Elementary School Performance Tasks for Mathematics

1. Elementary SchoolPerformance Tasks for Mathematics CFN 609Professional Development | March 8, 2012 RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services

2. Make 25¢ with: 1 coin 2 coins 3 coins 4 coins… …23 coins 24 coins 25 coins

3. Focus on Performance Tasks AGENDA • Tasks: Criteria for Judging Tasks • Metacognition • Aligning Tasks to the CCLS • Cognitive Demand • Task Implementation • Resources

4. Some Criteria for Considering Tasks • Level of challenge: accessible to the struggling, challenging enough for the advanced • Multiple points of entry • Various solution pathways • Identifying the math concept involved with, and strengthened by, working on the task

5. Fraction Value A and B are two different numbers selected from the first twenty counting numbers, 1 through 20 inclusive. What is the largest value that fraction A × B can have? A – B

6. Some Criteria for Considering Tasks • Opportunities to exercise the standards for mathematical practice • Opportunities to bring out student misconceptions, which can be identified and addressed

7. What number can replace the square to make the statement true? 5 × 11 = + 12

8. Research on Retention of Learning: Shell Center: Swan et al

9. Three Responses to a Math Problem • Answer getting • Making sense of the problem situation • Making sense of the mathematics you can learn from working on the problem

10. Bob, Jim and Cathy each have some money. The sum of Bob's and Jim's money is $18.00. The sum of Jim's and Cathy's money is $21.00. The sum of Bob's and Cathy's money is $23.00.

11. Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 5 Use appropriate tools strategically. 6 Attend to precision. 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning.

12. Math Olympiad for Elementary and Middle School

13. Math Olympiad 76-2 HE HE +HE AH Different letters represent different digits, and E is twice H. What two-digit number does AH, the sum, represent?

14. Working on a Task: Four Phases • Understand • Plan • Try it • Look back

15. Some Strategies for Approaching a Task • Make an organized list • Work backward • Look for a pattern • Make a diagram • Make a table • Use trial-and-error • Consider a related but simpler problem

16. Pizza Party Twelve friends were having a party and ordered seven small pizza pies. How can the pies be divided so that each friend gets exactly the same amount of pizza? No pie can be cut into more than four pieces.

17. And Some More Strategies • Consider extreme cases • Adopt a different point of view • Estimate • Look for hidden assumptions • Carry out a simulation

18. Metacognition

19. A Problem (DO NOT SOLVE) Make as many rectangles as you can with an area of 24 square units. Use only whole numbers for the length and width. Sketch the rectangles, and write the dimensions on the diagrams. Write the perimeter of each one next to the sketch. What questions do you ask yourself as you encounter this problem? How do these questions help you to develop a solution approach?

20. Meta-Cognition • Thinking about thinking. • The usually-unconscious process of cognition. • Habit of mind: taking the time to look back and reflect

21. Meta-cognition implications for lessons. • Make thinking public • Use multiple representations • Offer different approaches to solution • Ask questions about the problem posed. • Set a context, define the why of the problem • Focus students on their thinking, not the solution • Solve problems with partners • Prepare to present strategies

22. Aligning Tasks to the Common Core Learning Standards

23. Comparing Two Mathematical Tasks TASK A MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases. 1. Conjecture: The sum of any two odd numbers is ______? 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 3 + 5 = 8 201 + 305 = 506 2. Conjecture: The product of any two odd numbers is ____? 1 x 1 = 1 7 x 11 = 77 1 x 3 = 3 13 x 19 = 247 3 x 5 = 15 201 x 305 = 61,305

24. Comparing Two Mathematical Tasks TASK B MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases. Then explain why the conjecture is always true or show a case in which it is not true. 1. Conjecture: The sum of any two odd numbers is ______? 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 3 + 5 = 8 201 + 305 = 506 2. Conjecture: The product of any two odd numbers is ____? 1 x 1 = 1 7 x 11 = 77 1 x 3 = 3 13 x 19 = 247 3 x 5 = 15 201 x 305 = 61,305

25. Draw a Picture Every odd number (like 11 and 13) has one loner number. Add the two loner numbers and you will get an even number (24). Now add all together the loner numbers and the other two (now even) numbers.

26. Build a Model If I take the numbers 5 and 11 and organize the counters as shown, you can see the pattern. You can see that when you put the sets together (add the numbers), the two extra blocks will form a pair and the answer is always even. This is because any odd number will have an extra block and the two extra blocks for any set of two odd numbers will always form a pair.

27. Use a Pattern When we count, odd numbers alternate with even: 1, 2, 3, 4, 5… or odd, even, odd, even, odd…. If we start counting on an odd number and we count an even number of spaces forward, we land on another odd number, for example, start on 5 and count forward by 2 or 4: we get 7 or 9. But if we start on an odd number and count an odd number of spaces forward we always land on an even number, for example start on 5 and count forward by 3 or 5: we get 8 or 10.

28. Logical Argument An odd number = [an] even number + 1. e.g. 9 = 8 + 1 So when you add two odd numbers you are adding an even no. + an even no. + 1 + 1. So you get an even number. This is because it has already been proved that an even number + an even number = an even number. Therefore as an odd number = an even number + 1, if you add two of them together, you get an even number + 2, which is still an even number.

29. Use Algebra If a and b are odd integers, then a and b can be written a = 2m + 1 and b = 2n + 1, where m and n are other integers. If a = 2m + 1 and b = 2n + 1, then a + b = 2m + 2n + 2. If a + b = 2m + 2n + 2, then a + b = 2(m + n + 1). If a + b = 2(m + n + 1), then a + b is an even integer.

30. Comparing Two Mathematical Tasks How are the two versions of the task the same and how are they different?

31. Tasks A and B Same • Both ask students to complete a conjecture about odd numbers based on a set of finite examples that are provided Different • Task B asks students to develop an argument that explains why the conjecture is always true (or not) • Task A can be completed with limited effort; Task B requires considerable effort – students need to figure out WHY this conjecture holds up • The amount of thinking and reasoning required • The number of ways the problem can be solved • The range of ways to enter the problem

32. Standards for Mathematical Practice • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning Common Core State Standards for Mathematics, 2010, pp.6-7

33. Standards for Mathematical Practice • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning Common Core State Standards for Mathematics, 2010, pp.6-7

34. Button Pattern

35. Analysis of Tasks

36. Levels of Cognitive Demand Lower-level • Memorization • Procedures without connections Higher-level • Procedures with connections • Doing mathematics

37. Norman Webb’s Depth of Knowledge • Level 1: Recall and Reproduction • Level 2: Skills and Concepts • Level 3: Strategic Thinking • Level 4: Extended Thinking

39. Cognitive Complexity

40. What is Depth of Knowledge? • A language system used to describe different levels of complexity • A framework for evaluating curriculum, objectives, and assessments so they can be studied for alignment • Focuses on content and cognitive demand of test items, instructional strategies, and performance objectives

41. DOK Levels Level 1 measures Recall at a literal level. Level 2 measures a Skill or Concept at an interpretive level. Level 3 measures Strategic Thinking at an evaluative level. Level 4 measures Extended Thinking and Reasoning

42. DOK Level 1: Mathematics • Recall and recognize information such as facts, definitions, theorems, terms, formulas or procedures • Solve one-step problems, apply formulas, and perform well-defined algorithms • Demonstrate an understanding of fundamental math concepts

43. DOK 1 What is the place value of 9 in the number 74.295? A. hundreds B. tenths C. hundredths D. thousandths

44. DOK Level 2: Mathematics The cognitive demands are more complex than inLevel 1. Engage in mental processing beyond recall or habitual response: • Determine how to approach a problem • Solve routine multi-step problems • Estimate quantities, amounts, etc. • Use and manipulate multiple formulas, definitions, theorems, or a combination of these • Collect, organize, classify, display, and compare data • Extend a pattern

45. DOK 2 Draw the next figure in the following pattern:

46. DOK 2 On a road trip from Georgia to Oklahoma, Maria determined that she would cover about 918 miles. What speed would she need to average to complete the trip in no more than 15 hours of driving time?

47. DOK Level 3: Mathematics Engage in abstract, complex thinking • Determine which concepts to use in solving complex problems • Use multiple concepts to solve a problem • Reason, plan, and use evidence to explain and justify thinking • Make conjectures • Interpretinformation from complex graphs • Draw conclusions from logical arguments

48. DOK 3 Find the next three items in the pattern and give the rule for following the pattern of numbers: 1, 4, 3, 6, 5, 8, 7, 10…

49. DOK 3 A local bakery celebrated its one year anniversary on Saturday. On that day, every 4th customer received a free cookie. Every 6th customer received a free muffin. A. Did the 30th customer receive a free cookie, free muffin, both, or neither? Show or explain how you got your answer. B. Casey was the first customer to receive both a free cookie and a free muffin. What number customer was Casey? Show or explain how you got your answer. C. Tom entered the bakery after Casey. He received a free cookie only. What number customer could Tom have been? Show or explain how you got your answer. D. On that day the bakery gave away a total of 29 free cookies. What was the total number of free muffins the bakery gave away on that day? Show or explain how you got your answer.

50. DOK Level 4: Mathematics Extended Thinking/Reasoning requires complexreasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking.