1. Teaching through the Mathematical Processes Session 5: Assessing with the Mathematical Processes

2. Retention Timeline 0 2 4 6 8 10 years Pink: time students retain knowledge through procedures Yellow: time students retain knowledge through concepts Blue: time students retain knowledge through mathematical processes

3. What are we teaching? Reflect on the relevance of the video to retention and life-long learning. Share your ideas with a partner.

4. Assessing the Processes Assessment - Evaluation - Reporting What is Classroom Assessment? One possible answer: Classroom Assessment refers to the collection of information teachers use to monitor students’ learning, provide feedback and to make appropriate adjustments to instruction. Exploring Classroom Assessment in Math NCTM 1998

5. Name That Process… Identify the Mathematical Process that can be assessed for the given criteria. Share your choice with a partner.

6. Mathematics Processes Rubric Think → Pair → Share Reasoning and ProvingReflectingProblem SolvingConnectingCommunicatingRepresentingSelecting Tools and Computational Strategies

7. Generic Processes Rubric

8. The distance around a tennis ball can is less than the height of the can. Opposite Sides Agree / Disagree / Don’t Know! Justify your answer with your group. Those who ‘don’t know’ explore further.

9. h = 3 balls = 3 diameters C =  diameters The distance around a tennis ball can is less than the height of the can.

10. Tennis Ball Can Problem What Mathematical Processes were used in solving the problem? Which criteria could be used to assess the Mathematical Processes with this problem?

11. How do we effectively teach Mathematical Processes? Teach to the curriculum expectations. Do investigations and solve rich problems. Ask questions and provide feedback related to the Mathematical Processes. Explore the TIPS lessons that demonstrate how lessons can be adjusted to focus on particular Mathematical Processes.

12. Students learn content and problem solving by solving problems and sharing solutions

13. Problem Selection … Choose one of the following problems to form working groups.

14. Volume of Three Dimensional Shapes Develop, through investigation (e.g., using concrete materials) the formulas for the volume of a pyramid, a cone, and a sphere.

15. Volume of Three Dimensional Shapes Grade 9 students already know: area of rectangle, triangle, and circle volume of all prisms V = (area of base)(height)

16. =÷ 3OR Volume of a pyramid is one third the volume of the corresponding prism. Volume of a Pyramid

17. Developing the Formula for the Volume of a Sphere

18. Volume of a Sphere Using relational solids and pouring material we noted that the volume of a cone is the same as the volume of a hemisphere (with corresponding dimensions). Using “math language” Volume (cone) = ½ Volume (sphere) Therefore 2(Volume (cone) ) = Volume (sphere) = OR +

19. We already know the formula for the volume of a cone. =÷ 3OR Volume of a Sphere

20. AND we know the formula for the volume of a cylinder Base Height Volume of a Sphere

21. Summarizing: Volume (cylinder) = (Area Base) (height) Volume (cone) = Volume (cylinder) /3  Volume (cone) = (Area of Base) (height)/3 AND 2(Volume (cone) ) = Volume (sphere) Volume of a Sphere =÷ 3 2 X = >

22. 2(Volume (cone) ) = Volume (sphere) 2( ) (height) /3= Volume (sphere) 2( )(h)/3= Volume (sphere) BUT h = 2r 2(  r 2 )(2r)/3 = Volume (sphere) 4(  r 3 )/3 = Volume (sphere) Volume of a Sphere Area of Base r2r2 2 X = r r h

23. Volume of a Sphere

24. Painted Cube Problem A 3 x 3 x 3 cube made up of small cubes is dipped into a bucket of red paint and removed. a)How many small cubes will have 3 faces painted? b)How many small cubes will have 2 faces painted? c)How many small cubes will have 1 face painted? d)How many small cubes will have 0 faces painted? e)Generalize your results for an n x n x n cube.

25. 2 Faces Painted N 2 = 12(n – 2) Painted Cube Problem… Using Finite Differences

26. 1 Face Painted N 1 = 6(n – 2) 2 Painted Cube Problem… Using Finite Differences

27. 0 Faces Painted N 0 = (n – 2) 3 Painted Cube Problem… Using Finite Differences

28. Painted Cube Problem… Graphically using Fathom

29. Painted Cube Problem… Graphically using Excel

30. Geometrically, using cubes and patterns... 2 faces painted1 face painted Painted Cube Problem… 3 faces painted N 2 = 12(n – 2)N 1 = 6(n – 2) 2 N 3 = 8 8 Corners 12 Edges 6 Faces (n – 2) (n – 2)X(n – 2) “square”

31. Temperature Problem The inhabitants of Xenor use two scales for measuring temperature. On the A scale, water freezes at 0° and boils at 80°, whereas on the B scale, water freezes at -20° and boils at 120°. What is the equivalent on the A scale of a temperature of 15° on the B scale? The Math Forum@Drexel All rights Reserved.

32. Scale B Scale A Freeze Boil 15 20

33. Dart Board Problem This dart board is designed with a square inside a circle and a square outside the same circle. Assign numerical values of 2, 5, and 8 to the three coloured regions on the dart board such that regions with smaller areas are assigned higher scores. Justify your solution.

34. Deck Problem You have been hired to build a deck attached to the second floor of a cottage using exactly 30m of deck railing (note: the entire outside edge will have railing). Determine the dimensions of the deck that follow the specifications in the diagram and maximize the area of the deck.

35. Combination of Functions Card Game

36. Problem Selection … Consider your group problem to answer the question: “Which Mathematical Process(es) could be assessed?”

37. Assessing the Mathematical Processes Half of the group members “solve” the problem. The remaining members observe and assess the Mathematical Processes, using the given rubric. The observing members identify other Mathematical Processes that become apparent during the problem solving. Discuss the solution strategy and observations. Group members switch roles to solve the same or a different problem. Fishbowl Strategy

38. Debrief How did solving this problem provide opportunities for you to apply Mathematical Processes How did the rubric facilitate assessing the Mathematical Processes specifically? What other observations did you make?

39. Home Activity Journal Reflection: What role will the Generic Rubric for Mathematical Processes play in my teaching and assessment practices?