Grade 4 Mathematics Scope and Sequence Standards Trajectory Content Area Grade Level
Grade 4 Mathematics Scope and Sequence Standards Trajectory Content Area Mathematics Grade Level 4 Grade 4 Common Core State Standards Domain Cluster/Essential Learning Goal ELG.MA.4.OA.A: Use the four operations with whole numbers to solve problems. (Major) Operations and Algebraic Thinking (4.OA) ELG.MA.4.OA.B: Gain familiarity with factors and multiples. (Supporting) ELG.MA.4.OA.C: Generate and analyze patterns. (Additional) Number and Operations in Base Ten (4.NBT) ELG.MA.4.NBT.A: Generalize place value understanding for multidigit whole numbers. (Major) ELG.MA.4.NBT.B: Use place value understanding and properties of operations to perform multidigit arithmetic. (Major) ELG.MA.4.NF.A: Extend understanding of fraction equivalence and ordering. (Major) Number and Operations — Fractions (4.NF) ELG.MA.4.NF.B: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (Major) ELG.MA.4.NF.C: Understand decimal notation for fractions and compare decimal fractions. (Major) ELG.MA.4.MD.A: Solve problems involving measuring and converting measurements from larger to smaller units. (Supporting) Measurement and Data (4.MD) ELG.MA.4.MD.B: Represent and interpret data. (Supporting) ELG.MA.4.MD.C: Geometric measurement: Understand concept of angle and measure angles. (Additional) Geometry (4.G) ELG.MA.4.G.A: Draw and identify lines and angles and classify shapes by properties of their lines and angles. (Additional) Major clusters require greater emphasis based on depth of ideas, time they take to master, and their importance to future mathematics. An intense focus on these clusters allows in-depth learning carried out through the Standards for Mathematical Practice. Supporting clusters are closely connected to the major clusters and strengthen areas of major emphasis. Additional clusters may not tightly or explicitly connect to the major work of the grade. All standards should be taught as all will be assessed. Denver Public Schools 2014–2015 1 Grade 4 Mathematics Scope and Sequence Standards Trajectory Yearlong Focus Essential Learning Goals Number and Operations in Base Ten (4.NBT) ELG.MA.4.NBT.B: Use place value understanding and properties of operations to perform multidigit arithmetic. (Major) Colorado 21st Century Skills Mathematical Practices Critical Thinking and Reasoning: Thinking Deeply, Thinking Differently Invention Information Literacy: Untangling the Web Collaboration: Working Together, Learning Together Self-Direction: Own Your Learning Invention: Creating Solutions 1. 2. 3. 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique others’ reasoning. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Unit of Study Length of Unit Time Frame 1: Place Value and Addition and Subtraction of Multidigit Numbers 22 days August 25–September 24, 2014 2: Multiplication and Division 20 days September 25–October 24, 2014 3: Extended Multiplication 18 days October 27–November 21, 2014 4: Division 15 days December 1–19, 2014 5: Fractions 23 days January 6–February 6, 2015 6: Decimals 18 days February 9–March 6, 2015 7: Fractions and Decimals 10 days March 9–20, 2015 8: Perimeter and Area 12 days March 23–April 17, 2015 9: Geometric Figures, Measuring Angles, and Symmetry 15 days April 20–May 8, 2015 10: Weight, Volume, and Capacity 18 days May 11–June 5, 2015 End-of-Year Fluency Expectation Fluently add and subtract multidigit whole numbers using standard algorithm. (4.OA.4) Denver Public Schools 2014–2015 2 Grade 4, Unit 1: Place Value and Addition and Subtraction of Multidigit Numbers Unit of Study 1: Place Value and Addition and Subtraction of Multidigit Numbers Focusing Lenses Comparison and Structure Focus Essential Learning Goals Length of Unit 22 days (August 25–September 24, 2014) Generalize place value understanding for multidigit whole numbers. (ELG.MA.4.NBT.A) Use place value understanding and properties of operations to perform multidigit arithmetic. (ELG.MA.4.NBT.B) Content Standards Standards Number and Operations in Base Ten (4.NBT) Generalize place value understanding for multidigit whole numbers. (Major) [ELG.MA.4.NBT.A] 4.NBT.1: Recognize that in multidigit whole numbers, a digit in one place represent ten times what it represents in the place to its right (for example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division). 4.NBT.2: Read and write multidigit whole numbers using base-ten numerals, number names, and expanded form. Compare two multidigit numbers based on the meaning of the digit in each place using >, =, and < symbols to record comparison results. 4.NBT.3: Use place value understanding to round multidigit whole numbers to any place. Use place value understanding and properties of operations to perform multidigit arithmetic. (Major) [ELG.MA.4.NBT.B] 4.NBT.4: Fluently add and subtract multidigit whole numbers using standard algorithm. Standards for Mathematical Practice 1. 3. 4. 7. Make sense of problems and persevere in solving them. Construct viable arguments and critique others’ reasoning. Model with mathematics. Look for and make use of structure. Fluency Recommendation Fluently add and subtract multidigit whole numbers using standard algorithm. (4.OA.4) Inquiry Question What would change if we used a base-eight number system? What would our numbers look like? Concepts Addition, subtraction, place value, equivalency Denver Public Schools 2014–2015 3 Grade 4, Unit 1: Place Value and Addition and Subtraction of Multidigit Numbers Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual In multidigit whole numbers, a digit in one place represents ten times what it represents in the place to its right. (4.NBT.1) What determines digits’ values? What is the role of place value in comparing two multidigit numbers? How do digits’ values change when you change place value? The concept of place value allows us to write and describe numbers in a variety of equivalent forms. (4.NBT.2) How can we write numbers in expanded form? How can we use base-ten blocks to represent numbers? Why is the digit zero important in our number system? Increases to the number of digits in whole numbers always result in increases to the magnitude of the numbers. (4.NBT.2) How can we compare sizes of two multidigit numbers? How do we know for certain when multidigit numbers are larger, smaller, or equal to other multidigit numbers? Why do we compare whole numbers with the same numbers of digits starting on the left rather than the right of the numbers? Accurately rounding multidigit numbers depends on knowledge of place value and requires attention to context. (4.NBT.3) How are numbers rounded? How does context help us decide which number (which place value in multidigit numbers) to round? Why does the number five round up rather than down? The standard algorithm for adding and subtracting provides an efficient method to develop fluency with adding and subtracting multidigit numbers. (4.NBT.4) How does understanding place value support the standard algorithm for adding and subtracting? Why is fluency with multidigit addition and subtraction important? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Recognize that in multidigit whole numbers, a digit in one place represents ten times what it represents in the place to its right. (4.NBT.1) Read and write multidigit whole numbers using base-ten numerals, number names, and expanded form. (4.NBT.1) Round multidigit numbers depending on place value and context. (4.NBT.3) Fluently add and subtract multidigit whole numbers using the standard algorithm. (4.NBT.4) Denver Public Schools 2014–2015 4 Grade 4, Unit 1: Place Value and Addition and Subtraction of Multidigit Numbers WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. Multidigit numbers can be written in expanded form. The symbols <, >, = are used to show the comparison of multidigit numbers. Numbers can be represented with base-ten numerals, expanded form, or number names. Cross-Content Academic Words Technical Words Specific to Content Compare, efficient, equal to, generate, greater than, identify, larger, less than, pattern, precision, recognize, represent, smaller, table Addition, digit, equivalent form, equivalent name, expanded form, magnitude, multidigit, number name, place value, standard algorithm, subtraction, whole number Resources Core Lessons Everyday Mathematics Lessons 2*1–2*4 Lesson 2*4a: Rounding and Friendly Numbers (new lesson after Lesson 2*4) Lesson 2*7 Lesson 2*9 Unit Assessment Suggested Performance/ Learning Task To Regroup or Not to Regroup (end of unit) Technology Function Machine (explore concept of functions by putting values into the machine and observing its output) Literacy Connection 12 Ways to Get to 11 by Eve Merriam Students tend to have several misconceptions about writing numerals from verbal descriptions. Numbers, such as one thousand, do not cause a problem; however, numbers, such as one thousand two, cause problems. Many students understand the 1,000 and the 2 but rather than placing the 2 in the ones place, students write numbers as they hear them, or 10,002 (ten thousand two). Use multiple strategies, such as place value boxes, to help with this concept. Students assume the first digit of multidigit numbers indicates the “greatness” of numbers. They assume that 954 is greater than 1,002 because they focus on the first digit rather than the numbers as a whole. Misconceptions Essentials for Standards Implementation Denver Public Schools 4.NBT.3 refers to place value understanding, which extends beyond an algorithm or procedure for rounding. The expectation is that students have a deep understanding of place value and number sense and can explain and reason about answers they get when they round. Students should have numerous experiences using a number line and a hundreds chart as tools to support their work with rounding. 2014–2015 5 Grade 4, Unit 2: Multiplication and Division Unit of Study 2: Multiplication and Division Focusing Lenses Interpretation and Comparison Focus Essential Learning Goals Length of Unit 20 days (September 25–October 24, 2014) Use the four operations with whole numbers to solve problems. (ELG.MA.4.OA.A) Gain familiarity with factors and multiples. (ELG.MA.4.OA.B) Content Standards Standards Operations and Algebraic Thinking (4.OA) Use the four operations with whole numbers to solve problems. (Major) [ELG.MA.4.OA.A] 4.OA.1: Interpret multiplication equations as comparisons (for example, interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5). Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.2: Multiply or divide to solve word problems involving multiplicative comparison using drawings and equations with symbols for unknown numbers to represent problems, distinguishing multiplicative comparison from additive comparison. 4.OA.3: Solve multistep word problems posed 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 letters standing for unknown quantities. Assess reasonableness of answers using mental computation and estimation strategies including rounding. Gain familiarity with factors and multiples. (Supporting) [ELG.MA.4.OA.B] 4.OA.4: Find all factor pairs for whole numbers in the range 1–100. Recognize that whole numbers are multiples of each of their factors. Determine whether given whole numbers in the range 1–100 are multiples of given one-digit numbers. Determine whether given whole numbers in the range 1–100 are prime or composite. Generate and analyze patterns (Additional) [ELG.MA.4.OA.C] 4.OA.5: Generate number or shape patterns that follow given rules. Identify apparent pattern features not explicit in the rules themselves (for example, given the rule “add three” and the starting number one, generate terms in the resulting sequence and observe that terms appear to alternate between odd and even numbers, then explain informally why numbers will continue to alternate this way). Standards for Mathematical Practice 1. 2. 3. 4. 7. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique others’ reasoning. Model with mathematics. Look for and make use of structure. Fluency Recommendations N/A Inquiry Question Concepts Factoring, multiplication, division Denver Public Schools Why is one neither prime nor composite? 2014–2015 6 Grade 4, Unit 2: Multiplication and Division Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual What is a multiple? What is a factor? How can we use factors and multiples to determine whether numbers are prime or composite? Which characteristics can be used to classify numbers into different groups? Knowledge of factors and multiples allows us to classify all natural whole numbers greater than one as either prime or composite. (4.OA.4) Word problems typically contain unknown quantities (which can be represented by letters when solving) and provide opportunities to strengthen the use of mental strategies and estimation when assessing reasonableness of solutions. (4.OA.3) What is a variable? What does it mean to estimate? How is rounding used when estimating? Word problems involving dividing whole numbers sometimes result in remainders that must be interpreted to provide accurate solutions. (4.OA.3) What do remainders mean, and how are they used to solve word problems? How does knowledge of factors and multiples help solve multiplication and division problems? Why is every natural number a multiple of each of its factors? How does estimation support finding more precise solutions? When are “correct” answers not the most useful answers? Why are there multiple ways to interpret remainders? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Multiply or divide to solve word problems involving multiplicative comparisons using drawings and equations with symbols for unknown numbers to represent the problems. (4.OA.2) Solve multistep word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted and represent the word problems using equations with letters standing for unknown quantities. (4.OA.3) Assess reasonableness of answers using mental computation and estimation strategies, including rounding. (4.OA.3) Find all factor pairs for whole numbers in the range from 1–100. (4.OA.4) Recognize that a whole is a multiple of each of its factors. (4.OA.4) Determine whether given whole numbers in the range 1–100 are multiples of given one-digit numbers. (4.OA.4) Determine whether given whole numbers in the range 1–100 are prime or composite. (4.OA.4) Denver Public Schools 2014–2015 7 Grade 4, Unit 2: Multiplication and Division WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. How I solve the problem of how many buses are needed for 230 students: If each bus holds 40 students, I first estimate that I need less than 10, which holds 400 students, but more than five, which holds 200 students. Five buses leaves a remainder of 30 students. In this situation, I interpret the remainder to mean I need another bus or a total of six buses. Cross-Content Academic Words Technical Words Specific to Content Add, assess, classification, divide, factor, multiply, product, represent, solve, subtract, unknown, variable Algorithm, area model, composite, counting numbers, equation, estimation, mental strategy, multiple, natural number, prime, reasonableness, rectangular array, remainder, whole number, word problem Resources Core Lessons Everyday Mathematics Lessons 3*1–3*11 (use Lessons 3*3–3*5 only if needed) Lesson 3*2a: How Many Rectangles? (new lesson after Lesson 3*2) Unit Assessment Suggested Performance/ Learning Tasks Threatened and Endangered Comparing Money Raised Technology Stamps (computation in groups; multiplying with arrays) Product Game (fluency multiplying numbers 1–9) Factorize (dividing numbers into two factors and building arrays to represent each factorization) Literacy Connections Count Your Way through Africa by Jim Haskins Count Your Way through the Arab World by Jim Haskins Misconceptions When listing multiples of numbers, students do not list the number itself. Emphasize that the smallest multiple is the number itself. Students think larger numbers have more factors. Instruct students to share all factor pairs and ask them how they found the pairs. Review vocabulary so students understand terms, such as factor, product, multiple, and odd and even numbers. Students need to develop understanding of the concept of number theory, such as prime and composite numbers, including the relationship of factors and multiples. Multiplication and division are used to develop concepts of factors and multiples. Division problems resulting in remainders are used as counter-examples of factors. Using area models enables students to analyze numbers and understand whether numbers are prime or composite. Ask students to construct rectangles with areas equal to given numbers. Students should see associations between numbers of rectangles and given numbers for the areas as to whether the numbers are prime or composite. Definitions of prime and composite numbers should not be provided but determined by students using many strategies to find all possible Essentials for Standards Implementation Denver Public Schools 2014–2015 8 Grade 4, Unit 2: Multiplication and Division Denver Public Schools factors of numbers. Provide students with counters to find factors of numbers. Teach them to find ways to separate counters into equal subsets. For example, instruct them to find several factors of 10, 14, 25, or 32 and write multiplication expressions for the numbers. Students must develop strategies to determine whether numbers are prime or composite. Starting with a number chart of one to 20, use prime number multiples to eliminate later numbers in the chart. For example, two is prime, but four, six, eight, ten, and 12 are composite. Encourage rule development, which students can use to determine whether numbers are composite. For example, other than two, if numbers end in an even number (for example, zero, two, four, six, eight), they are composite. Determining factors and multiples is the foundation to find common multiples and factors, a sixth grade standard. One way to find number factors is to use arrays from square tiles drawn on grid paper. Instruct students to build rectangles with given numbers of squares (see Lesson 3*2, Teaching Masters). Writing multiplication expressions for numbers with several factors and numbers with few factors helps students make conjectures about the numbers. Students need to look for commonalities among numbers. Students’ algebraic thinking must be developed if they are to succeed later in the formal study of algebra. Understanding patterns is fundamental to algebraic thinking. Students should experience identifying arithmetic patterns, especially those in addition and multiplication tables. Students should also generate numerical or geometric patterns that follow given rules. They should look for relationships in patterns and describe and make generalizations. 2014–2015 9 Grade 4, Unit 3: Extended Multiplication Unit of Study 3: Extended Multiplication Focusing Lenses Interpretation and Comparison Focus Essential Learning Goals Length of Unit 18 days (October 27–November 21, 2014) Use the four operations with whole numbers to solve problems. (ELG.MA.4.OA.A) Generalize place value understanding for multidigit whole numbers. (ELG.MA.4.NBT.A) Use place value understanding and properties of operations to perform multidigit arithmetic. (ELG.MA.4.NBT.B) Content Standards Standards Operations and Algebraic Thinking (4.OA) Use the four operations with whole numbers to solve problems. (Major) [ELG.MA.4.OA.A] 4.OA.1: Interpret multiplication equations as comparisons (for example, interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5). Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.2: Multiply or divide to solve word problems involving multiplicative comparison using drawings and equations with symbols for unknown numbers to represent problems, distinguishing multiplicative comparison from additive comparison. 4.OA.3: Solve multistep word problems posed 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 letters standing for unknown quantities. Assess reasonableness of answers using mental computation and estimation strategies including rounding. Number and Operations in Base Ten (4.NBT) Generalize place value understanding for multidigit whole numbers. (Major) [ELG.MA.4.NBT.A] 4.NBT.1: Recognize that in multidigit whole numbers, a digit in one place represent ten times what it represents in the place to its right (for example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division). 4.NBT.2: Read and write multidigit whole numbers using base-ten numerals, number names, and expanded form. Compare two multidigit numbers based on the meaning of the digit in each place using >, =, and < symbols to record comparison results. 4.NBT.3: Use place value understanding to round multidigit whole numbers to any place. Use place value understanding and properties of operations to perform multidigit arithmetic. (Major) [ELG.MA.4.NBT.B] 4.NBT.5: Multiply whole numbers of up to four digits by one-digit whole numbers and multiply two, two-digit numbers using strategies based on place value and properties of operations. Illustrate and explain calculations using equations, rectangular arrays, and/or area models. Standards for Mathematical Practice 2. 4. 6. 7. Reason abstractly and quantitatively. Model with mathematics. Attend to precision. Look for and make use of structure. Fluency Recommendations N/A Inquiry Question Concepts Factoring, multiplication, estimation, distributive property Denver Public Schools Why don’t we classify decimals or fractions as prime or composite? 2014–2015 10 Grade 4, Unit 3: Extended Multiplication Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual Multiplicative comparisons enable the interpretation of word problems involving two quantities in which one is described as a multiple of the other. (4.OA.1) How do we write multiplicative comparisons as equations? How can we determine whether word problems involve multiplicative or additive comparisons? Word problems typically contain unknown quantities and provide opportunities to strengthen the use of mental strategies and estimation when assessing reasonableness of solutions. (4.OA.3) What is a variable? What does it mean to estimate? How is rounding used when estimating? Algorithms for multiplying multidigit numbers require applying distributive property and calculating partial products (visually represented in rectangular arrays and area models). (4.NBT.5) How can we use area models to explain multidigit multiplication? How can we make multiplying large numbers easy? Why do we break apart multidigit numbers when multiplying them? How is the concept of place value used when multiplying multidigit numbers? Strategies based on place value and properties of operations can be used to multiply whole numbers. (4.NBT.5) What does it mean to multiply with strategies based on place value and properties of operations? How does place value affect multiplication? Word problems involving multiplicative comparisons can be represented using drawings and equations. (4.OA.1) How would we represent multiplicative comparison problems using drawings or equations? Can we explain word problems using drawings or equations? How do multiplicative comparisons differ from additive comparisons? Why is division generally used to solve multiplicative comparisons and subtraction for additive comparisons? How does estimation support finding more precise solutions? When are “correct” answers not the most useful answers? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Interpret multiplication equations as comparisons; represent verbal statements of multiplicative comparisons as multiplication equations. (4.OA.1) Multiply or divide to solve word problems involving multiplicative comparisons using drawings and equations with symbols for unknown numbers to represent the problems. (4.OA.2) Distinguish between multiplicative and additive comparisons. (4.OA.2) Solve multistep word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted and represent the word problems using equations with letters standing for unknown quantities. (4.OA.3) Assess reasonableness of answers using mental computation and estimation strategies, including rounding. (4.OA.3) Multiply whole numbers of up to four digits by one-digit whole numbers and multiply two, two-digit numbers using strategies based on place value and properties of operations. (4.NBT.5) Illustrate and explain multiplication using equations, rectangular arrays, and area models. (4.NBT.5) Denver Public Schools 2014–2015 11 Grade 4, Unit 3: Extended Multiplication WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. I can use mental strategies to estimate reasonable products. Cross-Content Academic Words Technical Words Specific to Content Add, assess, classification, divide, factor, multiply, product, represent, solve, subtract, unknown, variable Algorithm, area model, equation, estimation, mental strategy, multiple, multiplicative comparison, reasonableness, rectangular array, remainder, rounding, whole number, word problem Resources Core Lessons Everyday Mathematics Lessons 5*1–5*6 Lesson 5*8 Lesson 5*10 Unit Assessment Suggested Performance/ Learning Tasks Comparing Money Raised (before Lesson 5*4) What’s My Number? (before Lesson 5*10) Numbers of Stadium Seats (after Lesson 5*10)—Click “Elementary school tasks” (left side), then “Numbers of stadium seats (grade 4).” Technology Rectangle Multiplication (visualize multiplying two numbers as an area) Rectangle Division (visualize and practice dividing numbers using area representations) Literacy Connections Ten Times Better by Richard Michelson Moira’s Birthday by Robert Munsch Students commonly misinterpret “B is three more than A” and “B is three times as much as A” in multiplicative comparisons. Visual models help students avoid these problems. In situations involving comparisons, one quantity is compared to another. For example: • John has three more seashells than Kim. John and Kim have 15 seashells altogether. Find the number of seashells John has. • Larry has three times as much money as Mary. Larry and Mary have $120 altogether. Find the amount of money Larry has. In the seashells problem, the number of seashells John has is compared to the number of seashells Kim has. In the money problem, the amount of money Larry has is compared to the amount of money Mary has. The first problem is an example of additive comparison, while the second problem is an example of multiplicative comparison. In additive comparison, one quantity is a certain amount more or less than another quantity. In multiplicative comparison, one quantity is a certain number of times another quantity. Students who develop flexibility in breaking numbers apart (decomposing numbers) better understand the importance of place value and Misconception Essentials for Standards Implementation Denver Public Schools 2014–2015 12 Grade 4, Unit 3: Extended Multiplication distributive property in multidigit multiplication. Students can use strategies, such as base-ten blocks, area models, partitioning, and compensation strategies when multiplying whole numbers and use words and diagrams to explain their thinking. They use the terms “factor” and “product” when communicating their reasoning. Multiple strategies enable students to develop fluency with multiplication and transfer that understanding to division. Fifth graders are expected to use the standard algorithm for multiplication and understand why it works. 4.OA.2 requires students to multiply numbers using a variety of strategies. Use situations for these problems in class. One Standard of Mathematical Practice asks students to interpret problems in and out of context. Most real-world problems we solve are in situations. Create situations for students to solve and ask them to create situations. If students can create situations for multiplication problems and the context is correct, they understand the concept behind multiplication. For example, each tray at the bakery has 22 cookies. I counted 23 trays. What is the total number of cookies at the bakery? The three examples below use base-ten blocks to solve the problem. Can you see how these strategies connect? Help students see connections between all these strategies, so they can construct viable arguments. 20 + 3 20 + 2 22 x 23 400 40 60 6 506 Denver Public Schools (20 x 20) (20 x 2) (3 x 20) (3 x 2) 2014–2015 13 Grade 4, Unit 3: Extended Multiplication Hundreds Denver Public Schools 100 100 100 100 Tens Ones 2014–2015 14 Grade 4, Unit 4: Division Unit of Study 4: Division Focusing Lens Units Focus Essential Learning Goals Length of Unit 15 days (December 1–19, 2014) Use the four operations with whole numbers to solve problems. (ELG.MA.4.OA.A) Use place value understanding and properties of operations to perform multidigit arithmetic. (ELG.MA.4.NBT.B) Content Standards Standards Operations and Algebraic Thinking (4.OA) Use the four operations with whole numbers to solve problems. (Major) [ELG.MA.4.OA.A] 4.OA.2: Multiply or divide to solve word problems involving multiplicative comparison using drawings and equations with symbols for unknown numbers to represent problems, distinguishing multiplicative comparison from additive comparison. 4.OA.3: Solve multistep word problems posed 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 letters standing for unknown quantities. Assess reasonableness of answers using mental computation and estimation strategies including rounding. Number and Operations in Base Ten (4.NBT) Use place value understanding and properties of operations to perform multidigit arithmetic. (Major) [ELG.MA.4.NBT.B] 4.NBT.6: Find whole number quotients and remainders with up to four-digit dividends and one-digit divisors using strategies based on place value, properties of operations, and/or the relationship between multiplication and division. Illustrate and explain calculations using equations, rectangular arrays, and/or area models. Standards for Mathematical Practice 1. 3. 5. 7. Make sense of problems and persevere in solving them. Construct viable arguments and critique others’ reasoning. Use appropriate tools strategically. Look for and make use of structure. Fluency Recommendations N/A Inquiry Question Concepts Decomposition, multiplication, division Denver Public Schools How do you decide when close is close enough? 2014–2015 15 Grade 4, Unit 4: Division Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual Algorithms for dividing multidigit numbers require applying distributive property and calculating partial quotients (visually represented in rectangular arrays and area models). (4.NBT.6) How can we use area models to explain multidigit division? How can we make dividing large numbers easy? Word problems involving dividing whole numbers sometimes result in remainders that must be interpreted to provide accurate solutions. (4.OA.3) What do remainders mean, and how are they used in solving word problems? Why do we break apart multidigit numbers when dividing them? How is the concept of place value used when dividing multidigit numbers? Why are there multiple ways to interpret remainders? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Multiply or divide to solve word problems involving multiplicative comparisons using drawings and equations with symbols for unknown numbers to represent the problems. (4.OA.2) Distinguish between multiplicative and additive comparisons. (4.OA.2) Solve multistep word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted and represent the word problems using equations with letters standing for unknown quantities. (4.OA.3) Assess reasonableness of answers using mental computation and estimation strategies, including rounding. (4.OA.3) Find whole number quotients and remainders with up to four-digit dividends and one-digit divisors using strategies based on place value, properties of operations, and/or the relationship between multiplication and division. (4.NBT.6) WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. I know multiplication is related to division. I can multiply to find quotients in division problems. Cross-Content Academic Words Technical Words Specific to Content Apply, demonstrate, describe, express, represent Dividend, division, divisor, multiplication, partial quotient, quotient, remainder Denver Public Schools 2014–2015 16 Grade 4, Unit 4: Division Resources Core Lessons Everyday Mathematics Lessons 6*1–6*4 Lesson 6*10 Instructional Task: A Trip to Adventure Land (use any time after Lesson 6*10) Unit Assessment Suggested Performance/ Learning Tasks Buses, Vans, and Cars (after Lesson 6*1)—Click “Elementary school tasks” (left side), then “Buses, vans, and cars (grade 4).” Ordering Juice Drinks (after Lesson 6*1)—Click “Elementary school tasks” (left side), then “Ordering juice drinks (grade 4).” Carnival Tickets (after Lesson 6*1) The Baker (after Lesson 6*1) Technology Rectangle Division (visualize and practice dividing numbers using area representations) Literacy Connection A Remainder of One by Elinor J. Pinczes Students often attempt to follow a procedure that means nothing to them. Classroom discourse reveals this misconception. Connecting multiplication to division through word problems may support their meaning making. In fourth grade, students build on their third grade work of division within 100. Students need to develop their understandings using problems in and out of context. For example: • A fourth grade teacher bought four new pencil boxes. She has 260 pencils. She wants to put the pencils in the boxes so each box has the same number of pencils. How many pencils will be in each box? Using base-ten blocks, students build 260 and distribute blocks into four equal groups. Some students may need to trade the two hundreds for tens, but others may easily recognize that 200 divided by four is 50. Model recording this thinking to help connect conceptual and abstract understanding. Using place value: 260 ÷ 4 = (200 ÷ 4) + (60 ÷ 4) Using multiplication: 4 x 50 = 200, 4 x 10 = 40, 4 x 5 = 20; 50 + 10 + 5 = 65, so 260 ÷ 4 = 65 • Kyle collects baseball cards. He has 348 baseball cards. If he puts 12 cards on each page of his collection notebook, how many pages does he need? Can you solve this problem using a T-table? Using partial quotient? Cards Pages 29 348 ? 12)348 12 1 –120 10 10 x 12 = 120 120 10 228 240 20 Student’s thinking: –120 10 10 x 12 = 120 360 30 360 is too many. 108 60 5 If 10 is 120, then 5 is 60. –60 5 5 x 12 = 60 200 25 So 25 is 360 – 60, or 300. 48 324 27 Two pages is 24 more pictures. –48 4 4 x 12 = 48 348 29 Two more pages 0 Kyle needs 29 pages. Misconception Essentials for Standards Implementation Denver Public Schools 2014–2015 17 Grade 4, Unit 4: Division • Denver Public Schools It is important to match the language to what you do. o How many pages do we need? o What does this number mean? [cards still needing pages] o What do these numbers represent? [number of pages we have used so far] o What have we found out? o What was the original question? Have we answered it? At the same time, we must teach formal terminology. 2014–2015 18 Grade 4, Unit 5: Fractions Unit of Study 5: Fractions Focusing Lenses Comparison and Structure Focus Essential Learning Goals Length of Unit 23 days (January 6–February 6, 2015) Extend understanding of fraction equivalence and ordering. (ELG.MA.4.NF.A) Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (ELG.MA.4.NF.B) Content Standards Standards Number and Operations — Fractions (4.NF) Extend understanding of fraction equivalence and ordering. (Major) [ELG.MA.4.NF.A] 4.NF.1: Explain why fraction a/b is equivalent to fraction (n x a)/(n x b) using visual fraction models, with attention to how number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2: Compare two fractions with different numerators and different denominators by creating common denominators or numerators or comparing to benchmark fractions, such as 1/2. Recognize that comparisons are valid only when two fractions refer to the same whole. Record results of comparisons with symbols >, =, or < and justify conclusions using visual fraction models. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (Major) [ELG.MA.4.NF.B] 4.NF.3: Understand fraction a/b with a > 1 as a sum of fraction 1/b. a. Understand adding and subtracting fractions as joining and separating parts referring to the same whole. b. Decompose fractions into sums of fractions with the same denominators in more than one way, recording each decomposition by an equation. Justify decompositions using visual fraction models (for example, 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8). c. Add and subtract mixed numbers with like denominators by replacing each mixed number with an equivalent fraction and/or using properties of operations and relationships between addition and subtraction. d. Solve word problems involving adding and subtracting fractions referring to the same whole and having like denominators using visual fraction models and equations to represent problems. 4.NF.4: Apply and extend previous understandings of multiplication to multiply fractions by whole numbers. a. Understand fraction a/b as a multiple of 1/b (for example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4)). b. Understand fraction a/b as a multiple of 1/b and use this understanding to multiply fractions by whole numbers (for example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5). (In general, n x (a/b) = (n x a)/b). c. Solve word problems involving multiplying fractions by whole numbers using visual fraction models and equations to represent the problems (for example, if each person at a party eats 3/8 of a pound of roast beef and five people will be at the party, how many pounds of roast beef are needed? The answer falls between which two whole numbers?). Measurement and Data (4.MD) Represent and interpret data. (Supporting) [ELG.MA.4.MD.B] 4.MD.4: Make line plots to display data sets of measurements in fractions of units (1/2, 1/4, 1/8). Solve problems involving adding and subtracting fractions using information presented in line plots (for example, from a line plot, find and interpret the difference in length between the longest and shortest specimens in an insect collection). Standards for Mathematical Practice 6. 7. 8. Denver Public Schools Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. 2014–2015 19 Grade 4, Unit 5: Fractions Fluency Recommendations N/A Inquiry Questions Concepts Equivalence, adding and subtracting fractions, fractions as numbers, comparison of fractions What would the world be like without fractions? Why are fractions useful? Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual Equivalent fractions describe the same part of a whole using different fractional parts. (4.NF.1) How can we show that and are equivalent fractions? Why (or when) are equivalent fractions necessary or helpful? Increasing or decreasing both fractions’ numerators and denominators by the same factors creates equivalent fractions. (4.NF.1) How can we justify two fractions are equivalent using visual fraction models? What happens when we multiply both numerators and denominators by same numbers? Why do the same fractions have multiple names? How can different fractions represent the same quantities? Why do we need to know equivalent fractions? What are examples of benchmark fractions, and how are they useful to compare fraction sizes? When do we need to find common denominators or numerators? Why is it possible to compare fractions with either common denominators or common numerators? How can we record fraction decompositions with equations? How can we justify fraction decompositions using visual fraction models? When we decompose fractions, why do we only break apart numerators and not denominators? How is decomposing fractions similar to and different from decomposing whole numbers? Decisions about fraction sizes relative to other fractions often involve comparing fractions’ denominators (if numerators are equal) or numerators (if denominators are equal) or creating common denominators or numerators for the fractions. (4.NF.2) As with whole numbers, we can compose fractions by joining or combining fractions (with the same denominators) as sums and decompose or separate fractions (with the same denominators) as differences in multiple ways. (4.NF.3a, 4.NF.3b, 4.NF.3c. 4.NF.3d) Denver Public Schools 2014–2015 20 Grade 4, Unit 5: Fractions Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Explain why fraction a/b is equivalent to fraction (n x a)/(n x b) using visual fraction models, with attention to how number and size of the parts differ even though the two fractions themselves are the same size. (4.NF.1) Generate equivalent fractions. (4.NF.1) Compare two fractions with different numerators and denominators by creating common denominators or numerators or comparing to benchmark fractions, record comparison results with symbols >, =, <, and justify conclusions. (4.NF.2) Decompose fractions into sums of fractions with same denominators in more than one way, recording each decomposition by an equation, and justify decompositions. (4.NF.3b) Add and subtract mixed numbers with like denominators. (4.NF.3c) Solve word problems involving adding and subtracting fractions referring to the same whole and having like denominators. (4.NF.3d) Understand fraction a/b as a multiple of 1/b and multiple of a/b as a multiple of 1/b. (4.NF.4a) Make line plots for measurement data using fractions of units. (4.MD.4) WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. I can compare two fractions by comparing each to a benchmark fraction, such as . I can compare two fractions with a common numerator by reasoning about the relative size of the denominators. Cross-Content Academic Words Technical Words Specific to Content Apply, compare, decreasing, estimation, explain, express, generate, increasing, model, part, understand, visual, whole Addition, benchmark fraction, common denominator, common numerator, decompose, denominator, equivalent, equivalent fractions, mixed number, multiple, numerator, solve, subtraction, sum, unit fraction Denver Public Schools 2014–2015 21 Grade 4, Unit 5: Fractions Resources Core Lessons Everyday Mathematics Lessons 7*1–7*2 Lessons 7*4–7*6 Lesson 7*5a: Line Plots (new lesson after Lesson 7*5—coming soon) Instructional Task: Picking Fractions (use any time after Lesson 7*6) Lesson 7*7 Lesson 7*7a: Mother’s Pizza (new lesson after Lesson 7*7) Lesson 7*7b: Recording Fractional Parts (new lesson after Lesson 7*7a) Lessons 7*8–7*10 Instructional Task: Queen Anne’s Dilemma (use any time after Lesson 7*10) Unit Assessment Suggested Performance/ Learning Tasks Explaining Fraction Equivalence with Pictures Using Benchmarks to Compare Fractions Writing a Mixed Number as an Equivalent Fraction Fractions — Comparing (compare different fractions that are equal) Fractions — Parts of a Whole (relate parts of a whole to written descriptions and fractions) Fraction Number Line Bars (divide fractions using number line bar) Fraction Game (explore relationships among fractions) Fraction Pieces (work with parts and wholes of shapes) Fractions — Equivalent (illustrate relationships between equivalent fractions) Equivalent Fractions (create equivalent fractions by dividing and shading squares or circles and match each fraction to its location on the number line) Gator Pie by Louise Matthews Technology Literacy Connection Misconceptions Students do not understand the need to use models that represent the same whole to find sums or differences of fractions. The same whole is also important when comparing fractions. Students think the smaller the numbers in the fraction, the larger the fraction. When comparing two fractions, students need to reason about the sizes of two fractions. Also students need to reason about what fraction of the whole is left when comparing the sizes of two fractions. Essentials for Standards Implementation Emphasize visual models. Students need to use a variety of models, such as area models, number lines, and fraction circles. Fourth grade expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Denver Public Schools 2014–2015 22 Grade 4, Unit 6: Decimals Unit of Study 6: Decimals Focusing Lenses Comparison and Structure Focus Essential Learning Goals Length of Unit 18 days (February 9–March 6, 2015) Understand decimal notation for fractions and compare decimal fractions. (ELG.MA.4.NF.C) Solve problems involving measuring and converting measurements from larger to smaller units. (ELG.MA.4.MD.A) Content Standards Standards Number and Operations — Fractions (4.NF) Understand decimal notation for fractions and compare decimal fractions. (Major) [ELG.MA.4.NF.C] 4.NF.5: Express fractions with denominator 10 as equivalent fractions with denominator 100 and use this technique to add two fractions with respective denominators 10 and 100 (for example, express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100). 4.NF.6: Use decimal notation for fractions with denominators 10 or 100 (for example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on number line diagram). 4.NF.7: Compare two decimals to hundredths by reasoning about their sizes. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record comparison results with the symbols >, =, or < and justify conclusions using visual models. Measurement and Data (4.MD) Solve problems involving measuring and converting measurements from larger to smaller units. (Supporting) [ELG.MA.4.MD.A] 4.MD.1: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in larger units in terms of smaller units. Record measurement equivalents in two-column tables (for example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36).) 4.MD.2: Use the four operations to solve word problems involving distances, time intervals, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals and problems that require expressing measurements given in larger units in terms of smaller units. Represent measurement quantities using diagrams, such as number line diagrams featuring measurement scales. Standards for Mathematical Practice 2. 4. 6. 8. Reason abstractly and quantitatively. Model with mathematics. Attend to precision. Look for and express regularity in repeated reasoning. Fluency Recommendation Fluently add and subtract multidigit whole numbers using standard algorithm. (4.OA.4) Inquiry Questions Is there a decimal closest to one? Why? What would change if we used a base-eight number system? What would our numbers look like? Concepts Addition, subtraction, place value, equivalency Denver Public Schools 2014–2015 23 Grade 4, Unit 6: Decimals Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual The standard algorithm for adding and subtracting provides an efficient method to develop fluency with adding and subtracting multidigit numbers. (4.NBT.4) How does understanding place value support the standard algorithm for adding and subtracting? How does the standard algorithm for adding and subtracting apply to decimals? Why is fluency with multidigit addition and subtraction important? Additional numbers to the right of the decimal do not necessarily increase decimals’ values in comparison to other decimals. (4.NF.7) How does comparing decimals differ from comparing whole numbers? How can you use the concept of equivalent fractions to compare two decimals? Why does placing zeros at the end of numbers with decimal places not change the numbers’ values? Why is every decimal easily written as a fraction but every fraction is not easily written as a decimal? How can numbers with greater decimal digits be less than ones with fewer decimal digits? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Use decimal notation for fractions with denominators 10 and 100. (4.NF.6) Compare two multidigit numbers based on the meaning of the digit in each place and compare two decimals to hundredths by reasoning about their size; record comparisons using <, =, or > symbols and justify comparisons. (4.NF.7) Recognize that comparisons are only valid when quantities refer to the same whole. (4.NF.7) Know relative sizes of measurement units and how to express larger units in terms of smaller units. (4.MD.1) WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. Multidigit numbers can be written in expanded form. The symbols <, >, = are used to compare multidigit numbers. Numbers can be represented with base-ten numerals, expanded form, or number names. Cross-Content Academic Words Technical Words Specific to Content Addition, compare, efficient, equal to, generate, greater than, identify, less than, precision, recognize, represent, table, whole Decimal, denominator, digit, equivalent forms, expanded form, fraction, hundredths, magnitude, multidigit, number name, numerator, place value, rounding, rule, standard algorithm, tenths, unit, whole number Denver Public Schools 2014–2015 24 Grade 4, Unit 6: Decimals Resources Core Lessons Everyday Mathematics Lessons 4*1–4*3 Instructional Task: Decimals (use any time after Lesson 4*3) Lessons 4*6–4*10 4*10a: Who is the tallest? (new lesson after Lesson 4*10) Unit Assessment Suggested Performance/ Learning Tasks Fraction Model (after Lesson 4*2) Margie Buys Apples (after Lesson 4*2) Technology Base Blocks Decimals (add and subtract decimal values; good visual, but not for core lessons, use as extension) Concentration (match fractions, decimals, multiplication, and more facts to equivalent representations) Coin Box (coin value, counting coins, making change) Literacy Connection If You Hopped Like a Frog by David M. Schwartz Students often treat decimals as whole numbers when comparing two decimals. They think the longer the number, the greater the value (for example, students think 0.03 is greater than 0.3).When reading decimals aloud, students should read numbers as fractions, such as “three hundredths” or “three tenths.” If students read decimals this way, they visualize the fractional pieces the decimals represent. Discourage students from reading decimal numbers “point 03.” The place value system for whole numbers extends to fractional parts represented as decimals, which connects to the metric system. The concept of one whole used in fractions extends to decimal models. Students can use base-ten blocks to represent decimals. A 10 x 10 block can be assigned the value of one whole to allow other blocks to represent tenths and hundredths. They can show decimal representations from the base-ten blocks by shading 10-by-10 grids. Students need to make connections between fractions and decimals. They should be able to write decimals for fractions with denominators of 10 or 100. Teach students to say fractions with denominators of 10 and 100 aloud (for example, 4/10 would be “four-tenths” or 27/100 would be “twenty-seven hundredths”). Also, instruct students to represent decimals in word form with digits and decimal place value, such as 4/10 would be “four tenths.” Students should express decimals to the hundredths as the sum of two decimals or fractions. This concept is based on understanding decimal place value (for example, 0.32 would be the sum of three tenths and two hundredths). Using this understanding, students can write 0.32 as the sum of two fractions (3/10 + 2/100). Students’ understanding of decimals to hundredths is important to perform decimal operations to hundredths in fifth grade. In decimal numbers, the value of each place is ten times the value of the place to its immediate right. Students need to understand decimal notations before doing metric system conversions. Understanding decimal place value system is important before students learn to move decimal points when performing decimal operations. Students extend fraction equivalence from third grade with denominators of two, three, four, six, and eight to fractions with denominators of ten and 100. Provide fraction models of tenths and hundredths so students can express fractions with a denominator of ten as equivalent fractions with a denominator of 100. Misconceptions Essentials for Standards Implementation Denver Public Schools 2014–2015 25 Grade 4, Unit 7: Fractions and Decimals Unit of Study Unit 7: Fractions and Decimals Focusing Lenses Comparison and Structure Focus Essential Learning Goals Length of Unit 10 days (March 9–20, 2015) Extend understanding of fraction equivalence and ordering. (ELG.MA.4.NF.A) Understand decimal notation for fractions and compare decimal fractions. (ELG.MA.4.NF.C) Content Standards Standards Number and Operations — Fractions (4.NF) Extend understanding of fraction equivalence and ordering. (Major) [ELG.MA.4.NF.A] 4.NF.1: Explain why fraction a/b is equivalent to fraction (n x a)/(n x b) using visual fraction models, with attention to how number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Understand decimal notation for fractions and compare decimal fractions. (Major) [ELG.MA.4.NF.C] 4.NF.5: Express fractions with denominator 10 as equivalent fractions with denominator 100 and use this technique to add two fractions with respective denominators 10 and 100 (for example, express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100). 4.NF.6: Use decimal notation for fractions with denominators 10 or 100 (for example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on number line diagram). 4.NF.7: Compare two decimals to hundredths by reasoning about their sizes. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record comparison results with the symbols >, =, or < and justify conclusions using visual models. Standards for Mathematical Practice 2. 4. 6. 7. Reason abstractly and quantitatively. Model with mathematics. Attend to precision. Look for and make use of structure. Fluency Recommendations N/A Inquiry Questions Concepts Place value, equivalency, comparison of fractions and decimals Denver Public Schools What are decimal fractions? Can we solve problems using equivalent decimals instead of fractions? Why? 2014–2015 26 Grade 4, Unit 7: Fractions and Decimals Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual Converting fractions with denominators of ten or 100 (or any power of ten) to decimals produces numbers that are ten or 100 times less than the numerators of the original fractions. (4.NF.5) How can we divide to convert fractions with denominators of ten or 100? How do equivalent fractions help explain different equivalent decimal forms of the same quantities? Additional numbers to the right of the decimal do not necessarily increase decimals’ values in comparison to other decimals. (4.NF.7) How does comparing decimals differ from comparing whole numbers? How can you use the concept of equivalent fractions to compare two decimals? Why does placing zeros at the end of numbers with decimal places not change the numbers’ values? Why is 0.7 equivalent to 0.70? Why is every decimal easily written as a fraction but every fraction is not easily written as a decimal? How can numbers with greater decimal digits be less than ones with fewer decimal digits? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Use decimal notation for fractions with denominators ten and 100. (4.NF.6) Compare two decimals. (4.NF.7) WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. I can write any number that is a decimal fraction (for example, 62/100) as a decimal. Cross-Content Academic Words Technical Words Specific to Content Convert, representation Decimal, decimal fraction, digit, hundredths, place value, ten times, tenths Denver Public Schools 2014–2015 27 Grade 4, Unit 7: Fractions and Decimals Resources Core Lessons Everyday Mathematics Lessons 9*1–9*3 Lesson 9*1a: Decimal Representations (new lesson after Lesson 9*1) Lesson 9*2a: The Clothesline (new lesson after Lesson 9*2) Lesson 9*2b: Objects in My Desk Line Plot (new lesson after Lesson 9*2a) Lesson 9*5 (focus on equivalent names for fractions section) Unit Assessment Suggested Performance/ Learning Task Expanded Fractions and Decimals Technology Percentages (discover relationships between fractions, percents, and decimals) Literacy Connection Piece = Part = Portion: Fractions = Decimals = Percents by Scott Gifford Students often treat decimals as whole numbers when comparing two decimals. They think the longer the number, the greater the value (for example, students think 0.03 is greater than 0.3). This standard continues the work of equivalent fractions by having students change fractions with ten in the denominator into equivalent fractions with 100 in the denominator. Experiences that allow students to shade decimal grids (10 x 10) prepare them to work with decimals. Student experiences should focus on working with grids rather than algorithms. Students can use base-ten blocks and other place value models to explore relationships between fractions with denominators of ten and 100. This work lays the foundation for performing operations with decimal numbers in fifth grade. Students make connections between fractions with denominators of ten and 100 and the place value chart. By reading fraction names, students say 32/100 as “thirty-two hundredths” and rewrite it as 0.32 or represent it on a place value model. Misconception Essentials for Standards Implementation Denver Public Schools 2014–2015 28 Grade 4, Unit 8: Perimeter and Area Unit of Study 8: Perimeter and Area Focusing Lens Units Focus Essential Learning Goal Length of Unit 12 days (March 23–April 17, 2015) Solve problems involving measuring and converting measurements from larger to smaller units. [this is ELG.MA.4.MD.A] Content Standards Standards Measurement and Data (4.MD) Solve problems involving measuring and converting measurements from larger to smaller units. (Supporting) [ELG.MA.4.MD.A] 4.MD.3: Apply area and perimeter formulas for rectangles in real-world and mathematical problems (for example, find the width of a rectangular room given the area of the flooring and the length by viewing the area formula as a multiplication equation with an unknown factor). Standards for Mathematical Practice 1. 4. 5. 6. Make sense of problems and persevere in solving them. Model with mathematics. Use appropriate tools strategically. Attend to precision. Fluency Recommendations N/A Inquiry Question Concepts Measurement, decomposition, perimeter, area, conversion How does what we measure influence how we measure? Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual Solutions to rectangular area and perimeter problems require knowing only two of given rectangles’ measurements of length, width, perimeter, or area. (4.MD.3) How do we find areas of rectangles? How do we find perimeters of rectangles? How can we find rectangles’ lengths if we know their areas and widths? Measurement systems embody varying size units, wherein larger units are multiples of smaller units in the system. (4.MD.1) Using multiplicative comparison, how many times larger is a foot than an inch? How can we convert from larger units of measurement to smaller ones? Denver Public Schools 2014–2015 Why do rectangles’ lengths and widths determine both their areas and perimeters? Why don’t rectangles’ perimeters determine their areas and vice versa? Why are metric system conversions easier than U.S. customary system conversions? Why are measurement conversions multiplicative rather than additive comparisons? Why do we convert units? 29 Grade 4, Unit 8: Perimeter and Area Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Apply area and perimeter formulas for rectangles in real-world and mathematical problems. (4.MD.3) Solve word problems involving measurement. (4.MD.2) Convert measurements within one system of units. (4.MD.1) WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. If I know a rectangle’s length and width, I can calculate its area and perimeter. Cross-Content Academic Words Technical Words Specific to Content Base, convert, distance, estimate, measure, solve Area, formula, height, length, perimeter, square unit, variable Resources Core Lessons Everyday Mathematics Lessons 8*1–8*5 Lesson 8*8 Unit Assessment Suggested Performance/ Learning Tasks Deer in the Park (after Lesson 8*5)—Click “Elementary school tasks” (left side), then “Deer in the Park (grade 4). Karl’s Garden (after Lesson 8*5)—Add to task by asking students to use drawings, equations, or words to explain their thinking. Technology Rectangle Multiplication (visualize multiplying two numbers as an area) Coloring Triangles and Squares Geoboard (explore two-dimensional shapes and their properties of area and perimeter and rational number concepts; click “Measures” button when shape is selected to see area and perimeter measurements) Literacy Connection Actual Size by Steve Jenkins Students confuse perimeter and area. Perimeter is a measure of length, or distance; area is a measure of surface. The word “perimeter” contains the word “rim.” Use Math Boxes and Mental Math and Reflexes from noncore lessons for ongoing practice. Give students multiple opportunities to find both perimeters and areas of the same rectangles. Provide situations for them to discuss comparing those two measurements. Misconception Essentials for Standards Implementation Denver Public Schools 2014–2015 30 Grade 4, Unit 9: Geometric Figures, Measuring Angles, and Symmetry Unit of Study 9: Geometric Figures, Measuring Angles, and Symmetry Focusing Lenses Form and Function Focus Essential Learning Goals Length of Unit 15 days (April 20–May 8, 2015) Geometric measurement: Understand concept of angle and measure angles. (ELG.MA.4.MD.C) Draw and identify lines and angles and classify shapes by properties of their lines and angles. (ELG.MA.4.G.A) Content Standards Standards Measurement and Data (4.MD) Geometric measurement: Understand concept of angle and measure angles. (Additional) [ELG.MA.4.MD.C] 4.MD.5: Recognize angles as geometric shapes formed wherever two rays share a common endpoint and understand concepts of angle measurement: a. Angles are measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. Angles that turn through 1/360 of a circle are called “one-degree angles,” and can be used to measure angles. b. Angles that turn through n one-degree angles are said to have angle measures of n degrees. 4.MD.6: Measure angles in whole number degrees using a protractor. Sketch angles of specified measures. 4.MD.7: Recognize angle measure as additive. When angles are decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on diagrams in real-world and mathematical problems using equations with symbols for unknown angle measures. Geometry (4.G) Draw and identify lines and angles and classify shapes by properties of their lines and angles. (Additional) [ELG.MA.4.G.A] 4.G.1: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines and identify in two-dimensional figures. 4.G.2: Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of specified sizes. Recognize right triangles as a category and identify right triangles. 4.G.3: Recognize lines of symmetry for two-dimensional figures as a line across the figures such that the figures can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. Standards for Mathematical Practice 1. 3. 4. 5. 6. Make sense of problems and persevere in solving them. Construct viable arguments and critique others’ reasoning. Model with mathematics. Use appropriate tools strategically. Attend to precision. Fluency Recommendations N/A Inquiry Question Concepts Angle measurement, polygon classification Denver Public Schools What symmetrical constructions are seen in the world? 2013–2014 31 Grade 4, Unit 9: Geometric Figures, Measuring Angles, and Symmetry Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual Points, lines, line segments, and rays designate locations in space and provide building blocks to create and understand shapes. (4.G.1) What are lines? What are line segments? What are rays? How are lines, line segments, and rays similar? How are they different? What real-world examples can we find of lines and line segments? Lines pointing in the same direction that share no points in common (parallel) and lines that share one point in common and form right angles (perpendicular) determine classifications of many geometric shapes. (4.G.1) How are parallel and perpendicular lines related? Why do angles matter when drawing perpendicular lines? How do people use parallel and perpendicular lines every day? Most basic geometric shapes possess lines of symmetry that divide the shapes into two mirror images. (4.G.3) What are lines of symmetry? What is congruency? How can mirrors help us find lines of symmetry? Why do circles have multiple (infinite) lines of symmetry? Where do lines of symmetry appear in nature? The rotation (or spread) from one ray to another ray sharing the same common endpoint determines angles’ sizes and classifications. (4.MD.5a, 4.MD.5b) How do we name angles? What are angles? How are angles formed? Why are angles described as measures of rotation? How do angle sizes change shapes’ forms? Angles (right, acute, obtuse) facilitate classifying and categorizing shapes. (4.G.2) What are the differences between acute, right, and obtuse angles? What is a right angle’s role in classifying triangles? Are squares still squares if they are tilted on their sides? How do perpendicular and parallel lines and angles classify and categorize shapes? Why is it helpful to classify angles and shapes? Why do angles with measures of zero and angles with measures of 360° look the same? What other angles look the same? Why are one-degree angles 1/360 of a rotation around a circle? Circles provide a reference from which to measure individual angles by locating angles’ vertices at the center. (4.MD.5a) How can we use fractions of the circular arcs between two rays to measure angles? What are the measures of angles that are a quarter turn of a circle? Half turn? One complete turn? Degrees of the circle visually represent all iterations of one-degree angles with measurements between 0 and 360 degrees. (4.MD.5a) What tool measures angles? Denver Public Schools 2013–2014 32 Grade 4, Unit 9: Geometric Figures, Measuring Angles, and Symmetry Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. (4.G.1) Identify points, line segments, angles, and perpendicular and parallel lines in two-dimensional figures. (4.G.1) Classify and identify two-dimensional figures according to attributes of line relationships (parallel, perpendicular) or angle size. (4.G.2) Recognize right triangles as a category and identify right triangles. (4.G.2) Recognize lines of symmetry for two-dimensional figures as a line across the figures such that the figures can be folded along the line into matching parts. (4.G.3) Identify line-symmetric figures and draw lines of symmetry. (4.G.3) WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. Right triangles are triangles with one right angle. Cross-Content Academic Words Technical Words Specific to Content Acute, analyze, angle, attribute, categorize, category, classify, construct, describe, determine, draw, exterior, identify, interior, point, ray, recognize, right Congruent, endpoint, intersect, kite, line, line of symmetry, line segment, obtuse, parallel, parallelogram, pentagon, perpendicular, polygon, quadrilateral, right angle, right triangle, rotation, side, symmetry, two-dimensional, vertices, vertex Resources Core Lessons Everyday Mathematics Lessons 1*1–1*5 Lesson 1*3a: Classifying Triangles (new lesson after Lesson 1*3—coming soon) Lessons 6*5–6*7 Lessons 10*1–10*2 Lesson 10*4 (symmetry) Unit Assessment Suggested Performance/ Learning Tasks Lesson 1*9 Open Response: Properties of Polygons Measuring Angles (after Lesson 6*7) Technology Congruent Triangles (build similar triangles by combining sides and angles) Angle Sums (visuals for three- to eight-sided figures with angles) Denver Public Schools 2013–2014 33 Grade 4, Unit 9: Geometric Figures, Measuring Angles, and Symmetry Literacy Connections Gregory and the Magic Line by Dawn Piggot Picture Pie: A Circle Drawing Book by Ed Emberly Sir Cumference and the Great Knight of Angleland by Cindy Neuschwander Students do not classify squares as rectangles. When working with two-dimensional figures and describing attributes of squares (having two opposite sides that are parallel and equal length and four right angles), students should classify squares as rectangles. Students might be confused about which number to use when determining measures of angles using protractors because most protractors have a double set of numbers. Students should decide first whether angles appear to be less than or greater than the measure of a right angle (90°). If angles appear to be less than 90°, they are acute angles and their measures range from 0° to 89°. If angles appear to be greater than 90°, they are obtuse angles and their measures range from 91° to 179°. Ask questions about angles’ appearances to help students determine which numbers to use. Misconceptions Essentials for Standards Implementation Denver Public Schools This unit uses several geometry terms, which should be discussed in the context of solving problems and related to students’ experiences. Teaching these terms in isolation or reducing them to a vocabulary list to memorize will not produce successful results in most cases. Students should have opportunities to work with each term and build models of, write, and discuss the terms. 2013–2014 34 Grade 4, Unit 10: Weight, Volume, and Capacity Unit of Study 10: Weight, Volume, and Capacity Focusing Lens Units Focus Essential Learning Goal Length of Unit 18 days (May 11–June 5, 2015) Solve problems involving measuring and converting measurements from larger to smaller units. (ELG.MA.4.MD.A) Content Standards Standards Measurement and Data (4.MD) Solve problems involving measuring and converting measurements from larger to smaller units. (Supporting) [ELG.MA.4.MD.A] 4.MD.1: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in larger units in terms of smaller units. Record measurement equivalents in two-column tables (for example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36).) 4.MD.2: Use the four operations to solve word problems involving distances, time intervals, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals and problems that require expressing measurements given in larger units in terms of smaller units. Represent measurement quantities using diagrams, such as number line diagrams featuring measurement scales. Standards for Mathematical Practice 1. 4. 5. 6. Make sense of problems and persevere in solving them. Model with mathematics. Use appropriate tools strategically. Attend to precision. Fluency Recommendations N/A Inquiry Question Concepts Measurement systems, conversion, intervals of time, masses and volumes of objects Denver Public Schools How can you describe geometric solids’ sizes? 2013–2014 35 Grade 4, Unit 10: Weight, Volume, and Capacity Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that… Factual Conceptual Measurement systems embody varying size units, wherein larger units are multiples of smaller units in the system. (4.MD.1) Using multiplicative comparison, how many times larger is a foot than an inch? How can we convert from larger units of measurement to smaller ones? Number line diagrams often provide an efficient way to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money. (4.MD.2) How can we represent quantities on number lines? What types of measurement problems are helpful to represent on number line diagrams? How can we use division to solve problems involving number of times given measurements fit into larger measurements? Why are metric system conversions easier than U.S. customary system conversions? Why are measurement conversions multiplicative rather than additive comparisons? Why do we convert units? Why are number line diagrams effective representations to solve measurement problems? Why can fractions be viewed as answers to division problems? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do)… Know relative sizes of measurements within a single measurement system. (4.MD.1) Convert measurements within one measurement system from larger units to smaller units and record measurement equivalents in two-column tables. (4.MD.1) 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 larger units in terms of smaller units. (4.MD.2) Represent measurement quantities using diagrams, such as number line diagrams featuring measurement scales. (4.MD.2) WIDA English Language Development (ELD) Standards 1: English language learners communicate for social and instructional purposes within the school setting. 3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics. Use WIDA Can-Do Descriptors to determine appropriate supports and scaffolds and differentiate appropriate outputs based on English proficiency levels. Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Grade 4 students demonstrate ability to apply and comprehend critical language through the following examples. When I convert from feet to inches, I need to multiply the number of feet by 12 because there are 12 times as many inches as feet, and I can show my conversions in a two-column table. Cross-Content Academic Words Technical Words Specific to Content Apply, centimeter, convert, demonstrate, describe, dimension, express, feet, foot, gram, inch, kilogram, kilometer, liter, measure, meter, milliliter, ounce, pound, record, represent, sketch Area, intersection, length, liquid volume, measurement scale, measurement system, object mass, perimeter, two-column table, vertex, width Denver Public Schools 2013–2014 36 Grade 4, Unit 10: Weight, Volume, and Capacity Resources Core Lessons Everyday Mathematics Lessons 11*1–11*2 Lesson 11*4 (background knowledge for volume in grade 5) Lesson 11*7 Unit Assessment Suggested Performance/ Learning Task Who is the tallest? Technology Cubes (determine box volume by filling with cubes, rows of cubes, or layers of cubes) Geometric Solids (manipulate various geometric solids and investigate their properties) Literacy Connection One Grain of Rice: A Mathematical Folktale by Demi Students believe larger units give larger measures. Give students multiple opportunities to measure the same objects with different measuring units. For example, instruct students to measure a room’s length with one-inch tiles, one-foot rulers, and yardsticks. Students should notice it takes fewer yardsticks to measure the room than rulers or tiles and explain their reasoning. This unit uses several geometry terms, which should be discussed in the context of solving problems and related to students’ experiences. Teaching these terms in isolation or reducing them to a vocabulary list to memorize will not produce successful results in most cases. Students should have opportunities to work with each term and build models of, write, and discuss the terms. Misconception Essentials for Standards Implementation Denver Public Schools 2013–2014 37
© Copyright 2024