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developing understanding and fluency with multi-digit multiplication and division.

develop an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions and whole numbers.

understanding geometric figures can be analyzed and classified based on their properties.

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draw and identify lines and angles, and classify shapes by properties of their lines and angles.

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Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

represent and interpret data.

Geometric measurement: understand concepts of angle and measure angles.

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extend understanding of fraction equivalence and ordering.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Understand decimal notation for fractions, and compare decimal fractions.

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generalize place value understanding for multi-digit whole numbers.

use place value understanding and properties of operations to perform multi-digit math.

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use the four operations with whole numbers to solve problems.

gain familiarity with factors and multiples

generate and analyze patterns.

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Create a frequency table and/or line plot to display measurement data.

Solve problems involving addition and subtraction by using information presented in a data display.

Analyze the data in a frequency table, line plot, bar graph or picture graph.

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Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

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Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

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| Know relative sizes of measurement units within one system of units.

Convert measurements in a larger unit in terms of a smaller unit

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Understand the concepts of angle and measure angles.

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Understand the concepts of angle and measure angles.

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Identify and estimate angles and their measure

Draw and measure angles in whole-number degrees using a protractor.

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Classify 2-dimensional shapes by properties of their lines and angles

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Classify 2-dimensional shapes by properties of their lines and angles

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Draw and identify points, lines, line segments, rays, angles, perpendicular lines and parallel lines

Classify two-dimensional shapes by their sides and/or angles.

Construct lines of symmetry for a two-dimensional figure.

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generate a number pattern that follows a given rule.

use words or math symbols to express a rule for a given pattern.

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recognize that a whole number is a multiple of each of it's factors

Determine if a whole number within 100 is composite or prime

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Use the four operations with whole numbers to solve problems 

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Use the four operations with whole numbers to solve problems 

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multiply or divide to solve problems

solve multi number division problems involving variables.

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understand decimal notation for fractions and compare decimal fractions.

read, write, and identify fractions to the hundredths place.

Compare fractions to the hundredths place.

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1. Connecting ratio and rate to whole number multiplication and division using concepts of ratio and rate to solve problems.
2. Completing Understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers.
3. Writing, interpreting, and using expressions and equations.
4. Developing understanding of statistical thinking.

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Understand and use ratios to solve problems.

Understand a ratio as a comparison of two quantities and represent these comparisons.

Understand the concept of a unit rate associated with a ratio, and describe the meaning of unit rate.

Solve problems involving ratios and rates.

a. Create tables of equivalent ratios, find missing values in the tables and plot the pairs of values on the Cartesian coordinate plane.
b. Solve unit rate problems.
c. Solve percent problems.
d. Convert measurement units within and between two systems of measurement.

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Ratios and Proportional Relationships (6.RP)

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Ratios and Proportional Relationships (6.RP)

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Understand ratio concepts and use ratio reasoning to solve problems.
1.  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
2.  Understand the concept of a unit rate a/b associated with a ratio  and use rate language in the context of a ratio relationship.
3.  Use ratio and rate reasoning to solve real-world and mathematical problems

Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

Solve unit rate problems including those involving unit pricing and constant speed.

Find a percent of a quantity as a rate per 100;  solve problems involving finding the whole, given a part and the percent.

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

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Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
1.  Interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions.
Compute fluently with multi-digit numbers and find common factors and multiples.
2.  Fluently divide multi-digit numbers using the standard algorithm.
3.  Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
4.  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

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Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Compute and interpret quotients of positive fractions. (Solve problems involving division of fractions by fractions.)

Compute with non-negative multi-digit numbers and find common factors and multiples.

Demonstrate fluency with division of multi-digit whole numbers.

Demonstrate fluency with addition, subtraction, multiplication and division of decimals.

Find common factors and multiples.

a. Find the greatest common factor (GCF) and the least common multiple (LCM).
b. Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers.
Apply and extend previous understandings of numbers to the system of rational numbers.

Use positive and negative numbers to represent quantities.

Locate a rational number as a point on the number line.

a. Locate rational numbers on a horizontal or vertical number line.
b. Write, interpret and explain problems of ordering of rational numbers.
c. Understand that a number and its opposite (additive inverse) are located on opposite sides of zero on the number line.

Understand that the absolute value of a rational number is its distance from 0 on the number line.

Extend prior knowledge to generate equivalent representations of rational numbers between fractions, decimals and percentages (limited to terminating decimals and/or benchmark fractions of 1/3 and 2/3).

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Explain that the graph of an equation in two variables is the set of all its solutions plotted in the Cartesian coordinate plane

Solve problems involving a system of linear inequalities.

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apply and extend previous understandings of arithmetic to algebraic expressions.

reason about and solve one-variable equations and inequalities.

represent and analyze quantitative relationships between dependent and independent variables

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Analyze the effect of translations and scale changes on functions.

Use arithmetic and geometric sequences.

Prove Geometric Theorems

find arc lengths and sectors of circles

use coordinates to prove geometric theorems algebraically.

apply geometric concepts to modeling.

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Data Analysis, Statistics, and Probability

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Data Analysis, Statistics, and Probability

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Interpret linear, quadratic and exponential functions in terms of the context.

Analyze linear, quadratic and exponential functions using different representations

Build new functions from existing functions (limited to linear, quadratic and exponential).

Construct and compare linear, quadratic and exponential models and solve problems.

Summarize, represent and interpret data.

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Represent and analyze patterns and relationships.

Investigate the relationship between two numeric patterns.

a. Generate two numeric patterns given two rules.
b. Translate two numeric patterns into two sets of ordered pairs.
c. Graph numeric patterns on the Cartesian coordinate plane.
d. Identify the relationship between two numeric patterns.

Write a rule to describe or explain a given numeric pattern.

Write and interpret numerical expressions.

Write, evaluate and interpret numeric expressions using the order of operations.

Translate written expressions into algebraic expressions.

Use the four operations to represent and solve problems.

Solve and justify multi-step problems involving variables, whole numbers, fractions and decimals

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Use place value system understanding to perform operations with multi-digit whole numbers to billions and decimals to thousandths.

Read, write and identify numbers from billions to thousandths using number names, base ten numerals and expanded form.

Compare two numbers from billions to thousandths using the symbols >, = or <, and justify the solution.

Understand that in a multi-digit number, a digit represents 1/10 times what it would represents in the place to its left.

Evaluate the value of powers of 10 and understand the relationship to the place value system.

Round numbers from billions to thousandths place.

Add and subtract multidigit whole numbers and decimals to the thousandths place, and justify the solution

Multiply multi-digit whole numbers and decimals to the hundredths place, and justify the solution

Divide multi-digit whole numbers and decimals to the hundredths place using up to two-digit divisors and four-digit dividends, and justify the solution.

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Number Sense and Operations in Fractions

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Number Sense and Operations in Fractions

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Understand the relationship between fractions and decimals (denominators that are factors of 100).

Understand that parts of a whole can be expressed as fractions and/or decimals.

Convert decimals to fractions and fractions to decimals.

Compare and order fractions and/or decimals to the thousandths place using the symbols >, = or <, and justify the solution.

Perform operations and solve problems with fractions and decimals.

Estimate results of sums, differences and products with fractions and decimals to the thousandths.

Justify the reasonableness of a product when multiplying with fractions.

a. Estimate the size of the product based on the size of the two factors.

b. Explain why multiplying a given number by a fraction greater than 1 results in a product larger than the given number.

c. Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number.

d. Explain why multiplying the numerator and denominator by the same number is equivalent to multiplying the fraction by 1.

Solve problems involving addition and subtraction of fractions and mixed numbers with unlike denominators, and justify the solution.

Extend the concept of multiplication to multiply a fraction or whole number by a fraction.

a. Recognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths.

b. Calculate and interpret the product of a fraction by a whole number and a whole number by a fraction.

c. Calculate and interpret the product of two fractions less than one

Extend the concept of division to divide unit fractions and whole numbers by using visual fraction models and equations.

a. Calculate and interpret the quotient of a unit fraction by a non-zero whole number.

b. Calculate and interpret the quotient of a whole number by a unit fraction

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Represent and analyze data.

Create a line graph to represent a data set, and analyze the data to answer questions and solve problems

Create a line plot to represent a given or generated data set, and analyze the data to answer questions and solve problems, recognizing the outliers and generating the median

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Classify two- and three-dimensional geometric shapes.

Understand that attributes belonging to a category of figures also belong to all subcategories.

Classify figures in a hierarchy based on properties.

Analyze and describe the properties of prisms and pyramids.

Understand and compute volume.

Understand the concept of volume and recognize that volume is measured in cubic units.

a. Describe a cube with edge length 1 unit as a “unit cube” and is said to have “one cubic unit” of volume and can be used to measure volume.
b. Understand that the volume of a right rectangular prism can be found by stacking multiple layers of the base

Apply the formulas V = l × w × h and V = B × h for volume of right rectangular prisms with whole-number edge lengths.

Graph points on the Cartesian coordinate plane within the first quadrant to solve problems.

Define a first quadrant Cartesian coordinate system.

a. Represent the axes as scaled perpendicular number lines that both intersect at 0, the origin.
b. Identify any point on the Cartesian coordinate plane by its ordered pair coordinates.
c. Define the first number in an ordered pair as the horizontal distance from the origin.
d. Define the second number in an ordered pair as the vertical distance from the origin.

Plot and interpret points in the first quadrant of the Cartesian coordinate plane.

Solve problems involving measurement and conversions within a measurement system.

Convert measurements of capacity, length and weight within a given measurement system.

Solve multi-step problems that require measurement conversions.

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1. Developing fluency with addition and subtraction of fractions and developing understanding of the multiplication and division of fractions in limited cases.
2. Extending division to two-digit divisors, integrating decimals fractions into the place-value system, developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations.
3. Developing understanding of volume.

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Operations and Algebraic Thinking (5.OA)

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Operations and Algebraic Thinking (5.OA)

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Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols.
2. Write simple expressions that record calculations with numbers and interpret numerical expressions without evaluating them.
Analyze patterns and relationships.
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

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Number and Operations in Base Ten (5.NBT)

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Number and Operations in Base Ten (5.NBT)

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Understand the place value system.
1. Recognize that in a multi-digit number, a digit in one place represents ten times as much as it represents in the place to its right and one-tenth of what it represents in the place to its left.
2. Explain patterns in the number of zeros of the product when multiplying a number by powers of ten and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of ten. Use whole-number exponents to denote powers of ten.
3. Read, write, and compare decimals to thousandths.

Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.

Compare two decimals to thousandths based on meanings of the digits in each place using >, =, < symbols to record the results of comparisons.

4. Use place value understanding to round decimals to any place.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluently multiply multi-digit  whole numbers using the standard algorithm.
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.
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.

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Use equivalent fractions as a strategy to add and subtract fractions.
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.
2.  Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3.  Interpret a fraction as division of the numerator by the denominator.  Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
4.  Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

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.

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.

5. Interpret multiplication as scaling (resizing) by:

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

Explaining why multiplying a given number by a fraction greater than 1 results in 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.

6.  Solve real world problems involving multiplication of fractions and mixed numbers.
7.  Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

Interpret division of a unit fraction by a non-zero whole number  and compute such quotients.

Interpret division of a whole number by a unit fraction, and compute such quotients.

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions

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Convert like measurement units within a given measurement system.
1. Convert among different-sized standard measurement units within a given measurement system  and use these conversions in solving multi-step, real world problems.
Represent and interpret data.
2. Make a line plot to display a data set of measurements in fractions of a unit. Use operations on fractions for this grade to solve problems involving information presented in line plots.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
3.  Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

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 solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

4.  Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5.  Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume

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.

Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and mathematical problems.

Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

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Graph points on the coordinate plane to solve real-world and mathematical problems.
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.
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.
Classify two-dimensional figures into categories based on their properties.
3.  Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that category.
4.  Classify two-dimensional figures in a hierarchy based on properties.

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Solve real-world and mathematical problems involving area, surface area, and volume.

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Extended understanding of operations on whole numbers to fraction operations 

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Extended understanding of operations on whole numbers to fraction operations 

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when you add, subtract, multiply or divide a fraction you are joining or separating parts referring to the same whole.

decomposing a fraction into multiple parts with the same denominator.

Adding and subtracting fractions with mixed numbers and like denominators.

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Rounding, and comparing numbers with a decimal point.

read, write, and identify multi digit whole numbers using number names, numerals, and expanded form.

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recognize, explain, and generate fraction equivalence.

compare and justify fractions

solve problems involving adding and subtracting mixed numbers with like denominators.

apply knowledge of multiplication to fractions and whole numbers

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develop understanding of statistical variability.

Summarize and describe distributions