TransMath Levels 1, 2, 3 Correlated with COMMON CORE STATE STANDARDS MATHEMATICS Grades 4-8

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1 Levels 1, 2, 3 Correlated with COMMON CORE STATE STANDARDS MATHEMATICS

2 How Voyager Addresses the Common Core Standards The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. Likewise, Voyager is dedicated to providing materials that are clearly written for students, as well as teachers, parents, classroom assistants and/or tutors who may be providing math instruction or support for the child. The Common Core State Standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. Voyager is focused on individualized instruction based on students developmental levels, as they work to meet the needs of all students, and particularly struggling students. This focus on developmental learning assures not only that the materials are robust and relevant for every student but provides a level of success in learning which is sorely lacking for the struggling student. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. Voyager is committed to providing students with meaningful skills to help move them forward in their education and in life. This is done by focusing on critical thinking skills that will serve them well in any arena. Products from Voyager place a strong emphasis on the process standards communication, connections, representations, reasoning and problem solving. These skills are critical for students to compete successfully in a global economy.

3 Grade Level: 4 Levels 1, 2, 3 Correlated with GRADE 4 Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems 1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 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. Unit 3, All (pp ) Students build a conceptual foundation for whole number multiplication. Unit 3, Lesson 14 (pp ) Students interpret multiplication as a comparison using data in a bar graph as a number of times bigger or smaller than other data in the same set. Unit 1, Lesson 2; Unit 2, Lessons 1-7; Unit 3, Lesson 8; and Unit 5, Lessons 6-8 (pp 18-22, , , and ) Students review the concept of multiplication, extend their knowledge to rational numbers, and interpret situations involving multiplicative comparisons. Unit 2, All (pp ) Students interpret a multiplication equation as a comparison; represent verbal statements of multiplicative comparisons as multiplication equations. Unit 6, Lessons 7-9 (pp ) Students are introduced to similarity; make comparisons using multiplicative relationships. Page 1 of 133

4 Grade Level: 4 Levels 1, 2, 3 Correlated with Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems (continued) 2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Units 3-4, All (pp ) Students multiply or divide whole numbers to solve word problems. Unit 2, All; Unit 3, Lessons 8-13; Unit 5, Lessons 5-14 (pp , , and ) Students multiply or divide rational numbers to solve word problems. Unit 2, All (pp ) Students use drawings such as pattern cards to solve problems involving multiplicative comparison; distinguish multiplicative comparisons from additive comparisons; use variables to write the unknown number in a multiplication equation. Page 2 of 133

5 Grade Level: 4 Levels 1, 2, 3 Correlated with Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems (continued) 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 a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Units 1-4, All (pp 8-475) Students solve multi-step word problems involving all 4 whole number operations, including problems in which remainders must be interpreted (Unit 4, Lessons 5-8 explore remainders); assess reasonableness of answers using mental computation and estimation strategies including rounding Unit 1, Lessons 10-14; Unit 2, Lessons 6-14; Unit 3, Lessons 9-13; Unit 4, Lessons 6-7 and (pp , , , , and ) These are the specific lessons involving estimation. Unit 1, Lessons 1-2 (pp 8-22) Students review whole number operations; use whole number operations to solve word problems. Unit 5, Lessons 1-9 (pp ) Students represent area problems using equations (formulas) with a letter standing for unknown quantities. Unit 2, All (pp ) Students are introduced to variables; represent problems using equations or expressions with a letter standing for the unknown quantity. Page 3 of 133

6 Grade Level: 4 Levels 1, 2, 3 Correlated with Operations and Algebraic Thinking 4.OA Gain familiarity with factors and multiples 4. Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. Unit 5, Lessons 1-4 (pp ) Students find all factor pairs for a whole number in the range Unit 7, Lessons 5-8 (pp ) Students recognize that a whole number is a multiple of each of its factors; determine whether a number is a multiple of a given 1-digit number. Unit 5, Lessons 5-10 (pp ) Students determine whether a given number is a prime or composite. Unit 5, Lessons (pp ) Students determine divisibility rules for 2, 3, 5, 6, and 10. Unit 1, Lessons (pp ) Students extend their understanding of multiples by using least common multiple (LCM) to find common denominators to add fractions. Unit 2, Lessons 5 & 12 (pp , ) Students extend their understanding of factors to simplify fractions using the concept of greatest common factor (GCF). Unit 9, Lesson 9 (pp ) Students review the concepts of prime and composite numbers; extend their understanding of number theory as it pertains to positive and negative integers. Unit 2, Lesson 12 (pp ) Students extend their understanding of factors as they use greatest common factor (GCF) to simplify ratios. Unit 4, Lesson 8 (pp ) Students extend their knowledge of prime numbers as they analyze predictable vs. non-predictable patterns involving different types of numbers, including odd/even, prime, square, and triangular. Page 4 of 133

7 Grade Level: 4 Levels 1, 2, 3 Correlated with Operations and Algebraic Thinking 4.OA Generate and analyze patterns 5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule Add 3 and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. Unit 6, Lessons 7-9; Unit 7, Lesson 1 (pp , ) Students generate a number or shape pattern that follows a given rule as they examine symbolic and visual examples of odd/even numbers, square numbers, and triangular numbers. Students describe the features of the pattern. Unit 9, Lessons 2-9 (pp ) Students extend their work with pattern analysis as they identify the features of the patterns demonstrated by the coordinates of vertices generated by shapes that have been translated, rotated or reflected on a coordinate grid. In Lesson 9, students review patterns observed in number theory topics and how the patterns change when you include positive and negative integers. Unit 2, All; Unit 4, Lesson 1-8 (pp , ) Students extend their work with pattern analysis to identify and generate numeric and non-numeric patterns, describe the rule for the pattern, and identify features of the pattern that were not explicit in the rule itself. Page 5 of 133

8 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 4.NBT Generalize place value understanding for multi-digit whole numbers 1. 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. For example, recognize that = 10 by applying concepts of place value and division. Unit 1, Lessons 1-2 (pp 8-25) Students 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. Unit 1, Lesson 1 (pp 8-14) Students review place value concepts and recognize the relationship from one place to the next on a place value chart. Unit 1, Lesson 1 (pp 8-16) Students review place value with whole numbers. Unit 4, Lesson 3 (pp ) Students extend their knowledge of the place value chart and the relationship from one place to the next to explore numbers to the right of the decimal point; use this information to understand why there is no one-ths place. Page 6 of 133

9 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 4.NBT Generalize place value understanding for multi-digit whole numbers (continued) 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. Unit 1, Lesson 1 (pp 8-16) Students read and write multidigit whole numbers using base ten numerals and number names. Unit 1, Lesson 10 (pp 80-90) Students compare whole numbers based on meanings of the digits using symbols and a number line. Unit 1, Lessons 1-2 (pp 8-22) Students review reading and writing multi-digit whole numbers using base ten numerals, number names, and expanded form. Unit 1, Lesson 1 (pp 8-18) Students review writing whole numbers in expanded form. Unit 1, Lessons 2-4, 6-7, 10, 12-13; Unit 2, Lessons 2-5, 10; Unit 3, Lessons 4-8 (pp 17-40, 47-68, 80-90, , , , and ) Students read and write multidigit whole numbers in expanded form. Page 7 of 133

10 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 4.NBT Generalize place value understanding for multi-digit whole numbers (continued) 3. Use place value understanding to round multi-digit whole numbers to any place. Unit 1, Lessons 10-14; Unit 2, Lessons 6-14; Unit 3, Lessons 9-13; Unit 4, Lessons 6-7 and (pp , , , , and ) Students use place value understanding to round multidigit whole numbers to any place. Unit 1, Lessons 1-2 (pp 8-22) Students review rounding whole numbers. Unit 1, Lesson 1 (pp 8-18) Students review rounding whole numbers. Page 8 of 133

11 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 4.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic 4. Fluently add and subtract multi-digit whole numbers using the standard algorithm. Unit 1, Lessons 8-14; Unit 2, Lessons 5-14 (pp , ) Students fluently add and subtract multi-digit whole numbers using the standard algorithm. Unit 1, Lessons 1-2 (pp 8-18) Students review standard and alternative algorithms for whole number operations. Unit 1, Lesson 1 (pp 8-18) Students review standard and alternative algorithms for whole number operations. 5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Unit 3, All (pp ) Students gain conceptual understanding of whole number multiplication and use strategies for multiplying whole numbers based on place value and properties of operations. Students use symbolic and visual models to illustrate and explain calculations involving multiplication. Unit 1, Lessons 1-2 (pp 8-18) Students review standard and alternative algorithms for whole number operations. Unit 1, Lesson 1 (pp 8-18) Students review standard and alternative algorithms for whole number operations. Page 9 of 133

12 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 4.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic (continued) 6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Unit 4, All (pp ) Students find whole-number quotients and remainders using strategies based on place value, properties of operations, and/or the relationship between multiplication and division. They illustrate and explain the calculations by using symbolic and visual models. Unit 1, Lessons 1-2 (pp 8-18) Students review standard and alternative algorithms for whole number operations. Unit 1, Lesson 1 (pp 8-18) Students review standard and alternative algorithms for whole number operations. Page 10 of 133

13 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Extend understanding of fraction equivalence and ordering 1. Explain why a fraction a/b is equivalent to a fraction (n a)/(n b) by using visual fraction models, with attention to how the 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. Unit 8, Lessons (pp ) Students understand the concept of equivalent fractions using fraction bars and other symbolic and nonsymbolic representations; understand how the number and size of the parts differ even though the 2 fractions themselves are equal; recognize and generate equivalent fractions. Unit 1, Lessons 7-14 (pp ) Students review finding equivalent fractions using various strategies, including multiplying by a fraction equal to 1 using the GCF, and comparing them using fraction bars. Unit 1, Lesson 2 (pp 19-28) Students review concept of equivalent fractions using various methods. Unit 9, Lessons 6-9 (pp ) Students explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b); multiply be a fraction equal to 1 to find common denominators. Page 11 of 133

14 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 4.NBT Generalize place value understanding for multi-digit whole numbers (continued) 2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Unit 8, Lessons 1, 6, & 10 (pp , , and ) Students compare and order fractions with different numerators and denominators using a variety of symbolic and visual models and realize that comparisons are only valid when you refer to the same whole. Students learn about important benchmark fractions. Unit 1, Lessons 3-6; Unit 3, Lesson 1 (pp 23-54, ) Students review comparing and ordering fractions using various symbolic and visual representations, including mixed numbers. Unit 1, Lessons 1-7 (pp 8-82) Students review comparing and ordering fractions. Page 12 of 133

15 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Unit 8, All (pp ) Students are introduced to the concept of part-to-whole relationships as represented by fractions, and explore the meanings of numerator and denominator, including fractions whose numerators are 1 (unit fractions). Unit 3, Lessons 7-8 (pp ) Students understand a fraction a/b with a>1 as a sum of fractions 1/b as they learn to convert improper fractions to mixed numbers. Unit 1, Lessons 1-6 (pp 8-70) Students review concept of fractions. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Unit 9, All (pp ) Students understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Unit 1, Lessons 9-14 (pp ) Students review addition and subtraction of fractions. Unit 1, Lessons 1-2 (pp 8-18) Students review addition and subtraction of fractions. Page 13 of 133

16 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = /8 = 8/8 + 8/8 + 1/8. Unit 8, All (pp ) Students are introduced to the concept of fractions, including counting by fractional parts; e.g., 1/8, 2/8, 3/8, Unit 3, Lessons 7-8 (pp ) Students decompose a fraction into a sum of fractions with the same denominator in more than one way as they explore conversions of improper fractions to mixed numbers; e.g., 5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 or 5/3 = 3/3 + 2/3 or 5/3 = 1 2/3. Unit 1, Lessons 1-6 (pp 8-70) Students review concept of fractions. Page 14 of 133

17 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (continued) c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Units 8 and 9, All (pp ) Students learn the concept of fractions, including equivalent fractions (Unit 8) and learn to add and subtract fractions (Unit 9). Unit 3, Lessons 1-7 (pp ) Students add and subtract mixed numbers using equivalent fractions, properties of operations, and/or the relationship between addition/subtraction. Unit 1, Lessons 1-2 (pp 8-18) Students review addition and subtraction of fractions. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Unit 9, All (pp -1026) Students solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators; using a variety of symbolic and visual representations. Unit 1, Lessons 9-14 (pp ) Students review fraction addition and subtraction by solving word problems using a variety of symbolic and visual representations. Unit 1, Lessons 1-2 (pp 8-18) Students review fraction addition and subtraction by solving word problems using a variety of symbolic and visual representations. Page 15 of 133

18 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (continued) 4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Unit 3, All (pp ) Students build an important conceptual foundation for multiplication by using various symbolic and visual representations of whole number multiplication. Unit 2, Lessons 1-7 (pp ) Students apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Unit 1, Lessons 3-4 (pp 29-50) Students review multiplication with fractions. a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 (1/4), recording the conclusion by the equation 5/4 = 5 (1/4). Units 8 and 9, All (pp ) Students build an important conceptual foundation for fractions; learn to add and subtract fractions. Unit 8, All (pp ) Students are introduced to the concept of fractions, including counting by fractional parts; e.g., 1/b, 2/b, 3/b, Unit 3, Lessons 7-8 (pp ) Students understand a fraction a/b as a multiple of 1/b. Students represent 5/4 as ¼ + ¼ + ¼ + ¼ + ¼ or 5 x (1/4) or 1 ¼ as they work with improper fraction conversions. Unit 1, Lessons 1-6 (pp 8-70) Students review concept of fractions. Page 16 of 133

19 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (continued) b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 (2/5) as 6 (1/5), recognizing this product as 6/5. (In general, n (a/b) = (n a)/b.) Unit 8, All (pp ) Students are introduced to the concept of part-to-whole relationships as represented by fractions, and explore the meanings of numerator and denominator, including fractions whose numerators are 1 (unit fractions). Unit 2, Lessons 1-7 (pp ) Students understand a multiple of a/b as a multiple of 1/b and use this understanding to multiply a fraction by a whole number. Unit 1, Lessons 3-4 (pp 29-50) Students review multiplication with fractions. Page 17 of 133

20 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (continued) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? Unit 3, All (pp ) Students build an important conceptual foundation for multiplication word problems by solving those involving whole number multiplications. Units 8 and 9, All (pp ) Students build important conceptual foundation for fraction multiplication word problems by solving those involving addition and subtraction with fractions. Unit 2, Lessons 1-7 (pp ) Students solve word problems involving multiplication of fractions using a variety of symbolic and visual representations. Unit 1, Lessons 3-4 (pp 29-50) Students review multiplication with fractions, including word problems. Page 18 of 133

21 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Understand decimal notation for fractions, and compare decimal fractions. 5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. Unit 9, Lessons 4-9 (pp ) Students express a fraction with a denominator of 10 as an equivalent fraction with denominator 100, and use this technique to add 2 fractions with respective denominators 10 and 100. Unit 1, Lessons 9-14 (pp ) Students review fraction addition and subtraction, including finding a common denominator. Unit 1, Lessons 1-2 (pp 8-18) Students review fraction addition and subtraction with unlike denominators. 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 a number line diagram. Unit 8, All (pp ) Students are introduced to part-to-whole relationships by working with fractions. Unit 4, Lessons 1-8 (pp ) Students use decimal notation for fractions with denominators 10 or 100. Unit 1, Lesson 7 (pp 71-82) Students review the concept of decimals, including using decimal notation for fractions with denominators of 10 or 100. Page 19 of 133

22 Grade Level: 4 Levels 1, 2, 3 Correlated with Number and Operations Fractions 4.NF Understand decimal notation for fractions, and compare decimal fractions. (continued) 7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Unit 8, All (pp ) Students build important foundations with part-to-whole relationships through conceptual understanding of fractions. Unit 4, All (pp ) Students compare 2 decimals by reasoning about their size; recognize that comparisons with decimals refer to the same whole and use various symbolic and visual models to represent decimals. Unit 1, Lessons 7-10 (pp ) Students review comparing and ordering decimals using various symbolic and visual models. Page 20 of 133

23 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 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 a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. 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),... Unit 3, Lessons 4-6 (pp ) Students know relative sizes of measurements of length in the metric system, including M, mm, and cm, and in the customary system, including ft, in, yd, and express measurements in larger/smaller units within the metric system. Unit 9, All (pp ) Students know relative sizes of measurement units within the customary system of measurement, including time, weight, capacity, and length. Students express measurements in a larger unit. Students reference conversion tables. Unit 2, Lesson 11 (pp ) Students know relative sizes of measurements of length within customary and metric systems. Unit 6, Lessons 2-3 (pp ) Students review relative sizes of measurement units in metric and customary systems as they learn about volume and its measurement in cubic units. Page 21 of 133

24 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit (continued) 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. Unit 3, Lessons 4-6 (pp ) Students use all 4 operations to solve word problems involving lengths measured in metric or customary units, including fractional parts of a unit. Unit 9, All (pp ) Students use the 4 operations to solve word problems involving conversions of measurement within the customary system, including situations resulting in remainders that call for expressing the answer in larger/smaller units of measure; solve problems involving intervals of time, liquid volumes and mass. Unit 2, Lesson 11 (pp ) Students use the 4 operations to sole word problems involving customary and metric measurement; learn about the inverse relationships between the unit of measure and the size of the measurement. Unit 4, Lessons (pp ) Students represent measurement quantities using diagrams, such as a decimal ruler which mimics a number line and features a measurement scale involving decimals. Unit 6, All (pp ) Students solve word problems involving volume; express the answers using cubic units. Page 22 of 133

25 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit (continued) 3. Apply the 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. Unit 4, Lessons (pp ) Students are introduced to the concept of area in real world problems; e.g., blueprints and floor plans for houses; compute area informally by counting squares. Unit 5, All (pp ) Students apply area formulas for rectangles and other shapes in real world and mathematical problems. Units 5 and 6, All (pp ) Students apply area formulas to find surface area and/or volume of 3-D shapes to solve real world and mathematical problems. Unit 5, All (pp ) Students learn important formulas for finding area and perimeter in real-world and mathematical problems. Page 23 of 133

26 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Represent and interpret data. 4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by 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. Unit 8, Lessons (pp ) Students make a line plot to display a data set of measurements in fractions of a unit; solve problems involving information presented in a line plot. Unit 1, Lesson 8 (pp 64-71) Students make a line plot to display data and solve problems involving the data. Unit 1, Lesson 13 (pp ) Students extend their knowledge of line plots as they draw a line of best fit on a scatter plot. Page 24 of 133

27 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Geometric measurement: understand concepts of angle and measure angles. 5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: Unit 6, Lessons 1-2 (pp ) Students build a foundation in 2-dimensional geometry as they examine attributes of shapes and classify them accordingly. Unit 2, Lessons 4-14 (pp ) Students recognize angles as geometric shapes that are formed wherever 2 rays share a common endpoint; understand concepts of angle measurement. Unit 7, Lessons 2, 4, and 7; Unit 8, All (pp , , , ) Students extend their knowledge of angles to include geometric constructions where they bisect an angle; learn properties and relationships of angles created by a transversal intersecting 2 parallel lines. Page 25 of 133

28 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Geometric measurement: understand concepts of angle and measure angles. (continued) a. An angle is 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. An angle that turns through 1/360 of a circle is called a one-degree angle, and can be used to measure angles. Unit 7, Lessons 8-9 (pp ) Students gain foundational knowledge about angle rotations as they identify and explore rotational symmetry in objects. Unit 2, Lessons (pp ) Students construct triangles (e.g., right triangle and equilateral triangle) in the middle of 2 intersecting circles using a compass and a straight edge in a geometric construction; understand the measures of angles with reference to the circle. Unit 3, Lessons 9-12 (pp ) Students learn about angle rotations; one turn is 1/360 of a circle and is called a onedegree angle; learn important benchmark angles. Unit 7, Lessons 2, 4, and 7; Unit 8, All (pp , , , ) Students extend their knowledge of angles to include geometric constructions where they bisect an angle; learn properties and relationships of angles created by a transversal intersecting 2 parallel lines. Page 26 of 133

29 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Geometric measurement: understand concepts of angle and measure angles. (continued) b. An angle that turns through n onedegree angles is said to have an angle measure of n degrees. Unit 7, Lessons 8-9 (pp ) Students gain foundational knowledge about angle rotations as they identify and explore rotational symmetry in objects. Unit 3, Lessons 9-12 (pp ) Students learn about angle rotations; one turn is 1/360 of a circle and is called a onedegree angle; learn important benchmark angles. Unit 7, Lessons 2, 4, and 7; Unit 8, All (pp , , , ) Students extend their knowledge of angles to include geometric constructions where they bisect an angle; learn properties and relationships of angles created by a transversal intersecting 2 parallel lines. 6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. Unit 6, Lessons 1-2 (pp ) Students build a foundation in 2-dimensional geometry as they examine attributes of shapes and classify them accordingly. Unit 2, Lessons 6-14 (pp ) Students measure angles in whole number degrees using a protractor; sketch angles of specified measure. Unit 7, Lessons 2, 4, and 7; Unit 8, All (pp , , , ) Students extend their knowledge of measuring angles to bisect an angle and understand angle relationships. Page 27 of 133

30 Grade Level: 4 Levels 1, 2, 3 Correlated with Measurement and Data 4.MD Geometric measurement: understand concepts of angle and measure angles. (continued) 7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping 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 a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. Unit 6, Lessons 1-2 (pp ) Students build a foundation in 2-dimensional geometry as they examine attributes of shapes and classify them accordingly. Unit 5, Lesson 8 (pp ) Students understand the additive nature of angle measurement; solve addition and subtraction problems to find unknown angles in diagrams. Unit 7, Lessons 2, 4, and 7; Unit 8, All (pp , , , ) Students extend their knowledge of the additive nature of angles as they look at supplementary, complementary, vertical and adjacent angles. Page 28 of 133

31 Grade Level: 4 Levels 1, 2, 3 Correlated with Geometry 4.G Draw and identify lines and angles, and classify shapes by properties of their lines and angles. 1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. Unit 6, Lessons 1-2 (pp ) Students build a foundation in 2-dimensional geometry as they examine attributes of shapes and classify them accordingly. Unit 6, Lessons 1-2 (pp ) Students build a foundation in 2-dimensional geometry as they examine attributes of shapes and classify them accordingly. Unit 2, Lessons 1-6 (pp ) Students draw points, lines, line segments, rays, angles (right, acute, obtuse). Unit 4, Lessons 3-8 (pp ) Students draw perpendicular and parallel lines; identify these in 2-dimensional figures. Unit 2, Lessons 6, (pp , ) Students recognize right triangles as a category and identify right triangles. Unit 4, All (pp ) Students classify 2-dimensional figures based on the presence or absence of parallel or perpendicular lines, or presence or absence of angles of a specified size. Units 7 and 8, All (pp ) Students learn and apply important properties of lines and angles. Unit 10, Lessons 1-3 (pp ) Students recognize right triangles as a category, identify right triangles, learn and apply important properties of right triangles. Page 29 of 133

32 Grade Level: 4 Levels 1, 2, 3 Correlated with Geometry 4.G Draw and identify lines and angles, and classify shapes by properties of their lines and angles. (continued) 3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. Unit 7, Lessons 4-7 (pp ) Students recognize a line of symmetry for a 2-D figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. Unit 3, Lesson 9; Unit 8, Lesson 12 (pp , ) Students review concept of symmetry; extend the concept to geometric transformations on a coordinate grid. Unit 9, Lesson 3 (pp ) Students extend the concept of symmetry to reflections on a coordinate grid. Page 30 of 133

33 Grade Level: 5 Levels 1, 2, 3 Correlated with GRADE 5 Operations and Algebraic Thinking 5.OA Write and interpret numerical expressions. 1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 ( ) is three times as large as , without having to calculate the indicated sum or product. Pre-Skill Unit 3, Lesson 1 (pp ) Students learn important concepts about the order in which expressions are solved; e.g., changing the order or the commutative property works for addition and multiplication but not division or subtraction. Pre-Skill Unit 1, Lesson 2; Unit 2, Lesson 1; Unit 3, Lesson 1, Unit 4, Lessons 2, 4, (pp 17-26, , , , , ) Students learn important preskills for interpreting numerical expressions without evaluating them as they solve extended facts based on their knowledge of basic facts. Pre-Skill Unit 5, Lesson 6 (pp ) Students use grouping symbols in the area formula for a trapezoid. Pre-Skill Unit 5, Lessons 1-9 (pp ) Students learn important preskills about writing simple expressions that record calculations as they derive area formulas for shapes from area formulas of other shapes; e.g., the area of a triangle is half that of a rectangle, A=1/2(b x h) Unit 5, Lesson 2-3 (pp ) Students use parentheses in numeric expressions and evaluate expressions based on order of operations. Unit 2, Lessons 1-2, 7-8; Unit 3, Lessons 1-3; Unit 5, Lesson 6; Unit 7, Lesson 9; Unit 8, Lessons 4-7 (pp , , , , , ) Students learn to write simple expressions to record calculations with numbers and variables; interpret numerical situations without evaluating them. Page 31 of 133

34 Grade Level: 5 Levels 1, 2, 3 Correlated with Operations and Algebraic Thinking 5.OA Analyze patterns and relationships. 3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. Unit 6, Lessons 7-8, Unit 7, Lessons 1, 5-8; Unit 9, Lesson 8 ( , , , ) Students identify relationships in numerical patterns, such as odd/even, square and triangular numbers; generate two numerical patterns as they list multiples for 2 numbers, find common multiples, and find common denominators. Unit 9, Lessons 2-5, 7-8 (pp , ) Students look at the numerical patterns in the coordinates of shapes transformed on a coordinate grid. Unit 9, All (pp ) Students work with functions using various representations graphical, tabular, symbolic and verbal. Page 32 of 133

35 Grade Level: 5 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 5.NBT Understand the place value system. 1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Unit 1, Lessons 1-2, Unit 7, Lesson 4 (pp 1-25, ) Students recognize that in a multidigit number, a digit in one place represent 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Unit 1, Lesson 2; Unit 2, Lesson 3, Unit 3, Lessons 1-3; Unit 4, Lessons 2-4 (pp 17-25, , , ) Students explain patterns in the number of zeros in products when multiplying by powers of 10; e.g., as in extended facts. Unit 7, All (pp ) Students use whole-number exponents to denote powers of 10 Unit 4, Lesson 3 (pp ) Students recognize that in a multi-digit number, a digit in one place represent 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left as they explore decimals in a place value chart and why there is no one-ths place. Unit 5, Lessons 8 and 13 (pp , ) Students explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Unit 1, Lesson 8 (pp 83-94) Students review concepts of place value. Unit 1, Lessons Students review multiplication and division of decimals; explain the patterns in the placement of the decimal point. Page 33 of 133

36 Grade Level: 5 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 5.NBT Understand the place value system. (continued) 3. Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., = (1/10) + 9 (1/100) + 2 (1/1000). Unit 8, All (pp ) Students build important foundations with part-to-whole relationships through conceptual understanding of fractions. Unit 8, All (pp ) Students build important foundations with part-to-whole relationships through conceptual understanding of fractions. Unit 8, All (pp ) Students build important foundations with part-to-whole relationships through conceptual understanding of fractions. Unit 4, All (pp ) Students read, write, and compare decimals to thousandths. Unit 4, All (pp ) Students read and write decimals using base-ten numerals, number names and expanded form. Unit 1, Lessons 7-10 (pp ) Students review concepts with decimals, including reading, writing and comparing. Unit 1, Lessons 7-10 (pp ) Students review concepts with decimals, including reading and writing in various forms. b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Unit 4, All (pp ) Students compare 2 decimals based on meanings of the digits in each place using various symbolic and visual representations. Unit 1, Lessons 7-10 (pp ) Students review concepts with decimals, including comparing them. Page 34 of 133

37 Grade Level: 5 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 5.NBT Understand the place value system. (continued) 4. Use place value understanding to round decimals to any place. Unit 1, Lessons 10-14; Unit 2, Lessons 6-14; Unit 3, Lessons 9-13; Unit 4, Lessons 6-7 and (pp , , , , and ) Students gain important foundational knowledge about rounding rational numbers by learning strategies for rounding whole numbers. Unit 4, Lessons 12-14; Unit 5, Lessons 2-3, 14 (pp , , and ) Students use place value understanding to round decimals to any place. Unit 1, Lessons 7-10 (pp ) Students review concepts with decimals, including rounding. Unit 8, All (pp ) Students build important foundations with part-to-whole relationships through conceptual understanding of fractions. Page 35 of 133

38 Grade Level: 5 Levels 1, 2, 3 Correlated with Number and Operations in Base Ten 5.NBT Perform operations with multi-digit whole numbers and with decimals to hundredths. 5. Fluently multiply multi-digit whole numbers using the standard algorithm. Unit 3, Lessons 8-14 (pp ) Students fluently multiply multidigit whole numbers using the standard algorithm. Unit 1, Lesson 2 (pp 15-22) Students review multiplication with multi-digit whole numbers using the standard algorithm. Unit 1, Lesson 1 (pp 8-18) Students review multiplication with multi-digit whole numbers using the standard algorithm. 6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Unit 4, All (pp 15-22) Students find whole-number quotients of whole numbers with multi-digit dividends and/or divisors, using strategies based on place value, the properties of operations, and/or relationship between multiplication and division using various models. Unit 1, Lesson 2 (pp 15-22) Students review division with multi-digit whole numbers using various models and strategies. Unit 1, Lesson 1 (pp 8-18) Students review division with multi-digit whole numbers using various models and strategies. 7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Unit 4, All (pp 15-22) Students review division with multi-digit whole numbers using various models and strategies. Unit 5, All (pp ) Students add, subtract, multiply, and divide decimals to hundredths, using various models and strategies; relate the strategy to a written method and explain the reasoning used. Unit 1, Lessons 9-14 (pp ) Students review operations with decimals. Page 36 of 133

39 Grade Level: 5 Levels 1, 2, 3 Correlated with Number and Operations Fractions 5.NF Use equivalent fractions as a strategy to add and subtract fractions. 1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) Unit 9, Lessons 4-9 (pp ) Students add and subtract fractions with unlike denominators. Unit 1, Lessons 9-14 (pp ) Students review addition and subtraction with fractions with unlike denominators. Unit 3, Lessons 2-7 (pp ) Students add and subtract mixed numbers. Unit 1, Lessons 1-2 (pp 8-17) Students review addition and subtraction with fractions; find common denominators. 2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Unit 9, Lessons 4-9 (pp ) Students solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators; use fraction bars or equations to represent the problem; use benchmark fractions and number sense to estimate mentally and assess reasonableness of answers. Unit 1, Lessons 9-14 (pp ) Students review addition and subtraction with fractions with unlike denominators in various situations, including word problems. Unit 3, Lessons 2-7 (pp ) Students add and subtract mixed numbers in various situations, including word problems. Unit 1, Lessons 1-2 (pp 8-17) Students review addition and subtraction with fractions in various problem situations, including word problems. Page 37 of 133