1. Content Deepening 6 th Grade Math January 24, 2014 Jeanne Simpson AMSTI Math Specialist

2.  Name  School  What are you hoping to learn today? Welcome 2

3. He who dares to teach must never cease to learn. John Cotton Dana 3

4. Goals for Today  Implementation of the Standards of Mathematical Practices in daily lessons  Understanding of what the CCRS expect students to learn blended with how they expect students to learn.  Student-engaged learning around high- cognitive-demand tasks used in every classroom.

5. Agenda  Statistics and Probability  Progression  Standards Analysis  Resources  High-Cognitive Demand Tasks  Expressions and Equations  Inequalities  Resources  Standards of Mathematical Practice  Fractions

6. acos2010.wikispaces.com  Electronic version of handouts  Links to web resources

7. Statistics and Probability

8. THE STRUCTURE IS THE STANDARDS The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.  http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422

9. KNOWLEDGE GAPS This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations. For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events.  http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422

10. Learning Progression Jigsaw  Read your assigned section  K-5 Data, pages 1-5  6-8 Overview, pages 6-7  Grade 6, pages 8-10  Grade 7, pages 11-14  Chart paper  Summarize what needs to be learned.  How can this document help you in your classroom?  Be prepared to share 10

11. Analysis Tool Content Standard Cluster Which Standards in the Cluster Are Familiar? What’s New or Challenging in These Standards? Which Standards in the cluster Need Unpacking or Emphasizing? How Is This Cluster Connected to the Other 6-8 Domains and Mathematical Practice? Develop understanding of statistical variability. 6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

12. What resources do you already have for teaching statistics? How do these match the standards?

13. SP Resources  Raisin Activity Raisin Activity  MARS – Mean, Median, Mode MARS – Mean, Median, Mode  CMP CMP  Data About Us  Common Core Investigations  Lessons for Learning  How MAD are You? How MAD are You?  Shakespeare vs. Rowling Shakespeare vs. Rowling

14. Raisin Activity  Count the number of raisins in your box.  Make a box plot for the number of raisins in each brand’s box.  Find the median, range, and interquartile range for each brand.  Make a dot plot of the data. Find the mean and the mean absolute deviation.

15. Mean, Median, Mode, and Range

16. Computer Games: Ratings P-16 Imagine rating a popular computer game. You can give the game a score of between 1 and 6.

17. Computer Games: Ratings P-17 Rate the game Candy Crush with a score between 1 and 6.

18. Bar Chart from a Frequency Table P-18 Mean score Median score Mode score Range of scores

19. Matching Cards 1. Each time you match a pair of cards, explain your thinking clearly and carefully. 2. Partners should either agree with the explanation or challenge it if it is unclear or incomplete. 3. Once agreed stick the cards onto the poster and write a justification next to the cards. 4. Some of the statistics tables have gaps in them and one of the bar charts is blank. You will need to complete these cards. P-19

20. Sharing Posters 1. One person from each group visit a different group and look carefully at their matched cards. 2. Check the cards and point out any cards you think are incorrect. You must give a reason why you think the card is incorrectly matched or completed, but do not make changes to the card. 3. Return to your original group, review your own matches and make any necessary changes using arrows to show if card needs to move. P-20

21. AMSTI Connected Math Unit

22. How MAD are You? (Mean Absolute Deviation)  Fist to Five…How much do you know about Mean Absolute Deviation?  0 = No Knowledge  5 = Master Knowledge

23. Create a distribution of nine data points on your number line that would yield a mean of 5.

24. Card Sort  Which data set seems to differ the least from the mean?  Which data set seems to differ the most from the mean?  Put all of the data sets in order from “Differs Least” from the mean to “Differs Most” from the mean.

25. The mean in each set equals 5. 3 3 3 3 2 1 1 4 6 Find the distance (deviation) of each point from the mean. Use the absolute value of each distance. Find the mean of the absolute deviations.

26.  How could we arrange the nine points in our data to decrease the MAD?  How could we arrange the nine points in our data to increase the MAD?  How MAD are you?

27. Shakespeare vs. Rowling

28. High Cognitive vs. Low Cognitive 28

29. An effective mathematical task is needed to challenge and engage students intellectually. 29

30. Read an excerpt from the article:

31. Comparing Two Mathematical Tasks Solve Two Tasks: Martha’s Carpeting Task The Fencing Task 31

32. How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? Comparing Two Mathematical Tasks 32

33. Similarities and Differences Similarities Both are “area” problems Both require prior knowledge of area Differences The amount of thinking and reasoning required The number of ways the problem can be solved Way in which the area formula is used The need to generalize The range of ways to enter the problem 33

34. Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or Why not? Comparing Two Mathematical Tasks 34

35. Does Maintaining Cognitive Demand Matter? YES

37. Criteria for low cognitive demand tasks Recall Memorization Low on Bloom’s Taxonomy

38. Developing Mathematical Practices Review each of the “higher demand” tasks: What mathematical practices do they engage students in using ?

39. Criteria for high cognitive demand tasks Requires generalizations Requires creativity Requires multiple representations Requires explanations (must be “worth explaining”)

40. Lower-Level Tasks Memorization What are the decimal equivalents for the fractions ½ and ¼? Procedures without connections Convert the fraction 3/8 to a decimal

41. Higher-Level Tasks Doing mathematics Shade 6 small squares in a 4 X 10 rectangle. Using the rectangle, explain how to determine: A. The decimal part of area that is that is shaded; B. The fractional part of area that is shaded

42. Higher-Level Tasks Procedures with Connections Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.

43. What causes high- level cognitive demand tasks to decline?

44. Stein & Lane, 2012 A. B. C. High Low HighLow Moderate High Low Task Set UpTask ImplementationStudent Learning Patterns of Set up, Implementation, and Student Learning 44

45. Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off- task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes (Stein, Grover & Henningsen, 2012) 45

46. Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore (Stein, Grover & Henningsen, 2012) Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands 46

47. “Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” (Stein, Smith, Henningsen, & Silver, 2011) “The level and kind of thinking in which students engage determines what they will learn.” (Hiebert et al., 2011)

48. Expressions and Equations

49. Activities  Inequalities Inequalities  Look at standards. What is required?  Illustrative Mathematics tasks  MARS – Laws of Arithmetic lesson MARS – Laws of Arithmetic lesson  Arithmetic with whole-number exponents  Order of operations  Finding area of compound rectangles by evaluating expressions  Math-Magic Math-Magic  Using variables

50. What are students asked to do with inequalities?  6.EE.5 – Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.5  6.EE.6 – Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.6  6.EE.7 – Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 6.EE.7  6.EE.8 – Write an inequality of the form x > c or x c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. 6.EE.8

51. Log Ride (6.EE.5) A theme park has a log ride that can hold 12 people. They also have a weight limit of 1500 lbs per log for safety reasons. If the average adult weights 100 lbs and the log itself weights 200, the ride can operate safely if the inequality 150A + 100C + 200 < 1500 is satisfied (A is the number of adults and C is the number of children in the log ride together). There are several groups of children of differing numbers waiting to ride. If 4 adults are already seated in the log, which groups of children can safely ride with them? Group 1: 4 children Group 2: 3 children Group 3: 9 children Group 4: 6 children Group 5: 5 children

52. Fishing Adventures (6.EE.8) Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can hold at most eight people. Additionally, each boat can only carry 900 pounds of weight for safety reasons. a. Let p represent the total number of people. Write an inequality to describe the number of people that a boat can hold. Draw a number line diagram that shows all possible solutions. b. Let w represent the total weight of a group of people wishing to rent a boat. Write an inequality that describes all total weights allowed in a boat. Draw a number line diagram that shows all possible solutions.

53. Laws of ArithmeticProjector Resources Laws of Arithmetic

54. Projector Resources Pre-Assessment Task

55. Laws of ArithmeticProjector Resources Pre-Assessment Task

56. Laws of ArithmeticProjector Resources Pre-Assessment Task

57. Laws of ArithmeticProjector Resources Teacher Notes

58. Laws of ArithmeticProjector Resources Writing Expressions P-58 3 4 5 Write an expression to represent the total area of this diagram

59. Laws of ArithmeticProjector Resources Compound Area Diagrams P-59 4 5 1 2 1 2 5 4 Area A Area C 2 1 4 5 Area B Which compound area diagram represents the expression: 5 + 4 x 2?

60. Laws of ArithmeticProjector Resources Matching Cards P-60 1.Take turns at matching pairs of cards that you think belong together. For each Area card there are at least two Expressions cards. 2.Each time you do this, explain your thinking clearly and carefully. Your partner should either explain that reasoning again in his/her own words or challenge the reasons you gave. 3.If you think there is no suitable card that matches, write one of your own on a blank card. 4.Once agreed, stick the matched cards onto the poster paper writing any relevant calculations and explanations next to the cards. You both need to be able to agree on and to be able to explain the placement of every card.

61. Laws of ArithmeticProjector Resources Sharing Work P-61 1.If you are staying at your desk, be ready to explain the reasons for your group’s matches. 2.If you are visiting another group: –Copy your matches onto your paper. –Go to another group’s desk and check to see which matches are different from your own. 3.If there are differences, ask for an explanation. If you still don’t agree, explain your own thinking. 4.Return to your original group, review your own matches and make any necessary changes using arrows to show that a card needs to move.

62. Variables and Expressions 6.EE.2 – Write, read, and evaluate expressions in which letters stand for numbers 6.EE.2a – Write expressions that record operations with numbers and with letters standing for numbers 6.NS.6 – Understand a rational number as a point on a number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

64. Math Magic – Trick #1  Pick a number… any number! (keep it a secret though)  Add 1 to that number  Multiply by 3  Subtract your ‘secret’ number  Add 5  Divide by 2  Subtract your secret number

65. Math Magic – Trick #1  Now map the digit you got to a letter in the alphabet. For example: A =1B=2 C=3D=4  Pick a name of a country in Europe that starts with that letter.

66. Countries in Europe  Albania  Andorra  Austria  Belarus  Belgium  Bosnia and Herzegovina  Bulgaria  Croatia  Cyprus  Czech Republic  Denmark  Estonia Finland France Germany Greece Hungary Iceland Ireland Italy Latvia Liechtenstein Lithuania Luxembourg Macedonia Malta Moldova Monaco Netherlands Norway Poland Portugal Romania Russia San Marino Serbia and Montenegro Slovakia (Slovak Republic) Slovenia Spain Sweden Switzerland Turkey Ukraine United Kingdom Vatican City

67. Math Magic – Trick #1  Take the second letter in the country's name, and pick and animal that starts with that letter.  Think of the color of that animal.

69. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2 b. Add 5 c. Multiply by 3 d. Subtract 3 e. Divide by 3 f. Subtract your original number

70. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2 b. Add 57 c. Multiply by 3 d. Subtract 3 e. Divide by 3 f. Subtract your original number

71. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2 b. Add 57 c. Multiply by 321 d. Subtract 3 e. Divide by 3 f. Subtract your original number

72. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2 b. Add 57 c. Multiply by 321 d. Subtract 318 e. Divide by 3 f. Subtract your original number

73. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2 b. Add 57 c. Multiply by 321 d. Subtract 318 e. Divide by 36 f. Subtract your original number

74. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2 b. Add 57 c. Multiply by 321 d. Subtract 318 e. Divide by 36 f. Subtract your original number4

75. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number27 b. Add 57 c. Multiply by 321 d. Subtract 318 e. Divide by 36 f. Subtract your original number4

76. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number27 b. Add 5712 c. Multiply by 32136 d. Subtract 31833 e. Divide by 3611 f. Subtract your original number44

77. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715 b. Add 5712 c. Multiply by 32136 d. Subtract 31833 e. Divide by 3611 f. Subtract your original number44

78. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715 b. Add 571220 c. Multiply by 3213660 d. Subtract 3183357 e. Divide by 361119 f. Subtract your original number444

79. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715n b. Add 571220 c. Multiply by 3213660 d. Subtract 3183357 e. Divide by 361119 f. Subtract your original number444

80. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715n b. Add 571220n + 5 c. Multiply by 3213660 d. Subtract 3183357 e. Divide by 361119 f. Subtract your original number444

81. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715n b. Add 571220n + 5 c. Multiply by 32136603n + 15 d. Subtract 3183357 e. Divide by 361119 f. Subtract your original number444

82. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715n b. Add 571220n + 5 c. Multiply by 32136603n + 15 d. Subtract 31833573n + 12 e. Divide by 361119 f. Subtract your original number444

83. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715n b. Add 571220n + 5 c. Multiply by 32136603n + 15 d. Subtract 31833573n + 12 e. Divide by 361119n + 4 f. Subtract your original number444

84. DirectionsHow The Numbers Change Algebraic Expressions a. Think of a number2715n b. Add 571220n + 5 c. Multiply by 32136603n + 15 d. Subtract 31833573n + 12 e. Divide by 361119n + 4 f. Subtract your original number4444

85. General MacArthur's # Game  Write down the number of the month you were born in  Double it  Add 5  Multiply by 50  Add in your age  Subtract 365  What is your number?

86. Your Game!  Create your own magic with math to share with your friends and family.  Complete the worksheet with directions you make up.  Try three different numbers.  Use algebraic expressions to show why the magic works.

87. Exit Question  Ricardo has 8 pet mice. He keeps them in two cages that are connected so that the mice can go back and forth between the cages. One of the cages is blue, and the other is green. Show all the ways that 8 mice can be in two cages.

88. Standards of Mathematical Practice

89. Students:(I) Initial(IN) Intermediate(A)Advanced 1a Make sense of problems Explain their thought processes in solving a problem one way. Explain their thought processes in solving a problem and representing it in several ways. Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways. 1b Persevere in solving them Stay with a challenging problem for more than one attempt. Try several approaches in finding a solution, and only seek hints if stuck. Struggle with various attempts over time, and learn from previous solution attempts. 2 Reason abstractly and quantitatively Reason with models or pictorial representations to solve problems. Are able to translate situations into symbols for solving problems. Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations. 3a Construct viable arguments Explain their thinking for the solution they found. Explain their own thinking and thinking of others with accurate vocabulary. Justify and explain, with accurate language and vocabulary, why their solution is correct. 3b Critique the reasoning of others. Understand and discuss other ideas and approaches. Explain other students’ solutions and identify strengths and weaknesses of the solution. Compare and contrast various solution strategies and explain the reasoning of others. 4 Model with Mathematics Use models to represent and solve a problem, and translate the solution to mathematical symbols. Use models and symbols to represent and solve a problem, and accurately explain the solution representation. Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem. 5 Use appropriate tools strategically Use the appropriate tool to find a solution. Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection. Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution. 6 Attend to precision Communicate their reasoning and solution to others. Incorporate appropriate vocabulary and symbols when communicating with others. Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas. 7 Look for and make use of structure Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7. Compose and decompose number situations and relationships through observed patterns in order to simplify solutions. See complex and complicated mathematical expressions as component parts. 8 Look for and express regularity in repeated reasoning Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns. Find and explain subtle patterns. Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such as discovering an underlying function. SMP Proficiency Matrix

90. SMP Instructional Implementation Sequence 1. Think-Pair-Share (1, 3) 2. Showing thinking in classrooms (3, 6) 3. Questioning and wait time (1, 3) 4. Grouping and engaging problems (1, 2, 3, 4, 5, 8) 5. Using questions and prompts with groups (4, 7) 6. Allowing students to struggle (1, 4, 5, 6, 7, 8) 7. Encouraging reasoning (2, 6, 7, 8)

91. Students:(I) Initial(IN) Intermediate(A)Advanced 1a Make sense of problems Explain their thought processes in solving a problem one way. Explain their thought processes in solving a problem and representing it in several ways. Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways. 1b Persevere in solving them Stay with a challenging problem for more than one attempt. Try several approaches in finding a solution, and only seek hints if stuck. Struggle with various attempts over time, and learn from previous solution attempts. 2 Reason abstractly and quantitatively Reason with models or pictorial representations to solve problems. Are able to translate situations into symbols for solving problems. Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations. 3a Construct viable arguments Explain their thinking for the solution they found. Explain their own thinking and thinking of others with accurate vocabulary. Justify and explain, with accurate language and vocabulary, why their solution is correct. 3b Critique the reasoning of others. Understand and discuss other ideas and approaches. Explain other students’ solutions and identify strengths and weaknesses of the solution. Compare and contrast various solution strategies and explain the reasoning of others. 4 Model with Mathematics Use models to represent and solve a problem, and translate the solution to mathematical symbols. Use models and symbols to represent and solve a problem, and accurately explain the solution representation. Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem. 5 Use appropriate tools strategically Use the appropriate tool to find a solution. Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection. Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution. 6 Attend to precision Communicate their reasoning and solution to others. Incorporate appropriate vocabulary and symbols when communicating with others. Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas. 7 Look for and make use of structure Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7. Compose and decompose number situations and relationships through observed patterns in order to simplify solutions. See complex and complicated mathematical expressions as component parts. 8 Look for and express regularity in repeated reasoning Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns. Find and explain subtle patterns. Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such as discovering an underlying function. SMP Proficiency Matrix Grouping/Engaging Problems Pair-Share Showing Thinking Questioning/Wait Time Questions/Prompts for Groups Pair-Share Grouping/Engaging Problems Questioning/Wait Time Grouping/Engaging Problems Allowing Struggle Grouping/Engaging Problems Showing Thinking Encourage Reasoning Grouping/Engaging Problems Showing Thinking Encourage Reasoning

92. Fractions

93. FINDING THE MISSING PIECES Middle Grades Fractions Jeanne Simpson NCTM 2014

94. Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens. National Mathematics Panel Report, 2008

95. WHY ARE FRACTIONS SO DIFFICULT?  There are many meanings of fractions (part-whole, measurement, division, operator, ratio).  Fractions are written in an unusual way.  Instruction does not focus on a conceptual understanding of fractions.  Students overgeneralize their whole-number knowledge. (McNamara & Shaughnessy, 2010) Van de Walle, Karp, & Bay-Williams, 2013

96. KEY IDEAS NEEDED FOR CONCEPTUAL UNDERSTANDING  The meaning of fractions  Partitioning  Unit fractions  Models  Number lines  Equivalent fractions  Comparing fractions

97. PARTITIONING

98. PARTITIONING IS KEY TO UNDERSTANDING AND GENERALIZING CONCEPTS RELATED TO FRACTIONS SUCH AS:  Identifying “fair shares”  Identifying fractional parts of an object  Identifying fractional parts of sets of objects  Comparing and ordering fractions  Locating fractions on number lines  Understanding the density of rational numbers  Evaluating whether two fractions are equivalent or finding equivalent fractions  Operating with fractions  Measuring

99. STAGES OF PARTITIONING  Sharing – two equal parts  Algorithmic halving – equal parts that are powers of two  Evenness – even numbers that have odd factors  Oddness – partitioning into an odd number of equal parts involves thinking about the relative size of each part to the whole before partitioning  Composition – using rows and columns (multiplicative)

100. DO I TEACH THESE STRATEGIES? NO!  Teachers should make intentional choices about which fractions they use to teach, reinforce, and strengthen concepts that can be built on understanding the impact of partitioning.  Provide students with a variety of models  Students should partition the models into a variety of fractional parts, starting with powers of two  Have students share their strategies so that all students are exposed to a variety of ways of thinking.  Over time, students will take on other strategies as they are ready.

102. UNIT FRACTIONS

103. CCSS 3.NF.A.1  Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.  (Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, 8.)

104. MODELS FOR FRACTIONS AreaLengthSet Whole = area of defined region Parts = equal areas Fraction = part of area covered Whole = Unit of distance or length Parts = equal distances or lengths Fraction = location in relation to other points on the line Whole = one set Parts = equal number of objects Fraction = the count of objects in the subset

105. THREE TYPES OF MODELS  Area models (regions, part to whole relationships)  Se  Set models (fractional part of a set of objects) Number lines (distance traveled or location)

106. MODELS  Learning is facilitated when students interact with multiple models that differ in perceptual features causing students to continuously rethink and ultimately generalize the concept.  Rectangles are better than circles. They are easier to partition equally, and they allow multiplicative reasoning.

107. MODELS DIFFER IN CHALLENGES  How the whole is defined  How “equal parts” are defined  What the fraction indicates ModelWholeEqual PartsFraction Indicates AreaDefined regionEqual areaThe part covered of whole unit of area SetWhat is in the set Equal number of objects The count of objects in the subset of the defined set of objects Number line Unit of distance or length (continuous) Equal distance The location of a point in relation to the distance from zero with regard to the defined unit

108. TO PREVENT OVER-RELIANCE…  Let students become comfortable with model.  Then give them a problem where the model is cumbersome.  Vary the model so students do not over- generalize.  The ultimate goal is a mental model.

110. RESEARCHERS SAY….  Over time, students should move from the need always to construct or use physical models to carrying the mental image of the model, while still being able to make a model as they learn new concepts or encounter a difficult problem. Petit, Laird, Marsden (2010)

111.  “Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.” Imm, Fosnot, Uittenbogaard (2012)

112. Jeanne Simpson UAHuntsville AMSTI jeanne.simpson@uah.edu acos2010@wikispaces.com Contact Information 112

113. Feedback 3 things I learned 2 things I liked 1 thing I want to know more about 113