correlation of math in focus™ - hmhco.com

Correlation of Math In Focus™ to the Common Core State Standards Attached are grade level correlations showing how closely Math In Focus™ covers the skills and concepts outlined in the Common Core State Standards. But it is equally important to recognize the parallel assumptions behind the Common Core and Math In Focus™. In fact, the Singapore curriculum was one of the 15 national curriculums examined by the committee and had a particularly important impact on the writers because Singapore is the top performing country in the world and the material is in English.

Overall, the CCSS are well aligned to Singapore’s Mathematics Syllabus. Policymakers can be assured that in adopting the CCSS, they will be setting learning expectations for students that are similar to those set by Singapore in terms of rigor, coherence and focus. – Achieve (achieve.org/CCSSandSingapore)

—Achieve*, (achieve.org/CCSSandSingapore) Here are the parallel assumptions: 1, Curriculum must be focused and coherent: Common Core State Standards:

For over a decade, research studies of mathematics education in high performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. (Common Core State Standards for Mathematics, 3)

Math In Focus™ is organized to teach fewer topics in each grade but to teach them thoroughly. When a concept appears in a subsequent grade level, it is always at a higher level. For instance, first grade does not address fractions, second grade covers what a fraction is, third grade covers equivalent fractions and fractions of a set, fourth grade deals with mixed fractions, and addition of simple fractions, while fifth grade teaches addition, subtraction, and multiplication of fractions as well as division of fractions by whole numbers. This is the coherence and focus that the standards call for.

2. Teach to mastery Common Core State Standards:

In grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes. (Common Core State Standards for Mathematics, 17) In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100;(2)developing understanding of fractions, especially unit fractions…;(3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing tw0-dimensional shapes (Common Core State Standards for Mathematics, 21)

Math In Focus™ has the identical structure. Rather than repeating topics, students master them in a grade level, and subsequent grades develop them to more advanced levels. Adding another digit is NOT an example. Moving from addition/subtraction in second grade to multiplication/division in third grade is such an example. Students continue to practice all the operations with whole numbers in every grade in the context of problem solving. 3. Focus on number, geometry and measurement in elementary grades Common Core State Standards:

Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to number than to other topics. (Common Core State Standards for Mathematics, 3)

Math In Focus™ emphasizes number and operations in every grade K-5 just as recommended in the CCSS. The textbook is divided into two books roughly a semester each. Approximately 75% of Book A is devoted to number and operations and 60-70% of Book B to geometry and measurement where the number concepts are practiced. The key number topics are in the beginning of the school year so students have a whole year to master them.

4. Organize content by big ideas such as place value Common Core State Standards:

These Standards endeavor to follow such a design, not only by stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value or the properties of operations to structure those ideas. (Common Core State Standards for Mathematics, 4)

Math In Focus™ is organized around place value and the properties of operations. The first chapter of each grade level from second to fifth begins with place value. In first grade, students learn the teen numbers and math facts through place value. In all the grades, operations are taught with place value materials so students understand how the standard algorithms work. Even the mental math that is taught uses understanding of place value to model how mental arithmetic can be understood and done. 5. Curriculum must include both conceptual understanding and procedural fluency. Common Core State Standards:

The Standards for Mathematical Content are a balanced combination of procedure and understanding (Common Core State Standards for Mathematics, 8)

Math In Focus™ is built around the Singapore Ministry of Education’s famous pentagon that emphasizes conceptual understanding, skill development, strategies for solving problems, attitudes towards math, and metacognition that enable students to become excellent problem solvers. The highly visual nature of the text and the consistent concrete to visual to abstract approach enables all students to both understand how procedures work and to fluently apply them to solve problems.

6. Mathematics is about reasoning Common Core State Standards:

These Standards define what students should understand and be able to do in their study of mathematics....One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity. (Common Core State Standards for Mathematics, 4)

Math In Focus™ is famous for its model drawing to solve problems and to enable students to justify their solutions. In addition to journal questions and other explicit opportunities to explain their thinking, students are systematically taught to use visual diagrams to represent mathematical relationships in such a way as to accurately solve problems, but also to explain their thinking. Works Cited:

1. "Common Core State Standards For Mathematics" Common Core State Standards Initiative | Home. 2 June 2010. Web. 26 July 2010. <http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf>.

1 Math in Focus is published by Marshall Cavendish International (Singapore) and exclusively distributed by Houghton Mifflin Harcourt

Common Core State Standards for Mathematics© Copyright 2010.

National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved

Houghton Mifflin Harcourt Specialized Curriculum Math in Focus, Grade 4 ©2013

Common Core Edition

correlated to the

Common Core State Standards for Mathematics Grade 4

Standards Descriptor Page Citations

Standards for Mathematical Practice MP.1 Make sense of problems and persevere in solving them. How Math in Focus Aligns: Math in Focus is built around the Singapore Ministry of Education’s mathematics framework pentagon, which places mathematical problem solving at the core of the curriculum. Encircling the pentagon are the skills and knowledge needed to develop successful problem solvers, with concepts, skills, and processes building a foundation for attitudes and metacognition. Math in Focus is based on the premise that in order for students to persevere and solve both routine and non-routine problems, they need to be given tools that they can use consistently and successfully. They need to understand both the how and the why of math so that they can self-monitor and become empowered problem solvers. This in turn spurs positive attitudes that allow students to solidify their learning and enjoy mathematics. Math in Focus teaches content through a problem solving perspective. Strong emphasis is placed on the concrete-to-pictorial-to-abstract progress to solve and master problems. This leads to strong conceptual understanding. Problem solving is embedded throughout the program.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 17, 19,22, 43, 62-63, 76, 109-115, 116, 128-129,

131, 134-139, 140-145, 150, 151, 158, 169, 173, 176-191, 204-214, 215, 259-269

Workbook 4A: 9A, 20A, 20B, 22A, 43B, 63A, 79C, 115A, 117,

139A, 150-150A, 151, 158D, 173B, 203A, 214–214C, 215A, 267–267B, 269A

SE/TE-4B: 26-30, 34, 48-49, 52, 76, 101, 121, 144, 148, 188-

189, 208, 210-215, 217, 235 Workbook 4B: 34A, 50, 76, 76A, 79B, 102, 121A, 121-B, 144A,

187A-187B, 189A-189B, 214, 235A

Houghton Mifflin Harcourt Specialized Curriculum Math in Focus, Grade 4 ©2013 – Common Core Edition correlated to the Common Core State Standards for Mathematics, Grade 4




Standards Descriptor Page Citations MP.2 Reason abstractly and quantitatively. How Math in Focus Aligns: Math in Focus’ concrete-pictorial-abstract progression helps students effectively contextualize and decontextualize situations by developing a deep mastery of concepts. Each topic is approached with the expectation that students will understand both how it works, and also why. Students start by experiencing the concept through hands-on manipulative use. Then, they must translate what they learned in the concrete stage into a visual representation of the concept. Finally, once they have gained a strong understanding, they are able to represent the concept abstractly. Once students reach the abstract stage, they have had enough exposure to the concept and they are able to manipulate it and apply it in multiple contexts. They are also able to extend and make inferences; this prepares them for success in more advanced levels of mathematics. They are able to both use the symbols and also understand why they work, which allows students to relate them to other situations and apply them effectively.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 32-43, 92-95, 105-108, 109, 140-150, 157-158,

224-229, 250-254, 255-258, 259-267, 268-269, Workbook 4A: 20B, 43A, 43B, 63A, 95A, 108A, 258A, 267A,

267B, 273, 273D, 273E SE/TE-4B: 47, 50, 93, 94-97, 100-101, 103-106, 152-153,

154-155, 157, 162, 176, 178-185, 187, 190 Workbook 4B: 71B, 93A, 162A, 169A, 169B





Standards Descriptor Page Citations MP.3 Construct viable arguments and critique the reasoning of others. How Math in Focus Aligns: As seen on the Singapore Mathematics Framework pentagon, metacognition is a foundational part of the Singapore curriculum. Students are taught to self-monitor, so they can determine whether or not their solutions make sense. Journal questions and other opportunities to explain their thinking are found throughout the program. Students are systematically taught to use visual diagrams to represent mathematical relationships in such a way as to not only accurately solve problems, but also to justify their answers. Chapters conclude with a Put on Your Thinking Cap! problem. This is a comprehensive opportunity for students to apply concepts and present viable arguments. Games, explorations, and hands-on activities are also strategically placed in chapters when students are learning concepts. During these collaborative experiences, students interact with one another to construct viable arguments and critique the reasoning of others in a constructive manner. In addition, thought bubbles provide tutorial guidance throughout the entire Student Book. These scaffolded dialogues help students articulate concepts, check for understanding, analyze, justify conclusions, and self-regulate if necessary.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 9, 17, 51, 84, 93, 131, 138, 145, 166, 170, 176,

190, 196, 202, 231, 240 Workbook 4A: 20B, 115B, 267B SE/TE-4B: 33, 39, 87, 91, 96, 99, 113, 117, 133, 135-136,

142, 156, 160, 174, 184, 199, 201, 206-207, 211, 225, 229, 233

Workbook 4B: 162A, 267A





Standards Descriptor Page Citations MP.4 Model with mathematics. How Math in Focus Aligns: Math in Focus follows a concrete-pictorial-abstract progression, introducing concepts first with physical manipulatives or objects, then moving to pictorial representation, and finally on to abstract symbols. A number of models are found throughout the program that support the pictorial stage of learning. Math in Focus places a strong emphasis on number and number relationships, using place-value manipulatives and place-value charts to model concepts consistently throughout the program. In all grades, operations are modeled with place-value materials so students understand how the standard algorithms work. Even the mental math instruction uses understanding of place value to model how mental arithmetic can be understood and done. These place-value models build throughout the program to cover increasingly complex concepts. Singapore math is also known for its use of model drawing, often called “bar modeling” in the U.S. Model drawing is a systematic method of representing word problems and number relationships that is explicitly taught beginning in Grade 2 and extends all the way to secondary school. Students are taught to use rectangular “bars” to represent the relationship between known and unknown numerical quantities and to solve problems related to these quantities. This gives students the tools to develop mastery and tackle problems as they become increasingly more complex.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 45, 78-82, 85, 86-88, 94, 109-113, 115, 118-

119, 126-133, 140-145, 146-147, 148-150, 154-156, 166, 176, 178-182, 184-186, 192-193, 204-205, 207, 214, 224-229, 230-232, 234-236, 237-242, 243-249, 250-252, 255-258, 259-260, 262-266, 268-269

Workbook 4A: 85A, 95A, 115A, 116A, 133A, 150A, 158C,

158D, 158E, 186A, 186B, 214B, 236-236A, 242A, 258A, 267B, 269A, 273C, 273D

SE/TE-4B: 6-9, 11-12, 14-18, 21-22, 25, 32, 35-41, 42-45,

56-59, 61-62, 65-66, 69, 73-75, Workbook 4B: 12A-12A, 23A, 37, 41





Standards Descriptor Page Citations MP.5 Use appropriate tools strategically. How Math in Focus Aligns: Math in Focus helps students explore the different mathematical tools that are available to them. New concepts are introduced using concrete objects, which help students break down concepts to develop mastery. They learn how to use these manipulatives to attain a better understanding of the problem and solve it appropriately. Math in Focus includes representative pictures and icons as well as thought bubbles that model the thought processes students should use with the tools. Several examples are listed below. Additional tools referenced and used in the program include clocks, money, dot paper, place-value charts, geometric tools, and figures.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 5-7, 9, 10, 71–75, 77-79, 86, 96-99, 131, 138, 196 Workbook 4A: 22A SE/TE-4B: 88-93, 94-97, 99, 102, 104, 106, 111-118, 121-

122, 124, 113, 135-136, 156, 168 Workbook 4B: 93A, 97A, 114A, 118A

MP.6 Attend to precision. How Math in Focus Aligns: As seen in the Singapore Mathematics Framework, metacognition, or the ability to monitor one’s own thinking, is key in Singapore math. This is modeled for students throughout Math in Focus through the use of thought bubbles, journal writing, and prompts to explain reasoning. When students are taught to monitor their own thinking, they are better able to attend to precision, as they consistently ask themselves, “does this make sense?” This questioning requires students to be able to understand and explain their reasoning to others, as well as catch mistakes early on and identify when incorrect labels or units have been used. Additionally, precise language is an important aspect of Math in Focus. Students attend to the precision of language with terms like factor, quotient, difference, and capacity.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 9, 17, 40, 51, 52, 84, 93, 107, 114, 131, 138, 145,

166, 170, 171, 176, 177, 180, 190, 195, 196, 200, 202, 231, 240

SE/TE-4B: 31, 32, 39, 40, 87, 91, 96, 99, 113, 117, 133, 135-

136, 142, 156, 160, 168, 174, 184, 185, 199, 201, 206-207, 211, 225, 226, 229, 233





Standards Descriptor Page Citations MP.7 Look for and make use of structure. How Math in Focus Aligns: The inherent pedagogy of Singapore math allows students to look for, and make use of, structure. Place value is one of the underlying principles in Math in Focus. Concepts in the program start simple and grow in complexity throughout the chapter, year, and grade. This helps students master the structure of a given skill, see its utility, and advance to higher levels. Many of the models in the program, particularly number bonds and bar models, allow students to easily see patterns within concepts and make inferences. As students progress through grade levels, this level of structure becomes more advanced.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 5-9, 10-13, 14-18, 21, 23-24, 70-75, 77-79, 86,

96-99, 127-133, 134-139, 142, 145, 149-151, 152, 155-156, 174-175, 177, 183, 186

Workbook 4A: 9A, 13A, 20A, 20B, 85B, 100A, 133A, 139A,

158B, 158C, 173A-173B, 186A, 273D, 273E SE/TE-4B: 24-29, 34, 35-41, 56-64, 65-67, 69-71, 11, 133-

135, 138, 142, 158-162, 167, 168, 170, 173, 177-178, 184, 186, 193, 208-215, 217

Workbook 4B: 34A, 61, 64A, 71A-71B, 139A, 150, 162-162A,

187A, 189A, 189B, 239B, 239G





Standards Descriptor Page Citations MP.8 Look for and express regularity in repeated reasoning. How Math in Focus Aligns: A strong foundation in place value, combined with modeling tools such as bar modeling and number bonds, gives students the foundation they need to look for and express regularity in repeated reasoning. Operations are taught with place value materials so students understand how the standard algorithms work in all grades. Even the mental math instruction uses understanding of place value to model how mental arithmetic can be understood and done. This allows students to learn shortcuts for solving problems and understand why they work. Additionally, because students are given consistent tools for solving problems, they have the opportunity to see the similarities in how different problems are solved and understand efficient means for solving them. Throughout the program, students see regularity with the reasoning and patterns between the four key operations. Students continually evaluate the reasonableness of solutions throughout the program; the consistent models for solving, checking, and self-regulation help them validate their answers.

This standard is covered throughout the program; the following are examples. SE/TE-4A: 77-85, 86-95, 96-100, 101-108, 109-115, 116-

119 Workbook 4A: 85A, 95A, 100A, 100B, 108A, 115A, 115B,

158B, 158D, 273B, 273D, 273F, 273G SE/TE-4B: 56-64, 65-71, 72-76, 77-79, 152-157, 161, 163-

165, 166-169, 170-175, 176-187, 188-189, 191-193,

Workbook 4B: 64A, 64B, 71A, 71B, 76A, 79B, 162, 162A,

1165A, 169A, 169B, 175A, 187A, 189A, 189B, 239B, 239C, 239D, 239F, 239G





Standards Descriptor Page Citations Standards for Mathematical Content 4.OA Operations and Algebraic Thinking Use the four operations with whole numbers to solve problems.

4.OA.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.

SE/TE-4A: 77-85, 86-88, 115, 116, 117 Workbook 4A: 85A, 115A, 115B

4.OA.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.

SE/TE-4A: 80-81, 86, 90, 109-116, 263-267 Workbook 4A: 115A-115B, 267A SE/TE-4A: Common Core Focus Lesson Appendix Chapter

3, Lesson 5.a

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 a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

SE/TE-4A: 41-43, 62-63, 109-116, 119, 204-214, 263-267 Workbook 4A: 43B, 63A, 115A-115B, 117, 119A, 214-214C,

267A SE/TE-4B: 72-75, 139-143, 176-187 Workbook 4B: 76, 143A-143B, 187A SE/TE-4A: Common Core Focus Lesson Appendix Chapter

3, Lesson 5.a





Standards Descriptor Page Citations Gain familiarity with factors and multiples.

4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

SE/TE-4A: 50-51, 53-54, 55, 65, 67-68, 44, 56-57 Workbook 4A: 54A, 61A, 68B, 68C, 273D

Generate and analyze patterns. 4.OA.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.

SE/TE-4A: 6, 8, 14-17, 19, 26 Workbook 4A: 9A, 20A, 20B, 63A, 63B SE/TE-4B: 26-30, 34, 52, 208, 210-215, 217, 220. 221-227,

228-235 Workbook 4B: 34A, 50, 227-227A, 235A-235C





Standards Descriptor Page Citations 4.NBT Number and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers.

4.NBT.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.

SE/TE-4A: 5-9, 10-13, 14-20, 21-22, 22A, 23-26, 32-34, 62, 77-79, 81-82, 86, 96-99

Workbook 4A: 13A, 20A, 20B, 22A, 68B, 273D SE/TE-4B: 6-9, 11-12, 14-19, 22-23, 28-29, 32, 34, 54-57,

59, 61-62, 65-66, 69 Workbook 4B: 12A, 23A, 34A

4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

SE/TE-4A: 5-9, 10-13, 14-20, 21-22, 23-26, 32-34, 62, 77-79, 81-82, 86, 96-99

Workbook 4A: 9A, 13A, 20A, 20B, 22A, 68B, 273D SE/TE-4B: 6-9, 11-12, 14-19, 22-23, 28-29, 32, 34, 54-57,

59, 61-62, 65-66, 69 Workbook 4B: 12A, 23A, 34A

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

SE/TE-4A: 32-33, 35-36, 43, 63, 64, 66-67, 93, 95, 116 Workbook 4A: 43A, 63A, 68B, 95A, 273F





Standards Descriptor Page Citations Use place value understanding and properties of operations to perform multi-digit arithmetic.

4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

SE/TE-4A: 32-35, 41-43, 64, 66-67, 68, 88-95 Workbook 4A: 43A, 43B, 68B, 95A, 100A-100B, 273D, 273F SE/TE-4B: 56-64, 64A-64B, 65-71, 71A-71B, 170-175,

175A SE/TE-4A: Common Core Focus Lesson Appendix Chapter

1, Lessons 2.a, 2.b

4.NBT.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.

SE/TE-4A: 78-85, 87-89, 91-95, 109, 116-119 Workbook 4A: 85A, 95A, 115A, 115B, 158B, 158D, 273B,

273C, 273G SE/TE-4B: 149-150 SE/TE-4A: Common Core Focus Lesson Appendix Chapter

3, Lessons 3.0, 3.1.a

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, 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.

SE/TE-4A: 96-100, 101-108, 109-113, 115, 117-119 Workbook 4A: 100A, 100B, 108A, 115A, 115B, 158B, 158D,

273D, 273F, 273G





Standards Descriptor Page Citations 4.NF Number and Operations – Fractions Extend understanding of fraction equivalence and ordering.

4.NF.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.

SE/TE-4A: 199, 202-203, 210-211, 224-229, 236, 250-254, 259-262, 269-270, 273

Workbook 4A: 203A, 214C, 249A, 267, 267A, 267B, 273B,

273C SE/TE-4B: 42-44, 47 Workbook 4B: 47, 239D

4.NF.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.

SE/TE-4A: Common Core Focus Lesson Appendix Chapter 6, Lesson 6.0

This standard covered to mastery in grade 3 See Grade 3: SE/TE-3B: 130-146 Workbook 3B: 146A, 147, 156A, 218B





Standards Descriptor Page Citations Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3.a Understand addition and subtraction of fractions as

joining and separating parts referring to the same whole.

SE/TE-4A: 224-229, 230, 232, 237-238, 250 251-253 259-260, 269

Workbook 4A: 236, 236A, 242A, 269A SE/TE-4B: 43-45

4.NF.3.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.

SE/TE-4A: 237-238, 243-248, 270 Workbook 4A: 242A, 249A

4.NF.3.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.

The skills and concepts presented on these pages prepare students to address the state standard in more depth at grade 5: SE/TE-4A: 140-144, 145-149, 151-153, 156, 159, 250-254 Workbook 4A: 144A, 149A, 153B, 153C, 153D, 155A, 203C,

203D, 315F, 315G

4.NF.3.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.

SE/TE-4A: 259-260 Workbook 4A: 267, 267A, 273D





Standards Descriptor Page Citations 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

4.NF.4.a Understand a fraction a/b as a multiple of 1/b. SE/TE-4A: 237-238, 243-248, 270 Workbook 4A: 242A, 249A

4.NF.4.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.

SE/TE-4A: 255-258, 263-267, 271, 273 Workbook 4A: 258A, 267A, 267B, 269A

4.NF.4.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.

SE/TE-4A: 255, 263-267, 271, 273 Workbook 4A: 267A, 267B SE/TE-4A: Common Core Focus Lesson Appendix Chapter

6, Lesson 7.a





Standards Descriptor Page Citations Understand decimal notation for fractions, and compare decimal fractions.

4.NF.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 100.

SE/TE-4B: 13–19, 62

4.NF.6 Use decimal notation for fractions with denominators 10 or 100.

SE/TE-4B: 4-9, 11, 12, 13-19, 22-23, 42-45, 47, 50-52 Workbook 4B: 12A, 23A, 79A, 239D

4.NF.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.

SE/TE-4B: 14-19, 25-26, 28-29, 32, 34, 50, 52 Workbook 4B: 34A, 79B





Standards Descriptor Page Citations 4.MD Measurement and Data Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

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 a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.

SE/TE-4A: 235, 261-262, 264, 267, 270 Workbook 4A: 236A, 267A, 269A, 273C, 273F SE/TE-4B: 9, 35, 37, 41, 76, 79, 129-130, 134, 137, 140-143,

144, 145, 147, 152-162, 163-165, 166-169, 170-173, 175, 176-187, 188-189, 191-193

Workbook 4B: 12A, 138A, 143B, 144A, 147B, 162A, 165A,

169A, 169B, 175A, 187A, 189A, 189B, 239B, 239C, 239D, 239E, 239G

SE/TE-4B: Common Core Focus Lesson Appendix Chapter

12, Lessons 12.0.a, 12.0.b, 12.0.c





Standards Descriptor Page Citations 4.MD.2 Use the four operations to solve word problems

involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

SE/TE-4A: 261-262, 264, 267, 270 Workbook 4A: 267A, 269A, 273C, 273F SE/TE-4B: 76, 78-79, 129-130, 134, 137, 140-143, 144, 145,

147, 152-162, 163-165, 166-169, 170-173, 175, 176-187, 188-189, 191-193

Workbook 4B: 12A, 76A, 79B, 138A, 143B, 144A, 147B, 162A,

165A, 169A, 169B, 175A, 187A, 189A, 189B, 239B, 239C, 239D, 239E, 239G

SE/TE-4B: Common Core Focus Lesson Appendix Chapter

12, Lesson 12.0.d

4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

SE/TE-4B: 152-157, 160-162, 163-165, 166-169, 170-175, 176, 178-187, 189, 191-193

Workbook 4B: 162A, 165A, 169A, 169B, 175A, 187A, 188,

189A, 189B, 239B, 239C, 239E, 239G





Standards Descriptor Page Citations Represent and interpret data.

4.MD.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.

SE/TE-4B: Common Core Focus Lesson Appendix Chapter 6, Lesson 8.a

This standard is covered to mastery in grade 3: See Grade 3: SE/TE-3B: 97-104, 106, 110, 111 Workbook 3B: 104A, 104B, 104C, 105A, 387F

Geometric measurement: understand concepts of angle and measure angles. 4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of

angle measurement: 4.MD.5.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.

SE/TE-4B: 88, 98, 100, 102, 104 SE/TE-4B: Common Core Focus Lesson Appendix Chapter

9, Lesson 3.a

4.MD.5.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

SE/TE-4B: 98-102, 104 Workbook 4B: 147B

4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

SE/TE-4B: 88-93, 94-97, 99, 102, 104 Workbook 4B: 93A, 97A





Standards Descriptor Page Citations 4.MD.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.

SE/TE-4B: 97, 98, 139-140, 147 Workbook 4B: 97A, 143A-143B, 147-147A SE/TE-4B: Common Core Focus Lesson Appendix Chapter

9, Lesson 3.b

4.G Geometry Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

SE/TE-4B: 87-93, 94-97, 99-100, 104, 111-114, 115-118, 121, 122, 124

Workbook 4B: 93A, 97A, 114A, 118A, 121A, 121B, 147A,

147B, 239D, 239E, 239F

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 a specified size. Recognize right triangles as a category, and identify right triangles.

SE/TE-4B: 114, 116, 129-132, 137-138, 145-147 Workbook 4B: 121B, 138A, 143B

4.G 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.

SE/TE-4B: 197-202, 209, 212-213, 215-216 Workbook 4B: 202A, 239B

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