1. Scottsboro City 6-8 Math November 14, 2014
Jeanne Simpson
AMSTI Math Specialist

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

3. He who dares to teach must never cease to learn.
John Cotton Dana
The new COS offers many opportunities for us to learn – new content, new teaching strategies, higher expectations for students, filling in gaps during the transition.

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

5. Major Work of the Grades

6. Major Work of Grade 6

7. Major Work of Grade 7

8. Major Work of Grade 8

9. 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).

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. Standards for Mathematical Practice
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

14. SMP Proficiency Matrix
Students:
(I) Initial
(IN) Intermediate
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.

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

16. SMP Proficiency Matrix
Students:
(I) Initial
(IN) Intermediate
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.
Pair-Share
Questioning/Wait Time
Grouping/Engaging Problems
Questioning/Wait Time
Grouping/Engaging Problems
Showing Thinking
Grouping/Engaging Problems
Grouping/Engaging Problems
Encourage Reasoning
Showing Thinking
Questioning/Wait Time
Grouping/Engaging Problems
Pair-Share
Questioning/Wait Time
Grouping/Engaging Problems
Grouping/Engaging Problems
Questions/Prompts for Groups
Showing Thinking
Grouping/Engaging Problems
Showing Thinking
Grouping/Engaging Problems
Showing Thinking
Allowing Struggle
Encourage Reasoning
Questions/Prompts for Groups
Allowing Struggle
Encourage Reasoning
Grouping/Engaging Problems
Allowing Struggle
Encourage Reasoning

19. EQuIP Rubric for Lessons & Units: Mathematics
Summary of the
EQuIP Rubric for Lessons & Units: Mathematics I.
Alignment to the Depth of the CCRS
II.
Key Shifts in
the CCRS
III.
Instructional Supports
IV.
Assessment
The letter and spirit of CCRS:
Depth of Standard
Mathematical Practice
Balance of procedure and understanding
Key Shifts in the CCRS:
Focus
Coherence
Rigor as application, understanding, and procedural skill and fluency
Student learning needs:
Guidance
Precise terminology
Student engagement
Productive struggle
Mathematical thinking
Differentiation
Intervention
Support learning styles
Student mastery of CCRS:
Direct, observable evidence
Formative feedback
Summative
Have them review the rubric as you briefly identify the big ideas of each dimension.
Note: Dimension I is non-negotiable. In order for the review to continue, a rating of 2 or 3 is required. If the review is discontinued, consider general feedback that might be given to developers/ teachers regarding next steps.

20. Resources

21. Illustrative Mathematics
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.

22. Chocolate Bar Sales

23. Who is the Better Batter?
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:
For each season, find the players’ batting averages. Who has better batting average?
For the combined and 1996 seasons, find the players’ batting averages. Who has the better batting average?
Are the answers to (a) and (b) consistent? Explain.
Season
Derek Jeter
David Justice
1995
12 hits in 48 at bats
104 hits in 411 at bats
1996
183 hits in 582 at bats
45 hits in 140 at bats

24. 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.

25. 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.

26. 6th 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

27. 7th Grade 8th 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
8th 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

28. 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.

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

30. 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
Handout 23-28

31. Create a distribution of nine data points on your number line that would yield a mean of 5.
Work in groups of 2-4.
Pass out copies so they can keep their handouts clean.
Share results. What might the data represent?
What limitations are there in only knowing the mean?

32. 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.
What are some possible contexts of data for the set?
How did you determine the order?
What is the best order?

33. The mean in each set equals 5.
Find the distance (deviation) of each point from the mean.
Use the absolute value of each distance. 3 3 1 3 2 1 3 4 6
Put the data in a table.
Use the MAD to re-order the data sets.
Find the mean of the absolute deviations.

34. 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?

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

36. Solving Proportions
Solve
Kanold, p. 94
If two pounds of beans cost $5, how much will 15 pounds of beans cost?
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.

37. 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.

38. 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?

39. 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?
made 7
Attempted 90
missed 3

40. made 7 9 Attempted 90 missed 3 90 ÷ 10 = 9
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?
made 7 9
Attempted 90
missed 3
90 ÷ 10 = 9

41. made 7 9 Attempted 90 missed 3 90 ÷ 10 = 9 9 x 7 = 63
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?
made 7 9
Attempted 90
missed 3
90 ÷ 10 = 9
9 x 7 = 63
Omar made 63 free throws.

42. 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?

43. 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?

44. Of the black and blue pens that Mrs
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?

45. Contact Information
Jeanne Simpson
UAHuntsville AMSTI