# Curriculum for Grade 4

## Driven by Pedagogy, Supported by Technology (and not vice versa),

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### MODULE 1. Place Value, Rounding, and Algorithms for Addition and Subtraction

Students work with whole numbers to 1,000,000 using number disks and the place value chart. They develop understanding of magnitudes of 10, as well as how multiplication and division relate to place value. Students also master the skill of recording large numbers, both in standard form with the appropriate commas and in expanded form.

A. Model multiplication by 10 using a disk model (Level 1)B. Model multiplication by 10 using a disk model (Level 2)C. Solve multiplication by 10 in unit form and number formD. Solve multiplication by 10 in unit form and number form and model on a place value chartE. Model division by 10 using a disk model (Level 1)F. Model division by 10 using a disk model (Level 2)G. Solve division by 10 in unit form and number formH. Solve division by 10 in unit form and number form and model on a place value chart

Students work with numbers in the thousands, ten thousands, and hundred thousands. Building upon previous knowledge, such as use of inequality symbols, they deepen their understanding of how to compare large numbers to several place values or numbers of different place value. As part of this understanding, they build their fluency in the base-10 system by finding 1,000, 10,000 and 100,000 more or less than a number.

A. Compare numbers in a place value chart (at their highest place value)B. Compare numbers in a place value chart (at a place value other than their highest)C. Compare numbers in a place value chart (at different place values)D. Complete inequalities comparing multi-digit numbersE. Order numbers in ascending and descending order using < and > (Part 1)F. Order numbers in ascending and descending order using < and > (Part 2)G. Order numbers in ascending and descending order using < and > (Part 3)H. Determine 1,000 more or less than a number using a place value chartI. Determine 10,000 more or less than a number using a place value chartJ. Determine 100,000 more or less than a number using a place value chartK. Complete a statement comparing numbers with and without a place value chart (1,000 or 10,000 more or less)L. Determine 1,000/10,000/100,000 more or less than a given numberM. Order numbers in a pattern increasing or decreasing by 1,000 or 10,000 (Level 1)N. Order numbers in a pattern increasing or decreasing by 1,000 or 10,000 (Level 2)

Using a number line to provide context, students first determine the midway point between two multi-digit numbers. They then progress to rounding using the number line and the midway point. Finally, students round 4-, 5-, and 6-digit numbers to any given place value without use of the number line.

A. Determine the midway point between two multi-digit numbers using a number lineB. Determine whether a number is greater or less than the midway point between two multi-digit numbers using a number lineC. Determine whether a multi-digit number rounds up or down to a given place value using a number line (Part 1)D. Determine the round numbers nearest a given multi-digit number and whether that number rounds up or down using a number line (Part 1)E. Determine whether a multi-digit number rounds up or down to a given place value using a number line (Part 2)F. Round a multi-digit number to the nearest thousandG. Determine the round numbers nearest a given multi-digit number and whether that number rounds up or down using a number line (Part 2)H. Determine whether a multi-digit number rounds up or down to a given place value (Part 1)I. Determine whether a multi-digit number rounds up or down to a given place value (Part 2)J. Round a multi-digit number to a given place value (Level 1)K. Round a multi-digit number to a given place value (Level 2)L. Round a multi-digit number to a given place value (Level 3)M. Round a multi-digit number to various place values

Students use column addition to add multi-digit numbers. They begin with the support of a disk model to illustrate the underlying concepts. They learn to record addition problems as column addition, how to regroup and record this action, how to line up numbers of varying length, and how to regroup to a place value higher than the original numbers.

A. Add two 4-digit numbers with and without regrouping in one place using a disk modelB. Add two 4-digit numbers with and without regrouping in two places using column addition and a disk modelC. Add multi-digit numbers with regrouping in multiple places using column additionD. Add multi-digit numbers using the standard algorithm

Students use column subtraction to subtract multi-digit numbers. They begin with the support of a disk model to illustrate the underlying concepts. They learn to record subtraction problems as column subtraction, how to regroup and record this action, how to line up numbers of varying length, and how to regroup across zeros.

A. Subtract two 4-digit numbers with and without regrouping in two places using column subtraction and a disk modelB. Subtract multi-digit numbers with regrouping in multiple places using column subtractionC. Subtract multi-digit numbers with regrouping across zeros using column subtractionD. Subtract multi-digit numbers using the standard algorithm

### MODULE 2. Multi-Digit Multiplication and Division

Students apply their understanding of measurement and area models to use the formulas for area and perimeter of a rectangle. They use the area model to find both the area and a missing side length. Students identify different ways of writing the same formula for both area and perimeter.

A. Determine the area of a rectangle by multiplying the lengths of its sidesB. Identify the formula for area of a rectangle and use it to solve a problemC. Determine the area of a rectangle using the formula A = l x wD. Determine the side length of a rectangle based on its area and width using the formulaE. Identify the formula for perimeter of a rectangle and use it to solve a problemF. Identify the formula for perimeter of a rectangleG. Determine the perimeter of a rectangle using the formula P = 2 x (l + w)H. Determine the area and perimeter of the same rectangle

Students extend their understanding of multiplication to include powers of ten. They begin each exercise with the assistance of a place value chart, which illustrates the meaning behind the operation. Then, they build their fact fluency by solving equations mentally.

A. Multiply by 10 with and without a disk modelB. Multiply by 100 with and without a disk modelC. Multiply by 1,000 with and without a disk modelD. Divide by 10 with and without a disk modelE. Divide by 100 with and without a disk modelF. Divide by 1,000 with and without a disk modelG. Multiply a 1-, 2-, or 3-digit number by 10, 100, or 1,000H. Multiply a 1-digit number by a round 2-digit number (Level 1)I. Multiply a 1-digit number by a round 3-digit number (Level 1)J. Multiply a 1-digit number by a round 2-digit number (Level 2)K. Multiply a 1-digit number by a round 3-digit number (Level 2)L. Multiply a 1-digit number by a round 2-digit number (Level 3)M. Multiply a 1-digit number by a round 3-digit number (Level 3)N. Solve multiplication equations with a 1-digit factor and a round 2- or 3-digit factorO. Multiply a 2-digit round number by a 2-digit round number

Students are gradually scaffolded through the steps of multiplying a multi-digit number by a single-digit number using the standard algorithm. They begin using a concrete disk model and a partial products method of recording. As they begin to work with the standard algorithm, they become more skilled and independent by regrouping, using zeros, using larger numbers, and receiving fewer prompts.

A. Solve a multiplication equation using partial products with and without and a disk modelB. Solve a multiplication equation with regrouping using partial products with and without and a disk modelC. Solve a multiplication equation including a zero using partial products with and without and a disk modelD. Solve a multiplication equation using the standard algorithm with and without and a disk modelE. Solve a multiplication equation with regrouping using the standard algorithm using a disk modelF. Solve a multiplication equation with regrouping using the standard algorithm (Level 1)G. Solve a multiplication equation with regrouping using the standard algorithm (Level 2)H. Solve a multiplication equation with regrouping using the standard algorithm (Level 3)I. Solve a multiplication equation with regrouping using the standard algorithm (Level 4)J. Solve a multiplication equation with regrouping while identifying steps of the standard algorithmK. Solve a multiplication equation with regrouping using the standard algorithm (Level 5)L. Solve a single multiplication equation using both partial products and the standard algorithmM. Multiply to find the area of a rectangleN. Multiply to find the area of a rectangle using partial products (Level 1)O. Multiply to find the area of a rectangle using partial products (Level 2)

Students divide 1- and 2-digit numbers to determine the number in each group or the number of groups. They use models to illustrate a word problem and equations to record their work. Students are introduced to the term "quotient" and a method for checking their answer. Students move from simple division to division with a remainder in this topic, and they learn to use long division notation.

A. Solve a division problem (number in each group) with a remainder based on a modelB. Solve a division problem (number of groups) with a remainder based on a modelC. Solve a division problem (number in each group) with a remainder using an array modelD. Solve a division problem (number of groups) with a remainder using an array modelE. Multiply to find multiples of a given numberF. List and identify multiples of a given numberG. Use multiples to find the quotient and remainder of a division problemH. Model division (number in each group) with a remainder using a tape diagramI. Model division (number of groups) with a remainder using a tape diagramJ. Solve a division problem with a remainder using the closest multiplication factK. Solve a division problem with a remainderL. Solve a division word problem (number of groups) with a remainder using a tape diagram and an equationM. Solve a division word problem (number in each group) with a remainder using a tape diagram and an equationN. Model and solve a division problem, and identify the divisorO. Model and solve a division problem using long division (single-digit quotient)P. Solve a division problem (number in each group) with a remainder using a disk modelQ. Model and solve a division problem using long division (two-digit quotient)R. Model and solve a division problem that involves regroupingS. Model and solve a division problem that involves regrouping using long division (two-digit quotient)T. Model and solve a division problem that involves regrouping using long division (two-digit quotient) (Level 2)U. Model and solve a division problem using long division by recording partial quotientsV. Use long division to solve problems with a 2-digit quotient

Students build a firm understanding of the concepts of factor, multiple, and divisible by, as well as the relationship among those concepts. They explore divisibility patterns/rules for 2, 3, 5, 6, 9, and 10. To build this understanding, students use manipulatives, arrays, long division, and the hundred chart.

A. Solve single-digit multiplication problemsB. Find factors of a given number by labeling arrays (Level 1)C. Find factors of a given number by labeling arrays (Level 2)D. Find factors of a given number by building and labeling arraysE. List factor pairs for a given number and identify the number as prime or compositeF. Determine whether a given number is a factor of another given numberG. Use long division to determine whether a given number is a factor of another given numberH. Use long division to show that if a number is a factor of another number, its factors are also factors of that numberI. Use properties of multiplication to show that if a number is a factor of another number, its factors are also factors of that numberJ. Determine multiples of a given factorK. Identify the relationship between factors, multiples, and divisible byL. Identify factors or multiples of a list of given numbersM. Use long division to determine whether a given number is a multiple of another given numberN. Use long division to show that if a number is a multiple of another number, it is also a multiple of that number's factorsO. Use properties of multiplication to show that if a number is a multiple of another number, it is also a multiple of that number's factorsP. Identify multiples of 2, 5, and 10 on a hundred chart and identify patterns in the ones place of the multiplesQ. Determine whether a given number is a multiple of 2, 5, or 10R. Determine whether a given number is even or oddS. Identify multiples of 3 and 9 on a hundred chart and identify patterns in the multiplesT. Determine whether a given number is a multiple of 3 or 9U. Identify factors (2, 3, 5, 10) of a given multipleV. Use a hundred chart to show that multiples of 2 and 3 are multiples of 6, and that multiples of 2 and 5 are multiples of 10

Students divide numbers in the hundreds and thousands by single-digit numbers. They move from unit notation to standard notation to help facilitate mental math with large numbers. Understanding of the standard algorithm is supported by familiar models - a disk model and tape diagrams. Students are supported in dealing with various division challenges from regrouping to remainders to working with zero.

A. Multiply using unit notation and standard notationB. Divide using a disk modelC. Divide using a disk model (with regrouping)D. Divide using unit notation and standard notation (Level 1)E. Divide using unit notation and standard notation (Level 2)F. Divide using standard notationG. Divide using a disk model (with regrouping) (Part 1)H. Divide using long division and a disk model (with regrouping and a remainder) (Part 1)I. Divide using a disk model (with regrouping) (Part 3)J. Divide using long division and a disk model (with regrouping and a remainder) (Part 2)K. Divide using long division and a disk model (with regrouping and a remainder) (Part 3)L. Divide using long division and a disk model (with regrouping and a remainder) (Part 4)M. Divide using long division with partial quotients and a disk modelN. Divide using long division with partial quotients (Level 1)O. Divide using long division with partial quotients (Level 2)P. Divide across a zero using a disk model (with regrouping and a remainder)Q. Divide across a zero using long division and a disk model (with regrouping and a remainder) (Part 1)R. Divide using a disk model with zero in the quotient (with regrouping)S. Divide using long division and a disk model with zero in the quotient (with regrouping)T. Divide across a zero using long division and a disk model (with regrouping and a remainder) (Part 2)U. Solve division problems with a quotient of zero (with a remainder) (Level 1)V. Solve division problems with a quotient of zero (with a remainder) (Level 2)W. Divide across a zero using long division with partial quotients and a disk model (with regrouping and a remainder) Part 1X. Divide across a zero using long division with partial quotients (with regrouping and a remainder) Part 1Y. Solve division word problems using long division and a tape diagram (with regrouping and a remainder)Z. Solve division word problems using long division and a tape diagram (with regrouping)AA. Divide using long division with partial quotients (Level 3)AB. Solve division word problems across zero using long division and a tape diagram (with regrouping)AC. Solve division word problems using long division and a tape diagram (with a remainder)

Students apply their prior knowledge of multiplication to multiply a 2-digit number by a 2-digit number. They use familiar tools and strategies, including a disk model, an area model, partial products, the distributive property, and the standard algorithm. To support their learning, students work extensively with multiples of 10.

A. Multiply a 2-digit number by 10B. Identify a round number as a multiple of 10C. Solve a word problem two different ways by regrouping factorsD. Multiply by splitting a round number into a multiple of 10 and regrouping factors based on a disk model (multiply by 10 last)E. Multiply by splitting a round number into a multiple of 10 and regrouping factors based on a disk model (multiply by 10 first)F. Multiply by splitting a round number into a multiple of 10 and regrouping factors (Level 1)G. Multiply by splitting a round number into a multiple of 10 and regrouping factors (Level 2)H. Rewrite an area model multiplication equation using the distributive propertyI. Multiply to find the area of a rectangle using the distributive propertyJ. Multiply using partial products and the standard algorithm (one round number)K. Multiply to find the area of a rectangle using the distributive property and the standard algorithmL. Multiply using partial products and the standard algorithmM. Multiply using the standard algorithm (one round number)N. Multiply using the standard algorithm with regrouping (one round number)O. Multiply using partial products and the standard algorithm with regrouping (Part 1)P. Multiply using partial products and the standard algorithm with regrouping (Part 2)Q. Multiply using the standard algorithm (one round number)

### MODULE 3. Fraction Equivalence, Ordering, and Operations

Students learn everything about what a fraction is, how it is written, what it represents, and what its parts are called. They work with fractions both less than and greater than 1 as they model, record, and rename fractions. They explore fractions as part of a whole and also as points on a number line.

A. Label a shaded figure using fraction notation and shade a given fraction of a figureB. Label a shaded figure using fraction notation and shade a given fraction of a figure (fractions greater than 1)C. Label a missing fraction on a labeled number lineD. Place a fraction on a number lineE. Identify numerator and denominator in a fractionF. Identify fractions with a given numerator or denominatorG. Model a fraction as the sum of its parts and record this as an equationH. Model a fraction as the sum of its parts and record this as an equation (fractions greater than one)I. Record a fraction as the sum of its partsJ. Record repeated addition of whole numbers as multiplicationK. Record repeated addition of fractions as multiplicationL. Record repeated addition of fractions as multiplication (fractions greater than 1)M. Identify the multiplication expression that matches a given fraction

Students compose equivalent fractions based on a model. They then label the fractions and identify the factor or divisor that relates one to the other.

A. Identify, label, and compare equivalent fractionsB. Divide a model in two different ways to show and label equivalent fractionsC. Multiply to find equivalent fractions based on a modelD. Multiply to find equivalent fractions with and without a modelE. Complete the numerator or denominator in a larger equivalent fractionF. Divide to find equivalent fractions based on a modelG. Divide to find equivalent fractions with and without a modelH. Complete the numerator or denominator in a smaller equivalent fractionI. Solve problems related to equivalent fractions and multiplier

Using familiar models and the number line, along with their ability to find equivalent fractions, students compare fractions. They explore strategies to find a common numerator or denominator, or to compare to a benchmark. Students work with fractions greater than and less than one.

A. Label and compare fractions with like denominators or like numerators based on a modelB. Place fractions with like denominators or like numerators on a number line and compareC. Order four fractions with like denominators or like numerators based on a modelD. Compare fractions with like denominators or like numeratorsE. Compare fractions to ½ with and without a number lineF. Solve word problems comparing a fraction to ½ using an equivalent fractionG. Compare fractions by comparing each one to ½H. Compare fractions by comparing each one to 1I. Choose a strategy and use it to compare fractionsJ. Compare fractions by comparing the remaining unit fraction using a number line and modelK. Compare fractions by finding a common numerator (when one numerator is a multiple of the other)L. Compare fractions by finding a common denominator (when one denominator is a multiple of the other)M. Compare fractions by finding a common denominator (when one denominator is not a multiple of the other)N. Compare fractions by finding a common numerator or denominator (when one is a multiple of the other)

Students apply their understanding of fraction basics to add and subtract fractions. They work with familiar models and the number line to build understanding of the concepts behind the operations. In solving addition and subtraction problems, students convert among equivalent fractions, mixed numbers, and improper fractions.

A. Identify and add fractions with a common denominator based on a modelB. Add fractions with a common denominator with and without a number lineC. Add fractions with a common denominatorD. Add fractions with a common denominator and convert the sum to a mixed number (Level 1)E. Subtract fractions with a common denominator with and without a number lineF. Subtract fractions with a common denominator with and without a number lineG. Subtract a fraction from 1H. Rename a mixed number as a fraction to subtract a fractionI. Subtract a fraction from a mixed number with and without renaming the mixed number as a fractionJ. Rename a fraction as an equivalent fractionK. Add fractions with different denominators by finding a common denominatorL. Add fractions with different denominators and rename the sum as a mixed number

To prepare for more complex work with addition and subtraction, students establish a firm understanding of mixed numbers and fractions greater than 1. They convert fractions to mixed numbers and vice versa. They find common denominators when one denominator is a factor of the other and when it is not.

A. Identify fractions as greater than, less than, or equal to 1B. Label a model with a mixed number and identify its written formC. Identify a mixed number on a number lineD. Label a model with a mixed number and a fraction greater than 1E. Rename a mixed number as a fraction greater than 1 based on a modelF. Rename a mixed number as a fraction greater than 1G. Rename a fraction greater than 1 as a mixed number based on a modelH. Rename a fraction greater than 1 as a mixed numberI. Match fraction addition to a mixed number and a fraction greater than 1J. Identify a fraction greater than 1 and rename it as a mixed numberK. Label models with mixed numbers and compare using <, =, or >L. Compare mixed numbers with different denominators (Part 1)M. Compare fractions greater than 1 by renaming them as mixed numbersN. Compare a fraction greater than 1 with a mixed number or a fraction greater than 1O. Compare mixed numbers with different denominators (Part 2)P. Compare mixed numbers with different denominators (Part 3)Q. Compare fractions greater than 1

Students rely heavily on their understanding of fractions and mixed numbers to complete operations. To add and subtract, they break apart, regroup, and rename numbers. They gradually move from a step-by-step guided strategy to solving problems mentally.

A. Round a mixed number to the nearest whole with and without a number lineB. Estimate the sum or difference of two mixed numbers by rounding them to the nearest wholeC. Round an improper fraction to the nearest whole by converting it to a mixed numberD. Round mixed numbers and improper fractions to the nearest wholeE. Estimate the sum or difference of a mixed number and an improper fraction by converting the fraction to a mixed numberF. Add a whole number to a mixed numberG. Add a mixed number to a fraction with the same denominatorH. Add a mixed number to a fraction with the same denominator and write the sum without an improper fractionI. Add a mixed number to a fraction with the same denominator by completing the whole (Level 1)J. Add a mixed number to a fraction with the same denominator by completing the whole (Level 2)K. Add two mixed numbers with the same denominator and write the sum without an improper fraction (Level 1)L. Add two mixed numbers with the same denominator and write the sum without an improper fraction (Level 2)M. Subtract a fraction from a mixed number with the same denominator (without converting to an improper fraction)N. Subtract a fraction from a mixed number with the same denominator by converting to an improper fraction (Level 1)O. Subtract a fraction from a mixed number with the same denominator by converting to an improper fraction (Level 2)P. Subtract mixed numbers with the same denominatorQ. Add and subtract mixed numbers with the same denominator

Students translate their understanding of addition of fractions to multiplication. They transition from work with unit fractions to other fractions and mixed numbers. In solving equations and word problems, students rename solutions as mixed numbers without a fraction greater than 1.

A. Write a fraction as a sum of unit fractionsB. Identify multiplication of a unit fraction by a whole number that matches a given fractionC. Multiply a whole number by a fraction by splitting it into a multiple of a unit fractionD. Multiply a whole number by a fractionE. Write repeated addition of fractions as a multiplication statement and rename the product as a mixed numberF. Multiply a whole number by a mixed numberG. Multiply a whole number by a mixed number and rename without a fraction greater than 1H. Solve a word problem by multiplying a whole number by a mixed number

### MODULE 4. Decimal Fractions

Students develop their understanding of the decimal form of fractions in the tenths. The rely on familiar representations such as the number line, place value chart, and fraction models to support understanding. Students convert fractions less than and greater than one from fraction form to decimal form and word form.

A. Relate fractions in tenths to decimals with and without a number lineB. Rewrite a fraction in tenths as a decimal, and vice versaC. Match fractions in tenths to their decimal form and word formD. Convert between fraction form, decimal form, and word form in tenthsE. Determine how many more tenths to make a whole using a number lineF. Represent mixed numbers in decimal form using a number lineG. Rewrite a mixed number with tenths as a decimal, and vice versaH. Match mixed numbers with tenths to their decimal form and word formI. Convert between fraction form, decimal form, and word form with mixed numbers with tenthsJ. Determine how many more tenths to make the next whole using a number lineK. Record a fraction model as a mixed number or a decimal with tenthsL. Relate tenths to one whole using a place value chartM. Record or represent a mixed number in decimal form using a place value chartN. Represent numbers greater than 10 tenths in decimal form

Students hone their understanding of tenths, hundredths, and the relationship between the two. Beginning with fractional parts only, students convert between fraction, decimal, and unit form. After mastering this, they work with mixed numbers. Along the way, their understanding is supported by area models, disk models, and number lines.

A. Identify hundredths using fraction and decimal notationB. Match hundredths fractions to decimal form and rewrite a hundredths fraction in decimal formC. Match hundredths decimals to unit form and rewrite a hundredths decimal in unit formD. Rewrite a hundredths fraction in decimal and unit formE. Match hundredths in fraction, decimal, and unit formF. Label tenths and hundredths on an area model using fraction and decimal formG. Label tenths and hundredths on an area model using decimal formH. Compare equivalent tenths and hundredths in decimal formI. Match equivalent tenths and hundredths in decimal and unit formJ. Rewrite tenths and hundredths in fraction, decimal, and unit formK. Show tenths and hundredths equivalencies using a disk modelL. Match and rewrite hundredths in fraction, decimal, and unit formM. Label a hundredths number based on a disk modelN. Represent a hundredths number using a disk modelO. Identify points in the hundredths on a number lineP. Label a mixed number on an area model using fraction and decimal formQ. Match mixed numbers in fraction and decimal form, and rewrite a mixed number fraction in decimal formR. Match mixed numbers in decimal form to unit form and rewrite a decimal in unit formS. Rewrite a mixed number fraction in decimal formT. Match mixed numbers in fraction, decimal, and unit formU. Label a mixed number based on a disk modelV. Represent a mixed number using a disk model

Students compare decimals in the hundredths using inequality signs and ordering. They rely on familiar representations of the area model, number line, and place value chart to build understanding. They also compare decimals to fractional numbers in unit form and fraction form.

A. Use <, =, and > to compare decimal measurements of length, mass, and volumeB. Use <, =, and > to compare decimal numbers based on an area modelC. Use <, =, and > to compare decimal numbers based on a number lineD. Use <, =, and > to compare decimal numbers based on a place value chartE. Use <, =, and > to compare decimal numbersF. Use <, =, and > to compare a decimal number to a mixed number or decimal in unit formG. Complete an inequality by writing a decimal numberH. Complete an inequality by choosing a mixed number, decimal, or decimal in unit formI. Complete a double inequality statement based on measurements of length, mass, and volumeJ. Order decimal numbers in a double inequality statement based on a number lineK. Order decimal numbers in a double inequality statementL. Order four decimal numbers in ascending orderM. Order decimals, fractions, and decimal numbers in unit form in a double inequality statement based on a number lineN. Order decimals, mixed numbers, and decimal numbers in unit form in a double inequality statementO. Order four decimals, mixed numbers, and decimal numbers in unit form in ascending orderP. Order five decimals, mixed numbers, and decimal numbers in unit form in ascending order

Students rely on their mastery of converting between decimals, fractions, and mixed numbers in order to add tenths and hundredths. They learn how to add different units (tenths to hundredths) in both fraction and decimal form. They also learn to record their sum without improper fractions.

A. Regroup to add tenths to hundredths with and without a disk modelB. Rewrite decimals in fraction form to addC. Rewrite mixed number decimals in fraction form to addD. Add tenths and hundredths fractions by making an equivalent fractionE. Rewrite decimals in fraction form and find an equivalent fraction to add