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**Habits Of Mind - Mathematical Practices of the Common Core**

Statewide Instructional Technology Project

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**How many of your feel the same?**

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Introduction An introductory webinar was offered March 6th. Please watch the recording for the explanation introducing Common Core and the changes to the AZ 2010 Math Standards.

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**To view the official ADE documents**

2010 Arizona Mathematics Standards Overview of the 2010 Mathematical Standards PDF Standards for Mathematical Practices PDF Mathematics Introduction (Coming Soon) Mathematics Glossary PDF Summary of Updates to Explanations and Examples PDF

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**Gauge the Audience What grade level do you teach? A = K-5 B = 6-8**

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**How does your classroom compare?**

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**Highlight the MP columns to show where they**

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**The Mathematical Practices**

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**The Mathematical Practices**

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**Productive Mathematical Thinker**

The Habits of Mind of a Productive Mathematical Thinker Make sense of problems and persevere in solving them 6. Attend to precision

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**MP1. Make sense of problems and persevere in solving them**

In Kindergarten, students begin to build the understanding that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” or they may try another strategy. In first grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They are willing to try other approaches.

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**MP1. Make sense of problems and persevere in solving them**

In first grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They are willing to try other approaches. In second grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They make conjectures about the solution and plan out a problem-solving approach.

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**MP1. Make sense of problems and persevere in solving them**

In second grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They make conjectures about the solution and plan out a problem-solving approach. In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

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**MP1. Make sense of problems and persevere in solving them**

In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. In 4th grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

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**MP1. Make sense of problems and persevere in solving them**

In 4th grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. In 5th grade, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, ―What is the most efficient way to solve the problem?, ―Does this make sense?, and ―Can I solve the problem in a different way?

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**Make sense of problems and persevere in solving them**

In 5th grade, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, ―What is the most efficient way to solve the problem?, ―Does this make sense?, and ―Can I solve the problem in a different way? In grade 6 grade, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.

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**MP1. Make sense of problems and persevere in solving them**

In grade 6 grade, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?” In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.

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**MP1. Make sense of problems and persevere in solving them**

In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”. In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”

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**MP1. Make sense of problems and persevere in solving them**

High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

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Good problems: Mathematics questions or tasks that are challenging enough so it becomes useful to take in ideas other than just one’s own to engage thinking. Productive Thinking vs Reproductive Thinking Connect to student experience or interest Incorporates rich mathematics Entry points solutions pathways are not obvious We will define it by explaining what it is not; we will contrast problems with exercises. Here is an exercise: What is 6 x 7? You got it? Easy. Here is another exercise: What is 6776 to the 766,667? That one was harder, but it was still an exercise because an exercise is a mathematical question that you immediately know how to answer. You may not answer it correctly. In fact, you may never answer it correctly. It may be nearly impossible, but there is no doubt about how to proceed. So what is a problem? In contrast, a problem is a mathematical question that you do not know how to answer, at least initially. …There is always a porous boundary between problem and exercise, but a problem by its very nature requires investigation, sometimes very intense and sustained investigation. The investigation of a problem employs strategies and tactics…”

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**MP1. Make sense of problems and persevere in solving them**

Mathematically proficient students make sense of problems: Mathematically proficient students persevere in problem solving: Make sense of problems Plan a solution pathway Explain the meaning of the problem to themselves Consider similar cases and alternate form Look for entry point Monitor progress and change course if necessary Analyze givens, constraints, relationships, goals Explain correspondence and search for trends Check their answers using alternate methods Continually ask themselves, “Does this make sense?” Listen to and work to understand the approaches of others

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**MP 6. Attend to precision Mathematically proficient students:**

Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning. State the meaning of the symbols they choose Specify units of measure, and labeling axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Give carefully formulated explanations to each other. In high school, they have learned to examine claims and make explicit use of definitions. Ask audience why they think the fifth bullet is gray instead of black. This is not the emphasis

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**Give participants time to solve the problem**

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**Technology Resources To solve this problem you might explore**

(Connecting Cubes) (Pattern Blocks) (Geoboards) (Graphs) Lillian with copy and past chate. Debbie will give 1-3 minutes showcase of these websites.

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**A Short Introduction to Using the Connecting Cubes eManipulatives**

Technology Resources Lillian with copy and past chate. Debbie will give 1-3 minutes showcase of these websites. A Short Introduction to Using the Connecting Cubes eManipulatives (Click to play)

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**A Short Introduction to Using the Virtual Pattern Blocks**

Technology Resources Lillian with copy and past chate. Debbie will give 1-3 minutes showcase of these websites. A Short Introduction to Using the Virtual Pattern Blocks (Click to play)

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**A Short Introduction to Using the Virtual Geoboard**

Technology Resources Lillian with copy and past chate. Debbie will give 1-3 minutes showcase of these websites. A Short Introduction to Using the Virtual Geoboard (Click to play)

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**Discuss how you solved the problem**

After solving the problem, complete the form linked ( XZDUmxOdzFENV93UkpqOXBMNmc6MQ#gid=0) in the chat area to answer the following questions: What answer did you get? Which technology tool did you use? Why did you choose this tool? How did this tool help you solve the problem?

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**Solutions to the Problem**

Click xuu3aQOMKNdHZQaXZDUmxOdzFENV93UkpqOX BMNmc to access the results of the form. Elicit the Habits of the Mathematical Thinker Make sense of problems Explain the meaning of the problem to themselves Look for entry points Analyze givens, constraints, relationships, goals

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**Problem Solving w/No Technology**

Elicit the Habits of the Mathematical Thinker Make sense of problems Explain the meaning of the problem to themselves Look for entry points Analyze givens, constraints, relationships, goals Some students may choose to draw the tables. Those who gradually "grow" the row, counting seats at each stage and keeping track of results in a list or table, will likely discover that each new table contributes three seats to the total. As a table with five places gets added, two seats are lost along the connected edges.

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**Problem Solving with Technology**

To help students find entry points: • did you try using pattern blocks, connecting cubes or drawing the tables? Pattern blocks, real or virtual, are good tools for modeling the situation and counting the number of seats.

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**Problem Solving with Technology**

Make sense of problems Explain the meaning of the problem to themselves Look for entry points Analyze givens, constraints, relationships, goals

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**Problem Solving with Technology**

Elicit the Habits of the Mathematical Thinker Make sense of problems Explain the meaning of the problem to themselves Look for entry points Analyze givens, constraints, relationships, goals

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**Problem Solving with Technology**

. In this problem, the number of seats is equivalent to the perimeter of the row of tables. Here’s another modeling strategy, one that lends itself to discussion of perimeter. .

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**Student Example: Problem Solving**

Student 1: List created after student modeled with pattern blocks. Number of Tables Number of Seats 1 5 2 8 3 11 4 14 17 … 20 62 As we elicit student solutions, it is important to respond in ways that help them to express how they went through the process, encourage each to defend their thinking, and to celebrate interesting insights and partial successes along the way. To help students find entry points: • did you try using pattern blocks, connecting cubes or drawing the tables? • did you make an organized list to help you see patterns? • did you check your arithmetic? Once solutions are offered, the work continues. Monitor progress and change course if necessary Explain correspondence and search for trends Check their answers using alternate methods Continually ask themselves, “Does this make sense?” Listen to and work to understand the approaches of others Ask - • is your explanation clear and complete? Does it make sense? How can you prove your solution works? • did you verify your answers with another method? • did you have any “Aha!” moments or notice any patterns? Describe them. I found there would be 17 seats at 5 tables. I noticed that each time I added a table, the number of seats increased by three. That is because we are adding five new places but losing two on the sides of the tables that connect. To answer question 2, I counted out 15 more trapezoids to make a total of 20 tables. Then I skip counted by threes for each new table, starting with 17, until I came to 62 seats for the 20th table.

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**Student Example: Problem Solving**

Student 2: (Draw a diagram, then direct calculation) I used pattern blocks to help me see the pattern. After adding several tables I discovered that each table added three seats to the total. At the ends there were always two more seats, no matter how long the row was. Here are my results: (organized list) To answer question 2, I figured out how many more tables I would need: = 15 more tables Each of those 15 new tables would add three seats to the row: * 3 = 45 more seats I added the seats from 5 tables and the new seats: = 62 total seats at 20 tables In the past we gave the students a problem and asked for the solution. If it was correct we were good to move on. In the new standards, it is more important for students to understand the process of solving the problem and articulate the method they used to reach the answer.

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**Student Example: Problem Solving**

Student 3: (Direct Calculation based on seats lost) I multiplied the number of tables by the number of people who can sit at one table: 5 * 5 = 25 For each of the places where two tables connect, we lose 2 seats. There are 4 of those places, one less than the total number of tables. 4 * 2 = 8 I subtracted the number of seats lost from the total places: 25 – 8 = 17 seats at 5 tables I did the same thing for 20 tables: 20 * 5 = 100 total seats 19 * 2 = 38 seats lost at the connections 100 – 38 = 62 seats at 20 tables What were their entry points? What ahah they had? What were the givens, relationships, and goals. Elicit the Habits of the Mathematical Thinker Make sense of problems Explain the meaning of the problem to themselves Look for entry points Analyze givens, constraints, relationships, goals In the past we gave the students a problem and asked for the solution. If it was correct we were good to move on. In the new standards, it is more important for students to understand the process of solving the problem and articulate the method they used to reach the answer.

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Use words or numbers and symbols to write a rule for calculating the number of volunteers that can sit at any given number of tables. How many tables would it take, arranged in a straight line, to seat 85 volunteers?

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Extend the Problem While the only arithmetic skills needed to solve the problem are counting and basic operations with whole numbers, the problem develops algebraic thinking by providing a concrete example of a linear function. Each new table added contributes three new seats to the total, analogous to slope, and every stage includes one seat at each end, analogous to the constant or y-intercept. Students ready for extra challenge can graph their results.

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Extend the Problem Extending the line…

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**Discuss your attention to precision**

Communicate terms and thinking process Link to Spreadsheet from form.

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**Mathematical Practices**

Expertise that we each seek to develop in our students What does it mean to do mathematics? What does it mean to understand mathematics? As teachers, our goal is to provide regular and consistent opportunities to develop and build these habits of mathematical thinking.

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**1st Steps to Implementation of Mathematical Practices**

How to transition Begin with the Mathematical Practices of CCS Look closely at the Critical Areas provided for each grade What shall I do first Focus on the Mathematical Practices How do your students model the Mathematical Practices? In what ways do your classroom strategies foster development of the Mathematical Practices? Implement the Critical Ideas Look at the ADE website for standards, crosswalk, and summary of changes.

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**Resources for Further Exploration**

The Illustrative Mathematics Project Math Common Core Coalition Achieve the Core National Council Teacher of Mathematics Guiding Principles for Mathematics Curriculum and Assessment Mathematics Problem Solving Mathematics Through Problem Solving

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**Resources for Further Exploration**

Inside Mathematics Curriculum Exemplars from EngageNY Indiana Dept of Ed Implementing the Standards for Mathematical Practice Indiana Dept of Ed - Implementing the Standards for Mathematical Practice StandforMath.html Tools for the Common Core Standards

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**Resources for Further Exploration**

New Jersey Center for Teaching & Learning Progressive: Mathematics Initiative Free digital course content for over twenty courses, these initiatives span K-12 mathematics and high school science. Rich problem source (middle/HS) materials/problem-bank.html Kenken Math Forum Online Pattern Blocks Wiki on Standards of Practice

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**To view the official ADE documents**

2010 Arizona Mathematics Standards Overview of the 2010 Mathematical Standards PDF Standards for Mathematical Practices PDF Mathematics Introduction (Coming Soon) Mathematics Glossary PDF Summary of Updates to Explanations and Examples PDF

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**More Questions Contact ADE Mary Knuck Mary.Knuck@azed.gov**

Suzi Mast

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