 (x) 2 x 2 10x 25 25 x 2 x 2 10x x 2 10x 0 x 0 Why does the last statement contradict the first? What is the error? How would you describe this type of error? MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 5 of 13 5. Challenge problem (if time permits): Find the error in the argument below. 24 24 16 40 36 60 16 40 25 36 60 25 4 2 2 4 5 52 62 2 6 5 52 (4 5) 2 (6 5) 2 4 5 65 46 6. Reasoningwith Magic Squares Place the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the square below so that the sum of each digits along the each row, column and diagonal is the same? MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 6 of 13 7. More Magic Squares Do the entries in a magic square have to be 1, 2, …, n2 ? What other sequences could you use? Can the set of numbers {1, 3, 5, 7, 8, 10, 12, 14, 16} be used to form a 3 X 3 magic square? Or {0, 2, 3, 4, 5, 7, 8, 9, 11}? In a 3X3 square, the center square is always ___________ of the magic sum. What is the general form of a 3X3 magic square if a is the value of the center? MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Mathematics as sensemaking (Schoenfeld, 1991): A. establishing subgoals B. working backwards C. focusing on key points D. exploiting extreme cases E. exploiting symmetry F. working forwards G. using systematic generating procedures 8. What sort of mathematics did you use? Handout - Page 7 of 13 MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 8 of 13 What types of mathematical reasoning did you use? (Think about last session on Mathematical Reasoning) 9. Prior to manipulating symbols, students must use reasoning in determining appropriate representations to _______________________________________________ ___________________________________________________________________________________. Many ideas in algebra can be expressed with multiple representations i.e. Rule of 3 + 1 10. Reasoning is often used in ________________________________________________ ________________________________________________________________________________. MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 9 of 13 11. The graph and table below show how values of two functions change. x y … … -3 1 -2 0 -1 -1 0 -2 1 -3 2 -4 3 -5 Choose functions from below that have a property in common with one or both of the functions above. Explain your point of view and find as many viewpoints as you can. 2 y x 3 1 y x Viewpoints: y x y 2x 1 y x2 1 y x 2 2 y x2 … … MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 10 of 13 12. Student Professor Problem Using the letter S to represent the number of students (at a university) and the letter P to represent the number of professors, write an equation that summarizes the following sentence “There are six times as many students as professors at this university”. 13. If a student answered 6S = P, how might you go about helping this student? What areas do you think the student needs help with? 14. Clement (1982) suggested that the two essential competencies for solving the problem are: Recognizing that the letters represented quantities (notion of variable) Creating a “hypothetical operation” to make the two quantities (such as the number of students and professors) equal. MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning 15. Let’s prove the following: 1 + 3 + 5 + … + (2n-1) = n2 Proof that proves: Handout - Page 11 of 13 MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Proof that explains: 1 + 3 + 5 + … + (2n-1) = n2 Why does this diagram justify the equation? What must you assume is true for the diagram? Handout - Page 12 of 13 MASTA Proof and Reasoning 9-12 Module Algebraic and Symbolic Reasoning Handout - Page 13 of 13 16. What equation is represented by the shapes below? Look closely at the symbols as well. 17. Baseline Assessment 18. Interview Protocol