Grade 5 Math Modules Module 1: Place Value and Decimal Fractions Module 2: Multi-­‐Digit Whole Number and Decimal Fraction Operations Module 3: Addition and Subtraction of Fractions Module 4: Multiplication and Division of Fractions and Decimal Fractions Module 5: Addition and Multiplication with Volume and Area Module 6: Problem Solving with the Coordinate Plane Summary of Year Fifth grade mathematics is about (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to two-­‐digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume. Key Areas of Focus for 3-­‐5: Multiplication and division of whole numbers and fractions—
concepts, skills, and problem solving Required Fluency: 5.NBT.5 Multi-­‐digit multiplication. Reporting Standards (Report Card) & Proficiency Levels CONCEPTS & PROCEDURES Understand the place value system Standards: 5.NBT.1, 5.NBT.2, 5.NBT.3, 5.NBT.4 Proficiency Level 4 Proficiency Level 3 a) In any multi-­‐digit a) In any multi-­‐digit number, determine that a number, determine that a digit in one place digit in one place represents 10 times as represents 10 times as much as it represents in much as it represents in the place to its right and the place to its right and 1/10 of what it represents 1/10 of what it represents in the place to its left. in the place to its left. ______________________ ______________________
____________ ____________ b) Explain patterns in the b) Explain patterns in the placement of the decimal number of zeros of the point when a decimal is product when multiplying multiplied or divided by a by powers of 10. power of 10. ______________________
______________________ ____________ ____________ c) Use whole number c) Use whole number exponents to denote exponents to denote powers of 10. powers of 10. Use symbols (>,<,=) to compare two powers of 10 expressed exponentially ______________________
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(compare 10 to 10 ). ____________ ______________________ Proficiency Level 2 a) In any multi-­‐digit number, determine that a digit in one place represents 10 times as much as it represents in the place to its right. ______________________
___________ b) Compare numbers based on the number of zeros. ______________________
____________ c) Use whole number exponents less than four to denote powers of ten. ______________________
____________ d) Read, write, and Proficiency Level 1 a) Using visual models and/or manipulatives, in any two-­‐ or three-­‐digit whole number, determine that a digit in one place represents ten times as much as it represents in the place to its right. ______________________
___________ b) Compare numbers using visual or concrete models. ______________________
____________ ______________________
____________ d) Read and write some multi-­‐digit whole numbers Grade 5 Math Modules ____________ d) Read, write, and compare decimals to any place using numerals, number names, expanded form, and inequality symbols (>,<,=). ______________________
____________ e) Round decimals to thousandths and determine which place value to round to based on the context of the problem or task. d) Read, write, and compare decimals to the thousandths using numerals, number names, expanded form, and inequality symbols (>,<,=). ______________________
____________ e) Round decimals to hundredths or tenths. compare decimals to the tenths using numerals, number names, expanded form, and inequality symbols (>,<,=). ______________________
____________ e) Round to the nearest whole number. using base-­‐ten numerals, number names and/or expanded form. ______________________
____________ e) Given visual models and/or manipulatives, round whole numbers to any place value. Perform operations with multi-­‐digit whole numbers and decimals to hundredths Standards: 5.NBT.5, 5.NBT.6, 5.NBT.7 Proficiency Level 4 Proficiency Level 3 Proficiency Level 2 a) Divide whole numbers a) Divide whole numbers a) Divide whole numbers with up to four-­‐digit with up to four-­‐digit with up to four-­‐digit dividends and two-­‐digit dividends and two-­‐digit dividends and two-­‐digit divisors using strategies divisors using strategies divisors which are based on place value, the based on place value, the multiples of ten using properties of operations, properties of operations, strategies based on place and/or the relationship and/or the relationship value, the properties of between multiplication between multiplication operations, and/or the and division. Identify and division. Check relationship between correspondences between reasonableness of answers multiplication and division. different approaches. using the standard Check reasonableness of algorithm for answers using the multiplication. ______________________
standard algorithm for ____________ multiplication. ______________________ ______________________ ____________ b) Add or subtract (with or ____________ without regrouping) two b) Add or subtract two decimals to hundredths b) Add or subtract two decimals to hundredths (decimals with the same decimals to hundredths using: concrete models, number of digits) using using: concrete models, drawings, or strategies concrete models, drawing, or strategies based on place value, drawings, or strategies based on place value, properties of operations, based on place value, properties of operations and/or the relationship properties of operations, and/or the relationship between addition and and/or the relationship between addition and subtraction. between addition and subtraction. subtraction. Relate the strategy to a Relate the strategy to a written equation, and ______________________
written equation, and explain the reasoning ____________ explain the reasoning used. used. ______________________ c) Multiply or divide (with ______________________ ___________ or without regrouping) two ____________ decimals to hundredths c) Multiply or divide two (decimals with the same c) Multiply or divide two decimals to hundredths number of digits) using decimals to hundredths using: concrete models, concrete models, Proficiency Level 1 a) Using visual models and/or manipulatives, divide whole numbers with up to two-­‐digit dividends and a one-­‐digit number. ______________________
____________ b) Using visual models and/or manipulatives, add or subtract two decimals to tenths (decimals with same number of digits). ______________________
____________ c) Using visual models and/or manipulatives, multiply or divide two decimals to tenths (decimals with same number of digits). Grade 5 Math Modules using: concrete models, drawings, or strategies based on place value, properties of operations, and/or the relationship between multiplication and division. Relate the representation to the written equation and explain the reasoning used. ______________________
___________ d) Apply this concept to a real-­‐world context, relate the strategy to a written method, and explain the reasoning used. drawings, or strategies based on place value, properties of operations, and/or the relationship between multiplication and division. Relate representation to the written equation and explain the reasoning used. drawings, or strategies based on place value, properties of operations, and/or the relationship between multiplication and division. Explain the reasoning used in the representation. Use equivalent fractions as a strategy to add and subtract fractions. Standards: 5.NF.1, 5.NF.2 Proficiency Level 4 Proficiency Level 3 Proficiency Level 2 a) Add and subtract two or a) Add and subtract a) Add fractions or mixed more fractions and mixed fractions or mixed numbers with unlike numbers with unlike numbers with unlike denominators using denominators in such a denominators using equivalent fractions. way as to produce an equivalent fractions. equivalent sum or difference with like ______________________
denominators. ______________________ ___________ ______________________ ____________ ____________ b) Solve word problems b) Solve word problems involving addition of b) Create and solve word involving addition and fractions using benchmark problems involving subtraction of fractions, fractions with unlike addition and subtraction of referring to the same denominators, referring to fractions, referring to the whole in cases of unlike the same whole by using same whole in cases of denominators by using visual fraction models or unlike denominators by visual fraction models or equations. using visual fraction equations. ______________________
models and equations. ______________________ ____________ ______________________ ____________ ____________ c) Assess reasonableness c) Assess and justify using benchmark fractions reasonableness using and number sense of benchmark fractions and fractions. number sense of fractions. Proficiency Level 1 a) Use visual models to add and subtract fractions with like denominators by joining and separating parts. ______________________
____________ b) Given visual models and/or manipulatives, compute sums of fractions in the context of some word problems. ______________________
____________ Grade 5 Math Modules Apply and extend previous understandings of multiplication and division to multiply and divide fractions Standards: 5.NF.3, 5.NF.4, 5.NF.5, 5.NF.6, 5.NF.7 Proficiency Level 4 Proficiency Level 3 Proficiency Level 2 Proficiency Level 1 a) Solve word problems a) Solve word problems a) Solve word problems a) Given visual models involving multiplication involving division of whole involving division of the and/or manipulatives, and division of fractions in numbers leading to whole numbers leading to multiply some fractions by all forms using abstract answers in the form of answers in the form of whole numbers to solve models and assess fractions or mixed fractions or mixed mathematical problems. reasonableness of numbers. numbers by using visual ______________________
solutions. models. ____________ ______________________ ______________________ ______________________ ____________ ___________ b) Multiply fractions by ____________ whole numbers using b) Interpret products and b) Multiply fractions by visual fraction models with b) Multiply fractions by quotients of fractions and whole numbers by support. whole numbers using mixed numerals. interpreting the product as visual fraction models. (e.g., ¾ x 3 = (3x3)/4 = 9/4) Explain why multiplying a Represent the product of given number by a fraction Multiply fractions by fractions as rectangular greater than 1 results in a fractions interpreting the areas. product greater than the product as (e.g., ¾ x ½ = given number and why (3x1) / (4x2) and relating multiplying a given to the areas of rectangles ______________________
number by a fraction less with fractional areas. ___________ ______________________
than 1 results in a product ___________ less than the given ______________________
c) Explain why multiplying number. ____________ a given number by a c) Explain why multiplying ______________________ fraction greater than 1 a given number by a ____________ c) Explain why multiplying results in a product greater fraction greater than 1 a given number by a than the given number and results in a product greater c) Explain why multiplying fraction greater than 1 why multiplying a given than the given number and a given number by a results in a product greater number by a fraction less why multiplying a given fraction greater than 1 than the given number and than 1 results in a product number by a fraction less results in a product greater why multiplying a given less than the given than 1 results in a product than the given number and number by a fraction less number. less than the given why multiplying a given than 1 results in a product ______________________
number. number by a fraction less less than the given ___________ ______________________
than 1 results in a product number. ____________ less than the given ______________________
number. ____________ d) Can define resize and ______________________ scaling. ____________ d) Can define resize and scaling. Use visual models to d) Solves real world understand that the size of problems involving resizing Understand that the size of the factors will help or scaling. the factors will help determine the size of the determine the size of the product. (e.g., If 2 x 4 = 8, product. (e.g., If 2 x 4 = 8, ______________________
then 2 x (2 x 4) = 2 x (8). then 2 x (2 x 4) = 2 x (8). ____________ Understand that scaling Understand that scaling e) Given a visual model involves multiplication. involves multiplication. and/or manipulatives, ______________________
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solve some word problems ___________ ______________________ ___________ involving the division of ________ whole numbers leading to e) Divide a unit fraction by e) Divide a unit fraction by answers in the form of a whole number or whole e) Divide a unit fraction by a whole number or whole fractions. number by a unit fraction a whole number of whole number by a unit fraction, using visual fraction Grade 5 Math Modules number by a unit fraction, and interpret the quotients to solve real-­‐
world problems. Solve real-­‐world problems by multiplying a mixed number by a fraction, a fraction by a fraction, and a whole number by a fraction; dividing a unit fraction by a whole number and whole number by a unit fraction using visual fraction models and creating context and equations including rectangular areas; and interpreting the product and/or quotient. and interpret the quotients and create a story context for the mathematics. Solve real-­‐world problems by multiplying a mixed number by a fraction, a fraction by a fraction, and a whole number by a fraction; dividing a unit fraction by a whole number and a whole number by a unit fraction using visual fraction models. models; including rectangular areas and recognize a story context for the mathematics with support. Solve real-­‐world problems by multiplying a mixed number by a fraction, a fraction by a fraction, and a whole number by a fraction; dividing a unit fraction by a whole number and a whole number by a unit fraction using visual fraction models, with support. Understand concepts of volume and relate volume to multiplication and addition Standards: 5.MD.3, 5.MD.4, 5.MD.5 Proficiency Level 4 Proficiency Level 3 Proficiency Level 2 a) Identify volume as an a) Identify volume as an a) Affirm that volume is attribute of solid figures, attribute of solid figures, measured using cubic units and affirm volume is and affirm volume is and can be found by measured using cubic units measured using cubic units packing a solid figure with and can be found by and can be found by unit cubes and counting packing a solid figure with packing a solid figure with them. unit cubes and counting unit cubes and counting them. them. ______________________
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____________ ______________________
____________ ____________ b) Given manipulatives, b) Show that the number determine the volume by b) Show that the number of unit cubes packed in a computing the base and of unit cubes packed in a rectangular prism is replicating the base n rectangular prism is equivalent to multiplying times (height). equivalent to multiplying the edge lengths. the edge lengths, which is also equivalent to Justify that two Determine that a multiplying the height by rectangular prisms with rectangular prism has the the area of the base. the same edge lengths but same volume regardless of different orientation have its orientation. Explain why two the same volume. rectangular prisms with different dimensions can ______________________
have the same volume. ______________________ ___________ ______________________ ____________ Given a visual model and ____________ the formulas for finding c) Solve real-­‐world the volume of a c) Create real-­‐world and mathematical problems by rectangular prism, solve mathematical problems applying the appropriate mathematical problems by that would be solved by formulas for volume (V = applying the formula V = finding volume. B’h and V = l’w’h). l’w’h for volume. Measure volumes of Measure volumes of Proficiency Level 1 a) Given manipulatives, determine volume by packing a solid figure with unit cubes and counting them. Grade 5 Math Modules composite figures composed of right rectangular prisms by adding the two separate volumes and applying this technique to solve real world problems. composite figures composed of right rectangular prisms by adding the two separate volumes. FLUENCY Master multiplication of multi-­‐digit numbers using the standard algorithm Standards: NBT.5.5 Proficiency Level 4 Proficiency Level 3 Proficiency Level 2 *Multiplies three-­‐digit by *Multiplies three-­‐digit by *Multiplies two-­‐digit by two-­‐digit whole numbers two-­‐digit whole numbers two-­‐digit whole numbers using the standard using the standard using the standard algorithm. algorithm. algorithm. Proficiency Level 1 *Inconsistently multiplies two-­‐digit by two-­‐digit whole numbers using the standard algorithm.