1. Decatur City 6-8 Math October 31, 2014 JEANNE SIMPSON AMSTI MATH SPECIALIST

3. Welcome Name School Classes you teach What do your students struggle to learn? 3

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

5. Agenda  Major Work of the Grades  Standards of Mathematical Practice  Doing Some Math

6. Major Work of the Grades

7. Major Work of Grade 6 7

8. Major Work of Grade 7 8

9. Major Work of Grade 8 9

10. Progressions Documents K–6 Geometry 6-8 Statistics and Probability 6–7 Ratios and Proportional Relationships 6–8 Expressions and Equations 6-8 Number System These are the documents currently available. They are working on documents for the other domains (Functions, Geometry 7-8). http://ime.math.arizona.edu/progressions/

11. Value of Learning Progressions/Trajectories to Teachers Know what to expect about students’ preparation. Manage more readily the range of preparation of students in your class. Know what teachers in the next grade expect of your students. Identify clusters of related concepts at grade level. Provide clarity about the student thinking and discourse to focus on conceptual development. Engage in rich uses of classroom assessment.

12. Standards for Mathematical Practice

13. Mathematically proficient students will: SMP1 - Make sense of problems and persevere in solving them SMP2 - Reason abstractly and quantitatively SMP3 - Construct viable arguments and critique the reasoning of others SMP4 - Model with mathematics SMP5 - Use appropriate tools strategically SMP6 - Attend to precision SMP7 - Look for and make use of structure SMP8 - Look for and express regularity in repeated reasoning 13

14. 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

15. 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)

16. 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. 8Look 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

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19. Resources

20. Illustrative Mathematics 20 Illustrative Mathematics provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards. http://www.illustrativemathematics.org/

21. Chocolate Bar Sales 21

22. Foxes and Rabbits Given below is a table that gives the population of foxes and rabbits in a national park over a 12 month period. Note that each value of t corresponds to the beginning of the month.

23. Who is the Better Batter? P-23 Below is a table showing the number of hits and the number of times at bat for two Major League Baseball players during two different season: a.For each season, find the players’ batting averages. Who has better batting average? b.For the combined 1995 and 1996 seasons, find the players’ batting averages. Who has the better batting average? c.Are the answers to (a) and (b) consistent? Explain. SeasonDerek JeterDavid Justice 199512 hits in 48 at bats104 hits in 411 at bats 1996183 hits in 582 at bats45 hits in 140 at bats

24. Shrinking Catrina read that a woman over the age of 40 can lose approximately 0.06 centimeters of height per year. a.Catrina’s aunt Nancy is 40 years old and is 5 feet 7 inches tall. Assuming her height decreases at this rate after the age of 40, about how tall will she be at age 65? (Remember that 1 inch = 2.54 centimeters.) b.Catrina’s 90-year-old grandmother is 5 feet 1 inch tall. Assuming her grandmother’s height had also decreased at this rate, about how tall was she at age 40?

25. Gotham City Taxis

26. US Garbage, Version 1

28. Mathematics Assessment Project Tools for formative and summative assessment that make knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. These tools comprise: Classroom Challenges: lessons for formative assessment, some focused on developing math concepts, others on non-routine problem solving. Professional Development Modules: to help teachers with the new pedagogical challenges that formative assessment presents. Summative Assessment Task Collection: to illustrate the range of performance goals required by CCSSM. Prototype Summative Tests: designed to help teachers and students monitor their progress, these tests provide a model for examinations that may replace or complement current US tests. 28 http://map.mathshell.org/

29. 6 th Grade  Proportional Reasoning  Laws of Arithmetic  Evaluating Statements About Number Operations  Interpreting Multiplication and Division  A Measure of Slope  Real-Life Equations  Using Coordinates to Interpret and Represent Data  Mean, Median, Mode, and Range  Representing Data Using Grouped Frequency Graphs and Box Plots

30. 7 th Grade  Proportion and Non-Proportion Situations  Using Positive and Negative Numbers in Context  Possible Triangle Constructions  Steps to Solving Equations  Applying Angle Theorems  Probability Games  Statements About Probability 8 th Grade  Applying Properties of Exponents  Estimating Length Using Scientific Notation  Interpreting Time-Distance Graphs  Classifying Solutions to Systems of Equations  Solving Linear Equations in One Variable  Representing and Combining Transformations

31. Playing Catch-up 8.EE.5 - Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationship represented in different ways. 31

32. Dan Meyer  Math Class Needs a Makeover Math Class Needs a Makeover  Three Act Math Tasks Three Act Math Tasks  Blog Blog

33. 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

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

35. 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.

36. 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.

37. 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?

38. Solving Proportions  If two pounds of beans cost $5, how much will 15 pounds of beans cost? 38

39. Solving Proportions Solve  The traditional method of creating and solving proportions by using cross-multiplication is de- emphasized (in fact it is not mentioned in the CCSS) because it obscures the proportional relationship between quantities in a given problem situation. Kanold, p. 94  If two pounds of beans cost $5, how much will 15 pounds of beans cost? 39

40. Implications for Instruction  Proportional reasoning is complex and needs to be developed over a long period of time.  The study of ratios and proportions should not be a single unit but a unifying theme throughout the middle school curriculum.  Students need time to explore a variety of multiplicative situations, to coordinate both additive and relative perspectives, to experience unitizing, and to explore informally the nature of ratio in different problem contexts.  Instruction should begin with physical experiments and situations that can be visualized and modeled.  The cross-product rule should be delayed until students understand and are proficient with informal and quantitative methods for solving proportion problems.

41. 1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throws at practice, how many free throws did Omar make? 41

42. 1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throw at practice, how many free throws did Omar make? 42 made 7 misse d 3 Attempte d 90

43. 1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throw at practice, how many free throws did Omar make? 43 made 7 misse d 3 Attempte d 90 90 ÷ 10 = 9 9 999 999999

44. 1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throw at practice, how many free throws did Omar make? 44 made 7 misse d 3 Attempte d 90 9 x 7 = 63 90 ÷ 10 = 9 Omar made 63 free throws. 9 999 999999

45. 2. At FDR High School, the ratio of seniors who attend college to those who do not is 5:2. If 98 seniors do not attend college, how many do? 45

46. 3. At Mesa Park High School, the ratio of students who have driver’s licenses to those who don’t is 8:3. If 144 students have driver’s licenses, how many students are enrolled at Mesa Park High School? 46

47. 4. Of the black and blue pens that Mrs. White has in a drawer in her desk, 18 are black. The ratio of black pens to blue pens is 2:3. When Mrs. White removes 3 blue pens, what is the new ratio of black pens to blue pens? 47 http://www.risd.k12.nm.us/instruction/mathdrawingbook.cfm

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