Middle School Performance Tasks and Student Thinking for Mathematics

1. Middle SchoolPerformance Tasks andStudent Thinking for Mathematics CFN 609Professional Development | March 29, 2012 RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services

2. Got change? Try to figure out a way to make change for a dollar that uses exactly 50 coins. Is there more than one way?

3. Performance Tasks and Student Thinking AGENDA • Reflection • Progressions • Properties • Misconceptions • Tasks • Looking at Student Work

4. Practice and Experience Have you tried out any of the strategies, tasks or ideas from our previous sessions, and if so, what were the results?

5. Reflection and Review What are one or two ways that US math instruction differs from that in higher-performing countries?

6. Reflection and Review Describe and list some of the Standards for Math Practice.

7. 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.

8. Expertise and Character Development of expertise from novice to apprentice to expert The Content of their mathematical Character

9. Reflection and Review How are levels of cognitive demand used in looking at math tasks?

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

11. Reflection and Review What do we mean by formative assessment and what are some strategies involved in it?

12. Some Habits of Mind • Visualization, including drawing a diagram • Explanation, using their own words • Reflection and metacognition • Consideration of strategies

13. A train one mile long travels at a rate of one mile per minute through a tunnel that is one mile long. How long will it take the train to pass completely through the tunnel?

14. Some More Habits of Mind • Listening to each other • Recognizing and extending patterns • Ability to generalize • Using logic • Mental math and shortcuts

15. Two Jars You have a 3-liter jar and a 5-liter jar. Neither of them have any markings and you do not have any extra jars. You can easily measure out exactly 3, 5 and 8 liters. Is it possible to measure out exactly 1 liter? If so, how? What about 4 liters? 6 liters? 7 liters?

16. Getting Comfortable with Non-Routine Problems

17. Tree A tree doubled in height each year until it reached its maximum height in 20 years. How many years did it take this tree to reach half its maximum height?

18. Pleasure in Problem-Solving

19. 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

20. Express this sum as a simple fraction in lowest terms: 1_+ 1_+ 1_+ 1_+ 1_+ 1_ 1×2 2×3 3×4 4×5 5×6 6×7

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

22. Tennis Tournament A tennis tournament has 50 contestants, with these rules: no tie games and the loser of each game is eliminated, the winner goes on to play in the next round. How many games are needed to determine a champion?

23. The most important ideas in the CCSS mathematics that need attention Properties of operations: their role in arithmetic and algebra Mental math and (algebra vs. algorithms) Units and unitizing Operations and the problems they solve Quantities-variables-functions-modeling Number-Operations-Expressions-Equations Modeling Practice Standards

24. Progression: quantities and measurement to variables and functions

26. Fractions Progression Understanding the arithmetic of fractions draws upon four prior progressions that inform the CCSS: • Equal partitioning • Unitizing • Number line and • Operations

28. Unitizing links fractions to whole number arithmetic • Students’ expertise in whole number arithmetic is the most reliable expertise they have in mathematics • It make sense to students • If we can connect difficult topics like fractions and algebraic expressions to whole number arithmetic, these difficult topics can have a solid foundation for students

29. Nine properties are the most important preparation for algebra Just nine: foundation for arithmetic Exact same properties work for whole numbers, fractions, negative numbers, rational numbers, letters, expressions. Same properties in 3rd grade and in calculus Not just learning them, but learning to use them

30. Using the properties • To express yourself mathematically (formulate mathematical expressions that mean what you want them to mean) • To change the form of an expression so it is easier to make sense of it • To solve problems • To justify and prove

31. Properties are like rules, but also like rights • You are allowed to use them whenever you want, it’s never wrong. • You are allowed to use them in any order • Use them with a mathematical purpose

34. Linking multiplication and addition: the ninth property Distributive property of multiplication over addition a × (b+c) = (a×b) + (a×c) a(b+c) = ab + ac

35. Find the properties in the multiplication table There are many patterns in the multiplication table, most of them are consequences of the properties of operations: Find patterns and explain how they come from the properties Find the distributive property patterns

37. What is an explanation? Why you think it’s true and why you think it makes sense. Saying “distributive property isn’t enough, you have to show how the distributive property applies to the problem.

38. Mental Math 72-29= ? In your head Composing and decomposing Partial products Place value in base 10 Factor x2 +4x + 4 in your head

39. Math Tasks

40. Bananas If three bananas are worth two oranges, how many oranges are 24 bananas worth?

41. NYSED Algebra Sample Tasks

42. Misconceptions How they arise and how to deal with them

43. Misconceptions about misconceptions • They weren’t listening when they were told • They have been getting these kinds of problems wrong from day one • They forgot • Their previous teachers didn’t know the math

44. More misconceptions about the cause of misconceptions • In the old days, students didn’t make these mistakes • They were taught procedures • They weren’t taught the right procedures • Not enough practice

45. Maybe • Teachers’ misconceptions perpetuated to another generation (where did the teachers get the misconceptions? How far back does this go?) • Mile-wide inch-deep curriculum causes haste and waste • Some concepts are hard to learn

46. Whatever the Cause • When students reach your class they are not blank slates • They are full of knowledge • Their knowledge will be flawed and faulty, half baked and immature; but to them it is knowledge • This prior knowledge is an asset and an interference to new learning

47. Dividing Fractions 1 3 1 2 ÷

48. Dividing Fractions “Ours is not to question why, just invert and multiply.” 1 3 1 2 ÷

49. Dividing Fractions WHY? 1 3 1 2 ÷

50. Hose A takes 30 minutes to fill a tub with water.