Georgia Standards of Excellence Curriculum Frameworks

Transcription

1 Georgia Standards of Excellence Curriculum Frameworks GSE Fourth Grade Unit 1: Whole Numbers, Place Value, and Rounding These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

2 Unit 1: WHOLE NUMBERS, PLACE VALUE, AND ROUNDING TABLE OF CONTENTS (*indicates a new addition) Overview..3 Standards for Mathematical Practice 3 Standards for Mathematical Content...4 Big Ideas...5 Essential Questions for the Unit...5 Concepts & Skills to Maintain.5 Strategies for Teaching and Learning..6 Selected Terms and Symbols...6 Tasks...7 *Intervention Table.11 Formative Assessment Lessons..12 TASKS What Comes Next?...13 Relative Value of Places.17 Number Scramble...23 Super Bowl Numbers.27 Ordering and Comparing Numbers...33 NFL Salaries..38 Nice Numbers 46 Estimation as a Check 50 Making Sense of the Algorithm.54 Reality Checking...58 Culminating Task: It s in the Numbers!...66 Classroom video available here: Standards/Pages/Implement-Math-Task-Classroom-Videos/What-Does-it-Look-Like-When-you- Implement-a-Task.aspx ***Please note that all changes made will appear in green. IF YOU HAVE NOT READ THE FOURTH GRADE CURRICULUM OVERVIEW IN ITS ENTIRETY PRIOR TO USE OF THIS UNIT, PLEASE STOP AND CLICK HERE: Standards/Frameworks/4th-Math-Grade-Level-Overview.pdf Return to the use of this unit once you ve completed reading the Curriculum Overview. Thank you. July 2016 Page 2 of 72

3 OVERVIEW Georgia Department of Education In this unit students will: read numbers correctly through the millions write numbers correctly through millions in standard form write numbers correctly through millions in expanded form identify the place value name for multi-digit whole numbers identify the place value locations for multi-digit whole numbers round multi-digit whole numbers to any place fluently solve multi-digit addition and subtraction problems using the standard algorithm solve multi-step problems using the four operations Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. The first unit should establish these routines, allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under Big Ideas be reviewed early in the planning process. A variety of resources should be utilized to supplement the tasks in this unit. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources. For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview for fourth grade. STANDARDS FOR MATHEMATICAL PRACTICE This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. This list is not exhaustive and will hopefully prompt further reflection and discussion. 1. Make sense of problems and persevere in solving them. Students make sense of problems involving place value and rounding in computation. 2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning about relative size of numbers. 3. Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding number strategies including addition and subtraction or rounding strategies. July 2016 Page 3 of 72

4 4. Model with mathematics. Students use base ten materials to demonstrate understanding of a multi-digit whole number. 5. Use appropriate tools strategically. Students select and use tools such as place value charts and base ten materials to identify patterns within the base ten system. 6. Attend to precision. Students attend to the language of real-world situations to determine if addition and subtraction answers are reasonable. 7. Look for and make use of structure. Students relate the structure of the base ten system to recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. 8. Look for and express regularity in repeated reasoning. Students relate the structure of the base ten system to explain that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. ***Mathematical Practices 1 and 6 should be evident in EVERY lesson. *** STANDARDS FOR MATHEMATICAL CONTENT Use the four operations with whole numbers to solve problems. MGSE4.OA.3 Solve multistep word problems with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a symbol or letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Generalize place value understanding for multi-digit whole numbers. MGSE4.NBT.1 Recognize that in a multi-digit whole number, a digit in any one place represents ten times what it represents in the place to its right. For example, recognize that = 10 by applying concepts of place value and division. MGSE4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. MGSE4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place. Use place value understanding and properties of operations to perform multi-digit arithmetic. MGSE4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. July 2016 Page 4 of 72

5 MGSE4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. BIG IDEAS The value of a number is determined by the place of its digits. Using rounding is an appropriate estimation strategy for solving problems and estimating. Rounded numbers are approximate and not exact. Exact answers can be rounded to different place values. A number can be written using digits in standard form, word, or expanded form. Larger numbers can be compared using the place value of the digits within the numbers. The relationship between the two numbers can be expressed using the symbols >, <, or =. ESSENTIAL QUESTIONS Choose a few questions based on the needs of your students. How does our base ten number system work? How does understanding the base-ten number system help us add and subtract? How does the value of a digit change if its location is changed in a large number? What determines the value of a digit? How does estimation help us understand large numbers? How are large numbers estimated? What conclusions can I make about the places within our base ten number system? What happens to a digit when it is multiplied and divided by 10? What effect does the location of a digit have on the value of the digit? How can we compare large numbers? What determines the value of a number? Why is it important for me to be able to compare numbers? What is a sensible answer to a real problem? What information is needed in order to round a whole number to any place? How can I ensure my answer is reasonable? How can rounding help me compute numbers? What effect does a remainder have on my rounded answer? What strategies can I use to help me make sense of a written algorithm? CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. Place value understanding for multi-digit whole numbers Round a whole number to the nearest ten or hundred Fluently add and subtract within 1000 using strategies July 2016 Page 5 of 72

6 Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number. Deep Understanding: Teachers teach more than simply how to get the answer and instead support students ability to access concepts from a number of perspectives. Therefore students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding. Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency. Fluency implies a much richer kind of mathematical knowledge and experience. Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context. Fluent students: flexibly use a combination of deep understanding, number sense, and memorization. are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them. are able to articulate their reasoning. find solutions through a number of different paths. For more about fluency, see: and: wpengine.netdna-ssl.com/wp-content/uploads/nctm-timed-tests.pdf STRATEGIES FOR TEACHING AND LEARNING Students should be actively engaged by developing their own understanding. Mathematics should be represented in as many ways as possible using graphs, tables, pictures, symbols, and words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher and peer feedback, as well as metacognition, which includes self-assessment and reflection. Students should write about the mathematical ideas and concepts they are learning. SELECTED TERMS AND SYMBOLS Note At the elementary level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. The writers of the Common Core Standards wrote a glossary of mathematical terms and it can be found at: July 2016 Page 6 of 72

7 The terms below are for teacher reference only and are not to be memorized by the students. algorithm digits estimate expanded form numbers numerals period place value rounding TASKS The following tasks represent the level of depth, rigor, and complexity expected of all fourth grade students. These tasks or tasks of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance tasks, they also may be used for teaching and learning. Scaffolding Task Constructing Task Practice Task Performance Task Culminating Task *Intervention Table Formative Assessment Lesson (FAL) CTE Classroom Tasks Tasks that build up to the learning task. Constructing understanding through deep/rich contextualized problemsolving tasks. Tasks that provide students opportunities to practice skills and concepts. Tasks, which may be a formative or summative assessment, that checks for student understanding/misunderstanding and or progress toward the standard/learning goals at different points during a unit of instruction. Designed to require students to use several concepts learned during the unit to answer a new or unique situation. Allows students to give evidence of their own understanding toward the mastery of the standard and requires them to extend their chain of mathematical reasoning. The Intervention Table provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealand s Numeracy Project. Thank you. Lessons that support teachers in formative assessment which both reveal and develop students understanding of key mathematical ideas and applications. These lessons enable teachers and students to monitor in more detail their progress towards the targets of the standards. Designed to demonstrate how the Career and Technical Education knowledge and skills can be integrated. The tasks provide teachers with realistic applications that combine mathematics and CTE content. July 2016 Page 7 of 72

8 3-Act Task A Three-Act Task is a whole-group mathematics task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three. More information along with guidelines for 3-Act Tasks may be found in the Guide to Three-Act Tasks on georgiastandards.org and the K-5 CCGPS Mathematics Wiki. July 2016 Page 8 of 72

9 Task Name Task Type/Grouping Strategy Georgia Department of Education Content Addressed Standard(s) Description of Task What Comes Next? Scaffolding Task Partner/Small Group Task Relative size of numbers MGSE4.NBT. 1 Students work with base ten materials to experience that a place value in a number is ten times more than the digit to its right. Relative Value of Places Constructing Task Partner/ Small Group Task Relative size of numbers MGSE4.NBT.2 MGSE4.NBT. 1 Students work with dotty array pieces to understand patterns within the base-ten number system. Students solve place value problems to show understanding of the patterns learned. Number Scramble Practice Task Individual/Partner Task Making and Naming Large Numbers MGSE4.NBT.2 Students create numbers given specific directions and write those numbers in standard, word and expanded form. Super Bowl Numbers 3 Act Task Individual/Partner Task Comparing Multi-digit Numbers, Adding Multidigit Numbers MGSE4.NBT.2 MGSE4.NBT.4 Students make connections between the base-ten number system and the Roman Numeral number system. Ordering and Comparing Numbers Practice Task Individual/Partner Task Ordering Larger Numbers MGSE4.NBT.2 Students order and compare numbers through a dice game played with a partner. July 2016 Page 9 of 72

10 NFL Salaries Nice Numbers 3 Act Task Individual/Partner Task Constructing Task Partner/Small group Task Georgia Department of Education Comparing Multi-digit Numbers, Adding Multidigit Numbers Rounding, Four Operations MGSE4.OA.3 MGSE4.NBT.4 MGSE4.OA.3 MGSE4.NBT.4 MGSE4.NBT.3 MGSE4.MD.2 Students compare salaries of football players to discuss why certain players are paid a ti l t f Students apply rounding concepts to find estimated solutions to word problems. Estimation as a Check Constructing Task Individual/ Partner Task Rounding, Adding, Subtracting multi-digit numbers MGSE4.NBT.4 MGSE4.NBT.3 Students find estimated solutions to problems. Making Sense of the Algorithm Constructing Task Individual/Partner Task Fluently subtracting multi-digit numbers MGSE4.NBT.4 Students write about strategies that are used for given problems in the task, leading to a discussion of procedures within the subtraction standard algorithm. Reality Checking Constructing Task Individual/ Partner Task Ordering, Adding, Subtracting and Rounding multi-digit numbers MGSE4.NBT.2 MGSE4.NBT.4 MGSE4.NBT.3 MGSE4.MD.2 Students apply knowledge of the addition and subtraction standard algorithm in order to balance a mock check registry. It s in the Numbers Culminating Task Individual Task Calculation and Estimation with Whole Numbers MGSE4.OA.3 MGSE4.NBT.2 MGSE4.NBT.3 MGSE4.MD.2 Students gather data about populations in the U.S. to draw conclusions about why people choose to live in certain regions of the country. Should you need further support for this unit, please view the appropriate unit webinar at : July 2016 Page 10 of 72

11 *INTERVENTION TABLE The Intervention Table below provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealand s Numeracy Project. Cluster of Standards Name of Intervention Snapshot of summary or Student I can statement... Materials Master Generalize place value understanding for multi-digit whole numbers. MGSE4.NBT.1 MGSE4.NBT.2 MGSE4.NBT.3 Use place value understanding and properties of operations to perform multi-digit arithmetic. MGSE4.NBT.4 Arrow Cards Number Hangman Place Value Houses Rocket-Where Will I Fit? When One Number is Near a Hundred Reversing Addition Checking Addition and Subtraction by Estimation Mental or Written? A Standard Written Form for Addition Identify all of the numbers in the range 0-1,000,000. Identify all of the numbers in the range 0-1,000,000. Identify all of the numbers in the range 0-1,000,000. Order whole numbers in the range 0-1,000,000. Solve addition and subtraction problems by compensating with tidy numbers. Solve subtraction problems by using addition. Solve addition and subtraction problems by using place value. Solve addition and subtraction problems using decomposition, leading to a written algorithm. Solve addition and subtraction problems using decomposition, leading to a written algorithm. MM 4-14 MM 4-1 MM 4-11 MM 8-1 *Only use problems with a focus on whole number computation MM 5-13 July 2016 Page 11 of 72

12 FORMATIVE ASSESSMENT LESSONS (FALS) Formative Assessment Lessons are designed for teachers to use in order to target specific strengths and weaknesses in their students mathematical thinking in different areas. A Formative Assessment Lesson (FAL) includes a short task that is designed to target mathematical areas specific to a range of tasks from the unit. Teachers should give the task in advance of the delineated tasks and the teacher should use the information from the assessment task to differentiate the material to fit the needs of the students. The initial task should not be graded. It is to be used to guide instruction. Teachers may use the following Formative Assessment Lessons (FALS) Chart to help them determine the areas of strengths and weaknesses of their students in particular areas within the unit. Formative Assessments FALS (Supporting Lesson Included) Content Addressed Pacing (Use before and after these tasks) Ordering 4-digit Numbers Ordering Larger Numbers What Comes Next Build 1,000 Relative Value of Places Number Scramble Ticket Master To Regroup or Not to Regroup Using the Addition Algorithm Nice Numbers Estimation as a Check Making sense of an Algorithm Reality Checking July 2016 Page 12 of 72

13 SCAFFOLDING TASK: What Comes Next? (Adapted from Teaching Student Centered Mathematics Volume 2) Back to Task Table TASK CONTENT: This task helps pre-assess students' previous knowledge and misconceptions about place value and number sense. STANDARDS FOR MATHEMATICAL CONTENT MGSE4.NBT.1 Recognize that in a multi-digit whole number, a digit in any one place represents ten times what it represents in the place to its right. For example, recognize that = 10 by applying concepts of place value and division. STANDARDS FOR MATHEMATICAL PRACTICE 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. BACKGROUND KNOWLEDGE This task helps pre-assess students' previous knowledge and misconceptions about place value and number sense. The strategies they use to solve the problem demonstrate students' understanding about a number of concepts including place value, grouping, computation, number sense, patterns, and mathematical communication. COMMON MISCONCEPTIONS Two important ideas developed for three-digit numbers should be extended to larger numbers. First, the grouping idea should be generalized. That is, ten in any position makes a single thing (group) in the next position, and vice versa. Second, the oral and written patterns for numbers in three digits are duplicated in a clever way for every three digits to the left. These two related ideas are not as easy for students to understand as adults seem to believe. Because models for large numbers are often difficult to demonstrate or visualize, textbooks frequently deal with these ideas in a predominantly symbolic manner. That is not sufficient! (Van de Walle, page 47-48) ESSENTIAL QUESTIONS How does our base ten number system work? How does understanding the base-ten number system help us add and subtract? July 2016 Page 13 of 72

14 MATERIALS Georgia Department of Education base-ten blocks base ten materials grid #14 from wps.ablongman.com: f GROUPING Students work in groups or pairs. NUMBER TALKS In the Fourth grade Grade Level Overview, the importance of giving students opportunities to mentally compute and explain computational strategies is discussed. Number Talks is an excellent way to do this. 5 to 15 minutes each day is dedicated to students sharing the ownership of determining whether answers are accurate, and given the expectation of thinking through all solutions and strategies carefully (Parrish, 2010). During the Number Talk, the teacher is not the definitive authority. The teacher is the facilitator and is listening and learning for and from the students natural mathematical thinking. The teacher gives a problem on the board in whole group or small group setting. The students mentally solve the problem and share with the whole group HOW they arrived at their answer. They must justify and defend their answer and their thinking. The teacher simply records the students thinking and poses extended questions to draw out deeper understanding for all. To let the teacher know they are ready, they make a fist and place it on their chest. If they have two strategies to share, they place two fingers on their chest and so on. Some lessons lend themselves to Number Talks better than others. You may modify/change them based on the needs of your students. Number Talks suggestions are included in tasks where appropriate. TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION Throughout the year, you will be incorporating Number Talks into your daily mathematical discussions. Number Talks is based on the premise of mental computations which means students will rely heavily on place value knowledge and understanding. The following task will cause the students to focus on number relationships and use these relationships to develop efficient, flexible strategies with accuracy (Parrish, page 13). It is important for students to realize that the number system does have a logical structure, is not totally arbitrary, and can be understood (Van de Walle, page 166). Students will be able to compose or make large numbers up to 1000 using base 10. This task will aid the students in their conceptual understanding of composing numbers which will later lead to them decomposing numbers when doing expanded notation and calculating mentally during the computation of addition, subtraction, multiplication, and division. This lesson by Van de Walle is full of rigor for the student and teacher and worth the conceptual understanding of a number. July 2016 Page 14 of 72

15 Students will use paper models of base-ten strips and squares to complete this activity. (See the attached website in the Materials section of the lesson to download the Blackline Master.) After passing out the materials, ask students, What piece of the base ten materials can be used to represent one? (The unit or ones piece is a one cm square and can be used to represent one.) Then ask students, What is next? Discuss with students that ten centimeter squares can be grouped to make the tens piece. The tens piece is a 10 cm by 1 cm strip. Ask students again, What is next? Discuss with students that ten strips that are 10 cm by 1 cm can be grouped together to form the hundreds piece. The hundreds piece is a 10 cm by 10 cm square. Again, pose the question, What is next? Allow students to turn and talk with a partner about what can be made next and how it will be made. (Placing ten 10 cm by 10 cm squares together is ten hundreds, or a thousand.) Ask students, What shape would a thousand be? Tape together a long strip made of ten paper hundreds squares. Continue the discussion by asking students, What comes next? Reinforce the idea of ten making one that has progressed to this point. Ten thousand strips would make a square measuring one meter on each side making a paper 10,000 model. As the class figures out the shape of each piece, continue to pose the question, What comes next? Let small groups work on the dimensions on a 100,000 piece. Ten ten-thousand squares (100,000) go together to make a huge strip. Draw this strip on a long sheet or roll of paper, and mark off the 10 squares that make it. It may be necessary to go out in the hall. FORMATIVE ASSESSMENT QUESTIONS What s a consistent pattern seen as the place values are being built? What is the meaning of the phrase base ten as a digit progresses from right to left in a larger number? Could there be another base besides 10? Justify or defend your thinking. DIFFERENTIATION Extension Collections- As a class or grade-level project, collect some type of object with the objective of reaching some specific quantity-for example, 1,000 or 10,000 buttons, walnuts, old pencils, jar lids, pieces of junk mail, soup labels, or cereal box tops. Illustrations- Sometimes it is easier to create large amounts. For example, start a project where students draw 100, 200 or even 500 dots on a sheet of paper. Allow students to use rulers to create the squares and rectangles in the activity. Students can draw the squares and rectangles on a large piece of chart or bulletin board paper. July 2016 Page 15 of 72

16 Intervention Using the base ten blocks, the students could build a number and have a peer determine what number was created and orally give the value of each place contingent upon the blocks. Intervention Table TECHNOLOGY Use this resource as a follow-up lesson to extend place value understanding. A lesson for your ActivSlate or SmartBoard to reinforce basic place value ideas through 100, This resource allows students to manipulate base ten blocks virtually. July 2016 Page 16 of 72

17 Constructing Task: Relative Value of Places Back to Task Table Adapted from Relative Value of Places, nzmaths, Adding, Subtraction and Place Value TASK CONTENT: In this task, students will explore patterns in the base-ten system. STANDARDS FOR MATHEMATICAL CONTENT MGSE4.NBT.1 Recognize that in a multi-digit whole number, a digit in any one place represents ten times what it represents in the place to its right. For example, recognize that = 10 by applying concepts of place value and division. MGSE4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. STANDARDS FOR MATHEMATICAL PRACTICE TO BE EMPHASIZED 2. Reason abstractly and quantitatively. 4. Model with mathematics 6. Attend to precision. 7. Look for and make use of structure. BACKGROUND KNOWLEDGE Students unfamiliar with dotty arrays will need to become familiar with the representation of 1, 10, 100, 1 000, and so on as arrays of single dots. This will help them to recognize the relative value of the places. July 2016 Page 17 of 72

18 ESSENTIAL QUESTIONS What conclusions can I make about the places within our base ten number system? What happens to a digit when multiplied and divided by 10? What effect does the location of a digit have on the value of the digit? MATERIALS Large dot arrays GROUPING Partner or small group NUMBER TALKS In her book Number Talks, Sherry Parrish discusses the importance of place value. She states,...the true test of whether students understand place value is if they can apply their understanding in computation. This lesson helps students understand what happens to numbers when they are multiplied or divided by 10. Students can begin using this skill to write numbers in expanded form and begin applying those skills to multiplying partial products. TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION Ask ten students to make a two-digit number, e.g., 37, using the dotty array pieces. Pose this problem: Imagine there are ten students and they each have 37 marbles/apples/dollars. Put the sets of 37 into a central space. Ask students, How many dots is that altogether? Some students are likely to have symbolic algorithms, such as add a zero, that enable them to get an answer of 370. Examine the actions on materials that explain the use of zero as a placeholder. For example: July 2016 Page 18 of 72

19 Using a place value chart, connect 37 with the result of 10 x 37: Ten Thousands Thousands Hundreds Tens Ones Students should notice that the digits have shifted one place to the left. Discuss with students and ask students, Would the digits shift to the left again if we tried another problem? Pose several other problems where ten students make numbers with dot array parts and look at the combined product. For each example, separate the place values to see what contribution they make to the whole product, and write the number and it s ten times equivalent on the place value chart. Further challenge the students by making a two-digit number and posing problems such as, Imagine that one hundred students had 42 marbles/apples/dollars each. How many would that be in total? Ask students how this might be modeled. In these cases, each of the ten students will need to create each number ten times. This is a useful generalization that shows that ten times ten times of any number is one hundred times that number. Transfer the focus to dividing by ten and by one hundred. Begin with a four-digit number like 3,800 (zero in the tens and ones places). Make this number with dot array pieces. Pose this problem: I have 3,800 marbles and I am going to share them equally among all ten of you. How many marbles will you get each? Ask the students to predict the result of the sharing, and then confirm it by modeling with the materials. July 2016 Page 19 of 72

20 The result of dividing 3,800 by ten can be shown on a place value chart as: Ten Thousands Thousands Hundreds Tens Ones The symbolic effect of dividing by ten is to shift the digits of the dividend (3,800) one place to the right. Ask the students to predict what the result would be if they shared 3,800 into one hundred equal sets. Expect them to realize that the shares would be one-tenth of 380, which is 38. This may need to be acted out by cutting the 3 hundreds in 30 tens and the 8 tens into 80 ones so the tenth shares can be established. Use the place chart to connect 3,800 and the result of 3, = 38. In this case, the symbolic effect is a two-place shift to the right. Pose problems like these below, expecting the students to reason the answers using place value understanding, not through the use of multiplication. The students must be able to justify their answers by explaining what occurs with the quantities involved boxes of 376 coins (37,600) boxes of 376 coins (3,760) skittles shared among 10 people (96) skittles shared among 1 people (960) sets of 40 pencils (400) sets of 40 pencils (4,000) sets of 40 pencils (40,000) 8. 4,300 movie tickets shared among 100 people (43) 9. 4,300 movie tickets shared among 10 people (430) 10. 1,000 sets of 56 marbles (56,000) ,000 sets of 56 marbles (560,000) FORMATIVE ASSESSMENT QUESTIONS What happens to the value of the digit in the ones place when the number is multiplied by 10? What happens to the value of the digit in the tens place when the number is divided by 10? What can be concluded about the value of a digit in the ones place compared to the value of that same digit in the tens place? What about the tens place and hundreds place? What about the hundreds place and thousands place? Which number is larger 960 or 96? How do you know? Why? DIFFERENTIATION Extension Have students explain what the value of a digit to the right of the ones place would be based on their conclusion about whole numbers. July 2016 Page 20 of 72

21 Georgia Department of Education Have students record the number sentences associated with each problem. Intervention Allow students to use base ten blocks to build numbers. Intervention Table TECHNOLOGY This serves as a means for students to practice place value understanding. Thousands and Millions of Fourth Graders is a task from the Illustrative Mathematics website that provides students with an opportunity to apply knowledge learned about the relationship between digits in a number. July 2016 Page 21 of 72

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23 Practice Task: Number Scramble Back to Task Table TASK CONTENT: In this task, students will manipulate the tens digit of the base ten-numeration system to complete various activities such as constructing large and small numbers and numbers with specific values in a given place. Students will also write numbers in expanded and standard form. STANDARDS FOR MATHEMATICAL CONTENT MGSE4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. STANDARDS FOR MATHEMATICAL PRACTICE TO BE EMPHASIZED 2. Reason abstractly and quantitatively. 4. Model with mathematics. 6. Attend to precision. BACKGROUND KNOWLEDGE Students should have prior experiences and/or instruction with ordering, writing numbers in expanded and standard form and comparing large numbers. ESSENTIAL QUESTIONS How do digit values change as they are moved around in large numbers? What determines the value of a digit? MATERIALS Scissors Number Scramble Recording Sheet Blank Place Value Chart GROUPING Individual/Partner Task NUMBER TALKS Now that students have explored what happens to a number when it is multiplied or divided by 10, they should be ready to attempt some mental calculations using partial products. For example, students could calculate 4 x 16 by first decomposing 16 into , then solving 4 x 10, and then 4 x 6. Finally they could combine the products of 40 and 24 for a final product of 64. Keep in mind that students may arrive at the product of 64 without using this particular strategy. July 2016 Page 23 of 72

24 Please refer to pgs in Number Talks by Sherry Parrish for more examples of number talks that will further develop this strategy. TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION Comments As students manipulate the numbers in this task, you will be able to see quickly which students have a good grasp of place value and the value of digit in a number. For example, in steps 1 and 2, if students randomly place their digits, they may need more practice to understand how the value of a number changes as its digits change. Task Directions Students will cut out number boxes (tiles) and use them to create numbers with the given requirements. 1. Make the largest whole number possible using 9 different tiles. Write your answer in standard form and expanded form. 2. Make the smallest whole number possible using 9 different tiles. Write your answer in standard form and expanded form. 3. Make a number worth more than two million, with a six in the ten-thousands place. Write the number in standard form and expanded form. Compare your number with your partner. 4. Make a number less than five million that has a two in the thousands place. Write the number in standard form and expanded form. Compare your number with your partner. 5. Make a number that has only odd digits in the thousands period of the place value chart. Write the number in words. 6. Look carefully at your answers to Questions 1 and 2. Find one digit that is in both of your answers. How does the value of this digit change from the way you used it in Question 1 to the way you used it in Question 2? Use complete sentences to explain how and why the value of the digit did or did not change between the two answers. FORMATIVE ASSESSMENT QUESTIONS Explain how you decided the order of the digits for each number you created. How can you tell which number is the largest or smallest? How does the value of a digit change when it is moved to the left on the place value chart? To the right? DIFFERENTIATION Extension Have students use all ten tiles to answer questions. Have students use tiles to create two more additional questions. Students will give their problems to a partner to solve using the tiles. Intervention Start students with building numbers in the hundreds, then the thousands, etc. July 2016 Page 24 of 72

25 Georgia Department of Education Allow students to use a blank place value chart and write the numbers in the chart, showing the correct placement of the digits. This cueing device may assist students in comparing digits in the same place in order to determine value. Have students to use base ten blocks to model the numbers made with the tiles. Intervention Table TECHNOLOGY CONNECTIONS This interactive flipchart lesson on comparing whole numbers, can be used n used with your ActivSlate or Smartboard. It can be used in whole group, small group or student led centers. This online activity has students comparing large numbers and determine their placement in a series of numbers. It can serve as a center activity for practice. July 2016 Page 25 of 72

26 Number Scramble Task Directions: Cut out the number boxes at the bottom of the page by cutting on the black lines. Use the digits to complete each task. 1.Make the largest whole number possible using 9 different tiles. Write your answer below. Standard Form Word Form Expanded Form 2. Make the smallest whole number possible using 9 different tiles. Write your answer below. Standard Form Word Form Expanded Form 3. Make a number larger than two million with a six in the ten-thousands place. What is your number? Expanded Form 4. Make a number smaller than five million with a two in the thousands place. What is your number? What is your partner s number? Expanded Form 5. Look carefully at your answers to Questions 1 and 2. Find one digit that is in BOTH of your answers. Write it here. How does the value of this digit change from the way you used it in Question 1 to the way you used it in Question 2? On the back of this sheet, use complete sentences to explain how and why the value of the digit did or did not change between the two answers. July 2016 Page 26 of 72

27 3 ACT TASK: Super Bowl Numbers Back to Task Table Adapted from TASK CONTENT: Comparing multi-digit whole numbers with the possibility of adding multidigit whole numbers. Approximate time, 1 class period. STANDARDS FOR MATHEMATICAL CONTENT MGSE4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. MGSE4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. STANDARDS FOR MATHEMATICAL PRACTICE 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 6. Attend to precision. 7. Look for and make use of structure. BACKGROUND KNOWLEDGE This task follows the 3-Act Math Task format originally developed by Dan Meyer. More information on this type of task may be found at A Three- Act Task is a whole-group mathematics task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three. More information along with guidelines for 3-Act Tasks may be found in the Guide to Three-Act Tasks on georgiastandards.org and the K-5 CCGPS Mathematics Wiki. Students should have a thorough knowledge by this time of how to compare and order whole numbers. Students must be able to articulate how they know the sizes of digits in a given number and how to equate any number with its word form and/or expanded form. This task provides an opportunity for students to connect numerals from other systems to our base ten system. Students should have flexibility in thinking about numbers and how they relate to the base ten system. ESSENTIAL QUESTIONS What kinds of things are large numbers used to measure? How can we tell which multi-digit number is the largest or smallest? July 2016 Page 27 of 72

28 MATERIALS Georgia Department of Education Super Bowl Numbers Student Recording Sheet Act 1 video Roman Numeral information (Act 2) Act 3 video GROUPING Individual/Partner Task TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION Comments In this task, students will watch a video and then tell what they noticed. Next, they will be asked to discuss what they wonder about or are curious about. Their curiosities will be recorded as questions on a class chart or on the board. Students will then use mathematics to answer their own questions. Students will be given information to solve the problem based on need. When they realize they don t have the information they need, and ask for it, it will be given to them. Task Directions Act I Whole Group - Pose the conflict and introduce students to the scenario by showing Act I video. 1. Students are shown the video: Students may need to view the video multiple times. If Windows is being used as the browser to open the video, there may be trouble when the video tries to open. Open the browser Chrome and copy and paste the aforementioned link in the address bar and click Enter to access the video. 2. Pass out the 3 Act recording sheet. 3. Ask students what they wonder about and what questions they have about what they saw. Students should share with each other first before sharing aloud and then record these questions on the recording sheet (think-pair-share). The teacher may need to guide students so that the questions generated are math-related. Anticipated questions: What was the order of Super Bowls in the video? What was the number of the 2014 Super Bowl? What will be the number of the 2015 Super Bowl? When did these Super Bowls happen? How is the Roman Numeral system like our place value system? How are the two different? 4. As the facilitator, you can select which question you would like every student to answer, have students vote on which question the class will answer or allow the students to pick July 2016 Page 28 of 72

29 which question they would like to answer. Once the question is selected ask students to estimate answers to their questions (think-pair-share). Students will write their best estimate. Act II Student Exploration - Provide additional information as students work toward solutions to their questions. 1. Ask students to determine what additional information they will need to solve their questions. The teacher provides that information only when students ask for it. Basic Roman Numerals I- 1 V- 5 X- 10 L- 50 Roman Numerals are written in a descending order of quantity. When a smaller symbol is written to the left of a larger symbol, you must subtract the lesser value from the greater value to determine the actual amount. XLVIII was the number for 2014 s Super Bowl C Ask students to work in small groups to answer the question(s) they created in Act I. The teacher provides guidance as needed during this phase by asking questions such as: Can you explain what you ve done so far? What strategies are you using? What assumptions are you making? What tools or models may help you? Why is that true? Does that make sense? Act III Whole Group - Share student solutions and strategies as well as Act III solution. 1. Ask students to present their solutions and strategies. 2. Share solution in Act III solution. If Windows is being used as the browser to open the video, there may be trouble when the video tries to open. Open the browser Chrome and copy and paste the aforementioned link in the address bar and click Enter to access the video. 3. Lead discussion to compare these, asking questions such as: How reasonable was your estimate? Which strategy was most efficient? Can you think of another method that might have worked? What might you do differently next time? July 2016 Page 29 of 72

30 Comments Act IV is an extension question or situation of the above problem. An Act IV can be implemented with students who demonstrate understanding of the concepts covered in acts II and III. The following questions and/or situations can be used as an Act IV: When you are 20 years old, what will the Roman numerals be for the Super Bowl will you watch? FORMATIVE ASSESSMENT QUESTIONS What models did you create? What organizational strategies did you use? DIFFERENTIATION Extension Work more with Roman Numerals: Roman numerals are used for numbering the Olympics, numbering the Super Bowls, writing outlines and often for copyright dates. Usually the pages before Chapter 1 in a book are written in lowercase Roman numerals. (i, ii, iii, iv, etc.) a. In 2012 the summer Olympics was in London. What was the Roman numeral representation for this event? b. In 2016 the summer Olympics will be in Rio de Janiero. What is this year in Roman numerals? c. A copy of Catcher in the Rye has the copyright date of MCMLI. When was the book written? d. The movie King Kong was made in MCMXXXIII. When was that? Intervention Intervention Table TECHNOLOGY This interactive flipchart lesson on comparing whole numbers, can be used with your ActivSlate or Smartboard. It can be used in whole group, small group or student led centers. This online activity has students comparing large numbers and determine their placement in a series of numbers. It can serve as a center activity for practice. July 2016 Page 30 of 72

31 Name: Adapted from Andrew Stadel Task Title: ACT 1 What questions come to your mind? Main question: Write an estimate for the question selected in Act 1. Explain your estimates. ACT 2 What information would you like to know or need to solve the MAIN question? Use this area for your work, tables, calculations, sketches, and final solution. July 2016 Page 31 of 72

32 ACT 3 Georgia Department of Education What was the result? ACT 4 (use this space when necessary) July 2016 Page 32 of 72

33 Practice Task: Ordering and Comparing Numbers Back to Task Table Classroom video available here: Standards/Pages/Implement-Math-Task-Classroom-Videos/What-Does-it-Look-Like-When-you- Implement-a-Task.aspx TASK CONTENT: In this task, students will create and order numbers with up to six digits using dice or spinners. Students will order and compare the numbers they have created. STANDARDS FOR MATHEMATICAL CONTENT MGSE4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. STANDARDS FOR MATHEMATICAL PRACTICE TO BE EMPHASIZED 2. Reason abstractly and quantitatively. 4. Model with mathematics. 6. Attend to precision. BACKGROUND KNOWLEDGE Students should have prior experiences and/or instruction with ordering large numbers. This activity may be used as an assessment or as an independent follow-up activity for reinforcement or review. ESSENTIAL QUESTIONS How are large numbers compared? What determines the value of a digit in a number? MATERIALS For Line Them Up: 0-9 die or spinner Line Them Up Recording Sheet For Dare to Compare: 1-6 die or spinner Deck of cards (face cards removed) Dare to Compare recording sheet GROUPING Partner Task July 2016 Page 33 of 72