Build whole numbers to 1 million by multiplying by 10 repeatedly.

Use whole numbers to denote powers of 10. Explain patterns in the number of zeros when multiplying any powers of 10 by any other powers of 10.

Explain patterns in the number of zeros of the product when multiplying a whole number by 10. Recognize that in a multi-digit whole number, a digit in any place represents 10 times as much as it represents in the place to its right.

Explain patterns in the number of zeros of the product when multiplying a whole number by powers of 10.

Explain patterns in the number of zeros of the quotient when dividing a whole number by 10. Recognize that in a multi-digit whole number, a digit in any place represents $${\frac{1}{10}}$$ as much as it represents in the place to its left.

Explain patterns in the number of zeros of the quotient when dividing a whole number by powers of 10.

Build decimal numbers to thousandths by dividing by 10 repeatedly.

Explain patterns in the placement of the decimal point when a decimal is multiplied by any power of 10. Recognize that in a multi-digit decimal, a digit in any place represents 10 times as much as it represents in the place to its right.

Explain patterns in the placement of the decimal point when a decimal is divided by a power of 10. Recognize that in a multi-digit decimal, a digit in any place represents $${\frac{1}{10}}$$ as much as it represents in the place to its left.

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

Compare multi-digit decimals to the thousandths based on meanings of the digits using $${>}$$, $${<}$$, or $$=$$ to record the comparison.

Use place value understanding to round decimals to the nearest whole.

Use place value understanding to round decimals to any place.

Evaluate numerical expressions involving addition, subtraction, multiplication, division, and grouping symbols.

Write expressions that record calculations with numbers, and interpret expressions without evaluating them.

Write expressions that represent real-world situations and evaluate them.

Multiply multiples of powers of ten. Estimate multi-digit products by rounding numbers to their largest place value.

Multiply two-digit, three-digit, and four-digit numbers by one-digit numbers.

Multiply two-digit numbers by two-digit numbers.

Multiply three-digit numbers by two-digit numbers.

Multiply four-digit numbers by two-digit numbers.

Multiply three- and four-digit numbers by three-digit numbers.

Multiply multi-digit numbers and assess the reasonableness of the product.

Divide multiples of powers of ten by multiples of ten without remainders. Estimate multi-digit quotients by rounding numbers to their largest place value.

Estimate multi-digit quotients using compatible numbers.

Divide two-digit, three-digit, and four-digit dividends by one-digit divisors.

Divide two- and three-digit dividends by multiples of 10 with one-digit quotients and remainders in the ones place.

Divide two-digit dividends by two-digit divisors with one-digit quotients and remainders in the ones place.

Divide three-digit dividends by two-digit divisors with one-digit quotients and remainders in the ones place.

Divide three-digit dividends by two-digit divisors with two-digit quotients, reasoning about the decomposition of a remainder in any place.

Divide four-digit dividends by two-digit divisors with two- and three-digit quotients, reasoning about the decomposition of a remainder in any place.

Divide multi-digit numbers by one- and two-digit divisors and assess the reasonableness of the quotient.

Solve word problems involving multi-digit multiplication and division.

Understand volume as an attribute of solid figures that is measured in cubic units. Find the volume of concrete three-dimensional figures.

Find the volume of pictorial three-dimensional figures.

Find the volume of a right rectangular prism by thinking about its layers.

Find the volume of a right rectangular prism using its side lengths.

Represent volume using expressions related to the formulas $$V = b \times h $$ and $$V = l \times w \times h$$. Use these formulas to find the volume of pictorial rectangular prisms.

Understand standard units for measuring volume, including cubic inches, cubic centimeters, cubic feet, and cubic meters. Solve real-world and mathematical problems involving these various units.

Solve more complex real-world and mathematical problems involving volume.

Understand that volume is additive. Find the volume of composite solid figures when all dimensions are given and their decomposition is already shown.

Understand that volume is additive. Find the volume of composite solid figures when not all dimensions are given and/or they must be decomposed.

Classify shapes as polygons versus non-polygons and classify polygons according to their number of sides.

Classify quadrilaterals based on the presence or absence of one pair of parallel sides. Define trapezoids as quadrilaterals with at least one pair of parallel sides.

Classify trapezoids based on the presence of one or two sets of parallel sides. Define parallelograms as trapezoids with two sets of parallel sides.

Classify parallelograms based on the presence or absence of right angles or based on the presence or absence of sides of equal length. Define* *rectangles as parallelograms with four right angles and rhombuses as parallelograms with four equal sides.

Classify rectangles based on the presence or absence of sides of equal length, and classify rhombuses based on the presence or absence of right angles. Define squares as quadrilaterals with sides of equal length and all right angles.

Classify triangles based on side and angle measures.

Recognize and generate equivalent fractions.

Add fractions with like denominators.

Subtract fractions with like denominators.

Add fractions with unlike denominators whose sum is less than 1.

Subtract fractions from fractions less than 1 with unlike denominators.

Add fractions with unlike denominators whose sum is less than 2.

Subtract fractions from fractions less than 2 with unlike denominators.

Add fractions with unlike denominators whose sum is greater than 2.

Subtract fractions from fractions greater than 2 with unlike denominators.

Use benchmark fractions and number sense to estimate mentally and assess the reasonableness of answers.

Add and subtract more than two fractions.

Solve two- and multi-step word problems involving addition and subtraction of fractions.

Add decimals.

Subtract decimals.

Solve two- and multi-step word problems involving addition and subtraction of decimals.

Relate equal shares of objects to division expressions and visual representations of fractions.

Write division expressions that represent fractions and vice versa.

Solve division problems when the quotient is a fraction or mixed number, including cases with larger values.

Multiply a fraction by a whole number where the product is a whole number.

Multiply a fraction by a whole number where the product is a whole number or a fraction.

Relate multiplication of a fraction by a whole number to multiplication of a whole number by a fraction and use this to develop a general method to multiply any fraction by any whole number (or vice versa).

Solve real-world problems involving multiplication of fractions and whole numbers and create real-world contexts for expressions involving multiplication of fractions and whole numbers.

Multiply a fraction by a fraction without subdivisions using tape diagrams and number lines.

Multiply a fraction by a fraction with subdivisions using tape diagrams and number lines.

Multiply a fraction by a fraction with more complicated subdivisions using an area model.

Develop a general method to multiply a fraction by a fraction.

Solve real-world problems involving multiplication of fractions with fractions and create real-world contexts for expressions involving multiplication of fractions with fractions.

Multiply mixed numbers by whole numbers.

Multiply mixed numbers by fractions.

Multiply mixed numbers by mixed numbers.

Develop a general method to multiply with mixed numbers.

Solve real-world problems involving multiplication with mixed numbers and create real-world contexts for expressions involving multiplication with mixed numbers.

Interpret multiplication as scaling.

Divide a unit fraction by a whole number.

Divide a whole number by a unit fraction.

Solve real-world problems involving division with fractions and create real-world contexts for expressions involving division with fractions.

Solve real-world problems involving multiplication and division with fractions.

Write and evaluate numerical expressions involving operations with fractions.

Create line plots.

Solve problems involving information presented in a line plot (dot plot).

Multiply a single-digit whole number by a decimal in cases that involve basic facts. Estimate the product of a single-digit whole number with a decimal by rounding numbers to their largest place value.

Multiply a single-digit whole number by a decimal.

Construct viable arguments and critique the reasoning of others regarding the placement of the decimal point in computations that involve multiplying a single-digit whole number by a decimal.

Multiply a multi-digit whole number by a decimal.

Multiply a decimal by a decimal in cases that involve basic facts. Estimate the product of two decimals by rounding numbers to their largest place value.

Multiply a decimal by a decimal.

Multiply with decimals in all cases, reasoning about the placement of the decimal point.

Divide a decimal by a single-digit whole number in cases that involve basic facts. Estimate quotients with single-digit divisors by rounding numbers to compatible numbers.

Divide a decimal by a single-digit whole number.

Divide a decimal by a single-digit whole number that requires decomposition in the smallest place value.

Divide a decimal by a two-digit whole number.

Divide a decimal by a two-digit whole number that requires decomposition in the smallest place value.

Divide a whole number or a decimal by 1 tenth or 1 hundredth.

Divide a whole number or a decimal by a decimal in cases that involve basic facts. Estimate quotients with decimal divisors by rounding numbers to compatible numbers.

Divide a whole number or a decimal by a decimal.

Divide with decimals in all cases, reasoning about the placement of the decimal point.

Solve real-world problems that require interpretation of the remainder involving all possible cases, including further decomposition into a decimal.

Solve real-world problems involving multiplication and division with decimals.

Write and evaluate numerical expressions involving multiplication and division with decimals.

Express measurements in a whole number of larger units or mixed units in terms of a smaller unit.

Express measurements in a fractional or decimal number of larger units in terms of a smaller unit.

Express measurements in a whole number of smaller units in terms of a larger unit.

Express measurements in a fractional or decimal number of smaller units in terms of a larger unit.

Solve real-world problems involving measurement conversions.

Define a coordinate plane and identify the coordinates of given points.

Construct a coordinate plane and plot points given their coordinates.

Construct a coordinate plane with non-unit intervals and use it to plot and identify points.

Determine the interval needed to plot various points and practice plotting and identifying points on coordinate grids with various axes.

Plot horizontal and vertical lines on a coordinate plane and find patterns in their coordinates.

Draw parallel lines on a coordinate plane (optional).

Draw perpendicular lines on a coordinate plane (optional).

Draw shapes on a coordinate plane.

Draw symmetric figures on a coordinate plane.

Understand that a coordinate plane can be used to represent real-world information and interpret coordinate values of points in the context of a situation.

Represent real-world problems by graphing information represented on a table in the coordinate plane and interpret coordinate values of points in the context of a situation.

Represent real-world problems by graphing information given as a description of a situation in the coordinate plane and interpret coordinate values of points in the context of a situation.

Generate two numerical patterns using two given rules, plotting the points, and identify multiplicative relationships between corresponding terms.

Generate two numerical patterns using two given rules, plotting the points, and identify more complex relationships between corresponding terms.

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