Using the Four Cs to Orchestrate the Common Core Classroom

UsingUsing thethe FourFour CsCs toto OrchestrateOrchestrate thethe CommonCommon CoreCore ClassroomClassroom

Matthieu Hamo

Glendale Unified School District

Gohar K. Hamo

Los Angeles Unified School District

California Math Council

2015 [email protected] [email protected] Presentation Objectives

1) Define the four Cs of 21st century learning

2) Create a guide to orchestrating the Common Core classroom.

3) Participate in tasks and activities that promote the four Cs

[email protected] [email protected] Common Core Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them. Mathematically proficient students start 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. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They justify their conclusions, communicate them to others, and respond to the arguments of others. 4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator. 6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. 7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. 8. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. As they work to solve a problem, mathematically proficient students

[email protected] [email protected] maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Source: http://www.corestandards.org/Math/Practice

[email protected] [email protected] Standards for Mathematical Practices Progression By Grade Levels Kindergarten SMP 1: Make sense of In Kindergarten, students begin to build the understanding problems and that doing mathematics involves solving problems and persevere in solving them. 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. SMP 2: Reason Younger students begin to recognize that a number represents a specific quantity. Then, abstractly and they connect the quantity to written symbols. Quantitative reasoning entails creating a quantitatively. representation of a problem while attending to the meanings of the quantities. SMP 3: Construct Younger students construct arguments using concrete referents, viable arguments and such as objects, pictures, drawings, and actions. They also begin to critique the reasoning of others. develop their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking. SMP 4: Model with In early grades, students experiment with representing problem mathematics. situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. SMP 5: Use Younger students begin to consider the available tools (including estimation) when appropriate tools solving a mathematical problem and decide when certain tools might be helpful. For strategically. instance, kindergarteners may decide that it might be advantageous to use linking cubes to represent two quantities and then compare the two representations side-by-side. SMP 6: Attend to As kindergarteners begin to develop their mathematical communication skills, they try precision. to use clear and precise language in their discussions with others and in their own reasoning. SMP 7: Look for and Younger students begin to discern a number pattern or structure. For instance, students make use of recognize the pattern that exists in the teen numbers; every teen number is written with structure. a 1 (representing one ten) and ends with the digit that is first stated. They also recognize that 3 + 2 = 5 and 2 + 3 = 5. SMP 8: Look for and In the early grades, students notice repetitive actions in counting and computation, etc. express regularity in For example, they may notice that the next number in a counting sequence is one more. repeated reasoning. When counting by tens, the next number in the sequence is “ten more”‖ (or one more group of ten). In addition, students continually check their work by asking themselves, “Does this make sense?”

Adapted from Arizona Department of Education Mathematics Standards-2010

[email protected] [email protected] Standards for Mathematical Practices Progression through Grade Levels 1st Grade SMP 1: Make sense of In first grade, students realize that doing mathematics problems and involves solving problems and discussing how they solved persevere in solving them. 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. SMP 2: Reason Younger students recognize that a number represents a specific quantity. They connect abstractly and the quantity to written symbols. Quantitative reasoning entails creating a quantitatively. representation of a problem while attending to the meanings of the quantities. SMP 3: Construct First graders construct arguments using concrete referents, such as viable arguments and objects, pictures, drawings, and actions. They also practice their critique the reasoning of others. mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” Explain your thinking,” and “Why is that true?” They not only explain their own thinking, but listen to others’ explanations. They decide if the explanations make sense and ask questions. SMP 4: Model with In early grades, students experiment with representing problem mathematics. situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. SMP 5: Use In first grade, students begin to consider the available tools (including estimation) appropriate tools when solving a mathematical problem and decide when certain tools might be helpful. strategically. For instance, first graders decide it might be best to use colored chips to model an addition problem. SMP 6: Attend to As young children begin to develop their mathematical communication skills, they try precision. to use clear and precise language in their discussions with others and when they explain their own reasoning. SMP 7: Look for and First graders begin to discern a number pattern or structure. For instance, if students make use of structure. recognize 12 + 3 = 15, then they also know 3 + 12 = 15. (Commutative property of addition.) To add 4 + 6 + 4, the first two numbers can be added to make a ten, so 4 + 6 + 4 = 10 + 4 = 14. SMP 8: Look for and In the early grades, students notice repetitive actions in counting and computation, etc. express regularity in When children have multiple opportunities to add and subtract “ten”‖ and multiples of repeated reasoning. “ten”‖ they notice the pattern and gain a better understanding of place value. Students continually check their work by asking themselves, “Does this make sense?”

[email protected] [email protected] Standards for Mathematical Practices Progression through Grade Levels 2nd Grade SMP 1: Make sense of In second grade, students realize that doing mathematics problems and involves solving problems and discussing how they solved persevere in solving them. 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. SMP 2: Reason Younger students recognize that a number represents a specific quantity. They abstractly and connect the quantity to written symbols. Quantitative reasoning entails creating a quantitatively. representation of a problem while attending to the meanings of the quantities. Second graders begin to know and use different properties of operations and relate addition and subtraction to length. SMP 3: Construct Second graders may construct arguments using concrete viable arguments and referents, such as objects, pictures, drawings, and actions. They critique the reasoning of others. practice their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?”, “Explain your thinking,” and “Why is that true?” They not only explain their own thinking, but listen to others’ explanations. They decide if the explanations make sense and ask appropriate questions. SMP 4: Model with In early grades, students experiment with representing problem mathematics. situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. SMP 5: Use In second grade, students consider the available tools (including estimation) when appropriate tools solving a mathematical problem and decide when certain tools might be better suited. strategically. For instance, second graders may decide to solve a problem by drawing a picture rather than writing an equation. SMP 6: Attend to As children begin to develop their mathematical communication skills, they try to use precision. clear and precise language in their discussions with others and when they explain their own reasoning. SMP 7: Look for and Second graders look for patterns. For instance, they adopt mental math strategies make use of structure. based on patterns (making ten, fact families, doubles). SMP 8: Look for and Students notice repetitive actions in counting and computation, etc. When children express regularity in have multiple opportunities to add and subtract, they look for shortcuts, such as repeated reasoning. rounding up and then adjusting the answer to compensate for the rounding. Students continually check their work by asking themselves, “Does this make sense?”

[email protected] [email protected] Using the Four Cs to Orchestrate the Common Core Classroom Matthieu Hamo & Gohar K. Hamo Links and Resources Content Standards Frameworks: http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp Standards Mappers: http://curtiscenter.math.ucla.edu/MapApp/prg_map.html http://achievethecore.org/coherence-map/?utm_source=close %20partners_claires_email&utm_medium=email&utm_campaign=Coherence%20Map SMPs: http://www.debbiewaggoner.com/math-practice-standards.html Van de Walle’s Blackline Masters: http://www.ablongman.com/vandewalleseries/  Mini ten frames: http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM17-18.pdf  Ten Frame and Double Ten Frame: http://www.ablongman.com/vandewalleseries/Vol_2_BLM_PDFs/BLM1-2.pdf Gohar’s CMC Pinterest Links: https://www.pinterest.com/gohar78/cmc-math-conference/

Classroom Activities 1. Mental Math/Number Talks a. Decomposing Numbers: Given a number of objects, students arrange into two groups. They show their decomposed numbers and compare with others at the table. b. Strategy Showcase: Given a mental math problem, students mentally solve and show one another; they communicate their solution method (strategy); they then move on to a different partner and receive a new problem.

2. Collaborative Tasks & Activities a. Place Value Mambo: Students work in groups to form a 4-digit number in multiple ways. b. Arrange the Dominoes: Given a number (5 - 9) students find the domino combinations whose sum equals that number. c. Different Ways to Show a Number: Ten-frame; Tallies; Standard Number; Base ten blocks; Picture; Number Bond.

3. Math Games a. Dice Games i. Ten-Frames Game; Roll the die and have it determine how many to put in the frame; Roll two dice and add up the numbers for a double ten-frame ii. Place Value Games: Roll 2 or 3 dice (6-sided or 10-sided); create largest number possible; play war; etc. b. Card games i. Tens Go Fish: Go Fish, but object is to pair numbers to make 10 (e.g. 2 and 8, etc.). Object is to get as many pairs as possible by “fishing.” ii. Target 20: Deal two cards to each player. Object is to get as close to twenty without going over. Students may wish to take a 3rd (or 4th or 5th card), but should not say anything if they go over. c. Finger games i. Quick Draw: played in 3s. Leader shows a number of fingers 1-9. Players A and B must show (with fingers) how many more to make 10. ii. Showdown: Two players-each player holds a number of fingers behind his/her back. At once they show their respective numbers and find the sum.

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