# Curriculum for Grade 5

Grade Description: Students dig more deeply into place value patterns and operations. Their understanding grows to include decimals in the hundredths and thousandths, operations with fractions and decimals, multi-digit multiplication and division with remainders, area models, properties of multiplication and division, estimation, and measuring volume. We introduce the coordinate plane (quadrant I), which students use to solve problems.

## MODULE 1. Place Value and Decimal Fractions

### Topic A: Multiplicative Patterns on the Place Value Chart

Relying on solid understanding of the place value chart and powers of ten, students multiply and divide to solve problems with a decimal number as a factor or quotient.

#### Multiply by 10 with and without a place value chart

#### Multiply by 100 with and without a place value chart

#### Multiply by 1,000 with and without a place value chart

#### Multiply by 10, 100, and 1,000

#### Complete a pattern of multiplication by 10 or 100

#### Divide by 10 with and without a place value chart

#### Divide by 100 with and without a place value chart

#### Divide by 1,000 with and without a place value chart

#### Divide by 10, 100, and 1,000

#### Identify and complete a pattern of division by 10 or 100

#### Factor out powers of 10 to multiply numbers with a single non-zero digit

#### Multiply numbers with a single non-zero digit

#### Record the relationship between hundredths and thousandths using a place value chart

#### Multiply a decimal number by 10 with and without a place value chart

#### Multiply a decimal number by 100 with and without a place value chart

#### Multiply a decimal number by 1,000 with and without a place value chart

#### Multiply a decimal number by 10, 100, and 1,000

#### Complete a pattern of multiplication by 10, 100, and 1,000

#### Convert between ones, tenths, hundredths, and thousandths in division problems

#### Divide a decimal number by 10 with and without a place value chart

#### Divide a decimal number by 100 with and without a place value chart

#### Divide a decimal number by 1,000 with and without a place value chart

#### Divide a number by 10, 100, and 1,000

#### Complete a pattern of division by 10, 100, and 1,000

#### Represent repeated multiplication of 10 as 10 with an exponent

#### Represent 10 with an exponent as repeated multiplication

#### Multiply a decimal number by 10 with an exponent

#### Divide a number by 10 with an exponent

### Topic B: Decimal Fractions and Place Value Patterns

Students use familiar symbols (<, =, >) to compare numbers to the thousandths place. They identify, compose, and arrange numbers of equal and unequal length.

#### Record a decimal number in expanded notation

#### Compare numbers to the hundredths place with and without a place value chart

#### Complete inequalities that compare numbers to the hundredths place

#### Arrange decimal numbers in order from smallest to greatest (Level 1)

#### Arrange decimal numbers in order from smallest to greatest (Level 2)

#### Arrange decimal numbers in order from smallest to greatest (Level 3)

### Topic C: Place Value and Rounding Decimal Fractions

Using a number line to provide context, students first determine the midway point between two decimal numbers. They then progress to rounding using the number line and the midway point. Finally, students round decimal numbers to any given place value without use of the number line.

#### Determine the midway point between two decimal numbers on a number line

#### Determine whether a decimal number is greater or less than halfway between two points on a number line

#### Compose and apply a rule for rounding numbers

#### Determine whether a decimal number rounds up or down

#### Round a given number by first determining its nearest tenths

#### Round a given number to the hundredths, tenths, and ones place

#### Round a decimal number to a given place value (Level 1)

#### Round a decimal number to a given place value (Level 2)

#### Round a decimal number to a given place value (Level 3)

#### Round a decimal number to a given place value (Level 4)

### Topic D: Adding and Subtracting Decimals

Based on the familiar disk model, students add and subtract numbers to the thousands place. They add and subtract in equations that require regrouping and learn to use the standard algorithm (column addition and column subtraction).

#### Add numbers to the hundredths place with regrouping using a disk model

#### Add numbers to the thousandths place with regrouping using a disk model

#### Add numbers to the thousandths place with regrouping using column addition and a disk model

#### Add numbers to the thousandths place with regrouping using column addition

#### Subtract numbers to the hundredths place with regrouping using a disk model

#### Subtract numbers to the thousandths place with regrouping using a disk model

#### Subtract numbers to the thousandths place with regrouping using column subtraction and a disk model

#### Subtract numbers of different lengths to the thousandths place with regrouping using column subtraction and a disk model

#### Subtract numbers to the thousandths place with regrouping using column subtraction

### Topic E: Multiplying Decimals

Students take the first steps in understanding multiplication of a decimal number. Working with a single-digit whole number and a decimal number, students work step-by-step through the process using the familiar disk model. They trade disks to model regrouping and explore the importance of decimal point placement.

#### Multiply a decimal number with regrouping using a disk model

#### Multiply a decimal number with regrouping using a disk model and the standard algorithm

#### Compare multiplying a whole number with multiplying a digit number with the same digits

#### Multiply a decimal number by converting it to a whole number and converting the product back to a decimal

### Topic F: Dividing Decimals

Students rely on familiar models and procedures to apply their division knowledge to decimals. Step-by-step, they learn to correctly work through regrouping, using a new place value, using zeros, and placing a decimal in their quotient.

#### Divide a decimal number by a single-digit number based on a model

#### Divide a decimal number by a single-digit number with and without regrouping based on a model

#### Divide a decimal number by a single-digit number with and without regrouping based on a model

#### Divide a decimal number by a single-digit number with regrouping to a new place value based on a model

#### Divide a decimal number by a single-digit number with and without regrouping using long division and a model

#### Use long division to divide a decimal number by a single-digit number with and without regrouping

#### Use long division to divide a decimal number by a single-digit number with quotients that include 0

#### Use long division to divide a decimal number by a single-digit number with regrouping to a new place value

## MODULE 2. Multi-Digit Whole Number and Decimal Fraction Operations

### Topic A: Mental Strategies for Multi-Digit Whole Number Multiplication

Students work with round numbers in the tens, hundreds, and thousands to see how factoring out can make mental multiplication easier. In doing so, they apply properties of multiplication.

#### Factor out 10, 100, or 1,000 from a given number

#### Solve multiplication problems using the associative and commutative properties

#### Solve a multi-digit multiplication problem using factoring out and properties of multiplication (Level 1)

#### Solve a multi-digit multiplication problem mentally

#### Solve a multi-digit multiplication problem using factoring out and properties of multiplication (Level 2)

#### Round multi-digit numbers to estimate the area of a rectangle

#### Round multi-digit numbers to estimate the product

### Topic B: The Standard Algorithm for Multi-Digit Whole Number Multiplication

Students hone their use of the standard algorithm for multiplication, working with multi-digit numbers. They master the use of regrouping and working across zeros. To further their conceptual understanding, the standard algorithm is presented alongside other strategies, such as the area model and factoring out powers of 10.

#### Identify sum, difference, product, and quotient in equations

#### Compose simple numerical expressions from text

#### Compose addition and multiplication expressions with multiple addends or factors

#### Compose complex numerical expressions based on a model (Part 1)

#### Compose complex numerical expressions based on a model (Part 2)

#### Compose complex numerical expressions based on a model (Part 3)

#### Multiply using the standard algorithm (1-digit by 3-digit)

#### Multiply using the standard algorithm (1-digit by 4-digit)

#### Factor out powers of 10 to simplify multiplication expressions

#### Solve related multiplication expressions after factoring out powers of 10

#### Multiply by a number with a power of 10 mentally

#### Multiply by a number with a power of 10 using the standard algorithm

#### Multiply using an area model and the standard algorithm

#### Multiply using the standard algorithm (3-digit by 3-digit) (Part 1)

#### Multiply using the standard algorithm (3-digit by 3-digit) (Part 2)

#### Multiply using the standard algorithm (3-digit by 4-digit) (Part 1)

#### Multiply using the standard algorithm (3-digit by 4-digit) (Part 2)

#### Multiply using the standard algorithm (2-digit by 5-digit)

### Topic C: Decimal Multi-Digit Multiplication

Students further their understanding multiplication of a decimal number by increasing the number of digits in factors. Working with a 2- or 3-digit whole number and a decimal number in the tenths or hundredths, students work step-by-step through the process using the standard algorithm. They learn how to place a decimal in their answer and how to estimate to check the reasonableness of their answer.

#### Multiply a 2-digit decimal number by a 2-digit whole number using partial products with an area model

#### Multiply a 2-digit decimal number by a 2-digit whole number using the standard algorithm (Level 1)

#### Multiply a 3-digit decimal number by a 2-digit whole number using the standard algorithm (Level 1)

#### Multiply a 2- or 3-digit decimal number by a 2-digit whole number using the standard algorithm (Level 2)

#### Multiply a 2- or 3-digit decimal number by a 2-digit whole number using the standard algorithm (Level 3)

#### Multiply a decimal number by a round number by factoring out powers of 10

#### Multiply a decimal number by a round number

#### Estimate the product of a decimal number and a 3-digit whole number (Part 1)

#### Estimate the product of a decimal number and a 3-digit whole number (Part 2)

#### Estimate the product of a decimal number and a 3-digit whole number then solve using the standard algorithm

### Topic D: Mental Strategies for Multi-Digit Whole Number Division

Students rely on their understanding of powers of 10, division, place value, and rounding to divide multi-digit whole numbers mentally.

#### Divide round numbers by powers of 10

#### Divide by factoring out powers of 10 from the divisor

#### Divide multi-digit round numbers mentally

#### Round a multi-digit number to a given place (Level 1)

#### Round a multi-digit number to a given place (Level 2)

#### List the multiples of a given 2-digit round number

#### Round a 3-digit number to the nearest multiple of a given 2-digit round number (Level 1)

#### Round a 3-digit number to the nearest multiple of a given 2-digit round number (Level 2)

#### Estimate a quotient by finding the nearest multiple

#### Estimate a quotient of a multi-digit division equation (Level 1)

#### Estimate a quotient of a multi-digit division equation (Level 2)

### Topic E: Partial Quotients and Multi-Digit Whole Number Division

Students look under the hood of division to really understand the concept of a remainder as well as the steps involved in long division. Students work with one- or two- and three-digit dividends and one- and two-digit divisors. They use the standard algorithm for long division, but approach it in different ways to build math flexibility.

#### Divide a group of objects into groups of given size with a remainder

#### Find the quotient and remainder by using multiples of the divisor

#### Find the quotient and remainder by subtracting the divisor from the dividend (Part 1)

#### Find the quotient and remainder by subtracting the divisor from the dividend (Part 2)

#### Estimate the quotient, then solve using long division with a 1-digit quotient (Part 1)

#### Estimate the quotient, then solve using long division with a 1-digit quotient (Part 2)

#### Solve using long division with a 1-digit quotient

#### Solve using long division with a 1-digit quotient (overestimate)

#### Solve using long division with a 1-digit quotient (underestimate)

#### Solve using long division with a 1-digit quotient

#### Solve using long division with a 2-digit quotient (Part 1)

#### Solve using long division with a 2-digit quotient (over and underestimate)

#### Solve using long division with a 2-digit quotient (Part 2)

#### Solve using long division with a 3-digit quotient

#### Solve using long division with a 3-digit quotient (over and underestimate) (Part 1)

#### Solve using long division with a 3-digit quotient (over and underestimate) (Part 2)

### Topic F: Partial Quotients and Multi-Digit Decimal Division

Students look under the hood of division to really understand the concept of a remainder as well as the steps involved in long division. Students work with one- or two- and three-digit dividends and one- and two-digit divisors. They use the standard algorithm for long division, but approach it in different ways to build math flexibility.

#### Rename decimal numbers in unit form

#### Rename a number in unit form as a decimal number

#### Rename a whole number in unit form and decimal form

#### Divide a decimal by a whole number (with and without using unit form)

#### Divide a decimal number by 10 or 100

#### Divide a decimal number by a multiple of 10 or 100

#### Estimate the quotient of a decimal number divided by a 2-digit number (Level 1)

#### Estimate the quotient of a decimal number divided by a 2-digit number (Level 2)

#### Solve long division to the tenths place

#### Solve long division to the hundredths place

#### Solve long division of a decimal number (Level 1)

#### Solve long division of a decimal number (Level 2)

#### Solve long division of a decimal number (over and underestimate)

#### Solve long division of a decimal number (Level 3)

#### Use long division to solve a word problem with a decimal quotient

#### Use long division to solve area problems with a decimal quotient

## MODULE 3. Addition and Subtraction of Fractions

### Topic A: Equivalent Fractions

Students are introduced to the basics of fraction equivalency. They learn to multiply to find an equivalent fraction and to divide to reduce a fraction. The concepts of common factors and greatest common factor are also introduced and applied.

#### Label the numerator and denominator in a fraction

#### Write an equivalent fraction (larger) based on a given multiple or denominator

#### Write an equivalent fraction (smaller) based on a given divisor, numerator, or denominator

#### Complete equivalent fractions

#### Identify common factors of two given numbers

#### Identify a common factor of two given numbers and divide both by the common factor

#### Reduce a fraction to its simplest form by dividing by the greatest common factor

### Topic B: Making Like Units Pictorially

Students rely on their knowledge of fraction equivalence to work through all aspects of adding and subtracting with fractions and mixed numbers. With plenty of visual support and step-by-step prompting, they work with like and unlike denominators. Students learn to convert between mixed numbers and fractions to solve problems, write their answer in simplest form, and solve word problems.

#### Add fractions with like denominators

#### Rewrite an improper fraction as a mixed number

#### Add fractions with like denominators and rewrite the sum as a mixed number

#### Divide shapes into an equal number of parts (precursor to finding common denominators)

#### Add fractions with unlike denominators (multiply denominators to find a common denominator)

#### Rename fractions with unlike denominators (Level 1)

#### Rename fractions with unlike denominators (Level 2)

#### Add fractions with unlike denominators and rewrite the sum as a mixed number

#### Add fractions with unlike denominators and rewrite the sum in simplest form (Part 1)

#### Add fractions with unlike denominators and rewrite the sum in simplest form (Part 2)

#### Subtract fractions with like denominators

#### Subtract fractions with unlike denominators (multiply denominators to find a common denominator)

#### Subtract fractions with unlike denominators

#### Rewrite a mixed number as an improper fraction

#### Subtract a fraction from a mixed number with like denominators (rename the mixed number as a fraction)

#### Subtract a fraction from a mixed number with unlike denominators (rename the mixed number as a fraction)

#### Subtract a fraction from a mixed number with unlike denominators and rewrite the difference in simplest form

#### Subtract a fraction from a mixed number with unlike denominators to solve a word problem

#### Subtract a fraction from 1 whole to solve a word problem

#### Subtract two fractions from 1 whole to solve a word problem

#### Add fractions to solve a word problem

### Topic C: Making Like Units Numerically

Students increase the complexity of the addition and subtraction operations they can solve by using mixed numbers. They reinforce and apply their understanding of number equivalence as they rename numbers to solve problems.

#### Add a mixed number and a whole number with like denominators

#### Subtract a fraction or a mixed number from a whole number with like denominators with and without a number line

#### Add a mixed number and a fraction with like denominators

#### Add mixed numbers with like denominators

#### Add mixed numbers with unlike denominators (Level 1)

#### Add mixed numbers with unlike denominators (Level 2)

#### Subtract mixed numbers with like denominators

#### Subtract mixed numbers with unlike denominators (Level 1)

#### Subtract a fraction from a mixed number with like denominators (rename the mixed number)

#### Subtract mixed numbers with unlike denominators (Level 2)

#### Subtract mixed numbers with unlike denominators (Level 3)

### Topic D: Further Applications

To support more complex understanding of operations with fractions, students are supported by visual models, the number line, and diagrams. They apply their knowledge of fractions to estimate sums and differences and solve multi-step word problems with mixed numbers.

#### Compare fractions with unlike denominators to 1/2 to estimate the sum

#### Compare fractions with unlike denominators to a missing unit to estimate the sum

#### Compare fractions with unlike denominators to estimate the difference of a mixed number minus a fraction

#### Add multiple mixed numbers and fractions

#### Solve a multi-step word problem with mixed numbers with unlike denominators (Part 1)

#### Solve a multi-step word problem with mixed numbers with unlike denominators (Part 2)

## MODULE 4. Multiplication and Division of Fractions and Decimal Fractions

### Topic A: Line Plots of Fraction Measurements

Students combine their knowledge of measurement, rounding, and mixed numbers to create line plots. They then analyze the data, requiring them to perform operations with mixed numbers.

#### Round to the nearest half inch on a ruler

#### Round to the nearest half inch on a ruler to create a line plot and analyze the data

#### Round to the nearest quarter inch on a ruler

#### Round to the nearest quarter inch on a ruler to create a line plot and analyze the data

#### Round to the nearest eighth of a foot on a ruler

#### Round to the nearest eighth of an inch on a ruler to create a line plot and analyze the data

### Topic B: Fractions as Division

To solidify their understanding of fractions as division, students work to convert between the two, using fractions greater than and less than one. They then work step by step to make sense of the remainder of a division problem as a fraction. Equipped with this understanding, students solve increasingly complex division word problems.

#### Rewrite division of whole numbers as a fraction based on a model

#### Rewrite division of whole numbers as a fraction and match a fraction to its related division equation

#### Relate division to fractions

#### Relate division to fractions greater than one

#### Divide and report the remainder as a fraction based on a model

#### Rename a fraction greater than one as a mixed number (Level 1)

#### Rename a fraction greater than one as a mixed number (Level 2)

#### Solve fraction word problems based on a model (Level 1)

#### Solve fraction word problems based on a model (Level 2)

#### Solve fraction word problems based on a model (Level 3)

#### Solve fraction word problems based on a model (Level 4)

#### Solve fraction word problems based on a model (Level 5)

### Topic C: Multiplication of a Whole Number by a Fraction

Students begin by finding a unit fraction of a whole number and other fractions of a whole number using familiar models. They then relate that process to multiplication. To multiply a whole number by a fraction, students learn to find common factors and reduce either before or after multiplying.

#### Determine fractions of a whole number based on a model (Level 1)

#### Determine fractions of a whole number based on a model to solve a word problem

#### Determine fractions of a whole number based on a model (Level 2)

#### Determine fractions of a whole number based on a model (Level 3)

#### Determine a whole number given a fraction of that number based on a model

#### Multiply a whole number by a fraction

#### Relate a fraction of a whole number to multiplication and solve

#### Determine a fraction of a measurement by converting units and multiplying

#### Determine common factors of two whole numbers

#### Divide whole numbers by a common factor

#### Reduce a fraction by dividing by a common factor

#### Reduce factors before multiplying a whole number by a fraction (Level 1)

#### Reduce factors before multiplying a whole number by a fraction (Level 2)

#### Multiply a whole number by a fraction by reducing before or after multiplying

### Topic D: Fraction Expressions and Word Problems

Students synthesize their understanding of fractions and operations to build and solve expressions. Familiar models support their learning, and word problems put the skills in context.

#### Match operations to the terms sum, difference, product, and quotient

#### Compose an expression based on text

#### Compose a compound expression based on text (Part 1)

#### Compose a compound expression based on text (Part 2)

#### Compose a compound expression based on text (Part 3)

#### Compose a compound equation based on a model and solve (Part 1)

#### Compose a compound equation based on a model and solve (Part 2)

#### Solve compound equations with parentheses and fractions

#### Solve a word problem by multiplying a whole number by a fraction (Level 1)

#### Solve a word problem by multiplying a whole number by a fraction (Level 2)

### Topic E: Multiplication of a Fraction by a Fraction

Students learn to multiply fractions by starting with unit fractions and the support of an area model. They then progress to multiplication that involves mixed numbers, reducing, and units of measure.

#### Multiply unit fractions using an area model

#### Multiply fractions using an area model

#### Multiply fractions (reduce before multiplying)

#### Multiply fractions (reduce before or after multiplying)

#### Solve word problems by multiplying fractions using an area model

#### Rename a mixed number as a fraction to multiply by a fraction

#### Convert measurements by multiplying a whole number by a fraction

#### Convert measurements by multiplying fractions

#### Convert multiplication by a decimal to multiplication by a fraction and solve

#### Multiply by 0.1 using a place value chart

#### Multiply by 0.01 using a place value chart

#### Multiply by 0.1 or 0.01

#### Decompose a decimal number to multiply

#### Determine the placement of the decimal point in the product of two decimal numbers

#### Multiply two decimal numbers using column multiplication (Part 1)

#### Multiply two decimal numbers using column multiplication (Part 2)

#### Multiply two decimal numbers using column multiplication (Part 3)

### Topic F: Multiplication with Fractions and Decimals as Scaling and Word Problems

Students explore the effects of multiplying by less than, equal to, and greater than 1. They solve multiplication problems, provide missing factors, and complete inequalities to solidify their understanding.

#### Multiply a fraction by a fraction equal to 1

#### Identify a missing factor as a fraction equal to 1

#### Identify patterns in multiplying by fractions less than, equal to, and greater than 1

#### Use <, =, and > to compare the result of multiplying by fractions less than, equal to, and greater than 1

#### Identify a fraction that will complete a multiplication inequality

#### Convert a fraction less than one to decimal form (Level 1)

#### Convert a fraction greater than one to decimal form

#### Convert a fraction less than one to decimal form (Level 2)

#### Convert a fraction less than one to decimal form using division

#### Convert a fraction greater than one to decimal form using division

#### Determine the outcome of multiplication using <, =, or > (Part 1)

#### Complete inequalities involving multiplication by a decimal number

#### Determine the outcome of multiplication using <, =, or > (Part 2)

#### Determine the outcome of multiplication using <, =, or > (Part 3)

### Topic G: Division of Fractions and Decimal Fractions

Students rely upon their understanding of multiplying fractions to learn how to divide fractions. With the assistance of familiar models such as tape diagrams and area models, they learn different ways to approach dividing with fractions. At this stage, they use unit fractions with whole numbers and unit fractions with unit fractions.

#### Rewrite an equation with multiplication by a fraction as a division equation based on a model

#### Compose a division equation with a fraction based on a model

#### Divide a whole number by a unit fraction based on a tape diagram

#### Relate division by a unit fraction to multiplication by the denominator

#### Divide a whole number by a unit fraction by multiplying by the denominator

#### Divide a whole number by a unit fraction

#### Divide a unit fraction by a whole number based on a tape diagram (Part 1)

#### Divide a unit fraction by a whole number based on a tape diagram (Part 2)

#### Divide a unit fraction by a whole number

#### Multiply a unit fraction by a unit fraction with and without an area model (Part 1)

#### Multiply a unit fraction by a unit fraction with and without an area model (Part 2)

#### Relate division by a whole number to multiplication by a unit fraction

#### Solve division equations with unit fractions and whole numbers

#### Multiply a decimal number by a power of ten

#### Simplify a fraction using the basic property of fractions

#### Convert a fraction to a given denominator using the basic property of fractions

#### Convert between division expressions and fractions

#### Complete fraction expressions using the basic property of fractions

#### Convert fractions with decimal denominators to fractions with whole number denominators (Level 1)

#### Convert fractions with decimal denominators to fractions with whole number denominators (Level 2)

#### Convert fractions with decimal denominators to fractions with whole number denominators (Level 3)

#### Convert fractions with decimal denominators to fractions with whole number denominators (Level 4)

#### Convert division expressions with decimal divisors to expressions with whole number divisors (Level 1)

#### Convert division expressions with decimal divisors to expressions with whole number divisors (Level 2)

## MODULE 5: Algebraic Expressions

### Topic A: Solving Complex Missing Addend Equations

Topic Description: Students solve increasingly complex missing addend equations. In these pre-algebra equations, the challenge is not in the number values but in the logic of how numbers and equations can be combined to solve for numbers represented by symbols.