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["The Arizona Mathematics Progression Documents are in first draft! Completed by the East Valley Math Specialist (EVCIA) in May, 2019, these documents are living documents and will be updated when feedback from the field is reviewed and agreed upon. The Arizona Mathematics Progression Documents will be updated each year with different resources and extensions as indicated by feedback from the field and through the East Valley Math Specialists group. Feedback would be wonderful! Please send feedback to K12Standardsinbox@azed.gov."]
[]
[]
[]
[]
[]
["Common Misconceptions - Resource Listed Below"]
[{"$f":"HYPERLINK(\"https://us.corwin.com/en-us/nam/the-common-core-mathematics-companion-the-standards-decoded-grades-3-5/book244986\",\"Gojak, L., & Miles, R. H. (2016). The common core mathematics companion: The standards decoded, grades K-2: What they say, what they mean, how to teach them. Thousand Oaks: Corwin Mathematics.\")","v":"Gojak, L., & Miles, R. H. (2016). The common core mathematics companion: The standards decoded, grades K-2: What they say, what they mean, how to teach them. Thousand Oaks: Corwin Mathematics."}]
[]
[]
["Links to Specific Spreadsheet"]
[null,null,"5.OA.A.1","5.NBT.A.1","5.NF.A.1","5.MD.A.1","5.G.A.1"]
[null,null,"5.OA.A.2","5.NBT.A.2","5.NF.A.2","5.MD.B.2","5.G.A.2"]
[null,null,"5.OA.B.3","5.NBT.A.3","5.NF.B.3","5.MD.C.3","5.G.B.3"]
[null,null,"5.OA.B.4","5.NBT.A.4","5.NF.B.4","5.MD.C.4","5.G.B.4"]
[null,null,null,"5.NBT.B.5","5.NF.B.5","5.MD.C.5"]
[null,null,null,"5.NBT.B.6","5.NF.B.6"]
[null,null,null,"5.NBT.B.7","5.NF.B.7"]
[]
["Universal Design for Learning"]

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[null,"Cluster"]
[null,"5.OA.A Write and interpret numerical expressions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["No prior standard connection.","5.OA.A.1\nUse parentheses and brackets in numerical expressions, and evaluate expressions with these symbols (Order of Operations).","5.OA.A.2\nWrite simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them (e.g., express the calculation \"add 8 and 7, then multiply by 2\" as 2 x (8 + 7). Recognize that 3 x (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product)."]
[null,null,"6.EE.A.2\nWrite, read, and evaluate algebraic expressions. a. Write expressions that record operations with numbers and variables. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, and coefficient); view one or more parts of an expression as a single entity. c. Evaluate expressions given specific values of their variables. Include expressions that arise from formulas used to solve mathematical problems and problems in real-world context. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations)."]
[null,null,"6.EE.A.3\nApply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x."]
[null,null,"6.EE.A.4\nIdentify when two expressions are equivalent. For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for."]
[null,null,"6.NS.B.4\nUse previous understanding of factors to find the greatest common factor and the least common multiple. Arizona Mathematics Standards 6 th Grade 5 multiples. a. Find the greatest common factor of two whole numbers less than or equal to 100. b. Find the least common multiple of two whole numbers less than or equal to 12. c. Use the distributive property to express a sum of two whole numbers 1 to 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9+2)."]
["Instructional Strategies"]
["Students should be given ample opportunities to explore numerical expressions with mixed operations. This is the foundation for evaluating numerical and algebraic expressions that will include whole-number exponents in Grade 6. There are conventions (rules) determined by mathematicians that must be learned with no conceptual basis. For example, multiplication and division are always done before addition and subtraction. Multiplication and division generally has a bigger impact on a problem than addition and subtraction. Begin with expressions that have two operations without any grouping symbols (multiplication or division combined with addition or subtraction) before introducing expressions with multiple operations. Use the same digits, with the operations in a different order, and have students evaluate the expressions, then discuss why the value of the expression is different. For example, have students evaluate 5 × 3 + 6 and 5 + 3 × 6. Discuss the rules (conventions) that must be followed. Have students insert parentheses around the multiplication or division section in an expression. A discussion should focus on the similarities and differences in the problems and the results. This should assist students in being able to solve problem situations which require an understanding of the order in which operations should take place. After students have evaluated expressions without grouping symbols, present problems parentheses. If you students are ready, you can move into using brackets and braces, but these must be done so students have an understanding of why these grouping symbols are being used. Have students write numerical expressions in words without calculating the value. This is the foundation for writing algebraic expressions. Then, have students write numerical expressions from phrases without calculating them."]
["Common Misconceptions"]
["Some students will have the misconception that all multiplications are calculated before divisions and additions are calculated before subtractions. Scaffold examples for students to practice solving multiplications and/or divisions in order from left to right and then additions and/or subtractions in order from left to right. Although parentheses are not necessary when the equation is written accurately, some students will find it helpful to add grouping symbols in order to solve equations and word problems."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/illustrations/555\",\"5.OA.A.1 Watch Out for Parentheses 1\")","v":"5.OA.A.1 Watch Out for Parentheses 1"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/OA/A/1/tasks/969\",\"5.OA.A.1 Bowling for Numbers\")","v":"5.OA.A.1 Bowling for Numbers"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/OA/A/1/tasks/1596\",\"5.OA.A.1 Using Operations and Parentheses\")","v":"5.OA.A.1 Using Operations and Parentheses"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.OA.A Write and interpret numerical expressions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["5.OA.A.1\nUse parentheses and brackets in numerical expressions, and evaluate expressions with these symbols (Order of Operations).","5.OA.A.2\nWrite simple expressions that record calculations with numbers, and interpret numerical expressions without\nevaluating them (e.g., express the calculation \"add 8 and 7, then multiply by 2\" as 2 x (8 + 7). Recognize that 3 x\n(18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product).","6.EE.A.2\nWrite, read, and evaluate algebraic expressions. a. Write expressions that record operations with numbers and variables. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, and coefficient); view one or more parts of an expression as a single entity. c. Evaluate expressions given specific values of their variables. Include expressions that arise from formulas used to solve mathematical problems and problems in real-world context. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations)."]
[null,null,"6.EE.A.3\nApply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x."]
[null,null,"6.EE.A.4\nIdentify when two expressions are equivalent. For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for."]
[null,null,"6.NS.B.4\nUse previous understanding of factors to find the greatest common factor and the least common multiple. Arizona Mathematics Standards 6 th Grade 5 multiples. a. Find the greatest common factor of two whole numbers less than or equal to 100. b. Find the least common multiple of two whole numbers less than or equal to 12. c. Use the distributive property to express a sum of two whole numbers 1 to 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9+2)."]
["Instructional Strategies"]
["Students should be given ample opportunities to explore numerical expressions with mixed operations. This is the foundation for evaluating numerical and algebraic expressions that will include whole-number exponents in Grade 6. There are conventions (rules) determined by mathematicians that must be learned with no conceptual basis. For example, multiplication and division are always done before addition and subtraction. Multiplication and division generally has a bigger impact on a problem than addition and subtraction. Begin with expressions that have two operations without any grouping symbols (multiplication or division combined with addition or subtraction) before introducing expressions with multiple operations. Use the same digits, with the operations in a different order, and have students evaluate the expressions, then discuss why the value of the expression is different. For example, have students evaluate 5 × 3 + 6 and 5 + 3 × 6. Discuss the rules (conventions) that must be followed. Have students insert parentheses around the multiplication or division section in an expression. A discussion should focus on the similarities and differences in the problems and the results. This should assist students in being able to solve problem situations which require an understanding of the order in which operations should take place. After students have evaluated expressions without grouping symbols, present problems parentheses. If you students are ready, you can move into using brackets and braces, but these must be done so students have an understanding of why these grouping symbols are being used. Have students write numerical expressions in words without calculating the value. This is the foundation for writing algebraic expressions. Then, have students write numerical expressions from phrases without calculating them."]
["Common Misconceptions"]
["Struggling students should begin with simple expressions. Because there may be several ways to write or read an expression, they should justify how they arrived at their answers."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/OA/A/2/tasks/139\",\"5.OA.A.2 Comparing Products\")","v":"5.OA.A.2 Comparing Products"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/OA/A/2/tasks/556\",\"5.OA.A.2 Words to Expressions 1\")","v":"5.OA.A.2 Words to Expressions 1"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/OA/A/2/tasks/590\",\"5.OA.A.2 Video Game Scores\")","v":"5.OA.A.2 Video Game Scores"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/OA/A/2/tasks/1222\",\"5.OA.A.2 Seeing is Believing\")","v":"5.OA.A.2 Seeing is Believing"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.OA.B Analyze patterns and relationships."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.OA.C.5\nGenerate a number pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself and explain the pattern informally (e.g., given the rule “add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers).","5.OA.B.3\nGenerate two numerical patterns using two given rules (e.g., generate terms in the resulting sequences). Identify and explain the 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 (e.g., given the rule \"add 3\" and the starting number 0, and given the rule \"add 6\" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence).","6.EE.A.2\nWrite, read, and evaluate algebraic expressions. a. Write expressions that record operations with numbers and variables. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, and coefficient); view one or more parts of an expression as a single entity. c. Evaluate expressions given specific values of their variables. Include expressions that arise from formulas used to solve mathematical problems and problems in real-world context. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations)."]
[null,null,"6.EE.C.9\nUse variables to represent two quantities that change in relationship to one another to solve mathematical problems and problems in real-world context. Write an equation to express one quantity (the dependent variable) in terms of the other quantity (the independent variable). Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation."]
[null,null,"6.RP.A.1\nUnderstand the concept of a ratio as comparing two quantities multiplicatively or joining/composing the two quantities in a way that preserves a multiplicative relationship. Use ratio language to describe a ratio relationship between two quantities. For example, \"There were 2/3 as many men as women at the concert.”"]
["Instructional Strategies"]
["Provide students with problems to solve in which students make a T-table and generate numerical patterns including probles with two rules or two patterns. Provide students with \"mathematical rules\" that give them additional opportunties to make a table and generate a sequence of numbers based on the rule."]
["Common Misconceptions"]
["Students often reverse coordinates when plotting them on a coordinate plane. They use the first number to count up the y-axis and then count over the second number in the ordered pair on the x-axis. Have students identify and label the x-axis and the y-axis on their coordinate planes. Remind them that since x is before y alphabetically, they move across the x-axis first and then up the y-axis. Students should also describe the difference of the location of (3,5) and (5,3) on the coordinate plane. Give students many opportunities to describe how to plot a point using a variety of ordered pairs."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/OA/B/3/tasks/1895\",\"5.OA.B.3 Sidewalk Patterns\")","v":"5.OA.B.3 Sidewalk Patterns"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.OA.B Analyze patterns and relationships."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.OA.B.4\nFind all factor pairs for a whole number in the range 1 to 100 and understand that a whole number is a multiple of each of its factors.","5.OA.B.4\nUnderstand primes have only two factors and decompose numbers into prime factors. AZ STANDARD","6.NS.B.4\nUse previous understanding of factors to find the greatest common factor and the least common multiple.\na. Find the greatest common factor of two whole numbers less than or equal to 100.\nb. Find the least common multiple of two whole numbers less than or equal to 12.\nc. Use the distributive property to express a sum of two whole numbers 1 to 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9+2)."]
["Instructional Strategies"]
["Definitions of prime and composite numbers should not be provided, but determined after many strategies have been used in finding all possible factors of a number.\nStudents investigate whether numbers are prime or composite by:\n- building rectangles (arrays) with the given area and finding which numbers have more than two rectangles (e.g. 7 can be made into only 2 rectangles, 1 x 7 and 7 x 1, therefore it is a prime number)\n- finding factors of the number\nStudents need to develop strategies for determining if a number is prime or composite, in other words, if a number has\na whole number factor that is not one or itself. Starting with a number chart of 1 to 20, use multiples of prime numbers\nto eliminate later numbers in the chart. For example, 2 is prime but 4, 6, 8, 10, 12 . . . are composite."]
["Common Misconceptions"]
["Students often confuse the terms factor and multiple. Emphasizing the term factor as one of the numbers multiplied to get a product throughout all of the work with multiplication, and expecting students to use that term, should help avoid confusion. Telling students they multiply to get a multiple or defining multiples of a number as products of the number is also helpful. The more experience students have with these terms, the more accurate they will become when using them.\n\nStudents may become confused about whether 1 is a prime or composite number, when actually it is neither prime nor composite because it has only one factor, itself. Developing precise definitions should help to eliminate this misconception."]
[null,"Resources","Site"]
[]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NBT.A Understand the place value system."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NBT.A.1\nApply concepts of place value, multiplication, and division to understand that in a multi-digit whole number, a\ndigit in one place represents ten times what it represents in the place to its right.","5.NBT.A.1\nApply concepts of place value, multiplication, and division to understand that in a multi-digit number, a digit in\none place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in\nthe place to its left.","5.NBT.A.2\nExplain 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."]
["4.NF.C.5\nExpress a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 (tenths) and 100 (hundredths). For example, express 3/10 as 30/100, and and 3/10 + 4/100 = 34/100. (Note: 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.)",null,"5NBT.A.3\nRead, write, and compare decimals to thousandths.\na. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.\nb. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons."]
["4.NF.C.6\nUse decimal notation for fractions with denominators 10 (tenths) or 100 (hundredths), and locate these decimals on a number line.",null,"5.NBT.A.4\nUse place value understanding to round decimals to any place."]
["4.NF.C.7\nCompare two decimals to hundredths by reasoning about their size. Understand that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, <.",null,"5.NBT.B.5\nFluently multiply multi-digit whole numbers using a standard algorithm."]
[null,null,"5.NBT.B.6\nApply and extend understanding of division to find whole-number quotients of whole numbers with up to fourdigit dividends and two-digit divisors."]
[null,null,"5.NBT.B.7\nAdd, subtract, multiply, and divide decimals to hundredths, connecting objects or drawings to strategies based on place value, properties of operations, and/or the relationship between operations. Relate the strategy to a\nwritten form."]
["Instructional Strategies"]
["In Grade 5, the concept of place value is extended to include decimal values to thousandths. The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers and decimal numbers. For example, students need to continue to represent, write and state the value of numbers, including decimal numbers. For students who are not able to read, write and represent multi-digit numbers, working with decimals could potentially be challenging. Money is a good medium to compare decimals (but keep in mind that 5th grade students are to extend their work into thousandths place so use other situations besides money). Present contextual situations that require the comparison of the cost of two items to determine which one is of a higher price or a lower price. Students should also be able to identify how many pennies, dimes, dollars and ten dollars, etc., are in a given value. Help students make connections between the number of each type of coin and the value of each coin, and the expanded form of the number. Build on the understanding that it always takes ten of the number to the right to make the number to the left. Number cards, number cubes, spinners and other manipulatives can be used to generate decimal numbers. For example, have students roll three number cubes and have them create the largest and smallest decimal number to the thousandths place. Ask students to represent the number with numerals and words, including standard and unit forms."]
["Common Misconceptions"]
["Students can get confused with the language describing the relationship between place values for whole numbers and decimal numbers. For example, when moving from ones to tens (one place to the left), the value is ten times greater, not ten greater or ten more. When moving from tenths to hundredths (one place to the right), the value is 1/10 the value and not ten less than. Working with concrete models and pictorial representations and practicing with appropriate language will help students avoid confusion."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/1/tasks/1562\",\"5.NBT.A.1 Kipton's Scale\")","v":"5.NBT.A.1 Kipton's Scale"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/1/tasks/1799\",\"5.NBT.A.1 Which Number is It?\")","v":"5.NBT.A.1 Which Number is It?"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/1/tasks/1800\",\"5.NBT.A.1 Tenths and Hundredths\")","v":"5.NBT.A.1 Tenths and Hundredths"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/1/tasks/1931\",\"5.NBT.A.1 Millions and Billions of People\")","v":"5.NBT.A.1 Millions and Billions of People"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NBT.A-Understand the place value system."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["5.NBT.A.1\nApply concepts of place value, multiplication, and division to understand that in a multi-digit number, a digit in\none place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in\nthe place to its left.","5.NBT.A.2\nExplain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain\npatterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.\n","6.EE.A.1\nWrite and evaluate numerical expressions involving whole-number exponents."]
["Instructional Strategies"]
["In Grade 5, the concept of place value is extended to include decimal values to thousandths. The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers and decimal numbers. For example, students need to continue to represent, write and state the value of numbers, including decimal numbers. For students who are not able to read, write and represent multi-digit numbers, working with decimals could potentially be challenging. Money is a good medium to compare decimals (but keep in mind that 5th grade students are to extend their work into thousandths place so use other situations besides money). Present contextual situations that require the comparison of the cost of two items to determine which one is of a higher price or a lower price. Students should also be able to identify how many pennies, dimes, dollars and ten dollars, etc., are in a given value. Help students make connections between the number of each type of coin and the value of each coin, and the expanded form of the number. Build on the understanding that it always takes ten of the number to the right to make the number to the left. Number cards, number cubes, spinners and other manipulatives can be used to generate decimal numbers. For example, have students roll three number cubes and have them create the largest and smallest decimal number to the thousandths place. Ask students to represent the number with numerals and words, including standard and unit forms."]
["Common Misconceptions"]
["It is of major importance that students understand the relationship between the number of zeros in the power of ten, or the exponent, and the number of zeros in the product beyond just noticing that they are the same. For example, when multiplying 62 x 100, students should be able to justify that the product represents 62 groups of 100, which is written as 6,200. This understanding is critical as students move to multiplying by decimals. The use of concrete materials and pictorial representations will help students to make these important connections.\n\nAs students begin to work with exponents they must understand that the exponent tells them the number of times the base (in this case 10) is used as a factor. Some students may think of addition (10^2 = 10 + 10 instead of 10 x 10). Students need a variety of experiences including concrete and pictorial representations connected to skill practice to build understanding and skill working with exponential notation.\n"]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/2/tasks/1524\",\"5.NBT.A.2 Marta's Multiplication Error\")","v":"5.NBT.A.2 Marta's Multiplication Error"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/2/tasks/1620\",\"5.NBT.A.2 Multiplying Decimals by 10\")","v":"5.NBT.A.2 Multiplying Decimals by 10"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NBT.A-Understand the place value system."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NBT.A.2\nRead 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.","5.NBT.A.3\nRead, write and compare decials to thousandths.\na. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.\nb. Compare two decimals to thousandths based on meaings of the digits in each place using >, =, < symbols to recod the results of comparisons.","5.NBT.A.4\nUse place value understanding to round decimals to any place."]
["4.NF.C.7\nCompare two decimals to hundredths by reasoning about their size. Understand that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <.\n"]
["Instructional Strategies"]
["In Grade 5, the concept of place value is extended to include decimal values to thousandths. The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers and decimal numbers. For example, students need to continue to represent, write and state the value of numbers, including decimal numbers. For students who are not able to read, write and represent multi-digit numbers, working with decimals could potentially be challenging. Money is a good medium to compare decimals (but keep in mind that 5th grade students are to extend their work into thousandths place so use other situations besides money). Present contextual situations that require the comparison of the cost of two items to determine which one is of a higher price or a lower price. Students should also be able to identify how many pennies, dimes, dollars and ten dollars, etc., are in a given value. Help students make connections between the number of each type of coin and the value of each coin, and the expanded form of the number. Build on the understanding that it always takes ten of the number to the right to make the number to the left. Number cards, number cubes, spinners and other manipulatives can be used to generate decimal numbers. For example, have students roll three number cubes and have them create the largest and smallest decimal number to the thousandths place. Ask students to represent the number with numerals and words, including standard and unit forms."]
["Common Misconceptions"]
["5.NBT.A.3a - It is imperative that students read decimal numbers correctly to reinforce the meaning of the decimal and its place value. For example, 1.12 should be read as “one and twelve hundredths” and not “one point twelve.” Teachers should model this and the expectation should be clear. This not only reinforces the value of the decimal number but also explicitly connects decimal numbers to fraction numbers.\n\n5.NBT.A.3b - When comparing two decimals with different place values, students may have the misconception that 0.345 > 0.57 because 345 is greater than 57 without considering the value of each place after the decimal point. Modeling these numbers using drawings or base-ten blocks will help students to see that 0.57 is greater because there are more tenths. They should have many opportunities to use models and drawings and explain their thinking. Once students demonstrate understanding using concrete materials, they should begin to explore why 0.57 is equivalent to 0.570 using models, place value understanding, and previous experience with equivalent fractions. If both decimal numbers have the same number of places, thousandths for example, students can then compare the number of thousandths in each decimal number to determine which is greater and which is less."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/3/tasks/1801\",\"5.NBT.A.3 Drawing Pictures to Illustrate Decimal Comparisons\")","v":"5.NBT.A.3 Drawing Pictures to Illustrate Decimal Comparisons"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/3/tasks/1802\",\"5.NBT.A.3 Comparing Decimals on a Number Line\")","v":"5.NBT.A.3 Comparing Decimals on a Number Line"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/3/tasks/1803\",\"5.NBT.A.3 Placing Thousandths on a Number Line\")","v":"5.NBT.A.3 Placing Thousandths on a Number Line"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/3/tasks/1813\",\"5.NBT.A.3 Are These Equivalent to 9.52?\")","v":"5.NBT.A.3 Are These Equivalent to 9.52?"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NBT.A-Understand the place value system."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NBT.A.3\nUse place value understanding to round multi-digit whole numbers to any place.","5.NBT.A.4\nUse place value understanding to round decimals to any place.","No future alignment."]
["5.NBT.A.1\nApply concepts of place value, multiplication, and division to understand that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left."]
["5.NBT.A.3\nRead, write, and compare decimals to thousandths.\na. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.\nb. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons."]
["Instructional Strategies"]
["In Grade 5, the concept of place value is extended to include decimal values to thousandths. The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers and decimal numbers. For example, students need to continue to represent, write and state the value of numbers, including decimal numbers. For students who are not able to read, write and represent multi-digit numbers, working with decimals could potentially be challenging. Money is a good medium to compare decimals (but keep in mind that 5th grade students are to extend their work into thousandths place so use other situations besides money). Present contextual situations that require the comparison of the cost of two items to determine which one is of a higher price or a lower price. Students should also be able to identify how many pennies, dimes, dollars and ten dollars, etc., are in a given value. Help students make connections between the number of each type of coin and the value of each coin, and the expanded form of the number. Build on the understanding that it always takes ten of the number to the right to make the number to the left. Number cards, number cubes, spinners and other manipulatives can be used to generate decimal numbers. For example, have students roll three number cubes and have them create the largest and smallest decimal number to the thousandths place. Ask students to represent the number with numerals and words, including standard and unit forms."]
["Common Misconceptions"]
["Students who are taught to round decimals by using a rule rather than place value understanding have difficulty determining places when rounding up or down. This is true with both whole numbers and decimals. For example, when rounding to the nearest tenth, a student might round 15.28 to 15.38. When using a number line model, students need to determine the numbers that the given number falls between. In the previous case it would be between 15.2 and 15.3. Using benchmark numbers such as 15.25, which falls exactly in the middle, will help students determine the closest tenth. By plotting the given point on the number line, students can determine to which tenth it is closer. Scaffold examples for students who are struggling with this concept."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/A/4/tasks/1804\",\"5.NBT.A.4 Rounding to Tenths and Hundredths\")","v":"5.NBT.A.4 Rounding to Tenths and Hundredths"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NBT.B Perform operations with multi-digit whole numbers and with decimals to hundredths."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NBT.B.4\nFluently add and subtract multi-digit whole numbers using a standard algorithm.","5.NBT.B.5\nFluently multiply multi-digit whole numbers using a standard algorithm.","5.NBT.B.6\nApply and extend understanding of division to find whole-number quotients of whole numbers with up to fourdigit dividends and two-digit divisors."]
["4.NBT.B.5\nMultiply 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.",null,"6.NS.B.3\nFluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation."]
["Instructional Strategies"]
["Because students have used various models and strategies to solve problems involving multiplication with whole numbers, they should be able to transition to using standard algorithms with understanding effectively. With guidance from the teacher, they should understand the connection between the standard algorithm and other algorithms, methods, and strategies. Connections between the different algorithms for multiplying multi-digit whole numbers and strategies are necessary for students’ understanding."]
["Common Misconceptions"]
["Students who become confused with regrouping in multi-digit multiplication need additional experiences with the partial product algorithm. Once they are proficient multiplying using partial products, they can begin to consider how using regrouping can save several steps. Scaffold examples for these students and give them time to understand how both the partial product and the regrouping algorithms are alike."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/B/5/tasks/1812\",\"5.NBT.B.6 Elmer's Multiplication Error\")","v":"5.NBT.B.6 Elmer's Multiplication Error"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NBT.B Perform operations with multi-digit whole numbers and with decimals to hundredths."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NBT.B.4\nFluently add and subtract multi-digit whole numbers using a standard algorithm.","5.NBT.B.6\nApply and extend understanding of division to find whole-number quotients of whole numbers with up to fourdigit\ndividends and two-digit divisors.","6.NS.B.2\nFluently divide multi-digit numbers using a standard algorithm."]
["4.NBT.B.6\nDemonstrate understanding of division by finding whole-number quotients and remainders with up to four-digit\ndividends and one-digit divisors.",null,"6.NS.B.3\nFluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation."]
["Instructional Strategies"]
["Because students have used various models and strategies to solve problems involving multiplication with whole numbers, they should be able to transition to using standard algorithms with understanding effectively. With guidance from the teacher, they should understand the connection between the standard algorithm and other algorithms, methods, and strategies. Connections between the different algorithms for multiplying multi-digit whole numbers and strategies are necessary for students’ understanding."]
["Common Misconceptions"]
["Division is a complex operation, and students who depend on following rote steps cannot determine whether their answer is reasonable. Emphasis on place value and connections to multiplication will help students to develop a deeper understanding of division. All division experience should be developed in the context of asking question such as “How many groups of 20 can you make from 700?” and then allowing students to estimate and identify the number of objects (for example, If I make 30 groups of 20 that would be 600, and if I make 40 groups that would be 800, and that is too high). Such reasoning will help students to hone in on a good estimate and use partial products to determine the exact quotient.\n\nStudents may still need additional experiences with the meaning of the remainder built on previous work in Grade 4. Problems in which the remainder is the answer, in which the remainder is dropped, or in which the quotient should be one more because of a remainder should all be included in division problems students are asked to solve."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/B/6/tasks/878\",\"5.NBT.B.6 Minutes and Days\")","v":"5.NBT.B.6 Minutes and Days"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NBT.B Perform operations with multi-digit whole numbers and with decimals to hundredths."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NBT.B.4\nFluently add and subtract multi-digit whole numbers using a standard algorithm.","5.NBT.B.7\nAdd, subtract, multiply, and divide decimals to hundredths, connecting objects or drawings to strategies based\non place value, properties of operations, and/or the relationship between operations. Relate the strategy to a\nwritten form.","5.MD.A.1\nConvert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real-world problems."]
["5.NF.A.1\nAdd 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 enominators (e.g., 2/3 + 5/4 = 8/12 + 15/12 = 23/12).",null,"6.NS.B.3\nFluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation."]
["5.NF.B.3\nInterpret a fraction as the number that results from dividing the whole number numerator by the whole number denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers."]
["5.NF.B.4\nApply and extend previous understandings of multiplication to multiply a fraction by a whole number and a fraction by a fraction.\na. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation.\nb. Interpret the product of a fraction multiplied by a fraction (a/b) x (c/d). Use a visual fraction model and create\na story context for this equation. For example, use a visual fraction model to show (2/3) x (4/5) = 8/15, and\ncreate a story context for this equation. In general, (a/b) x (c/d) = ac/bd.\nc. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit\nfraction side lengths, and show that the area is the same as would be found by multiplying the side lengths.\nMultiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas."]
["5.NF.B.7\nApply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.\na. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. Use the\nrelationship between multiplication and division to justify conclusions.\nb. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to justify conclusions (e.g., 4 ÷ (1/5) = 20 because 20 x (1/5) = 4).\nc. Solve problems in real-world context involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, using a variety of representations."]
["Instructional Strategies"]
["Because students have used various models and strategies to solve problems involving multiplication with whole numbers, they should be able to transition to using standard algorithms with understanding effectively. With guidance from the teacher, they should understand the connection between the standard algorithm and other algorithms, methods, and strategies. Connections between the different algorithms for multiplying multi-digit whole numbers and strategies are necessary for students’ understanding."]
["Common Misconceptions"]
["Student misconceptions when working with decimal numbers are usually based on place value. Simply telling students to line up the decimal points when adding and subtracting decimals does not build the important understanding that similar place values are to be added or subtracted and can lead to errors such as the following:\n2.5\n-1.75\n0.85\nBuilding on whole-number experiences using concrete materials and place value charts will help students to relate previous work with composing and decomposing whole numbers to composing and decomposing decimals.\n\nSince the “rules” for multiplication and division of decimals are much easier to teach than developing place value understanding, we are often tempted to provide students with these rules at some point. Don’t do it! Students need time to see the structure of multiplication and division of decimals and how it relates to whole-number multiplication and division. Without this foundation, students may move decimal points when it is convenient rather than when it is necessary. Place value understanding also allows students to determine whether answers are reasonable. It is far more meaningful to students when they can generalize rules after many experiences and good questions from the teacher."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/B/7/tasks/292\",\"5.NBT.B.7 What is 23 /5?\")","v":"5.NBT.B.7 What is 23 /5?"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NBT/B/7/tasks/1293\",\"5.NBT.B.7 The Value of Education\")","v":"5.NBT.B.7 The Value of Education"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NF.A-Use equivalent fractions to add and subtract fractions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NF.A1\nExplain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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 understand and generate equivalent fractions.","5.NF.A.1\nAdd 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 (e.g., 2/3 + 5/4 = 8/12 + 15/12 = 23/12).","5.NBT.B.7\nAdd, subtract, multiply, and divide decimals to hundredths, connecting objects or drawings to strategies based on place value, properties of operations, and/or the relationship between operations. Relate the strategy to a written form."]
["4.NF.B.3\nUnderstand a fraction a/b with a > 1 as a sum of unit fractions (1/b). a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way (e.g., 3/8 = 1/8 + 1/8+1/8; 3/8 = 2/8 + 1/8; 2 1/8 = 1 + 1 + 1/8 + or 2 1/8 = 8/8 + 8/8 + 1/8). c. Add and subtract mixed numbers with like denominators (e.g., by using properties of operations and the relationship between addition and subtraction and/or by replacing each mixed number with an equivalent fraction). d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.",null,"5.NF.A.2\nSolve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators by using a variety of representations, equations, and visual models to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (e.g. recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2)."]
["4.MD.A.2\nUse the four operations to solve word problems and problems in real-world context involving distances, intervals of time (hr, min, sec), liquid volumes, masses of objects, and money, including decimals and problems involving fractions with like denominators, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using a variety of representations, including number lines that feature a measurement scale.",null,"6.EE.B.7\nSolve mathematical problems and problems in real-world context by writing and solving equations of the form x + p = q, x - p = q, px = q, and x/p = q for cases in which p, q and x are all non-negative rational numbers."]
["4.MD.B.4\nMake 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.",null,"6.NS.B.3\nFluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation."]
["Instructional Strategies"]
["To add or subtract fractions with unlike denominators, students use their understanding of equivalent fractions to create fractions with the same denominators. Start with problems that require the changing of one of the fractions and progress to changing both fractions. Allow students to add and subtract fractions using different strategies, such as number lines, area models, fraction bars or strips. Have students share their strategies and discuss commonalities among the strategies to begin making generalizations. Students need to develop an understanding that when adding or subtracting fractions, the fractions must refer to the same whole. All models used must refer to the same whole. Students may find that a circular model might not be the best model when adding or subtracting fractions because of the difficulty in partitioning the pieces so they are equal. It is essential that instruction regularly includes word problems involving addition or subtraction of fractions. The concept of adding or subtracting fractions with unlike denominators will develop through solving problems where the students make sense of the fractional pieces. Mental computations and estimation strategies should be used to determine the reasonableness of answers. Students need to prove or disprove whether an answer provided for a problem is reasonable. Estimation is about getting useful answers, it is not about getting the right answer. Students need to be able to explain their reasoning."]
["Common Misconceptions"]
["Watch for students who have surface understanding of the necessity for finding common denominators when adding and subtracting fractions and mixed numbers. Consistent practice in the form of number talks or using formative assessment tasks coupled with students explaining their thinking and considering the reasonableness of their solutions will help students to see the importance of thinking about the value of the numbers rather than using random calculations (add the numerators, add the denominators). Relating the fractions to benchmark numbers (0, ½, 1) will help students to determine whether their answer is reasonable.\n\nTwo areas that should be explicit in providing meaningful situations include considering the size of the piece (that is, how many pieces make one whole or the denominator) and that the fractions must refer to the same size whole. Students must always consider that adding ½ of a small candy bar with ½ of a large candy bar will not produce 1 whole candy bar."]
[null,"Resources","Site"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/839\",\"5.NF.A.1 Eygptian Fractions\")","v":"5.NF.A.1 Eygptian Fractions"},{"$f":"HYPERLINK(\"https://www.illustrativemathematics.org/\",\"Illustrative Mathematics\")","v":"Illustrative Mathematics"}]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/847\",\"5.NF.A.1 Mixed Numbers with Unlike Fractions\")","v":"5.NF.A.1 Mixed Numbers with Unlike Fractions"},"Illustrative Mathematics"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/848\",\"5.NF.A.1 Finding Common Denominators to Add\")","v":"5.NF.A.1 Finding Common Denominators to Add"},"Illustrative Mathematics"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/855\",\"5.NF.A.1 Jog-a-Thon\")","v":"5.NF.A.1 Jog-a-Thon"},"Illustrative Mathematics"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/859\",\"5.NF.A.1 Finding Common Denominators to Subtract\")","v":"5.NF.A.1 Finding Common Denominators to Subtract"},"Illustrative Mathematics"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/861\",\"5.NF.A.1 Making S'Mores\")","v":"5.NF.A.1 Making S'Mores"},"Illustrative Mathematics"]
[null,{"$f":"HYPERLINK(\"https://tasks.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/1563\",\"5.NF.A.1 Fractions on a Line Plot\")","v":"5.NF.A.1 Fractions on a Line Plot"},"Illustrative Mathematics"]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NF.A Use equivalent fractions to add and subtract fractions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NF.A.2\nCompare two fractions with different numerators and different denominators (e.g., by creating common\ndenominators or numerators and by comparing to a benchmark fraction).\na. Understand that comparisons are valid only when the two fractions refer to the same size whole.\nb. Record the results of comparisons with symbols >, =, or <, and justify the conclusions.","5.NF.A.2  Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators by using a variety of representations, equations, and visual models to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (e.g. recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2).","5.MD.B.2\nMake a line plot to display a data set of measurements in fractions of a unit (1/8, 1/2, 3/4). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally."]
["5.NF.A.1\nAdd 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 (e.g., 2/3 + 5/4 = 8/12 + 15/12 = 23/12).",null,"6.SP.A.1\nRecognize a statistical question as one that anticipates variability in the data related to the question and accounts for variability in the answers. For example, \"How old am I?\" is not a statistical question, but \"How old are the students in my school?\" is a statistical question because one anticipates variability in students' ages.Make a line plot to display a data set of measurements in fractions of a unit (1/8, 1/2, 3/4). Use operations on\nfractions for this grade to solve problems involving information presented in line plots. For example, given\ndifferent measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the\ntotal amount in all the beakers were redistributed equally."]
[null,null,"6.SP.A.2\nUnderstand that a set of data collected to answer a statistical question has a distribution whose general characteristics can be described by its center, spread, and overall shape."]
[null,null,"6.SP.B.4\nDisplay and interpret numerical data by creating plots on a number line including histograms, dot plots, and box plots."]
["Instructional Strategies"]
["To add or subtract fractions with unlike denominators, students use their understanding of equivalent fractions to create fractions with the same denominators. Start with problems that require the changing of one of the fractions and progress to changing both fractions. Allow students to add and subtract fractions using different strategies, such as number lines, area models, fraction bars or strips. Have students share their strategies and discuss commonalities among the strategies to begin making generalizations. Students need to develop an understanding that when adding or subtracting fractions, the fractions must refer to the same whole. All models used must refer to the same whole. Students may find that a circular model might not be the best model when adding or subtracting fractions because of the difficulty in partitioning the pieces so they are equal. It is essential that instruction regularly includes word problems involving addition or subtraction of fractions. The concept of adding or subtracting fractions with unlike denominators will develop through solving problems where the students make sense of the fractional pieces. Mental computations and estimation strategies should be used to determine the reasonableness of answers. Students need to prove or disprove whether an answer provided for a problem is reasonable. Estimation is about getting useful answers, it is not about getting the right answer. Students need to be able to explain their reasoning."]
["Common Misconceptions"]
["Students who struggle to determine the appropriate operation to solve a problem need more experience with the problem situations for addition and subtraction (see Table 1 from the Arizona 5th Grade Math Standards). They need to use strategies such as act it out, draw a picture, write an equation, or make a model to determine how to best approach a problem. Give students opportunities to explain their thinking as they read the problem and use models to determine the correct operation. Make connections to earlier experiences  with whole numbers that will help students to think of addition and subtraction in a particular situation. Once students determine the correct operation, they can use fractions and mixed numbers to solve the problem."]
[null,"Resources","Site"]
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[]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NF.B Use previous understandings of multiplication and division to multiply and divide fractions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["3.OA.A.1\nInterpret products of whole numbers as the total number of objects in equal groups (e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each).","5.NF.B.3 Interpret a fraction as the number that results from dividing the whole number numerator by the whole number denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.","6.RP.A.2\nUnderstand the concept of a ratio as comparing two quantities multiplicatively or joining/composing the two quantities in a way that preserves a multiplicative relationship. Use ratio language to describe a ratio relationship between two quantities. For example, \"There were 2/3 as many men as women at the concert.”"]
["3.OA.A.2\nInterpret whole number quotients of whole numbers (e.g., interpret 56 ÷ 8 as the number of objects in each group when 56 objects are partitioned equally into 8 groups, or as a number of groups when 56 objects are partitioned into equal groups of 8 objects each). See Table 2.",null,"7.NS.A.2\nMultiply and divide integers and other rational numbers.\na. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world context.\nb. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by\ndescribing real-world context.\nc. Apply properties of operations as strategies to multiply and divide rational numbers.\nd. Convert a rational number to decimal form using long division; know that the decimal form of a rational number terminates in 0’s or eventually repeats.\n"]
["3.OA.B.6\nUnderstand division as an unknown-factor problem (e.g., find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8)."]
["4.MD.A.1 Know relative sizes of measurement units within one system of units which could include 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 and in a smaller unit in terms of a larger unit."]
["4.OA.A.1 Represent verbal statements of multiplicative comparisons as multiplication equations. Interpret a multiplication equation as a comparison (e.g., 35 is the number of objects in 5 groups, each containing 7 objects, and is also the number of objects in 7 groups, each containing 5 objects)."]
["4.OA.A.2 Multiply or divide within 1000 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 Table 2."]
["4.OA.A.3 Solve multistep word problems using the four operations, including problems in which remainders must be interpreted. Understand how the remainder is a fraction of the divisor. Represent these problems using equations with a letter standing for the unknown quantity."]
["Instructional Strategies"]
["Connect the meaning of multiplication and division of fractions with whole-number multiplication and division. Consider area models of multiplication; and both sharing and measuring models for division. When connecting the understanding students already have of whole number multiplication and division, ask questions such as, “What does 2 × 3 mean?” and “What does 12 ÷ 3 mean?” Then, follow those with questions for multiplication of fractions, such as, “What does 3/4 × 1/3 mean?” ( 3/4 of 1/3 ); “What does 3/4 × 7 mean?” (3/4 of 7); and “What does 3/4 x 7 mean?” (3/4 of a set of 7)\n\nWhen examining the meaning of division, ask “What does 4 ÷ 1/2 mean?” (How many 1/2 are in 4?) and “What does 1/2 ÷ 4 mean?? (How many groups of 4 are in 1/2 ?).\n\nStudents should use drawings and models to show their understanding of multiplication and division of fractions. Begin with students modeling multiplication and division with whole numbers. Have them explain how they used the model or drawing to arrive at the solution. Models to consider when multiplying or dividing fractions include, but are not limited to: area models (rectangles), linear models (fraction strips/bars, Cuisenaire rods, and number lines) and set models (counters). Use calculators or models to explain what happens to the result when dividing a unit fraction by a non-zero whole number (1/8 ÷ 4, 1/8 ÷ 8, 1/8 ÷ 1/6 , …) and what happens to the result when dividing a whole number by a unit fraction (4 ÷ 1/4 , 8 ÷ 1/4 , 12 ÷ 1/4 , …). Present problem situations and have students use models and equations to solve the problem. It is important for students to develop understanding of multiplication and division of fractions through contextual situations."]
["Common Misconceptions"]
["Students may initially think that you cannot divide a “smaller number by a bigger number” since this will be a new situation for them to consider. Provide them with good problems to solve and give them many opportunities to explore with models so that they are developing conceptual understanding. It is important that they understand this concept in a way that makes sense to them rather than be shown how to do it. The role of the teacher is to provide sensible problem situations, ask supporting questions, and facilitate conversations in which the students are making sense of the situation and why their answers make sense.\n\nStudents who struggle with interpreting the remainder of division examples need more experience solving problems using concrete models so they understand that the remainder tells what part of a group is left over. Asking questions such as “How many are left?” and “How many would it take to make another full group?” and modeling what part of a full group is left over will help them to understand the meaning of the remainder when it is expressed as a fraction."]
[null,"Resources","Site"]
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[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NF.B Use previous understandings of multiplication and division to multiply and divide fractions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["3.MD.C.7\nRelate area to the operations of multiplication and addition.\na. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.\nb. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving realworld and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.\nc. Use tiling to show that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.\nd. Understand that rectilinear figures can be decomposed into non-overlapping rectangles and that the sum of the areas of these rectangles is identical to the area of the original rectilinear figure. Apply this technique to solve problems in real-world contexts.","5.NF.B.4 (Parts A & B ONLY)\nApply and extend previous understandings of multiplication to multiply a fraction by a whole number and a fraction by a fraction.\na. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation.\nb. Interpret the product of a fraction multiplied by a fraction (a/b) x (c/d). Use a visual fraction model and create a story context for this equation. For example, use a visual fraction model to show (2/3) x (4/5) = 8/15, and create a story context for this equation. In general, (a/b) x (c/d) = ac/bd.","5.NBT.B.7\nAdd, subtract, multiply, and divide decimals to hundredths, connecting objects or drawings to strategies based on place value, properties of operations, and/or the relationship between operations. Relate the strategy to a written form."]
["4.NF.B.4\nBuild fractions from unit fractions.\na. Understand a fraction 𝑎/𝑏 as a multiple of a unit fraction 1/𝑏 . In general, 𝑎/𝑏 = a x 1/𝑏 .\nb. Understand a multiple of 𝑎/𝑏 as a multiple of a unit fraction 1 /𝑏 , and use this understanding to multiply a whole number by a fraction. In general, n x 𝑎/𝑏 = (𝑛𝑥𝑎)/𝑏.\nc. Solve word problems involving multiplication of a whole number by a fraction. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?",null,"5.NF.B.5\nInterpret multiplication as scaling (resizing), by:\na. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.\nb. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number; explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence 𝑎/𝑏 = (𝑛 𝑥 𝑎)/ ( 𝑛 𝑥 𝑏) to the effect of multiplying 𝑎/𝑏 by 1."]
[null,null,"6.EE.B.7\nSolve mathematical problems and problems in real-world context by writing and solving equations of the form x + p = q, x - p = q, px = q, and x/p = q for cases in which p, q and x are all non-negative rational numbers."]
[null,null,"6.G.A.1\nFind the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques to solve mathematical problems and problems in real-world context."]
["Instructional Strategies"]
["Connect the meaning of multiplication and division of fractions with whole-number multiplication and division. Consider area models of multiplication; and both sharing and measuring models for division. When connecting the understanding students already have of whole number multiplication and division, ask questions such as, “What does 2 × 3 mean?” and “What does 12 ÷ 3 mean?” Then, follow those with questions for multiplication of fractions, such as, “What does 3/4 × 1/3 mean?” ( 3/4 of 1/3 ); “What does 3/4 × 7 mean?” (3/4 of 7); and “What does 3/4 x 7 mean?” (3/4 of a set of 7)\n\nWhen examining the meaning of division, ask “What does 4 ÷ 1/2 mean?” (How many 1/2 are in 4?) and “What does 1/2 ÷ 4 mean?? (How many groups of 4 are in 1/2 ?).\n\nStudents should use drawings and models to show their understanding of multiplication and division of fractions. Begin with students modeling multiplication and division with whole numbers. Have them explain how they used the model or drawing to arrive at the solution. Models to consider when multiplying or dividing fractions include, but are not limited to: area models (rectangles), linear models (fraction strips/bars, Cuisenaire rods, and number lines) and set models (counters). Use calculators or models to explain what happens to the result when dividing a unit fraction by a non-zero whole number (1/8 ÷ 4, 1/8 ÷ 8, 1/8 ÷ 1/6 , …) and what happens to the result when dividing a whole number by a unit fraction (4 ÷ 1/4 , 8 ÷ 1/4 , 12 ÷ 1/4 , …). Present problem situations and have students use models and equations to solve the problem. It is important for students to develop understanding of multiplication and division of fractions through contextual situations."]
["Common Misconceptions"]
["5.NF.B.4a - Students may see the pattern and see that to multiply fractions you “simply” multiply the numerators and multiply the denominators. This is the correct algorithm or procedure. However, only references to real-life situations and using models and visual representation will help students develop a conceptual understanding of what is actually happening when they multiply fractions.\n\n5.NF.B.4b - Watch for students who have difficulty determining the part of the unit square. Thinking in terms of the whole rectangle will help them define the number of parts when the dimensions are fractional parts of the whole. Reinforcing when they multiply a fraction be a fraction they are taking a part will help students to see that the “overlap” is the number of pieces (or numerator), and the total number of pieces in the whole is the denominator.\n\n5.NF.B.4c - Watch for students who have difficulty determining the part of the unit square. THinking in terms of the whole rectangle will help them define the number of parts when the dimensions are fractional parts of the whole. Reinforcing when they multiply a fraction by a fraction they are taking a part of a part will help students to see that the “overlap” is the number of pieces (or numerator), and the total number of pieces is the denominator.\n\nIf students are struggling to see how to find their answer when the fractions are overlapping, have students model each fraction separately to clarify how the unit tiles are created."]
[null,"Resources","Site"]
[]
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[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NF.B Use previous understandings of multiplication and division to multiply and divide fractions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.MD.A.2\nUse the four operations to solve word problems and problems in real-world context involving distances, intervals of time (hr, min, sec), liquid volumes, masses of objects, and money, including decimals and problems involving fractions with like denominators, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using a variety of representations, including number lines that feature a measurement scale.","5.NF.B.5\nInterpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number; explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = n x a/n x b to the effect of multiplying a/b by 1.","6.RP.A.1\nUnderstand the concept of a ratio as comparing two quantities multiplicatively or joining/composing the two quantities in a way that preserves a multiplicative relationship. Use ratio language to describe a ratio relationship between two quantities."]
["4.OA.A.3\nSolve multistep word problems using the four operations, including problems in which remainders must be interpreted. Understand how the remainder is a fraction of the divisor. Represent these problems using equations with a letter standing for the unknown quantity."]
["4.NF.A.1\nExplain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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 understand and generate equivalent fractions."]
["4.OA.A.1\nRepresent verbal statements of multiplicative comparisons as multiplication equations. Interpret a multiplication equation as a comparison (e.g., 35 is the number of objects in 5 groups, each containing 7 objects, and is also the number of objects in 7 groups, each containing 5 objects)."]
["4.OA.A.2\nMultiply or divide within 1000 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 Table 2."]
["Instructional Strategies"]
["Encourage students to use models or drawings to multiply or divide with fractions. Begin with students modeling multiplication and division with whole numbers. Have them explain how they used the model or drawing to arrive at the solution. Models to consider when multiplying or dividing fractions include, but are not limited to: area models (rectangles), linear models (fraction strips/bars, Cuisenaire rods, and number lines) and set models (counters)."]
["Common Misconceptions"]
["5.NF.B.5a - Students will likely have many misconceptions about what happens to the product when one or both factors are scaled. For example, if both the length and width of a rectangle are doubled, some students will assume the product (area) is doubled. When they test their conjecture by drawing a picture, they will see that the product is actually four times greater. Allow students to explore a variety of multiplication scaling situations by drawing pictures and making models that will help them to make conjectures as to why the results are true, which is less likely to happen if they simply multiply.\n\n5.NF.B.5b - Students are often puzzled when they find that the product is less than one or both of the factors. In previous work with multiplication of whole numbers, the product was always greater than both factors. Give students many opportunities to use visual models to “see” what is happening when they multiply with fractions. Discussions in which students explain their thinking will also help to identify and address misconceptions."]
[null,"Resources","Site"]
[]
[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NF.B Use previous understandings of multiplication and division to multiply and divide fractions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.MD.A.2\nUse the four operations to solve word problems and problems in real-world context involving distances, intervals of time (hr, min, sec), liquid volumes, masses of objects, and money, including decimals and problems involving fractions with like denominators, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using a variety of representations, including number lines that feature a measurement scale.","5.NF.B.6\nSolve problems in real-world contexts involving multiplication of fractions, including mixed numbers, by using a variety of representations including equations and models.","6.SP.A.1\nRecognize a statistical question as one that anticipates variability in the data related to the question and accounts for variability in the answers."]
["4.OA.A.3\nSolve multistep word problems using the four operations, including problems in which remainders must be interpreted. Understand how the remainder is a fraction of the divisor. Represent these problems using equations with a letter standing for the unknown quantity.",null,"6.SP.A.2\nUnderstand that a set of data collected to answer a statistical question has a distribution whose general characteristics can be described by its center, spread, and overall shape."]
["4.OA.A.1\nRepresent verbal statements of multiplicative comparisons as multiplication equations. Interpret a multiplication equation as a comparison (e.g., 35 is the number of objects in 5 groups, each containing 7 objects, and is also the number of objects in 7 groups, each containing 5 objects).",null,"6.SP.B.4\nDisplay and interpret numerical data by creating plots on a number line including histograms, dot plots, and box plots"]
["4.OA.A.2\nMultiply or divide within 1000 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 Table 2."]
["Instructional Strategies"]
["Connect the meaning of multiplication and division of fractions with whole-number multiplication and division. Consider area models of multiplication; and both sharing and measuring models for division. When connecting the understanding students already have of whole number multiplication and division, ask questions such as, “What does 2 × 3 mean?” and “What does 12 ÷ 3 mean?” Then, follow those with questions for multiplication of fractions, such as, “What does 3/4 × 1/3 mean?” ( 3/4 of 1/3 ); “What does 3/4 × 7 mean?” (3/4 of 7); and “What does 3/4 x 7 mean?” (3/4 of a set of 7)\n\nWhen examining the meaning of division, ask “What does 4 ÷ 1/2 mean?” (How many 1/2 are in 4?) and “What does 1/2 ÷ 4 mean?? (How many groups of 4 are in 1/2 ?).\n\nStudents should use drawings and models to show their understanding of multiplication and division of fractions. Begin with students modeling multiplication and division with whole numbers. Have them explain how they used the model or drawing to arrive at the solution. Models to consider when multiplying or dividing fractions include, but are not limited to: area models (rectangles), linear models (fraction strips/bars, Cuisenaire rods, and number lines) and set models (counters). Use calculators or models to explain what happens to the result when dividing a unit fraction by a non-zero whole number (1/8 ÷ 4, 1/8 ÷ 8, 1/8 ÷ 1/6 , …) and what happens to the result when dividing a whole number by a unit fraction (4 ÷ 1/4 , 8 ÷ 1/4 , 12 ÷ 1/4 , …). Present problem situations and have students use models and equations to solve the problem. It is important for students to develop understanding of multiplication and division of fractions through contextual situations."]
["Common Misconceptions"]
["Watch for misconceptions from previous multiplication standards. Students who struggle understanding why they should multiply in these problems need more experience using visual representations. It is helpful to have them break the problem into smaller parts and explain their thinking as they complete each part of the problem."]
[null,"Resources","Site"]
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[]
[]
["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.NF.B Use previous understandings of multiplication and division to multiply and divide fractions."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.NF.B.4\nBuild fractions from unit fractions. a. Understand a fraction 𝑎𝑎 𝑏𝑏 as a multiple of a unit fraction 1 𝑏𝑏 . In general, 𝑎𝑎 𝑏𝑏 = a x 1 𝑏𝑏 . b. Understand a multiple of 𝑎𝑎 𝑏𝑏 as a multiple of a unit fraction 1 𝑏𝑏 , and use this understanding to multiply a whole number by a fraction. In general, n x 𝑎𝑎 𝑏𝑏 = 𝑛𝑛 𝑥𝑥 𝑎𝑎 𝑏𝑏 . c. Solve word problems involving multiplication of a whole number by a fraction. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?","5.NF.B.7\nApply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. Use the relationship between multiplication and division to justify conclusions. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to justify conclusions (e.g., 4 ÷ (1/5) = 20 because 20 x (1/5) = 4). c. Solve problems in real-world context involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, using a variety of representations.","6.NS.A.1\nInterpret and compute quotients of fractions to solve mathematical problems and problems in real-world context involving division of fractions by fractions using visual fraction models and equations to represent the problem. For example, create a story context for 2/3 ÷ 3/4 and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3. In general, a/b ÷ c/d = ad/bc."]
[null,null,"6.RP.A.2\nUnderstand the concept of a unit rate a/b associated with a ratio a : b with b ≠ 0, and use rate language (e.g., for every, for each, for each 1, per) in the context of a ratio relationship. (Complex fraction notation is not an expectation for unit rates in this grade level.)"]
["Instructional Strategies"]
["Connect the meaning of multiplication and division of fractions with whole-number multiplication and division. Consider area models of multiplication; and both sharing and measuring models for division. When connecting the understanding students already have of whole number multiplication and division, ask questions such as, “What does 2 × 3 mean?” and “What does 12 ÷ 3 mean?” Then, follow those with questions for multiplication of fractions, such as, “What does 3/4 × 1/3 mean?” ( 3/4 of 1/3 ); “What does 3/4 × 7 mean?” (3/4 of 7); and “What does 3/4 x 7 mean?” (3/4 of a set of 7)\n\nWhen examining the meaning of division, ask “What does 4 ÷ 1/2 mean?” (How many 1/2 are in 4?) and “What does 1/2 ÷ 4 mean?? (How many groups of 4 are in 1/2 ?).\n\nStudents should use drawings and models to show their understanding of multiplication and division of fractions. Begin with students modeling multiplication and division with whole numbers. Have them explain how they used the model or drawing to arrive at the solution. Models to consider when multiplying or dividing fractions include, but are not limited to: area models (rectangles), linear models (fraction strips/bars, Cuisenaire rods, and number lines) and set models (counters). Use calculators or models to explain what happens to the result when dividing a unit fraction by a non-zero whole number (1/8 ÷ 4, 1/8 ÷ 8, 1/8 ÷ 1/6 , …) and what happens to the result when dividing a whole number by a unit fraction (4 ÷ 1/4 , 8 ÷ 1/4 , 12 ÷ 1/4 , …). Present problem situations and have students use models and equations to solve the problem. It is important for students to develop understanding of multiplication and division of fractions through contextual situations."]
["Common Misconceptions"]
["5.NF.B.7a - Dividing a fraction by a whole number is likely to cause students initial confusion around understanding how you can possibly divide a fraction (part of a whole) by a whole number. One misconception is that you always have to “divide the bigger number by the smaller number.” Connect to work with earlier standards in this domain in which students interpreted a fraction such as ¾ to also mean 3 divided by 4. It is important to give students many opportunities to solve problems with visual representations to develop understanding that this is the same as the sharing situations they used when dividing whole numbers. Do not rush students into writing equations. Allow students to write their own problems modeled after those you have given. This will help them to think about the situation and when it makes sense to divide a fraction by a whole number. Do not give them the traditional rule for division of fractions. Rather, take time for classroom discussions in which students explain their thinking and work to make sense out of the solution process and to determine the reasonableness of their answers. The role of the teacher is to clarify student thinking by posing good questions.\n\n5.NF.B.7b - Watch for students who are having difficulty identifying what operation to use in solving problems with fractions. Using keywords is not helpful and removes making sense from the process. Rather, have students model the problem using pictures and ask supporting questions, such as “What do you know? What do you want to find out? How can you show that in your picture?” As students solve mixed problems, adapt your questions to help students think about the meaning of the operations and how it can help them determine which operation to use.\n\nGive students a variety of problems and ask them to model and write an expression they would use to solve the problem. Ask them to explain their model and expression.\n\n5.NF.B.7c - Students may struggle determining which number goes where in the division problem. “Am I dividing the fraction by the whole number or the whole number by the fraction?” Drawing a picture using the information in the problem and focusing on what they want to find out will help. Model asking questions and encourage them to ask themselves similar questions, such as:\n• What is being divided or broken up?\n• Am I trying to determine how much in a group or how many groups?\n• What visual representations can I use to show the action of the problem?"]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.MD.A Convert like measurement units within a given measurement system"]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.MD.A.1\nKnow relative sizes of measurement units within one system of units which could include km, m, cm; kg, g; lb, oz.;\nl, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a\nsmaller unit and in a smaller unit in terms of a larger unit. For example, know that 1 ft is 12 times as long as 1 in.\nExpress the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number\npairs (1,12), 2,24), (3,36).","5.MD.A.1\nConvert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real-world problems.","6.RP.A.3\nUse ratio and rate reasoning to solve mathematical problems and problems in real-world context (e.g., by reasoning about data collected from measurements, tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).\na. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.\nb. Solve unit rate problems including those involving unit pricing and constant speed. \nc. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity). Solve percent problems with the unknown in all positions of the equation.\nd. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities."]
["4.OA.A.2\nMultiply or divide within 1000 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 Table 2."]
["4.MD.A.2\nUse the four operations to solve word problems and problems in real-world context involving distances, intervals of time (hr, min, sec), liquid volumes, masses of objects, and money, including decimals and problems involving fractions with like denominators, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using a variety of representations, including number lines that feature a measurement scale."]
["Instructional Strategies"]
["Students should gain ease in converting units of measures in equivalent forms within the same system. To convert from one unit to another unit, the relationship between the units must be known. In order for students to have a better understanding of the relationships between units, they need to use measuring tools in class. The number of units must relate to the size of the unit. For example, students should have discovered that there are 12 inches in 1 foot and 3 feet in 1 yard. This understanding is needed to convert inches to yards. Using 12-inch rulers and yardsticks, students can see that three of the 12-inch rulers are equivalent to one yardstick (3 × 12 inches = 36 inches; 36 inches = 1 yard). Using this knowledge, students can decide whether to multiply or divide when making conversions. Once students have an understanding of the relationships between units and how to do conversions, they are ready to solve multi-step problems that require conversions within the same system. Allow students to discuss methods used in solving the problems. Begin with problems that allow for renaming the units to represent the solution before using problems that require renaming to find the solution."]
["Common Misconceptions"]
["Some students may not pay attention to the unit of measurement when subtracting. For example, when subtracting 5 inches from 2 feet (2 ft. - 5 in.), students may incorrectly think the answer is 1 ft. and 5 inches instead of 1 foot and 7 inches. To address this misconception, talk about and show the example using 2 twelve-inch rulers, then subtract."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.MD.B-Represent and interpret data"]
["Prerequisite Standards","Focus Standard","Future Learning"]
["4.MD.B.4\nMake 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.","5.MD.B.2\nMake a line plot to display a data set of measurements in fractions of a unit (1/8, 1/2, 3/4). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.","6.SP.A.1\nRecognize a statistical question as one that anticipates variability in the data related to the question and accounts for variability in the answers. For example, \"How old am I?\" is not a statistical question, but \"How old are the students in my school?\" is a statistical question because one anticipates variability in students' ages."]
[null,null,"6.SP.A.2\nUnderstand that a set of data collected to answer a statistical question has a distribution whose general characteristics can be described by its center, spread, and overall shape."]
[null,null,"6.SP.B.4\nDisplay and interpret numerical data by creating plots on a number line including histograms, dot plots, and box plots."]
["Instructional Strategies"]
["Using data displays to solve problems involving operations with unit fractions includes addition, subtraction, multiplication, and division. Revisit using a number line to solve multiplication and division problems with whole numbers. In addition to knowing how to use a number line to solve problems, students also need to know which operation to use to solve problems. Use the tables for common addition and subtraction, and multiplication and division situations (Table 1 and Table 2 in the Arizona 5th Grade Math Standards) as a guide to the types of problems students need to solve without specifying the type of problem. Allow students to share methods used to solve the problems. Also have students create problems to show their understanding of the meaning of each operation."]
["Common Misconceptions"]
["Some students may not know what measurement to use if the object measures between ⅛ and ¼ inch. To address this, help students understand that approximations can be used to measure to the closest ⅛ inch and ¼ inch."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.MD.C Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["3.MD.C.5\nUnderstand area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.","5.MD.C.3\nRecognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.","5.MD.C.4\nMeasure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units."]
[null,null,"5.MD.C.5\nRelate volume to the operations of multiplication and addition and solve mathematical problems and problems in real-world contexts involving volume.\na. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes (e.g., to represent the associative property of multiplication).\nb. Understand and use the formulas V = l x w x h and V = B x h, where in this case B is the area of the base (B = l x w), for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths to solve mathematical problems and problems in real-world contexts.\nc. Understand volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms, applying this technique to solve mathematical problems and problems in real-world contexts."]
["Instructional Strategies"]
["Volume refers to the amount of space that an object takes up and is measured in cubic units, such as cubic inches or cubic centimeters. Provide students with opportunities to find the volume of rectangular prisms by counting unit cubes, in metric and standard units of measure, before the formula is generated.  Multiple opportunities are needed for students to develop the formula for the volume of a rectangular prism with activities similar to the one described below. Give students one block (a 1- or 2- cubic centimeter or cubic-inch cube), a ruler with the appropriate measure based on the type of cube, and a small rectangular box. Ask students to determine the number of cubes needed to fill the box. Have students share their strategies with the class using words, drawings or numbers. Allow them to confirm the volume of the box by filling the box with cubes of the same size. By stacking geometric solids with cubic units in layers, students can begin understanding the concept of how addition plays a part in finding volume. This will lead to an understanding of the formula for the volume of a right rectangular prism, b x h, where b is the area of the base. A right rectangular prism has three pairs of parallel faces that are all rectangles. Have students build a prism in layers. Then, have students determine the number of cubes in the bottom layer and share their strategies. Students should use multiplication based on their knowledge of arrays and its use in multiplying two whole numbers. Ask what strategies can be used to determine the volume of the prism based on the number of cubes in the bottom layer. Expect responses such as “adding the same number of cubes in each layer as were on the bottom layer” or multiply the number of cubes in one layer times the number of layers."]
["Common Misconceptions"]
["Some students may think only about one of the dimensions needed to find volume. Some students may believe that because an object is tall, it will have lots of volume, ignoring the other two dimensions. Provide additional experiences for students to measure and compare a variety of objects by using all three dimensions to address this misconception."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.MD.C Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.","5.MD.C.4\nMeasure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.","5.MD.C.5 Relate volume to the operations of multiplication and addition and solve mathematical problems and problems in real-world contexts involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes (e.g., to represent the associative property of multiplication). b. Understand and use the formulas V = l x w x h and V = B x h, where in this case B is the area of the base (B = l x w), for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths to solve mathematical problems and problems in real-world contexts. c. Understand volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms, applying this technique to solve mathematical problems and problems in real-world contexts."]
["Instructional Strategies"]
["Volume refers to the amount of space that an object takes up and is measured in cubic units, such as cubic inches or cubic centimeters. Provide students with opportunities to find the volume of rectangular prisms by counting unit cubes, in metric and standard units of measure, before the formula is generated. . Multiple opportunities are needed for students to develop the formula for the volume of a rectangular prism with activities similar to the one described below. Give students one block (a 1- or 2- cubic centimeter or cubic-inch cube), a ruler with the appropriate measure based on the type of cube, and a small rectangular box. Ask students to determine the number of cubes needed to fill the box. Have students share their strategies with the class using words, drawings or numbers. Allow them to confirm the volume of the box by filling the box with cubes of the same size. By stacking geometric solids with cubic units in layers, students can begin understanding the concept of how addition plays a part in finding volume. This will lead to an understanding of the formula for the volume of a right rectangular prism, b x h, where b is the area of the base. A right rectangular prism has three pairs of parallel faces that are all rectangles. Have students build a prism in layers. Then, have students determine the number of cubes in the bottom layer and share their strategies. Students should use multiplication based on their knowledge of arrays and its use in multiplying two whole numbers. Ask what strategies can be used to determine the volume of the prism based on the number of cubes in the bottom layer. Expect responses such as “adding the same number of cubes in each layer as were on the bottom layer” or multiply the number of cubes in one layer times the number of layers."]
["Common Misconceptions"]
["Some students may think only about one of the dimensions needed to find volume. Some students may believe that because an object is tall, it will have lots of volume, ignoring the other two dimensions. Provide additional experiences for students to measure and compare a variety of objects by using all three dimensions to address this misconception."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.MD.C Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["3.OA.B.5\nApply properties of operations as strategies to multiply and divide. Properties include commutative and associative properties of multiplication and the distributive property. (Students do not need to use the formal terms for these properties.)","5.MD.C.5\nRelate volume to the operations of multiplication and addition and solve mathematical problems and problems in real-world contexts involving volume.\na. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes (e.g., to represent the associative property of multiplication).\nb. Understand and use the formulas V = l x w x h and V = B x h, where in this case B is the area of the base (B = l x w), for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths to solve mathematical problems and problems in real-world contexts.\nc. Understand volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms, applying this technique to solve mathematical problems and problems in real-world contexts.","6.G.A.2\nFind the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Understand and use the formula V = B · h, where in this case, B is the area of the base (B = l x w) to find volumes of right rectangular prisms with fractional edge lengths in mathematical problems and problems in real-world context."]
["4.MD.A.3\nApply the area and perimeter formulas for rectangles in mathematical problems and problems in real-world contexts including problems with unknown side lengths. See Table 2."]
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["Instructional Strategies"]
["Volume refers to the amount of space that an object takes up and is measured in cubic units, such as cubic inches or cubic centimeters. Provide students with opportunities to find the volume of rectangular prisms by counting unit cubes, in metric and standard units of measure, before the formula is generated.  Multiple opportunities are needed for students to develop the formula for the volume of a rectangular prism with activities similar to the one described below. Give students one block (a 1- or 2- cubic centimeter or cubic-inch cube), a ruler with the appropriate measure based on the type of cube, and a small rectangular box. Ask students to determine the number of cubes needed to fill the box. Have students share their strategies with the class using words, drawings or numbers. Allow them to confirm the volume of the box by filling the box with cubes of the same size. By stacking geometric solids with cubic units in layers, students can begin understanding the concept of how addition plays a part in finding volume. This will lead to an understanding of the formula for the volume of a right rectangular prism, b x h, where b is the area of the base. A right rectangular prism has three pairs of parallel faces that are all rectangles. Have students build a prism in layers. Then, have students determine the number of cubes in the bottom layer and share their strategies. Students should use multiplication based on their knowledge of arrays and its use in multiplying two whole numbers. Ask what strategies can be used to determine the volume of the prism based on the number of cubes in the bottom layer. Expect responses such as “adding the same number of cubes in each layer as were on the bottom layer” or multiply the number of cubes in one layer times the number of layers."]
["Common Misconceptions"]
["Some students may think only about one of the dimensions needed to find volume. Some students may believe that because an object is tall, it will have lots of volume, ignoring the other two dimensions. Provide additional experiences for students to measure and compare a variety of objects by using all three dimensions to address this misconception."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.G.A Graph points on the coordinate plane to solve mathematical problems as well as problems in realworld context."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["3.MD.B.4\nGenerate measurement data by measuring lengths using rulers marked with halves and fourths of an inch to the\nnearest quarter-inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate\nunits— whole numbers, halves, or quarters.","5.G.A.1\nUnderstand and describe a coordinate system as perpendicular number lines, called axes, that intersect at the origin (0 , 0). Identify a given point in the first quadrant of the coordinate plane using an ordered pair of numbers, called coordinates. Understand that the first number (x) indicates the distance traveled on the horizontal axis, and the second number (y) indicates the distance traveled on the vertical axis.","6.NS.C.6\nUnderstand a rational number can be represented as a point on the number line. Extend number line diagrams\nand coordinate axes familiar from previous grades to represent points on the line and in the plane with negative\nnumber coordinates.\na. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line;\nrecognize that the opposite of the opposite of a number is the number itself and that 0 is its own opposite.\nb. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane;\nrecognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections\nacross one or both axes.\nc. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and\nposition pairs of integers and other rational numbers on a coordinate plane."]
["3.NF.A.1 Understand a fraction (1/b) as the quantity formed by one part when a whole is partitioned into b equal parts;\nunderstand a fraction a/b as the quantity formed by a parts of size 1/b."]
["3.NF.A.2\nUnderstand a fraction as a number on the number line; represent fractions on a number line diagram.\na. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and\npartitioning it into b equal parts. Understand that each part has size 1/b and that the end point of the part based\nat 0 locates the number 1/b on the number line.\nb. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Understand that the\nresulting interval has size a/b and that its endpoint locates the number a/b on the number line including values\ngreater than 1.\nc. Understand a fraction 1/b as a special type of fraction that can be referred to as a unit fraction (e.g. 1/2, 1/4)."]
["Instructional Strategies"]
["Students need to understand the underlying structure of the coordinate system and how axes make it possible to locate points anywhere on a coordinate plane. This is the first time students are working with coordinate planes, and only in the first quadrant. It is important that students create the coordinate grid themselves. This can be related to two number lines and have the students make connections to previous experiences with moving along a number line.\nCoordinate grids are fun! Multiple experiences with plotting points are needed. Provide points plotted on a grid and have students name and write the ordered pair. Have students describe how to get to the location. Encourage students to articulate directions, attending to precision as they plot points.\nPresent real-world and mathematical problems and have students graph points in the first quadrant of the coordinate plane. Gathering and graphing data is a valuable experience for students. It helps them to develop an understanding of coordinates and what the overall graph represents. Students also need to analyze the graph by interpreting the coordinate values in the context of the situation."]
["Common Misconceptions"]
["The x and y axes may confuse students, because this is the first time they are working with coordinated using the first quadrant of a coordinate grid. Some students may not understand the first number of the ordered pair (x-axis) indicates how far to move horizontally from the origin and must occur first. Some students may not think the order is critical. To address the misconception, students can plot points so that the position of the coordinates is switched. For example, have students plot (4,5) and (5,4) and ask them to explain and discuss the importance of order used to plot the points."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.G.A Graph points on the coordinate plane to solve mathematical problems as well as problems in realworld context."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["3.MD.B.4\nGenerate measurement data by measuring lengths using rulers marked with halves and fourths of an inch to the\nnearest quarter-inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate\nunits— whole numbers, halves, or quarters.","5.G.A.2 Represent real-world 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.","6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques to solve mathematical problems and problems in a real-world context."]
["3.NF.A.1 Understand a fraction (1/b) as the quantity formed by one part when a whole is partitioned into b equal parts;\nunderstand a fraction a/b as the quantity formed by a parts of size 1/b.",null,"6.NS.C.8 Solve mathematical problems and problems in real-world context by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate."]
["3.NF.A.2\nUnderstand a fraction as a number on the number line; represent fractions on a number line diagram.\na. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and\npartitioning it into b equal parts. Understand that each part has size 1/b and that the end point of the part based\nat 0 locates the number 1/b on the number line.\nb. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Understand that the\nresulting interval has size a/b and that its endpoint locates the number a/b on the number line including values\ngreater than 1.\nc. Understand a fraction 1/b as a special type of fraction that can be referred to as a unit fraction (e.g. 1/2, 1/4).",null,"6.RP.A.3"]
["Instructional Strategies"]
["Students need to understand the underlying structure of the coordinate system and how axes make it possible to locate points anywhere on a coordinate plane. This is the first time students are working with coordinate planes, and only in the first quadrant. It is important that students create the coordinate grid themselves. This can be related to two number lines and have the students make connections to previous experiences with moving along a number line.\nCoordinate grids are fun! Multiple experiences with plotting points are needed. Provide points plotted on a grid and have students name and write the ordered pair. Have students describe how to get to the location. Encourage students to articulate directions, attending to precision as they plot points.\nPresent real-world and mathematical problems and have students graph points in the first quadrant of the coordinate plane. Gathering and graphing data is a valuable experience for students. It helps them to develop an understanding of coordinates and what the overall graph represents. Students also need to analyze the graph by interpreting the coordinate values in the context of the situation."]
["Common Misconceptions"]
["Fifth graders may not realize how important the order is in plotting a coordinate point. A reminder of the order in ordered pairs may help. In labeling points, the x-coordinate is always given first, then the y-coordinate. Additional experiences for the students to plot the coordinates correctly may be necessary. Some students may not realize that a coordinate system differs from a grid system. In a coordinate system, the lines, not the area surrounding the lines, are labeled. Providing the opportunity to talk about this will help."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.G.B Classify two-dimensional figures into categories based on their properties."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["3.G.A.1\nUnderstand that shapes in different categories (e.g., rhombuses, rectangles, and others)may share attributes\n(e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals).\nRecognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples quadrilaterals\nthat do not belong to any of these subcategories.","5.G.B.3\nUnderstand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.","6.G.A.4\nRepresent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the\nsurface area of these figures. Apply these techniques to solve mathematical problems and problems in real-world\ncontext."]
["4.G.A.2\nClassify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the\npresence or absence of angles of a specified size (e.g., understand right triangles as a category, and identify right\ntriangles)."]
["4.MD.C.5\nRecognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and\nunderstand concepts of angle measurement:\na. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by\nconsidering the fraction of the circular arc between the points where the two rays intersect the circle. An angle\nthat turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.\nb. An angle that turns through n one-degree angles is said to have an angle measure of n degrees"]
["Instructional Strategies"]
["This cluster builds from Grade 3 when students described, analyzed and compared properties of two-dimensional shapes. They compared and classified shapes by their sides and angles, and connected these with definitions of shapes. In Grade 4 students built, drew and analyzed two-dimensional shapes to deepen their understanding of the properties of two-dimensional shapes. They looked at the presence or absence of parallel and perpendicular lines or the presence or absence of angles of a specified size to classify two-dimensional shapes. Now, students classify two-dimensional shapes in a hierarchy based on properties. Details learned in earlier grades need to be used in the descriptions of the attributes of shapes. The more ways that students can classify and discriminate shapes, the better they can understand them. Shapes are not limited to quadrilaterals. Students can use graphic organizers, such as flow charts or T-charts, to compare and contrast the attributes of geometric figures. Have students create a T-chart with a shape on each side. Have them list attributes of the shapes, such as number of side, number of angles, types of lines, etc. They need to then discuss and determine what is alike or different about the two shapes to determine the categories for the shapes and be able to explain their properties. Pose questions such as, “Why is a square always a rectangle?” and “Why is a rectangle not always a square?” Expect students to use precision in justifying and explaining their reasoning."]
["Common Misconceptions"]
["Some students may think that when describing geometric shapes and placing them into subcategories, the last category is the only classification that can be used. As students are creating a chart with subcategories, have them talk about why attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]

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[null,"Cluster"]
[null,"5.G.B Classify two-dimensional figures into categories based on their properties."]
["Prerequisite Standards","Focus Standard","Future Learning"]
["5.G.B.3\nUnderstand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.","5.G.B.4\nClassify two-dimensional figures in a hierarchy based on properties.","No future alignment."]
["Instructional Strategies"]
["This cluster builds from Grade 3 when students described, analyzed and compared properties of two-dimensional shapes. They compared and classified shapes by their sides and angles, and connected these with definitions of shapes. In Grade 4 students built, drew and analyzed two-dimensional shapes to deepen their understanding of the properties of two-dimensional shapes. They looked at the presence or absence of parallel and perpendicular lines or the presence or absence of angles of a specified size to classify two-dimensional shapes. Now, students classify two-dimensional shapes in a hierarchy based on properties. Details learned in earlier grades need to be used in the descriptions of the attributes of shapes. The more ways that students can classify and discriminate shapes, the better they can understand them. Shapes are not limited to quadrilaterals. Students can use graphic organizers, such as flow charts or T-charts, to compare and contrast the attributes of geometric figures. Have students create a T-chart with a shape on each side. Have them list attributes of the shapes, such as number of side, number of angles, types of lines, etc. They need to then discuss and determine what is alike or different about the two shapes to determine the categories for the shapes and be able to explain their properties. Pose questions such as, “Why is a square always a rectangle?” and “Why is a rectangle not always a square?” Expect students to use precision in justifying and explaining their reasoning."]
["Common Misconceptions"]
["Some students may not be able to just look at pictures of shapes and talk about their properties. To address this, some students may need to touch and turn the shapes to help determine the properties. Focus should be on quadrilaterals, although other polygons can be included as well."]
[null,"Resources","Site"]
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["Return to Intro Grade 5"]