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Title: Properties of Multiplication
Description:
This lesson will introduce students to three of the four properties of multiplication. It will provide practice for students to identify and distinguish between the properties, and solve multiplication problems using each property.
This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.
Standard(s): [TC2] (35) 1: Use input and output devices of technology systems. [TC2] (35) 5: Practice safe use of technology systems and applications. [TC2] (35) 6: Describe social and ethical behaviors related to technology use. [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4]
Subject: Mathematics (3), or Technology Education (3  5)
Title: Properties of Multiplication
Description: This lesson will introduce students to three of the four properties of multiplication. It will provide practice for students to identify and distinguish between the properties, and solve multiplication problems using each property.
This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.
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Title: A Math and Language Arts lesson: Introduction to Division
Description:
The lesson will help students develop an initial understanding of division and clarify how the four operations of addition, subtraction, multiplication, and division relate to and are separate from each other. The lesson begins with a brainstorming discussion which builds background and fosters comprehension. A big book, The Doorbell Rang, by Pat Hutchins, is used along with manipulatives to provide instruction at concrete and pictorial levels. Students will demonstrate what they have learned by writing a short story incorporating simple division.
Standard(s): [ELA2013] (3) 2: Recount stories, including fables, folktales, and myths from diverse cultures; determine the central message, lesson, or moral and explain how it is conveyed through key details in the text. [RL.3.2] [ELA2013] (3) 1: Ask and answer questions to demonstrate understanding of a text, referring explicitly to the text as the basis for the answers. [RL.3.1] [ELA] (3) 3: Use a wide range of strategies, including using context clues and predicting outcomes, to comprehend thirdgrade recreational reading materials in a variety of genres. [ELA] (3) 4: Use a wide range of strategies and skills, including retelling information, using context clues, and making inferences to identify main idea, to comprehend thirdgrade informational and functional reading materials. [ELA] (3) 7: Compare fictional characters and events to reallife experiences. [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 2: Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. [3OA2] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4] [MA2013] (3) 5: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) [3OA5]
Subject: English Language Arts (3), or English Language Arts (3), or Mathematics (3)
Title: A Math and Language Arts lesson: Introduction to Division
Description: The lesson will help students develop an initial understanding of division and clarify how the four operations of addition, subtraction, multiplication, and division relate to and are separate from each other. The lesson begins with a brainstorming discussion which builds background and fosters comprehension. A big book, The Doorbell Rang, by Pat Hutchins, is used along with manipulatives to provide instruction at concrete and pictorial levels. Students will demonstrate what they have learned by writing a short story incorporating simple division.
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Title: Let's Go Shopping!
Description:
This technologybased lesson provides students practice in collecting and organizing data through the use of grocery store circulars.
Standard(s): [TC2] (35) 11: Use digital tools to analyze authentic problems. [MA2013] (4) 11: Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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. [4NBT6] [MA2013] (5) 9: Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit 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. [5NBT6] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3]
Subject: Mathematics (3  5), or Technology Education (3  5)
Title: Let's Go Shopping!
Description: This technologybased lesson provides students practice in collecting and organizing data through the use of grocery store circulars.
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Title: It’s My Party!
Description:
Students will learn the value of money by planning their own birthday party. By using the Internet and a calculator, students will be given the opportunity for a practical reallife opportunity to solve a problem related to money exchange.
Standard(s): [TC2] (35) 8: Collect information from a variety of digital sources. [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 8: Solve twostep word problems using the four operations. 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. (This standard is limited to problems posed with whole numbers and having wholenumber answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).) [3OA8] [MA2013] (4) 8: Use place value understanding to round multidigit whole numbers to any place. [4NBT3] [MA2013] (4) 9: Fluently add and subtract multidigit whole numbers using the standard algorithm. [4NBT4] [MA2013] (4) 18: 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. [4NF7] [MA2013] (5) 6: Read, write, and compare decimals to thousandths. [5NBT3] [MA2013] (5) 7: Use place value understanding to round decimals to any place. [5NBT4] [MA2013] (5) 16: Solve realworld problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. [5NF6]
Subject: Mathematics (3  5), or Technology Education (3  5)
Title: It’s My Party!
Description: Students will learn the value of money by planning their own birthday party. By using the Internet and a calculator, students will be given the opportunity for a practical reallife opportunity to solve a problem related to money exchange.
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Title: Budgeting for the Oregon Trail
Description:
As a part of the online collaborative unit, Westward Ho!, students will work in cooperative groups of 45 students to decide what items they need to take on the Oregon Trail, determine the cost of the items, and complete a ledger to inventory (keep up with) items used on the trail. Information will then be used as part of a multimedia slideshow journal describing the journey.
Standard(s): [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (4) 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. (See Appendix A, Table 2.) [4OA2]
Subject: Mathematics (3  4)
Title: Budgeting for the Oregon Trail
Description: As a part of the online collaborative unit, Westward Ho!, students will work in cooperative groups of 45 students to decide what items they need to take on the Oregon Trail, determine the cost of the items, and complete a ledger to inventory (keep up with) items used on the trail. Information will then be used as part of a multimedia slideshow journal describing the journey.
Thinkfinity Lesson Plans
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Title: Bag the Beans
Description:
In this Science NetLinks lesson, students work with manipulatives (beans) to create and solve math problems, some of which have more than one correct answer. This lesson addresses the common misconception that there is only one correct way to solve any math problem.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3]
Subject: Mathematics Title: Bag the Beans
Description: In this Science NetLinks lesson, students work with manipulatives (beans) to create and solve math problems, some of which have more than one correct answer. This lesson addresses the common misconception that there is only one correct way to solve any math problem. Thinkfinity Partner: Science NetLinks Grade Span: 3,4,5
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Title: A Brownie Bake
Description:
This lesson, one of a multipart unit from Illuminations, focuses on student organization, preparation, and presentation of some simple foods as a way to apply various mathematical concepts, with problemsolving techniques being a central theme. Students prepare, after determining minimum amounts of ingredients required, a commercial brownie mix and serve equal portions to all class members.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 13: Understand a fraction ^{1}/_{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction ^{a}/_{b} as the quantity formed by a parts and size ^{1}/_{b}. [3NF1] [MA2013] (4) 12: Explain why a fraction ^{a}/_{b} is equivalent to a fraction ^{nxa}/_{nxb} 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. [4NF1] [MA2013] (5) 12: 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. [5NF2]
Subject: Mathematics Title: A Brownie Bake
Description: This lesson, one of a multipart unit from Illuminations, focuses on student organization, preparation, and presentation of some simple foods as a way to apply various mathematical concepts, with problemsolving techniques being a central theme. Students prepare, after determining minimum amounts of ingredients required, a commercial brownie mix and serve equal portions to all class members. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
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Title: Multiplying Meanings
Description:
In this eightlesson unit, from Illuminations, students explore the many meanings of multiplication (skip counting, repeated addition, sets, arrays, number line, cross products of sets and balanced equations). They also learn about the order property, the multiplicative inverse and the results of multiplying by 1 and by 0. In addition, students write story problems in which the operation of multiplication is required. Opportunities to connect with literature are presented in four of the lessons.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 5: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) [3OA5] [MA2013] (3) 7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. [3OA7]
Subject: Language Arts,Mathematics Title: Multiplying Meanings
Description: In this eightlesson unit, from Illuminations, students explore the many meanings of multiplication (skip counting, repeated addition, sets, arrays, number line, cross products of sets and balanced equations). They also learn about the order property, the multiplicative inverse and the results of multiplying by 1 and by 0. In addition, students write story problems in which the operation of multiplication is required. Opportunities to connect with literature are presented in four of the lessons. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
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Title: Modeling Multiplication with Rows and Columns
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This lesson, one of a multipart unit from Illuminations, encourages students to explore the array model of multiplication. Students make arrays with counters and then create a second concrete example of the array model using toothpicks. They also write problems that involve multiplication. A children s book is suggested to connect multiplication with children s literature. Several pieces of literature appropriate for use with this lesson are suggested.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (4) 10: Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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. [4NBT5]
Subject: Language Arts,Mathematics Title: Modeling Multiplication with Rows and Columns
Description: This lesson, one of a multipart unit from Illuminations, encourages students to explore the array model of multiplication. Students make arrays with counters and then create a second concrete example of the array model using toothpicks. They also write problems that involve multiplication. A children s book is suggested to connect multiplication with children s literature. Several pieces of literature appropriate for use with this lesson are suggested. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
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Title: Multiplication Stories
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In this lesson, one of a multipart unit from Illuminations, students practice memorization of multiplication facts where one factor is 6 or 7. They create multiplication stories, play a multiplication game, and then record their current level of mastery of the multiplication facts.
Standard(s): [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4]
Subject: Mathematics Title: Multiplication Stories
Description: In this lesson, one of a multipart unit from Illuminations, students practice memorization of multiplication facts where one factor is 6 or 7. They create multiplication stories, play a multiplication game, and then record their current level of mastery of the multiplication facts. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
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Title: Times Six, Times Seven
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In this sixlesson unit, from Illuminations, students who have previously studied the meanings of multiplication use its properties to help them memorize the products to 70, where 6 or 7 is a factor. Retention is fostered through class discussions, preparation of a personal multiplication chart, and game playing. The unit is most appropriate for students who understand the process of multiplication and have mastered the multiplication facts through the 5 table.
Standard(s): [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. [3OA7] [MA2013] (4) 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. (See Appendix A, Table 2.) [4OA2]
Subject: Mathematics Title: Times Six, Times Seven
Description: In this sixlesson unit, from Illuminations, students who have previously studied the meanings of multiplication use its properties to help them memorize the products to 70, where 6 or 7 is a factor. Retention is fostered through class discussions, preparation of a personal multiplication chart, and game playing. The unit is most appropriate for students who understand the process of multiplication and have mastered the multiplication facts through the 5 table. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
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Title: The Variable Machine
Description:
This Illuminations lesson provides an introduction to the use of variables. Students explore the idea of variable as a symbol that can stand for any member of a set of numbers, and they substitute numbers for variables (letters) to discover unknown values.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 15: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. [3NF3] [MA2013] (4) 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. (See Appendix A, Table 2.) [4OA2] [MA2013] (4) 7: Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. [4NBT2] [MA2013] (4) 13: 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. [4NF2] [MA2013] (4) 18: 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. [4NF7] [MA2013] (4) 25: Recognize angle measure as additive. When an angle is 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 a diagram in realworld or mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. [4MD7] [MA2013] (5) 1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. [5OA1] [MA2013] (5) 6: Read, write, and compare decimals to thousandths. [5NBT3]
Subject: Mathematics Title: The Variable Machine
Description: This Illuminations lesson provides an introduction to the use of variables. Students explore the idea of variable as a symbol that can stand for any member of a set of numbers, and they substitute numbers for variables (letters) to discover unknown values. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
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Title: Clothing Sizes
Description:
This reproducible worksheet, from an Illuminations lesson, prompts students to analyze measurements of their body, specifically as they relate to clothing sizes.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 17: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm^{3} and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve onestep word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (Excludes multiplicative comparison problems (problems involving notions of "times as much").) (See Appendix A, Table 2.) [3MD2] [MA2013] (3) 19: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate units — whole numbers, halves, or quarters. [3MD4] [MA2013] (3) 20: Recognize area as an attribute of plane figures, and understand concepts of area measurement. [3MD5] [MA2013] (4) 19: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and 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 twocolumn table. [4MD1] [MA2013] (4) 20: 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. [4MD2] [MA2013] (4) 22: 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. [4MD4] [MA2013] (4) 23: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. [4MD5] [MA2013] (5) 18: Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, realworld problems. [5MD1] [MA2013] (5) 19: Make a line plot to display a data set of measurements in fractions of a unit (^{1}/_{2}, ^{1}/_{4}, ^{1}/_{8}).
Use operations on fractions for this grade to solve problems involving information presented in line plots. [5MD2] [MA2013] (5) 20: Recognize volume as an attribute of solid figures, and understand concepts of volume measurement. [5MD3]
Subject: Mathematics Title: Clothing Sizes
Description: This reproducible worksheet, from an Illuminations lesson, prompts students to analyze measurements of their body, specifically as they relate to clothing sizes. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
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Title: Making Your First Million
Description:
In this lesson, one of a multipart unit from Illuminations, students attempt to identify the concept of a million by working with smaller numerical units, such as blocks of 10 or 100, and then expanding the idea by multiplication or repeated addition until a million is reached. Additionally, they use critical thinking to analyze situations and to identify mathematical patterns that enable them to develop the concept of very large numbers.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 9: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. [3OA9] [MA2013] (4) 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. (See Appendix A, Table 2.) [4OA2] [MA2013] (5) 5: 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 wholenumber exponents to denote powers of 10. [5NBT2] [MA2013] (7) 2: Recognize and represent proportional relationships between quantities. [7RP2] [MA2013] (7) 9: Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate, and assess the reasonableness of answers using mental computation and estimation strategies. [7EE3] [MA2013] (7) 11: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. [7G1] [MA2013] (8) 6: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. [8EE4]
Subject: Mathematics Title: Making Your First Million
Description: In this lesson, one of a multipart unit from Illuminations, students attempt to identify the concept of a million by working with smaller numerical units, such as blocks of 10 or 100, and then expanding the idea by multiplication or repeated addition until a million is reached. Additionally, they use critical thinking to analyze situations and to identify mathematical patterns that enable them to develop the concept of very large numbers. Thinkfinity Partner: Illuminations Grade Span: 3,4,5,6,7,8
ALEX Podcasts
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Title: Math Tricks: Always a 4
Overview:
This podcast was created to motivate students with a math trick that they can share with friends and family. Students are practicing several math concepts while they preform the trick. Pi the Math Guy, will lead your students through the steps. A transcript and a handout of the steps are included. Don't forget to check out the next episode for a new fun math trick. Standard(s):
[MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4] [MA2013] (3) 7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. [3OA7] [MA2013] (3) 8: Solve twostep word problems using the four operations. 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. (This standard is limited to problems posed with whole numbers and having wholenumber answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).) [3OA8] [MA2013] (3) 11: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. [3NBT2] [MA2013] (4) 1: Interpret a multiplication equation as a comparison, e.g., 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. [4OA1] [MA2013] (4) 9: Fluently add and subtract multidigit whole numbers using the standard algorithm. [4NBT4] [MA2013] (4) 10: Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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. [4NBT5]
Math Tricks: Always a 4 Overview: This podcast was created to motivate students with a math trick that they can share with friends and family. Students are practicing several math concepts while they preform the trick. Pi the Math Guy, will lead your students through the steps. A transcript and a handout of the steps are included. Don't forget to check out the next episode for a new fun math trick.
Web Resources
Lesson Plans
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Title: Family Time: What Does it Cost? PBL
Description:
This is a unit plan that requires students to create a family vacation plan, a budget, and a schedule. Minilessons are included for prerequisite skills. Rubrics are also included.
Standard(s): [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 11: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. [3NBT2] [MA2013] (3) 16: Tell and write time to the nearest minute, and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. [3MD1] [MA2013] (4) 9: Fluently add and subtract multidigit whole numbers using the standard algorithm. [4NBT4] [MA2013] (4) 10: Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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. [4NBT5] [MA2013] (4) 20: 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. [4MD2]
Family Time: What Does it Cost? PBL
http://wveis.k12.wv....
This is a unit plan that requires students to create a family vacation plan, a budget, and a schedule. Minilessons are included for prerequisite skills. Rubrics are also included.
Learning Activities
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Title: Math Word Problems (Kidzone)
Description:
The word problems on this page are listed by grade and, within each grade, by theme.
Standard(s): [MA2013] (1) 1: Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 1.) [1OA1] [MA2013] (1) 2: Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. [1OA2] [MA2013] (2) 1: Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 1.) [2OA1] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (4) 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. (See Appendix A, Table 2.) [4OA2]
Math Word Problems (Kidzone)
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The word problems on this page are listed by grade and, within each grade, by theme.
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Title: Number and Operations Web Links
Description:
This collection of Web links, reviewed and presented by Illuminations, offers teachers and students information about and practice in concepts related to arithmetic. Users can read the Illuminations Editorial Board's review of each Web site, or choose to link directly to the sites.
Standard(s): [MA2013] (0) 1: Count to 100 by ones and by tens. [KCC1] [MA2013] (0) 2: Count forward beginning from a given number within the known sequence (instead of having to begin at 1). [KCC2] [MA2013] (0) 3: Write numbers from 0 to 20. Represent a number of objects with a written numeral 020 (with 0 representing a count of no objects). [KCC3] [MA2013] (0) 4: Understand the relationship between numbers and quantities; connect counting to cardinality. [KCC4] [MA2013] (0) 5: Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 120, count out that many objects. [KCC5] [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 2: Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. [3OA2] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4] [MA2013] (3) 5: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) [3OA5] [MA2013] (3) 6: Understand division as an unknownfactor problem. [3OA6] [MA2013] (3) 7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. [3OA7] [MA2013] (3) 8: Solve twostep word problems using the four operations. 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. (This standard is limited to problems posed with whole numbers and having wholenumber answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).) [3OA8] [MA2013] (3) 9: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. [3OA9] [MA2013] (3) 10: Use place value understanding to round whole numbers to the nearest 10 or 100. [3NBT1] [MA2013] (3) 11: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. [3NBT2] [MA2013] (3) 12: Multiply onedigit whole numbers by multiples of 10 in the range 10  90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations. [3NBT3] [MA2013] (3) 13: Understand a fraction ^{1}/_{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction ^{a}/_{b} as the quantity formed by a parts and size ^{1}/_{b}. [3NF1] [MA2013] (3) 14: Understand a fraction as a number on the number line; represent fractions on a number line diagram. [3NF2] [MA2013] (3) 15: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. [3NF3] [MA2013] (3) 16: Tell and write time to the nearest minute, and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. [3MD1] [MA2013] (3) 17: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm^{3} and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve onestep word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (Excludes multiplicative comparison problems (problems involving notions of "times as much").) (See Appendix A, Table 2.) [3MD2] [MA2013] (3) 18: Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one and twostep "how many more" and "how many less" problems using information presented in scaled bar graphs. [3MD3] [MA2013] (3) 19: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate units — whole numbers, halves, or quarters. [3MD4] [MA2013] (3) 20: Recognize area as an attribute of plane figures, and understand concepts of area measurement. [3MD5] [MA2013] (3) 21: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). [3MD6] [MA2013] (3) 22: Relate area to the operations of multiplication and addition. [3MD7] [MA2013] (3) 23: Solve realworld and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. [3MD8] [MA2013] (3) 24: Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. [3G1] [MA2013] (3) 25: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. [3G2] [MA2013] (4) 1: Interpret a multiplication equation as a comparison, e.g., 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. [4OA1] [MA2013] (4) 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. (See Appendix A, Table 2.) [4OA2] [MA2013] (4) 3: Solve multistep word problems posed with whole numbers and having wholenumber 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. [4OA3] [MA2013] (4) 4: Find all factor pairs for a whole number in the range 1100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1100 is prime or composite. [4OA4] [MA2013] (4) 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. [4OA5] [MA2013] (4) 6: Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. [4NBT1] [MA2013] (4) 7: Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. [4NBT2] [MA2013] (4) 8: Use place value understanding to round multidigit whole numbers to any place. [4NBT3] [MA2013] (4) 9: Fluently add and subtract multidigit whole numbers using the standard algorithm. [4NBT4] [MA2013] (4) 10: Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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. [4NBT5] [MA2013] (4) 11: Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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. [4NBT6] [MA2013] (4) 12: Explain why a fraction ^{a}/_{b} is equivalent to a fraction ^{nxa}/_{nxb} 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. [4NF1] [MA2013] (4) 13: 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. [4NF2] [MA2013] (4) 14: Understand a fraction ^{a}/_{b} with a > 1 as a sum of fractions ^{1}/_{b}. [4NF3] [MA2013] (4) 15: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. [4NF4] [MA2013] (4) 16: 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. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.) [4NF5] [MA2013] (4) 17: Use decimal notation for fractions with denominators 10 or 100. [4NF6] [MA2013] (4) 18: 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. [4NF7] [MA2013] (4) 19: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and 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 twocolumn table. [4MD1] [MA2013] (4) 20: 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. [4MD2] [MA2013] (4) 21: Apply the area and perimeter formulas for rectangles in realworld and mathematical problems. [4MD3] [MA2013] (4) 22: 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. [4MD4] [MA2013] (4) 23: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. [4MD5] [MA2013] (4) 24: Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. [4MD6] [MA2013] (4) 25: Recognize angle measure as additive. When an angle is 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 a diagram in realworld or mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. [4MD7] [MA2013] (4) 26: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. [4G1] [MA2013] (4) 27: Classify twodimensional 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. [4G2] [MA2013] (4) 28: Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify linesymmetric figures and draw lines of symmetry. [4G3] [MA2013] (5) 1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. [5OA1] [MA2013] (5) 2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. [5OA2] [MA2013] (5) 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. [5OA3] [MA2013] (5) 4: Recognize that in a multidigit 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. [5NBT1] [MA2013] (5) 5: 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 wholenumber exponents to denote powers of 10. [5NBT2] [MA2013] (5) 6: Read, write, and compare decimals to thousandths. [5NBT3] [MA2013] (5) 7: Use place value understanding to round decimals to any place. [5NBT4] [MA2013] (5) 8: Fluently multiply multidigit whole numbers using the standard algorithm. [5NBT5] [MA2013] (5) 9: Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit 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. [5NBT6] [MA2013] (5) 10: 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. [5NBT7] [MA2013] (5) 11: 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. [5NF1] [MA2013] (5) 12: 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. [5NF2] [MA2013] (5) 13: Interpret a fraction as division of the numerator by the denominator (^{a}/_{b} = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. [5NF3] [MA2013] (5) 14: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. [5NF4] [MA2013] (5) 15: Interpret multiplication as scaling (resizing), by: [5NF5] [MA2013] (5) 16: Solve realworld problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. [5NF6] [MA2013] (5) 17: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general by reasoning about the relationship between multiplication and division. However, division of a fraction by a fraction is not a requirement at this grade.)
[5NF7] [MA2013] (5) 18: Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, realworld problems. [5MD1] [MA2013] (5) 19: Make a line plot to display a data set of measurements in fractions of a unit (^{1}/_{2}, ^{1}/_{4}, ^{1}/_{8}).
Use operations on fractions for this grade to solve problems involving information presented in line plots. [5MD2] [MA2013] (5) 20: Recognize volume as an attribute of solid figures, and understand concepts of volume measurement. [5MD3] [MA2013] (5) 21: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. [5MD4] [MA2013] (5) 22: Relate volume to the operations of multiplication and addition, and solve realworld and mathematical problems involving volume. [5MD5] [MA2013] (5) 23: Use a pair of perpendicular number lines, called axes, to define a coordinate system with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., xaxis and xcoordinate, yaxis and ycoordinate). [5G1] [MA2013] (5) 24: Represent realworld and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. [5G2] [MA2013] (5) 25: Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that category. [5G3] [MA2013] (5) 26: Classify twodimensional figures in a hierarchy based on properties. [5G4]
Subject: Mathematics Title: Number and Operations Web Links
Description: This collection of Web links, reviewed and presented by Illuminations, offers teachers and students information about and practice in concepts related to arithmetic. Users can read the Illuminations Editorial Board's review of each Web site, or choose to link directly to the sites. Thinkfinity Partner: Illuminations Grade Span: K,1,2,3,4,5,6,7,8,9,10,11,12
Thinkfinity Interactive Games
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Title: Communicating about Mathematics Using Games: Playing Fraction Tracks
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Mathematical games can foster mathematical communication as students explain and justify their moves to one another. In addition, games can motivate students and engage them in thinking about and applying concepts and skills. This eexample from Illuminations contains an interactive version of a game that can be used in the grades 35 classroom to support students' learning about fractions. eMath Investigations are selected eexamples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The eexamples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the eexamples are natural companions to the iMath investigations.
Standard(s): [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 2: Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. [3OA2] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4] [MA2013] (3) 5: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) [3OA5] [MA2013] (3) 6: Understand division as an unknownfactor problem. [3OA6] [MA2013] (3) 7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. [3OA7] [MA2013] (3) 8: Solve twostep word problems using the four operations. 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. (This standard is limited to problems posed with whole numbers and having wholenumber answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).) [3OA8] [MA2013] (3) 9: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. [3OA9] [MA2013] (3) 10: Use place value understanding to round whole numbers to the nearest 10 or 100. [3NBT1] [MA2013] (3) 11: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. [3NBT2] [MA2013] (3) 12: Multiply onedigit whole numbers by multiples of 10 in the range 10  90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations. [3NBT3] [MA2013] (3) 13: Understand a fraction ^{1}/_{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction ^{a}/_{b} as the quantity formed by a parts and size ^{1}/_{b}. [3NF1] [MA2013] (3) 14: Understand a fraction as a number on the number line; represent fractions on a number line diagram. [3NF2] [MA2013] (3) 15: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. [3NF3] [MA2013] (3) 16: Tell and write time to the nearest minute, and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. [3MD1] [MA2013] (3) 17: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm^{3} and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve onestep word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (Excludes multiplicative comparison problems (problems involving notions of "times as much").) (See Appendix A, Table 2.) [3MD2] [MA2013] (3) 18: Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one and twostep "how many more" and "how many less" problems using information presented in scaled bar graphs. [3MD3] [MA2013] (3) 19: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate units — whole numbers, halves, or quarters. [3MD4] [MA2013] (3) 21: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). [3MD6] [MA2013] (3) 22: Relate area to the operations of multiplication and addition. [3MD7] [MA2013] (3) 23: Solve realworld and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. [3MD8] [MA2013] (3) 24: Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. [3G1] [MA2013] (3) 25: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. [3G2] [MA2013] (4) 1: Interpret a multiplication equation as a comparison, e.g., 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. [4OA1] [MA2013] (4) 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. (See Appendix A, Table 2.) [4OA2] [MA2013] (4) 3: Solve multistep word problems posed with whole numbers and having wholenumber 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. [4OA3] [MA2013] (4) 4: Find all factor pairs for a whole number in the range 1100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1100 is prime or composite. [4OA4] [MA2013] (4) 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. [4OA5] [MA2013] (4) 6: Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. [4NBT1] [MA2013] (4) 7: Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. [4NBT2] [MA2013] (4) 8: Use place value understanding to round multidigit whole numbers to any place. [4NBT3] [MA2013] (4) 9: Fluently add and subtract multidigit whole numbers using the standard algorithm. [4NBT4] [MA2013] (4) 10: Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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. [4NBT5] [MA2013] (4) 11: Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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. [4NBT6] [MA2013] (4) 12: Explain why a fraction ^{a}/_{b} is equivalent to a fraction ^{nxa}/_{nxb} 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. [4NF1] [MA2013] (4) 13: 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. [4NF2] [MA2013] (4) 14: Understand a fraction ^{a}/_{b} with a > 1 as a sum of fractions ^{1}/_{b}. [4NF3] [MA2013] (4) 15: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. [4NF4] [MA2013] (4) 16: 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. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.) [4NF5] [MA2013] (4) 17: Use decimal notation for fractions with denominators 10 or 100. [4NF6] [MA2013] (4) 18: 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. [4NF7] [MA2013] (4) 19: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and 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 twocolumn table. [4MD1] [MA2013] (4) 20: 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. [4MD2] [MA2013] (4) 21: Apply the area and perimeter formulas for rectangles in realworld and mathematical problems. [4MD3] [MA2013] (4) 22: 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. [4MD4] [MA2013] (4) 23: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. [4MD5] [MA2013] (4) 24: Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. [4MD6] [MA2013] (4) 25: Recognize angle measure as additive. When an angle is 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 a diagram in realworld or mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. [4MD7] [MA2013] (4) 26: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. [4G1] [MA2013] (4) 27: Classify twodimensional 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. [4G2] [MA2013] (4) 28: Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify linesymmetric figures and draw lines of symmetry. [4G3] [MA2013] (5) 1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. [5OA1] [MA2013] (5) 2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. [5OA2] [MA2013] (5) 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. [5OA3] [MA2013] (5) 4: Recognize that in a multidigit 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. [5NBT1] [MA2013] (5) 5: 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 wholenumber exponents to denote powers of 10. [5NBT2] [MA2013] (5) 6: Read, write, and compare decimals to thousandths. [5NBT3] [MA2013] (5) 7: Use place value understanding to round decimals to any place. [5NBT4] [MA2013] (5) 8: Fluently multiply multidigit whole numbers using the standard algorithm. [5NBT5] [MA2013] (5) 9: Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit 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. [5NBT6] [MA2013] (5) 10: 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. [5NBT7] [MA2013] (5) 11: 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. [5NF1] [MA2013] (5) 12: 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. [5NF2] [MA2013] (5) 13: Interpret a fraction as division of the numerator by the denominator (^{a}/_{b} = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. [5NF3] [MA2013] (5) 14: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. [5NF4] [MA2013] (5) 15: Interpret multiplication as scaling (resizing), by: [5NF5] [MA2013] (5) 16: Solve realworld problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. [5NF6] [MA2013] (5) 17: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general by reasoning about the relationship between multiplication and division. However, division of a fraction by a fraction is not a requirement at this grade.)
[5NF7] [MA2013] (5) 18: Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, realworld problems. [5MD1] [MA2013] (5) 19: Make a line plot to display a data set of measurements in fractions of a unit (^{1}/_{2}, ^{1}/_{4}, ^{1}/_{8}).
Use operations on fractions for this grade to solve problems involving information presented in line plots. [5MD2] [MA2013] (5) 20: Recognize volume as an attribute of solid figures, and understand concepts of volume measurement. [5MD3] [MA2013] (5) 21: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. [5MD4] [MA2013] (5) 22: Relate volume to the operations of multiplication and addition, and solve realworld and mathematical problems involving volume. [5MD5] [MA2013] (5) 23: Use a pair of perpendicular number lines, called axes, to define a coordinate system with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., xaxis and xcoordinate, yaxis and ycoordinate). [5G1] [MA2013] (5) 24: Represent realworld and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. [5G2] [MA2013] (5) 25: Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that category. [5G3] [MA2013] (5) 26: Classify twodimensional figures in a hierarchy based on properties. [5G4]
Subject: Mathematics Title: Communicating about Mathematics Using Games: Playing Fraction Tracks
Description: Mathematical games can foster mathematical communication as students explain and justify their moves to one another. In addition, games can motivate students and engage them in thinking about and applying concepts and skills. This eexample from Illuminations contains an interactive version of a game that can be used in the grades 35 classroom to support students' learning about fractions. eMath Investigations are selected eexamples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The eexamples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the eexamples are natural companions to the iMath investigations. Thinkfinity Partner: Illuminations Grade Span: 3,4,5
Thinkfinity Learning Activities
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Title: Concentration
Description:
This student interactive from Illuminations allows students to play the classic game of Concentration by themselves or against a friend. The interactive can be used to review numbers, shapes, multiplication facts, fractions, decimals, and percents.
Standard(s): [MA2013] (1) 21: Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. [1G3] [MA2013] (2) 4: Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. [2OA4] [MA2013] (2) 26: Partition circles and rectangles into two, three, or four equal shares; describe the shares using the words halves, thirds, half of, a third of, etc.; and describe the whole as two halves, three thirds, or four fourths. Recognize that equal shares of identical wholes need not have the same shape. [2G3] [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4] [MA2013] (3) 9: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. [3OA9] [MA2013] (3) 14: Understand a fraction as a number on the number line; represent fractions on a number line diagram. [3NF2] [MA2013] (3) 15: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. [3NF3] [MA2013] (3) 24: Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. [3G1] [MA2013] (3) 25: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. [3G2] [MA2013] (4) 18: 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. [4NF7] [MA2013] (4) 20: 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. [4MD2] [MA2013] (4) 23: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. [4MD5]
Subject: Mathematics Title: Concentration
Description: This student interactive from Illuminations allows students to play the classic game of Concentration by themselves or against a friend. The interactive can be used to review numbers, shapes, multiplication facts, fractions, decimals, and percents. Thinkfinity Partner: Illuminations Grade Span: K,PreK,1,2,3,4,5
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Title: Product Game
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This interactive game, from Illuminations, exercises students skills with factors and multiples. Players take turns moving a marker along a number line to create products, which are then colored in on the game board. The object of the game is to get four squares in a row.
Standard(s): [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4] [MA2013] (3) 5: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) [3OA5] [MA2013] (3) 7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. [3OA7] [MA2013] (4) 1: Interpret a multiplication equation as a comparison, e.g., 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. [4OA1]
Subject: Mathematics Title: Product Game
Description: This interactive game, from Illuminations, exercises students skills with factors and multiples. Players take turns moving a marker along a number line to create products, which are then colored in on the game board. The object of the game is to get four squares in a row. Thinkfinity Partner: Illuminations Grade Span: 3,4,5,6,7,8
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Title: Factorize
Description:
This student interactive, from Illuminations, allows students to visually explore the concept of factors by creating rectangular arrays with an area equal to the product of the factors. They can choose whether or not to use the commutative property when identifying factors.
Standard(s): [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3OA3] [MA2013] (3) 4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3OA4] [MA2013] (4) 1: Interpret a multiplication equation as a comparison, e.g., 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. [4OA1] [MA2013] (4) 4: Find all factor pairs for a whole number in the range 1100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1100 is prime or composite. [4OA4] [MA2013] (4) 21: Apply the area and perimeter formulas for rectangles in realworld and mathematical problems. [4MD3] [MA2013] (6) 5: Fluently divide multidigit numbers using the standard algorithm. [6NS2]
Subject: Mathematics Title: Factorize
Description: This student interactive, from Illuminations, allows students to visually explore the concept of factors by creating rectangular arrays with an area equal to the product of the factors. They can choose whether or not to use the commutative property when identifying factors. Thinkfinity Partner: Illuminations Grade Span: 3,4,5,6,7,8

