Title Math Grade 5 Type Essential Document Map Authors Melissa Cosgrove, Jennifer Hallet, Jen Khomyak Subject Mathematics Course Math Grade 5 Grade(s) 05 Location District Curriculum Writing History Notes Attachments Page: 1 of 15
September/Week 2 - October/Week 8 September October November December January February March April May June 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Understanding the Place Value System November/Week 9 - December/Week 15 Geometric Measure and Understanding Volume December/Week 16 - February/Week 22 Operations with Multi-Digit Whole Numbers, Decimals and Fractions February/Week 23 - April/Week 29 Fraction Multiplication by a Whole Number and Scaling April/Week 30 - May/Week 36 Shape and Coordinate Geometry Page: 2 of 15
Duration: September/Week 2 - October/Week 8 UNIT NAME: Understanding the Place Value System Enduring Understandings Essential Questions Knowledge Skills Standards Any number, measure, numerical expression, or equation can be represented in a variety of ways that can have the same value. Numbers can represent quantity, position, location, and relationships, and symbols may be used to express these relationships. The four operations are interrelated, and the properties of each may be used to understand the others. Although standard algorithms exist for most mathematical computations, a variety of methods exist for solving the same problem. Mathematical expressions represent relationships. Each place value to the left of another is tens times greater than the one to the right. Decimal place value is an extension of whole number place value. The base-ten numeration system extends infinitely to very large and very small numbers (e.g., millions and millionths). How can an understanding of patterns, models and relationships lead to an understanding of simple equations and how can this pattern be generalized into an equation? How can we decide what operation to use when presented with a problem? How can place value properties aid in computation? (powers of ten) How can decimals be modeled and compared? How are common fractions and decimals alike? How can relationships be expressed symbolically? What strategies and models help us understand how to solve multiplication and division problems and how multiplication and divsion are related/connected? What relationships exist between various multiplication and division algorithms? Vocabulary: parentheses, brackets, braces, numerical expressions, place value, decimal, decimal point, patterns, multiply, divide, tenths, hundredths, thousandths, greater than, less than, equal to, <, >, =, compare, comparison, round, exponent, multiplication/multiply, divide/division, product, quotient, factor, dividend, divisor, rectangular array, area model, properties (rules about how numbers work) Mathematically, there cannot be brackets or braces in a problems that does not have parentheses. Mathematically, there cannot be braces in a problem that does not have both parentheses and brackets. Expressions are a series of numbers and symbols (+, -, x, ) without an equals sign. The tens place is ten times as much as the ones place. The ones place is 1/10th the size of the tens place. Benchmarks are convenient numbers for comparing and rounding numbers (e.g., 0, 0.5, 1, 1.5). Evaluate (simplify) expressions with parentheses ( ), brackets [ ], and braces { }. Write a numerical expression when given a word problem or a scenario in words. (e.g., Write an expression for the steps "double the 5 and then add 26." (2 x 5) + 26) Interpret numerical expressions verbally or in writing. Explain the "ten times" or 1/10 relationships for place values in multi-digit numbers. Recognize and explain patterns of the number of zeros and the placement of the decimal point in a product or quotient when a number is multiplied or divided by a power of ten. Write decimals to thousandths in expanded form with fractions. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. 5.OA.1-Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. (05) 5.OA.2-Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. (05)[State:New Jersey CCSS] 5.NBT.1-Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. (05)[State:New Jersey CCSS] 5.NBT.2-Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. (05)[State:New Jersey CCSS] Page: 3 of 15
Powers of 10 are important benchmarks in our numeration system. There are patterns when multiplying and dividing whole numbers and decimals by powers of ten. Flexible methods of computation involve grouping numbers in strategic ways. Proficiency with basic facts aids estimation and computation of larger and smaller numbers. Plans: Compare two decimals to thousandths based on the value of the digits in each place using the symbols <, >, = when presented as base-ten numerals, number names or expanded form. Round a decimal to any place. Calculate whole number quotients with 4-digit dividends and 2-digit divisors and explain answers with equations, rectangular arrays, and area models. Use the standard algorithm to multiply 3-digit whole numbers by 1-digit whole numbers. 5.NBT.3.a-Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 100 + 4 10 + 7 1 + 3 (1/10) + 9 (1/100) + 2 (1/1000). (05) 5.NBT.3.b-Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (05)[State:New 5.NBT.4-Use place value understanding to round decimals to any place. (05) 5.NBT.5-Fluently multiply multidigit whole numbers using the standard algorithm. (05) 5.NBT.6-Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (05)[State:New Jersey CCSS] Page: 4 of 15
Duration: November/Week 9 - December/Week 15 UNIT NAME: Geometric Measure and Understanding Volume Enduring Understandings Essential Questions Knowledge Skills Standards Objects can be described and compared using their geometric attributes. Objects have distinct attributes that can be measured. Standard units provide common language for communicating measurements. Two measurement systems, metric and standard, have been developed to quantify objects' attributes. The choice of measurement tools depends on the measureable attribute and the degree of precision desired. What are tools of measurement and how are they used? What is the purpose of standard units of measurement? What is volume? What strategies can we use to find the volume of a shape? How are the area of a right rectangular prim and the volume of a right rectangular prism related? How can we describe twodimensional and threedimensional shapes? How can putting shapes together and breaking large shapes into smaller shapes help us understand them? How can we use 2- and 3-D shapes and attributes to describe real world solids and solve problems? Vocabulary: measurement, attribute, volume, solid figure, right rectangular prism, unit, unit cube, gap, overlap, cubic units (cubic cm, cubic in., cubic ft, nonstandard cubic units), multiplication, addition, edge lengths, height, area of base 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. A cubic unit is written with an exponent of 3 (e.g., in 3, m 3 ). 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. Volume is additive. Measure volume by counting the total number of same size cubic units required to fill a figure without gaps or overlaps. Choose an appropriate cubic unit based on the attributes of the 3- dimensional figure you are measuring. Show that the volume of a right rectangular prism found by counting all the unit cubes is the same as the formulas V = l x w x h or V = b x h. Explain how both volume formulas relate to counting the cubes in one layer and multiplying that value by the number of layers (height). Find the volume of a composite solid figure composed of two nonoverlapping right rectangular prisms. Apply formulas to solve real world and mathematical problems involving volumes of right rectangual prisms and composites of same. 5.MD.3-Recognize volume as an attribute of solid figures and understand concepts of volume measurement. (05)[State:New 5.MD.3.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. (05) 5.MD.3.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. (05)[State:New Jersey CCSS] 5.MD.4-Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. (05) 5.MD.5.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 wholenumber products as volumes, e.g., to represent the associative property of multiplication. (05)[State:New 5.MD.5.b-Apply the formulas V Page: 5 of 15
Plans: = l w h and V = b h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems. (05)[State:New 5.MD.5.c-Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. (05) 5.MD.5-Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. (05) Page: 6 of 15
Duration: December/Week 16 - February/Week 22 UNIT NAME: Operations with Multi-Digit Whole Numbers, Decimals and Fractions Enduring Understandings Essential Questions Knowledge Skills Standards Each place value to the left of another is tens times greater than the one to the right. Decimal place value is an extension of whole number place value. The base-ten numeration system extends infinitely to very large and very small numbers (e.g., millions and millionths). Powers of 10 are important benchmarks in our numeration system. Flexible methods of computation involve grouping numbers in strategic ways. Proficiency with basic facts aids estimation and computation of larger and smaller numbers. The magnitude of numbers affects the outcome of operations on them. Standard units provide common language for communicating measurements. The choice of measurement tools depends on the measureable attribute and the degree of precision desired. How does understanding the structure of the number system help you solve problems? How does the size of a number affect the outcome of the operation? How can identification of patterns assist me when multiplying/dividing decimals? How do units within a system relate to each other? (time, length, volume, weight) Why can't you add and subtract fractions with unlike denominators? How do you determine which form for a number is most appropriate? What makes a computational strategy both effective and efficient? Vocabulary: place value, Describe the place value of decimal, decimal point, numeral digits relative to patterns, multiply, divide, both the place to the right tenths, hundredths, billions and the place to the left The tens place is ten times (decimal to hundredths and as much as the ones place. whole numbers to billions). The ones place is 1/10th Add, subtract, multiply and the size of the tens place. divide decimals to Vocabulary Measurement: hundredths using concrete conversion/convert, metric models or drawings and and standard measurement strategies based on placevalue, properties of Vocabulary from previous grades: relative size, liquid operations, and/or the volume, mass, length, relationship between kilometer (km), meter (m), addition and subtraction. centimeter (cm), kilogram Explain the reasoning used (kg), gram (g), liter (l), to solve the above milliliter (ml), inch (in.), foot problems. (ft), yard (yd), mile (mi), Convert standard ounce (oz.), pound (lb), cup measurement units within (c), pint (pt), quart (qt), the same system (e.g., gallon (gal), hour, minute, centimeters to meters) to second solve multi-step problems. Vocabulary Fractions: Add and subtract fractions fraction, equivalent, addition/ (including mixed numbers) add, sum, subtraction/ with unlike denominators. subtract, difference, unlike, Solve word problems denominator, numerator, involving adding and benchmark fraction, subtracting fractions estimate, reasonableness, including unlike mixed numbers denominators. Determine if the answer to the word problem is reasonable, using estimations with benchmark fractions. 5.NBT.1-Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. (05)[State:New Jersey CCSS] 5.NBT.7-Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. (05) 5.MD.1-Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. (05)[State:New 5.NF.1-Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In Page: 7 of 15
Two measurement systems, metric and standard, have been developed to quantify objects' attributes. Computational fluency includes understanding not only the meaning but also the appropriate use of numerical operations. Interpret a fraction as a division of the numerator by the denominator. (a/b = a b) Solve word problems where division of whole numbers leads to fractional or mixed number answers (e.g., by using visual fraction models or equations to represent the problem). Multiply multi-digit whole numbers using the standard algorithm. general, a/b + c/d = (ad + bc)/ bd.) (05)[State:New Jersey CCSS] 5.NF.2-Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. (05) 5.NF.3-Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50- pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? (05) Page: 8 of 15
Plans: 5.NBT.5-Fluently multiply multidigit whole numbers using the standard algorithm. (05) Page: 9 of 15
Duration: February/Week 23 - April/Week 29 UNIT NAME: Fraction Multiplication by a Whole Number and Scaling Enduring Understandings Essential Questions Knowledge Skills Standards The magnitude of numbers affects the outcome of operations on them. Fractions express a relationship between two numbers. A fraction describes division. The four operations are interrelated, and the properties of each may be used to understand the others. Any number, measure, numerical expression, or equation can be represented in a variety of ways that have the same value. Although standard algorithms exist for most mathematical computations, a variety of methods exist for solving the same problem. How does the size of the number affect the outcome of the operation? How do you determine which form of a number is most appropriate? How are multiplication and division related? Why is the product greater than the given number when you multiply the given number by a fraction greater than 1? Why is the product smaller than the given number when you multiply the given number by a fraction less than 1? How can we decide what operation to use when presented with a problem? Vocabulary: fraction, Multiply fractions by whole equivalent, numerator, numbers and draw visual denominator, mixed models or create story numbers, multiply, factor, contexts. product, scaling, unit Interpret the product (a/b) x fraction, divide q as a parts of a whole In general, in the partitioned into b equal expression (a/b) x q, if q is parts added q times. a fraction c/d, then (a /b) x Find the area of a rectangle (c/d) = a(1/b) x c(1/d) = ac x with fractional side lengths (1/b)(1/d) = ac x (1/bd) = ac/ by tiling it with unit squares bd of the appropriate unit A unit fraction is a fraction fraction side lengths. that has one in the Show that the tiled area is denominator. the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles. Represent fraction products as rectangular areas. Explain how a product is related to the magnitude of the factors. Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number. Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. 5.NF.4.a-Interpret the product (a/ b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. For example, use a visual fraction model to show (2/3) 4 = 8/3, and create a story context for this equation. Do the same with (2/3) (4/5) = 8/15. (In general, (a/b) (c/d) = ac/bd.) (05) 5.NF.4.b-Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. (05) 5.NF.5-Interpret multiplication as scaling (resizing), by: (05) 5.NF.5.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. (05)[State:New 5.NF.5.b-Explaining why multiplying a given number by a fraction greater than 1 results in Page: 10 of 15
Relate the principle of fraction equivalence a/b = (n x a)/(n x b) to the effect of multiplying a/b by 1. Solve real world problems involving multiplication of fractions (including mixed numbers), using visual fraction models or equations to represent the problem. Divide a unit fraction by a non-zero fraction and interpret by creating a story context or visual fraction model. Divide a whole number by a unit fraction and interpret by creating a story context or visual fraction model. Solve real world problems involving division of unit fractions by whole numbers or whole numbers by unit fractions. a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); 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 a)/(n b) to the effect of multiplying a/b by 1. (05) 5.NF.6-Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. (05) 5.NF.7.a-Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. (05) 5.NF.7.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 explain that 4 (1/5) = 20 Page: 11 of 15
Plans: because 20 (1/5) = 4. (05) 5.NF.7.c-Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? (05)[State:New Page: 12 of 15
Duration: April/Week 30 - May/Week 36 UNIT NAME: Shape and Coordinate Geometry Enduring Understandings Essential Questions Knowledge Skills Standards The four operations are interrelated, and the properties of each may be used to understand the others. Any number, measure, numerical expression, or equation can be represented in a variety of ways that have the same value. Although standard algorithms exist for most mathematical computations, a variety of methods exist for solving the same problem. The magnitude of numbers affects the outcome of operations on them. Decimal place value is an extension of whole number place value. The coordinate system is a scheme that uses two perpendicular lines intersecting at zero to tell the location of points in the plane. Ordered pairs show an exact location on a coordinate plane. Graphs convey data in a concise way. Objects can be described and compared using their geometric attributes. What is a decimal? How can identification of patterns assist me when multiplying/dividing decimals? How does the size of the number affect the outcome of the operation? How does understanding the structure of the number system help you solve problems? How can you describe the location of a point on a coordinate plane? How is this useful? What aspects of a graph help people understand and interpret the data easily? What kinds of questions can and cannot be answered from a graph? How can plane and solid figures be described? What types of problems are solved with measurement? What are tools of measurement and how are they used? What is the purpose of standard units of measurement? How do you determine which form of a number is most appropriate? Vocabulary: place value, Add, subtract, multiply and decimal, decimal point, divide decimals to patterns, multiply, divide, hundredths, using concrete tenths, hundredths, models or drawings and thousandths strategies based on place Vocabulary Geometry: value, properties of coordinate system, operations, and/or the coordinate plane, first relationship between quadrant, points, lines, axis/ addition subtraction, axes, x-axis, y-axis, multiplication, and division. horizontal, vertical, Plot points on a coordinate intersection of lines, origin, graph (first quadrant). ordered pairs, coordinates, Represent real world and x-coordinate, y-coordinate, mathematical problems by numerical patterns, rules, graphing points in the first attribute, category, quadrant of the coordinate subcategory, heirarchy, plane. properties (rules about how Interpret coordinate values numbers work), twodimensional the problem or situation. of points in the context of Vocabulary from previous Generate two numerical grades: polygon, rhombus/ patterns using two given rhombi, rectangle, square, rules. triangle, quadrilateral, Identify apparent pentagon, hexagon, cube, relationships between trapezoid, circle corresponding terms for the In an ordered pair, the first two numerical patterns. number indicates how far to Form ordered pairs travel from the origin in the consisting of corresponding direction of one axis (x-axis). terms from the two patterns, In an ordered pair, the and graph the ordered pairs second number indicates on a coordinate graph. how far to travel in the Identify attributes of a twodimensional shape based direction of the second axis (y-axis). on attributes of the groups and categories in which the shape belongs. 5.NBT.7-Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. (05) 5.G.1-Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x- axis and x-coordinate, y-axis and y-coordinate). (05) 5.G.2-Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate Page: 13 of 15
There is more than one way to classify most shapes and solids. Objects have distinct attributes that can be measured. Standard units provide common language for communicating measurement. The choice of measurement tools depends on the measureable attribute and the degree of precision desired. Computational fluency includes understanding not only the meaning but also the appropriate use of numerical operations. Attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. (For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.) Vocabulary Measurement: line plot, length, mass, liquid volume Classify two-dimensional figures in a hierarchy based on properties. Measure objects to the nearest 1/2, 1/4, and 1/8 unit (length, mass, liquid volume). Make a line plot to display the data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions to solve problems involving information presented in line plots. Fluently multiply multi-digit whole numbers using the standard algorithm. values of points in the context of the situation. (05)[State:New 5.OA.3-Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. (05) 5.G.3-Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. (05) 5.G.4-Classify two-dimensional figures in a hierarchy based on properties. (05)[State:New 5.MD.2-Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this Page: 14 of 15
Plans: 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. (05) 5.NBT.5-Fluently multiply multidigit whole numbers using the standard algorithm. (05) Page: 15 of 15