Balancing Rigor Holding Tank

Balancing the Rigor of Mathematics (5.NBT.2)

• Conceptual Understanding
• Understand that when multiplying a whole number by a power of 10, the product shows an increased number of zeroes in relationship to the exponent
• Understand how multiplication and division by powers of ten impacts the number of zeros in the product
• Understand the pattern that exists when multiplying/dividing by a power of 10 and how that relates to the placement of the decimal point
• Understand that an exponent indicates the number of times a base is multiplied by itself

• Express powers of 10 using exponents.
• Explain the pattern for the number of zeros in a product and how it relates to the power of 10
• Explain a pattern for how multiplying/dividing any whole number by a power of 10 impacts the
  placement of the decimal point
• Multiply/Divide by powers of 10

• Apply understanding of exponents to real-world examples involving scientific notation

Balancing the Rigor of Mathematics (5.NBT.3)

• Understand that numbers can be represented in many different ways
• Understand how to use the value of the digits to compare numbers
• Understand why one number is larger/smaller than another using manipulatives/models
• Understand the meaning of comparison symbols and how to read them correctly
• Understand that expanded form is represented by each digit being multiplied by its place value and then being added back together

• Read and write decimals to the thousandths place in word form, base-ten form, and expanded form
• Compare two decimals to the thousandths based on the meaning of the digits in each place (using <, >, =)
• Use manipulatives to model and compare numbers

• Accurately explain how benchmark numbers can be used to compare decimals
• Solve word problems or use problem solving tasks involving comparing numbers with decimals

Balancing the Rigor of Mathematics (5.NBT.4)

• Understand that rounding is not the same as estimating
• Understand place value concepts
• Understand that “rounding” means to identify the multiple of .01, 1, 10, 100, etc. that is closest to a given number
• Understand that numbers exactly between two multiples of .01, 1, 10, 100, etc. round to the larger multiple

• Round whole numbers to the nearest .01, 1, 10, 100, etc.
• Represent rounding using tools such as a number line, base ten blocks, or hundreds charts

• Accurately explain why a number rounds to another number based on the number’s relationship to the nearest multiple of .01, 1, 10, 100, etc.
• Solve real-world problems or use problem solving tasks that involve rounding to the nearest .01, 1, 10, 100, etc.
• Identify numbers that can round to a given number (multiple of .01, 1, 10, 100, etc.)
• Solve word problems or use problem solving tasks involving rounding
• Explain how rounding is helpful in computation estimation

Balancing the Rigor of Mathematics (5.NBT.5)

• Understand why the standard algorithm works based on place value ideas
• Understand place value to 1,000,000

• Multiply multi-digit numbers using the standard algorithm

• Explain the steps in the standard algorithm, showing a place value understanding of why the strategy works
• Solve word problems or use problem solving tasks that involve multiplication of multi-digit numbers
• Decide if an answer is reasonable using mental math, estimation and/or rounding

Balancing the Rigor of Mathematics (5.NBT.6)

• Understand the concept of division
• Understand that division can be partitive and quotative
• Understand the inverse relationship between multiplication and division
• Understand there are multiple ways to divide multi-digit numbers
• Understand the role of place value in division strategies
• Understand how to interpret the remainder in a division problem based on the context of the problem

• Use multiple strategies to solve multi-digit division problems
• Represent multi-digit division with manipulatives, pictures and equations

• Solve the same division problem using more than one strategy
• Accurately explain the selected division strategy
• Decide if an answer is reasonable using mental math, estimation and/or rounding
• Solve real-world problems or use problem solving tasks involving multi-digit division
• Apply known strategies to larger numbers
• Explain how changing the value of the divisor affects the quotient (i.e. 350 ÷ 5 vs. 350 ÷ 50)

Balancing the Rigor of Mathematics (5.NBT.7)

• Understand place value concepts
• Understand properties of operations including commutative, associative and distributive
• Understand the inverse relationship of addition/subtraction and multiplication/division
• Understand whole number computation for addition, subtraction, multiplication and division
• Understand patterns in decimal computation (i.e. when multiplying two numbers less than one, a product may result that is less than either factor)

• Add, subtract, multiply, and divide decimals to the hundredths
• Division of decimals should be limited to simple situations that do not require the standard algorithm (i.e. 12÷.5, .5÷.1, etc)
• Explain decimal computation and the role place value plays in the computation

• Recognize and use the patterns in decimal computation for use with larger numbers
• Explain the connections between whole number computation and decimal computation
• Accurately explain the critical role of estimation in decimal computation
• Solve real-world problems or use problem solving tasks involving decimal computation

Balancing the Rigor of Mathematics (5.NF.1)

• Understand factors and multiples
• Understand how to find equivalent fractions
• Understand how to convert mixed numbers to improper fractions and improper fractions to mixed numbers
• Understand that denominators tell the size of the parts and having same size parts makes adding and subtracting fractions easier
• Understand that when adding and subtracting fractions, there is an underlying assumption that the wholes are the same size

• Use models/manipulatives to represent conversions (between mixed and improper), equivalent fractions, and computation
• Create equivalent fractions with common denominators
• Add and subtract fractions including mixed numbers

• Use estimation and compare to actual computations
• Flexibly manipulate numbers to make situations true (i.e. use specific digits to form fractions with a specific sum)
• Solve real-world problems or use problem solving tasks involving addition and subtraction of fractions

Balancing the Rigor of Mathematics (5.NF.2)

• Understand how to find equivalent fractions
• Understand how to convert mixed numbers to improper fractions and improper fractions to mixed numbers
• Understand that denominators tell the size of the parts and having same size parts makes adding and subtracting fractions easier
• Understand that when adding and subtracting fractions, there is an underlying assumption that the wholes are the same size
• Understand that real-world scenarios occur and require adding and subtracting fractions

• Accurately solve word problems involving addition and subtraction of fractions
• Use visual models and/or equations to represent the problem
• Estimate the sum or difference and then use fraction sense to evaluate the reasonableness of calculations

• Write a fraction word problem for a given equation/sum/difference
• Make connections between fraction computation and decimal computation

Balancing the Rigor of Mathematics (5.NF.3)

• Understand that a fraction represents the division of one whole number by another whole number
• Understand that the numerator is the same as the dividend and the denominator is the same as the divisor

• Represent a fraction as division using visual models
• Explain how fractions represent division
  Write a remainder as a fraction

• Solve real-world word problems that involve division of whole numbers and interpret the quotient in the context of the problems

Balancing the Rigor of Mathematics (5.NF.4)

• Understand the concept of multiplication and how it can apply to fractions
• Understand that when multiplying fractions, the product is not always greater than the factors
• Understand that an equation such as ¼ x 8 is said as one-fourth of eight or interpreted as ¼ of 8 pies
• Understand the area model for multiplication
• Understand estimation of products
• Understand area as tiling with unit squares of equal fraction and or whole measure
• Understand the formula for area

Procedural Understanding

• Multiply a fraction by a whole number
• Multiply a fraction by a fraction
• Multiply fractional side lengths to find areas of rectangles
• Use a model to show a fraction multiplied by a fraction
• Use multiplication to find area

• Represent the problem in multiple ways
• Decide if an answer is reasonable using mental math, estimation and/or rounding
• Use a number line to show multiplication of a whole number by a fraction and fraction by a whole number
• Use a model to show multiplication of fractions and fraction by a whole number
• Create story contexts for problems involving multiplication of fractions and whole numbers or multiplication of two fractions

Balancing the Rigor of Mathematics (5.NF.5)

• Understand the relative size of two factors
• Understand scaling (resizing)
• Understand the identity property of multiplication
• Understand estimation of products

• Tell about the size of a product based on the factors (relative to 1)
• Estimate the product based on the factors

• Solve problems and tasks involving scaling
• Create story problems that represent a scaling situation
• Model multiplication equations using arrays and other representations that have at least 1 factor less than 1
• Model story problems, tasks, and/or problems involving scaling (with products less than 1)

Balancing the Rigor of Mathematics (5.NF.6)

Conceptual Understanding

• Understand all story structures and when to add, subtract, multiply, or divide to solve problems

• Write equations to reflect a given story problem (structure)
• Solve equations that multiply fractions and mixed numbers
• Explain or illustrate solution using fraction models or equations

• Solve problems and tasks that multiply fractions and mixed numbers
• Solve problems and/or tasks that involve multiplying fractions and mixed numbers using models

Balancing the Rigor of Mathematics (5.NF.7)

• Understand that multiplication and division are inverse operations
• Understand the connection between whole number division and division involving fractions in that it involves grouping (quotative model) or fair share (partitive model)
• Understand properties of operations (commutative, associative, identity, etc..)

• Create story contexts for problems involving division of a fraction by a whole number and division of a whole number by a fraction
• Solve problems involving division of fractions

Application

• Explain and represent whole number divided by a fraction and a fraction divided by a whole number using models, drawings, and equations

Balancing the Rigor of Mathematics (5.MD.1)

• Understand the relative sizes of measurement units within a system
• Understand the relationship between the units being converted

• Write and solve an equation to show conversion of units

• Make reasonable estimates when converting
• Explain how measurement units are similar and different (for example: How are a meter and yard alike? How are
  they different?)
• Represent the relationship between units in a function table and/or tape diagram
• Use tools, models and symbols to accurately convert from one unit of measure to another
• Solve problems and tasks using measurement conversions

Balancing the Rigor of Mathematics (5.MD.2)

• Understand how to measure lengths including fractional increments
• Understand the purpose of a line plot
• Understand computation of fractions

• Measure lengths including fractional increments
• Create line plots with fraction measurements
• Interpret line plots to solve multi-step problems using data from line plots

• Recognize and explain the relationship between line plots and number lines
• Using technology, research real-world examples of line plots in use

Balancing the Rigor of Mathematics (5.MD.3)

• Understand volume as an attribute of a solid figure
• Understand that volume can be measured in cubic units
• Understand that a unit cube with side length of one is made up of six identical square faces and used to measure volume
• Understand that unit cubes fill a container without gaps or overlaps to measure volume

• Identify volume as an attribute of a solid figure
• Recognize that a cube with a 1 unit side length is "one cubic unit" of volume
• Find volume by counting cubic units
• Explain how to find volume of a figure

• Explain why volume is measured in cubic units
• Explain why you would want to know the volume of an object
• Solve problems and tasks that involve volume
• Make a unit cube from a net of six identical square faces with the measure of one

Balancing the Rigor of Mathematics (5.MD.4)

• Understand that a cubic unit can be in inches, centimeters, feet, etc..)
• Understand that a unit cube with side length of one is made up of six identical square faces and used to measure volume
• Understand that unit cubes fill a container without gaps or overlaps to measure volume

• Measure the volume of a solid figure by filling it with cubes and counting the number of cubes
• Count units in three-dimensional pictures using different measures to find volume

• Build or draw two different figures with the same volume
• Determine the dimensions of a rectangular prism given the volume

Balancing the Rigor of Mathematics (5.MD.5)

• Understand that volume can be found by multiplying the dimensions of a figure
• Understand the associative property
• Understand the mathematical operation for determining volume in a right rectangular prism using whole numbers
• Understand real-world situations by recognizing that volume is the number of cubic units needed to fill a solid figure
• Understand the volume of two or more solid figures added together equals the composite volume of the complete figure

• Use unit cubes to determine the volume of a rectangular prism
• Explain why multiplying the area of the base times the height will result in the volume
• Write and solve an equation(s) that will determine the volume of a figure
• Relate finding the product of three numbers to finding volume and explain how it is related to the associative property
• Use a formula for finding the volume of a rectangular prism
• Decompose irregular figures into rectangular prisms

• Build the volume of each piece in a composite figure to determine the total volume
• Solve problems and tasks that involve volume
• Determine the dimensions (factors) of a figure given its volume

Balancing the Rigor of Mathematics (5.G.1)

• Understand that coordinate grids can be used for maps
• Understand that a coordinate grid is composed of perpendicular lines that create an x-axis and y-axis
• Understand that the horizontal axis is the x-axis and that the vertical axis is the y-axis
• Understand horizontal and vertical number lines
• Understand that an ordered pair is a set of numbers that indicate direction and length of movement
• Understand the location and ordered pair (0,0) associated with the origin

• Construct a coordinate system and recognize the origin
• Recognize the x-axis and y-axis
• Identify an ordered pair
• Explain the relationship of an ordered pair and the location on the coordinate plane

• Plot points and show movement on a coordinate grid
• Solve problems and tasks involving coordinate grids

Balancing the Rigor of Mathematics (5.G.2)

• Understand how coordinate grids are used to solve real-world problems
• Understand that an ordered pair is a set of numbers that represent a mathematical problem/equation

• Interpret and label the x and  y-axis in relationship to the problem/task being solved
• Graph points in the first quadrant of a coordinate plane that represents real-world data
• Graph shapes in quadrant one of a coordinate grid

• Find the distance between two coordinate points
• Discuss the consequence of enlarging or decreasing coordinates of a specific shape
• Extend understanding to graphing points in quadrants 2, 3, and 4

Balancing the Rigor of Mathematics (5.G.3)

• Understand defining attributes of two-dimensional shapes
• Understand the similarities and differences among two-dimensional figures
• Understand that figures can be sorted into categories with subcategories

• Classify two-dimensional figures by their attributes
• Explain how two-dimensional attributes can belong to several two-dimensional figures
• Identify subcategories using two-dimensional attributes

• Draw or compose figures based on definitions, attributes, or given categories
• Define figures given category titles and decide if additional figures belong

Balancing the Rigor of Mathematics (5.G.4)

• Understand that figures can be sorted into categories with subcategories

• Group shapes that share a single property, and then among these shapes, group those that share a second property, etc.
• Use a graphic organizer to organize and compare shapes

Application

• Explain how a set of two-dimensional shapes are alike and different

Balancing the Rigor of Mathematics (5.OA.1)

• Understand that an expression names a quantity
• Understand that an equation describes a relationship between two expressions
• Understand that a variable is a quantity that can change or vary
• Understand that variables are often represented with a letter such as, x, n
• Understand that an expression can describe a scenario
• Understand that the word “then” names a series of operations in order
• Understand that the order of operations helps to evaluate expressions

• Write an expression to describe/reflect a word problem/real-life situation
• Use parenthesis and brackets to group an expression within a multi-step expression
• Evaluate expressions with parenthesis and brackets
• Solve a multi-step word problem

• Explain and model the difference between two equations
• Write a numerical expression given the solution
• Explain why we need an order of operations
• Solve word problems and tasks involving parentheses, brackets, and or braces in expressions
• Model, illustrate, and/or diagram the grouping of objects

Balancing the Rigor of Mathematics (5.OA.2)

• Understand that a numeric expression represents the value of a number
• Understand that an equation describes a relationship between two expressions
• Understand that a variable is a quantity that can change or vary, and can be represented with a letter/symbol
• Understand that parentheses indicate which operation to perform first

• Use an expression to show a calculation described verbally.
• Write simple expressions that describe word problems or scenarios
• Write a given expression in words
• Evaluate a numerical expression without evaluating (solving)
• Use numbers and symbols appropriately in simple expressions (variables, parentheses, etc…)

• Evaluate expressions
• Extend understanding of simple expressions to more complex expressions
• Write a word problem/scenario for a given expression

Balancing the Rigor of Mathematics (5.OA.3)

• Understand numerical patterns
• Understand how patterns can be represented on a coordinate grid

• Make two numerical patterns with the same starting number for 2 different given rules
• Explain the relationship between each of the corresponding terms from a pattern
• Make ordered pairs with the corresponding terms in a pattern
• Graph ordered pairs (a pattern) on a coordinate plane

• Create an input/output chart for two different two-operation rules
• Create story problems from a given set of ordered pairs or from a given pattern
• Solve word problems and tasks that involve patterns and ordered pairs
• Use models of a coordinate plane to extend a pattern