Decimals and place value
Students read and write numbers as small as thousandths and as large as the millions. They start comparing decimals and fractions on a number line and matching them to grids, rulers, and money.
This is the year math stretches past whole numbers into the smaller pieces between them. Students work with fractions and decimals as different ways to name the same amount, add and subtract them with unlike denominators, and divide larger numbers by one- and two-digit divisors. Students also find the volume of a box by counting unit cubes and start spotting patterns on a coordinate grid. By spring, students can add 2.45 and 1.7 on paper and explain what a decimal place is worth.
Students read and write numbers as small as thousandths and as large as the millions. They start comparing decimals and fractions on a number line and matching them to grids, rulers, and money.
Students divide larger numbers by one and two-digit numbers and decide what to do with the leftover. A remainder might stay a whole number, become a fraction, or turn into a decimal, depending on the problem.
Students add and subtract fractions and mixed numbers, even when the bottom numbers are different. They use the same skills with decimals to handle money, measurements, and recipes.
Students use rules and tables to spot patterns and predict what comes next. They plot points on a coordinate grid and work with expressions that use letters in place of numbers.
Students sort triangles and 3D shapes like cubes and pyramids, then figure out how many unit cubes fit inside a box. They measure length and angles and convert between inches, feet, centimeters, and meters.
Students collect data and find the mean, median, mode, and range. They build line graphs and double-bar graphs to compare groups and spot what stands out.
Students read, write, and compare fractions and decimals, including switching between the two forms. They use both in everyday situations, like splitting a bill or measuring ingredients.
Students use grids, base-ten blocks, and number lines to show how fractions like 1/10 and 1/100 connect to decimals and whole numbers.
Students read, write, and represent numbers with decimals out to the thousandths place (like 0.357) and whole numbers up to the millions. They connect what each digit means to its position in the number.
Students put decimals, fractions, and mixed numbers in order from smallest to largest and mark them on a number line. This includes fractions smaller than one, like 3/4, and numbers like 2 1/2.
Students convert fractions, mixed numbers, and decimals into different forms that mean the same amount, such as seeing that 1/2 and 0.5 are equal. They also show these equivalencies using pictures, number lines, or other models.
Students divide large numbers by hand and use that skill to solve everyday problems, like splitting a bill or figuring out how many boxes are needed to pack a set number of items.
Students check whether a division answer makes sense by estimating first. For example, before dividing 4,782 by 6, they round to get a ballpark answer, then use that estimate to catch mistakes in their final calculation.
Dividing a large number by a one- or two-digit number, like splitting 4,368 into equal groups of 12. Students use what they know about place value to work through the steps, which may include the standard long division method.
When dividing numbers with leftovers, students decide how to express the remainder: as a whole number, a fraction, or a decimal. The answer changes based on what makes sense in context, like splitting money versus splitting groups of people.
Students use drawings, equations, and physical objects to solve problems with large numbers, then check whether the answer actually makes sense given the situation.
Students practice adding and subtracting fractions, mixed numbers, and decimals, even when the fractions have different bottom numbers. They use these skills to solve problems they might actually encounter outside of school.
Before doing the full math, students check whether their answer is in the right ballpark by estimating fraction and decimal sums or differences. This helps them catch mistakes before they happen.
Students use diagrams, number lines, and fraction strips to show how to add and subtract fractions, mixed numbers, and decimals, including fractions with different denominators.
Students add and subtract fractions, mixed numbers, and decimals to solve problems involving money, measurement, and shapes. Denominators may be the same or different, and students choose the strategy that works best for each problem.
Students mentally count up or down by a tenth, hundredth, or thousandth without writing anything down. For example, they recognize that 0.01 less than 4.235 is 4.225 without doing the calculation on paper.
| Standard | Definition | Code |
|---|---|---|
| Read, write, represent | Students read, write, and compare fractions and decimals, including switching between the two forms. They use both in everyday situations, like splitting a bill or measuring ingredients. | 5.N.1 |
| Represent decimal fractions | Students use grids, base-ten blocks, and number lines to show how fractions like 1/10 and 1/100 connect to decimals and whole numbers. | 5.N.1.1 |
| Read, write, and represent decimals using place value to describe decimal… | Students read, write, and represent numbers with decimals out to the thousandths place (like 0.357) and whole numbers up to the millions. They connect what each digit means to its position in the number. | 5.N.1.2 |
| Compare and order decimals and fractions, including mixed numbers and fractions… | Students put decimals, fractions, and mixed numbers in order from smallest to largest and mark them on a number line. This includes fractions smaller than one, like 3/4, and numbers like 2 1/2. | 5.N.1.3 |
| Recognize and generate equivalent terminating decimals, fractions, mixed numbers | Students convert fractions, mixed numbers, and decimals into different forms that mean the same amount, such as seeing that 1/2 and 0.5 are equal. They also show these equivalencies using pictures, number lines, or other models. | 5.N.1.4 |
| Divide multi-digit numbers and solve real-world and mathematical problems using… | Students divide large numbers by hand and use that skill to solve everyday problems, like splitting a bill or figuring out how many boxes are needed to pack a set number of items. | 5.N.2 |
| Estimate solutions to division problems to assess the reasonableness of results | Students check whether a division answer makes sense by estimating first. For example, before dividing 4,782 by 6, they round to get a ballpark answer, then use that estimate to catch mistakes in their final calculation. | 5.N.2.1 |
| Divide multi-digit numbers, by one- and two-digit divisors, based on knowledge… | Dividing a large number by a one- or two-digit number, like splitting 4,368 into equal groups of 12. Students use what they know about place value to work through the steps, which may include the standard long division method. | 5.N.2.2 |
| Recognize that remainders can be represented in a variety of ways, including a… | When dividing numbers with leftovers, students decide how to express the remainder: as a whole number, a fraction, or a decimal. The answer changes based on what makes sense in context, like splitting money versus splitting groups of people. | 5.N.2.3 |
| Construct models to solve multi-digit whole number problems requiring addition… | Students use drawings, equations, and physical objects to solve problems with large numbers, then check whether the answer actually makes sense given the situation. | 5.N.2.4 |
| Add and subtract fractions with like and unlike denominators, mixed numbers | Students practice adding and subtracting fractions, mixed numbers, and decimals, even when the fractions have different bottom numbers. They use these skills to solve problems they might actually encounter outside of school. | 5.N.3 |
| Estimate sums and differences of fractions with like and unlike denominators… | Before doing the full math, students check whether their answer is in the right ballpark by estimating fraction and decimal sums or differences. This helps them catch mistakes before they happen. | 5.N.3.1 |
| Illustrate addition and subtraction of fractions with like and unlike… | Students use diagrams, number lines, and fraction strips to show how to add and subtract fractions, mixed numbers, and decimals, including fractions with different denominators. | 5.N.3.2 |
| Add and subtract fractions with like and unlike denominators, mixed numbers | Students add and subtract fractions, mixed numbers, and decimals to solve problems involving money, measurement, and shapes. Denominators may be the same or different, and students choose the strategy that works best for each problem. | 5.N.3.3 |
| Apply mental math and knowledge of place value | Students mentally count up or down by a tenth, hundredth, or thousandth without writing anything down. For example, they recognize that 0.01 less than 4.235 is 4.225 without doing the calculation on paper. | 5.N.3.4 |
Students look at a number pattern, describe how the values grow or shrink, and plot those changes on a graph to see the shape of the pattern.
Students follow a rule (like "multiply by 2, then add 1") to complete a table of numbers, spot how the pattern grows, and predict what comes next.
Students use a rule or table to create pairs of numbers, then plot those pairs as points on a grid. They learn how the grid's center point and labeled lines connect to the numbers they plotted.
Students read and write math sentences that use letters as stand-ins for unknown numbers, then use those sentences to solve real problems. They also figure out whether two amounts are equal or which one is larger.
Students learn that the order and grouping of numbers in a problem can change without changing the answer. They practice rewriting and solving whole-number expressions using rules like multiplying before adding.
Students plug a number into an equation or inequality to check whether it makes the statement true or false. For example, they test whether a value like 4 actually works in something like x + 3 = 7.
Given a variable like x or n, students substitute the assigned number, calculate the result, and find the value of the expression. Think of it as filling in a blank and then doing the math.
| Standard | Definition | Code |
|---|---|---|
| Describe and graph patterns of change created through numerical patterns | Students look at a number pattern, describe how the values grow or shrink, and plot those changes on a graph to see the shape of the pattern. | 5.A.1 |
| Use tables and rules with up to two operations to describe patterns of change… | Students follow a rule (like "multiply by 2, then add 1") to complete a table of numbers, spot how the pattern grows, and predict what comes next. | 5.A.1.1 |
| Use a rule or table to represent ordered pairs of whole numbers and graph these… | Students use a rule or table to create pairs of numbers, then plot those pairs as points on a grid. They learn how the grid's center point and labeled lines connect to the numbers they plotted. | 5.A.1.2 |
| Understand and interpret expressions, equations | Students read and write math sentences that use letters as stand-ins for unknown numbers, then use those sentences to solve real problems. They also figure out whether two amounts are equal or which one is larger. | 5.A.2 |
| Generate equivalent numerical expressions and solve problems using number sense… | Students learn that the order and grouping of numbers in a problem can change without changing the answer. They practice rewriting and solving whole-number expressions using rules like multiplying before adding. | 5.A.2.1 |
| Determine whether an equation or inequality involving a variable is true or… | Students plug a number into an equation or inequality to check whether it makes the statement true or false. For example, they test whether a value like 4 actually works in something like x + 3 = 7. | 5.A.2.2 |
| Evaluate expressions involving variables when values for the variables are… | Given a variable like x or n, students substitute the assigned number, calculate the result, and find the value of the expression. Think of it as filling in a blank and then doing the math. | 5.A.2.3 |
Students sort and build flat and solid shapes by describing what makes each one distinct: how many sides, angles, faces, or edges it has.
Triangles get sorted by their sides and angles. Students learn the difference between a triangle where all sides match, one with a right-angle corner, and ones where some or no sides are equal.
Students sort and describe 3D shapes like cubes, prisms, and pyramids by counting their edges, corners, and flat faces and naming the shape of each face.
Students unfold a 3-D shape in their mind, then draw the flat pattern of faces that would fold back up into that shape. They practice with cubes, boxes, and pyramids.
Students find the volume of a box by multiplying its length, width, and height. Then they compare different boxes that hold the same amount of space but have different shapes.
Students figure out the volume of a box by counting the unit cubes that fill it, then by multiplying its length, width, and height. Two boxes with different dimensions can hold the exact same volume.
Students estimate the distance around polygons and explain why their answer is a reasonable guess. For shapes with curved edges, they build an argument for what the perimeter could be.
Students measure things like angles, distances, and weights using the right tool for the job: a protractor for angles, a ruler for length, a scale for weight. Then they use those measurements to solve everyday problems.
Students measure angles with a protractor and compare which ones are wider or narrower. This builds the foundation for geometry work in middle school.
Students measure real objects with a ruler, reading the result to the nearest centimeter or to the nearest sixteenth of an inch. The focus is on choosing the right tool and reading the measurement accurately.
Students measure real objects and convert between inches, feet, and yards to solve problems, like figuring out how many feet are in a given number of yards or comparing two lengths measured in different units.
Students measure real objects and convert between millimeters, centimeters, and meters to solve problems. Think of it as knowing when to use a ruler marked in centimeters versus one marked in smaller millimeter lines.
Students practice guessing measurements before grabbing a ruler or tape measure, using familiar reference points like the width of a thumb or the length of a shoe to get close to the right answer in both inches and centimeters.
| Standard | Definition | Code |
|---|---|---|
| Describe, identify, classify | Students sort and build flat and solid shapes by describing what makes each one distinct: how many sides, angles, faces, or edges it has. | 5.GM.1 |
| Describe, identify, classify | Triangles get sorted by their sides and angles. Students learn the difference between a triangle where all sides match, one with a right-angle corner, and ones where some or no sides are equal. | 5.GM.1.1 |
| Describe, identify, and classify three-dimensional figures | Students sort and describe 3D shapes like cubes, prisms, and pyramids by counting their edges, corners, and flat faces and naming the shape of each face. | 5.GM.1.2 |
| Recognize and draw a net for a three-dimensional figure | Students unfold a 3-D shape in their mind, then draw the flat pattern of faces that would fold back up into that shape. They practice with cubes, boxes, and pyramids. | 5.GM.1.3 |
| Determine volume using the object's dimensions | Students find the volume of a box by multiplying its length, width, and height. Then they compare different boxes that hold the same amount of space but have different shapes. | 5.GM.2 |
| Determine the volume of rectangular prisms by the number of unit cubes | Students figure out the volume of a box by counting the unit cubes that fill it, then by multiplying its length, width, and height. Two boxes with different dimensions can hold the exact same volume. | 5.GM.2.1 |
| Estimate the perimeter of polygons and create arguments for reasonable… | Students estimate the distance around polygons and explain why their answer is a reasonable guess. For shapes with curved edges, they build an argument for what the perimeter could be. | 5.GM.2.2 |
| Understand angle, length, weight | Students measure things like angles, distances, and weights using the right tool for the job: a protractor for angles, a ruler for length, a scale for weight. Then they use those measurements to solve everyday problems. | 5.GM.3 |
| Measure and compare angles according to size using various tools | Students measure angles with a protractor and compare which ones are wider or narrower. This builds the foundation for geometry work in middle school. | 5.GM.3.1 |
| Measure the length of an object to the nearest whole centimeter or up to 1/16… | Students measure real objects with a ruler, reading the result to the nearest centimeter or to the nearest sixteenth of an inch. The focus is on choosing the right tool and reading the measurement accurately. | 5.GM.3.2 |
| Apply the relationship between inches, feet | Students measure real objects and convert between inches, feet, and yards to solve problems, like figuring out how many feet are in a given number of yards or comparing two lengths measured in different units. | 5.GM.3.3 |
| Apply the relationship between millimeters, centimeters | Students measure real objects and convert between millimeters, centimeters, and meters to solve problems. Think of it as knowing when to use a ruler marked in centimeters versus one marked in smaller millimeter lines. | 5.GM.3.4 |
| Estimate lengths and geometric measurements to the nearest whole unit, using… | Students practice guessing measurements before grabbing a ruler or tape measure, using familiar reference points like the width of a thumb or the length of a shoe to get close to the right answer in both inches and centimeters. | 5.GM.3.5 |
Students collect data, then find the middle value, the most common value, and the average. They also calculate the spread from lowest to highest. These are the tools used to summarize and compare sets of numbers.
Students find the mean, median, and mode of a data set, plus the range from lowest to highest value. Mean gets special attention: it's the number you'd land on if you spread all the values out evenly.
Reading and building graphs where the scale counts by fractions or decimals, not just whole numbers. Students make these graphs from data and answer questions about what the graph shows.
| Standard | Definition | Code |
|---|---|---|
| Create and analyze data to find the range and measures of central tendency | Students collect data, then find the middle value, the most common value, and the average. They also calculate the spread from lowest to highest. These are the tools used to summarize and compare sets of numbers. | 5.D.1 |
| Find the measures of central tendency | Students find the mean, median, and mode of a data set, plus the range from lowest to highest value. Mean gets special attention: it's the number you'd land on if you spread all the values out evenly. | 5.D.1.1 |
| Create and analyze line and double-bar graphs with increments of whole numbers… | Reading and building graphs where the scale counts by fractions or decimals, not just whole numbers. Students make these graphs from data and answer questions about what the graph shows. | 5.D.1.2 |
Students should divide larger numbers, add and subtract fractions with different bottom numbers, and work with decimals down to the thousandths place. They should also find the volume of a box, measure angles, and read line and bar graphs.
Cooking is the easiest way in. Halve a recipe, double it, or ask how much is left when two thirds of a cup is poured out. Folding paper into equal parts also helps students see why one half and two fourths are the same amount.
Money and measuring are the two best tools. Read prices out loud, compare which is cheaper per item, and let students measure things with a ruler that shows tenths of a centimeter. Ask which decimal is bigger and why.
Start with an estimate before dividing. Ask about how many times the smaller number fits into the bigger one, then check after. Talking about the leftover amount as a real thing, like leftover slices or dollars, makes remainders make sense.
Build estimation and place value sense first, then move to one-digit divisors before two-digit divisors. Spend real time on what the remainder means in context, since students often compute correctly but answer the wrong question.
Adding and subtracting fractions with unlike bottom numbers tends to need the most time, along with comparing decimals past the tenths place. Volume often needs a second pass once students confuse it with area.
Plan units that move between the two forms in the same lesson. A ten by ten grid, a meter stick, and a number line all show that 0.5 and one half live in the same spot. Students who see the connection early stop treating decimals as a new topic.
Look for students who can divide fluently, add and subtract fractions with unlike bottom numbers, and explain a rule from a table of numbers. They should also plot points on a coordinate grid and find the mean of a small data set without prompting.