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

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

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.

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

Major Work of the Grades

Major Work of Grade 6

Major Work of Grade 7

Major Work of Grade 8

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/

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.

Standards for Mathematical Practice

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

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.

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)

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

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.

Resources

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. http://www.illustrativemathematics.org/

Chocolate Bar Sales

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 1995 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

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.

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. http://map.mathshell.org/

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

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

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.

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

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

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?

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?

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.

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?

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

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.

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.

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?

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

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

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.

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?

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?

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? http://www.risd.k12.nm.us/instruction/mathdrawingbook.cfm

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