5.NBT.A | Understand the place value system.

5.NBT.B | Perform operations with multi-digit whole numbers and with decimals to hundredths.

5.MD.A | Convert like measurement units within a given measurement system.

Welcome to the UnboundEd Mathematics Guide series! These guides are designed to explain what new, high standards for mathematics say about what students should learn in each grade, and what they mean for curriculum and instruction. This guide, the first for Grade 5, includes three parts. The first part gives a “tour” of the standards in the Number & Operations in Base Ten domain using freely available online resources that you can use or adapt for your class. The second part shows how Number & Operations in Base Ten relates to other concepts in Grade 5. And the third part explains some important progressions of learning that lead to (and extend from) these standards in Grade 5. Throughout all of our guides, we include a large number of sample math problems. We strongly suggest tackling these problems yourself to help best understand the methods and strategies we’re covering, and the potential challenges your students might face.

Part 1: What do the standards say?

In Grade 5, the Number & Operations in Base Ten (NBT) domain is composed of two clusters; each cluster has associated standards and is part of the major work of Grade 5, as indicated by the green square.¹ The Common Core State Standards for Mathematics (CCSSM) are organized into major, additional, and supporting clusters in the Focus by Grade Level documents from Student Achievement Partners. The Common Core State Standards for Mathematics (CCSSM) are organized into major, additional, and supporting clusters in the Focus by Grade Level documents from Student Achievement Partners. The Common Core State Standards for Mathematics (CCSSM) are organized into major, additional, and supporting clusters in the Focus by Grade Level documents from Student Achievement Partners. A large majority of time should be spent teaching the major work of the grade. The other cluster discussed in this guide comes from the Measurement & Data (MD) domain. This is a supporting cluster (blue square). Supporting clusters enhance the focus and coherence of the major clusters. The standards in this MD cluster can help students deepen their understanding of the standards in the NBT clusters. We will talk about these standards in detail in Part 2.

Beginning the year with study of the NBT standards is just one of several options in Grade 5, but it has its advantages. For example, starting off working with the four operations serves as a nice “refresher” for students, as they have been learning about addition and subtraction since Kindergarten and multiplication and division since Grade 3. It also connects closely with work on place value in Grade 4. Finally, students will be using the four operations all year in a variety of contexts, so starting off with some experiences with them will support later learning.

Throughout the guide, we’ll look at examples of tasks and lessons that focus on students’ abilities to look for and make use of structure (MP.7), as students understand and explain place value and use place value to perform operations.

The first cluster in the Number & Operations in Base Ten domain has four standards. Let’s begin by reading these standards, and then we’ll think through what they mean and how they look in practice.

5.NBT.A | Understand the place value system

There’s a lot in there about place value, decimals and more! In cluster NBT.A, it’s important to note the emphasis on conceptual understanding. A lot of the language looks familiar (“reading,” “writing,” “comparing,” “rounding,” “digits,” “decimals”) but the cluster heading (“Understand the place value system”) and some of the key verbs (“recognize,” “explain”) help us see that getting students to a deep understanding of place value is our central goal. Not only is this challenging for us as teachers, it’s also different from the way that many of us learned place value and operations in base ten. We’ll revisit this idea as we discuss some examples.

Place value

Place value understanding in Grade 5 is the culmination of a great deal of prior work with place value, fractions and decimals. We will elaborate on this prior work in Part 3 of this guide; however, in this case it’s important to know what understandings students cultivated in Grade 4, so that the Grade 5 expectations are clear and so that instruction can coherently draw on prior learning in these three areas:

• Place Value: Students learn that for multi-digit whole numbers, a digit in one place represents 10 times as much as it represents in the place to its right (e.g., In the number 3,910, the value of the digit in the hundreds place is 10 times as much the value of the digit in the tens place in the number 391, as 900 ÷ 90 = 10). (4.NBT.A.1)
• Fractions: Students multiply fractions by whole numbers (e.g., 3 × (1/10) = 3/10 and 9 × (1/100) = 9/100) (4.NF.B.4)
• Decimals: Students use decimal notation for fractions with denominators of 10 and 100 (e.g. 3/10 = 0.3 and 9/100 = 0.09). (4.NF.C.6)

Combining these leads students to the critical understanding in Grade 5 that:

• For multi-digit numbers (now including decimals), a digit in one place represents 10 times as much as it represents in the place to its right, and represents 1/10 times as much as it represents in the place to its left. (5.NBT.A.1)

For example, let’s look closely at 4.2 and 42. The value of the digit 2 in 42 is 10 times the value of the digit 2 in 4.2 (i.e., 0.2 x 10 = 2), and the value of the digit 2 in 4.2 is 1/10 the value of the digit 2 in 42 (i.e., 2 x (1/10) = 0.2).

The task below is meant to help students consider the relationship between digits in different places, namely that one place is 10 times the place to its right or 1/10 times the place to its left.

Let’s also note a few characteristics of this task that work to build conceptual understanding:

• Students are asked to respond to another students’ reasoning. (Part a: “Is Netta correct?”)
• Students are asked to explain their thinking. (Part b: “What does a big square represent? Explain.”)
• Students are asked to interpret visual models in different ways (Part a: interpreting Netta’s model; and Part b: Name some other numbers this picture could represent) and draw visual models. (Part c: Draw a picture to represent 0.047)

Justifying, explaining and using models are some common structures that serve to deepen students’ understanding of concepts and should be a big part of what students do with place value in Grade 5.

Reading, writing and comparing decimals

In addition to understanding place value, students in Grade 5 will read and write decimals to the thousandths and represent them with base-ten numerals, number names, and expanded form. (5.NBT.A.3.A) They’ll also compare decimals to the thousandths. (5.NBT.A.3.B) The following task asks students to compare decimals to the thousandths and use explanations and representations to support their reasoning. Note again the inclusion of the directives to explain and create drawings; in this case, the drawings help reinforce the relationship between each decimal place.

Rounding

Lastly, students build on their understanding of rounding, which begins in Grade 3 with rounding whole numbers to the nearest 10 and 100 (3.NBT.A.1) and continues in Grade 4 with rounding multi-digit whole numbers to any place. (4.NBT.A.3) In Grade 5, students are expected to round decimals to any place using place value understanding. (5.NBT.A.4) The inclusion of place value understanding here is important to note; many of us learned to round by memorizing rules or even rhymes (“5 or more, Raise the score, 4 or less, Give it a rest!”). However, memorizing rules without developing understanding is not enduring for most students because these tricks have little substantive meaning, are easy to forget, and are often misapplied. In contrast to memorizing rules, using a number line to teach rounding helps students visualize how numbers are closer to a particular ten or hundred; it is strategy based on students’ existing number sense. While we certainly wouldn’t expect students to draw a number line every time they round for the duration of their mathematical lives, the visual representation helps to form a solid conceptual basis for rounding. The example below asks students to use a number n and determine which number to round it to.

Since students don’t know the exact value of n, they have to use its position on the number line to make a determination. The very fact that the task uses “n” instead of a number serves both to rely on and build conceptual understanding; students can’t apply a simple rule or rhyme to find the answer.

Powers of 10

A key advancement in Grade 5 is learning the meaning and notation for an exponent as it denotes powers of 10 (e.g., 103 = 10 x 10 x 10). (Work with powers of numbers other than 10 begins in Grade 6. (6.EE.A.1)) Students should understand patterns in the number of 0s in the product when multiplying a whole number by a power of 10, and the patterns in the placement of the decimal point when multiplying or dividing a decimal by a power of 10. Both patterns can be explained using place value understanding. Let’s take a closer look at these patterns; we’ll consider the pattern in the number of 0s first.

Patterns with 0s

Students know that when multiplying a whole number by 10, each digit in the product is ten times as large and consequently shifts one place to the left (e.g., 71 x 10 = 710). This is because the value of each place in the base-ten system is 10 times as large as the value of the place to its right (e.g., 700 ÷ 70 = 10 and 10 ÷ 1 = 10). In Grade 5, students observe these patterns based on multiplication by powers of 10. For example:

71 x 100 = 71 x 10 x 10 = 71 x 102 = 7100

71 x 1000 = 71 x 10 x 10 x 10 = 71 x 103 = 71,000.

The pattern that emerges with the 0s is that when multiplying a whole number by a power of 10, the product has the same digits as the whole number, which are then followed by the number of 0s denoted by the power of 10. For example, from examples like those above, students will likely predict that, for example, when multiplying 71 x 104 the product will have the digits 7 and 1 followed by four 0s: 710,000. This pattern also accounts for multiplying a multiple of 10 by a power of 10. For example, when multiplying 70 x 104 the product has the digits 7 and 0 followed by four 0s: 700,000.² Progressions for the Common Core State Standards in Mathematics (draft): Number and Operations in Base Ten, K-5, p. 18. Progressions for the Common Core State Standards in Mathematics (draft): Number and Operations in Base Ten, K-5, p. 18. Progressions for the Common Core State Standards in Mathematics (draft): Number and Operations in Base Ten, K-5, p. 18.

Placement of the decimal point

With respect to patterns with the placement of the decimal point, students extend the same place value understanding used with whole numbers to decimals: when multiplying a decimal by 10, each digit in the product is 10 times as large and shifts one place to the left, this includes the digits in the places to the right of the decimal point. For example, multiplying 6.421 by 102 is the same as multiplying 6.421 by 10 two times. The digits in the product are 100 times as large and shift two places to the left (e.g., 6.421 x 102 = 6.421 x 10 x 10 = 642.1). The pattern that emerges is that the digits shift to the left by the number of places indicated by the power of 10 (in the case of 102, two places), and the decimal point is positioned that many places to the right (from 6.421 to 642.1). Students should also explore dividing by powers of 10, (i.e., the digits in the quotient are shifted to the right and consequently the decimal point is positioned to the left). Thus making the quotient decreased by the power of 10, so 6.421102 is the same as dividing 6.421 by 10 twice which yields 0.06421. The following task requires students to reason about patterns with 0s and patterns with the decimal point; note again the inclusion of “explain” in this task.

This task highlights the common misconception that students have when multiplying a decimal (rather than a whole number) by a power of 10. Students will sometimes append a zero to the right of the decimal point, but not move the decimal point to account for the power of 10, as illustrated by Marta’s response in the task.They think that adding a 0 is sufficient, but forget that the patterns in use are based on place value; appending a zero in this position does not change the value of any of the digits. Students should note that 84.15 x 10 is not 84.150, but actually 841.5, and they should be able to explain why using place value understanding. When appending a 0 to a whole number the value of the number changes because each digit shifts one place to the left. However, when appending a 0 to a decimal the value of the number does not change. As the solution to the task notes, this could also be illustrated using expanded form and the distributive property.

Operations with whole numbers and decimals

The second cluster in the Number & Operations in Base Ten domain deals with performing operations with whole numbers and decimals. Let’s read the standards associated with this cluster, and then we’ll think through what they mean and how they look in practice.

5.NBT.B | Perform operations with multi-digit whole numbers and with decimals to hundredths.

Multiplication and division with whole numbers

By the end of Grade 5, students are expected to fluently multiply multi-digit whole numbers using the standard algorithm. (5.NBT.B.5) The progression toward this fluency begins in Grade 3 and is rooted in developing meaning for multiplication and division. The example below shows how students might support multi-digit multiplication using an area model and the connection of that area model to the standard algorithm.

Examples like these are an important part of the work in Grade 5; students become proficient with the standard algorithm by connecting it to prior work with visual models and other strategies.

Also in Grade 5, students extend their work with division of whole numbers to 2-digit divisors (roughly speaking, the number you “divide by”) with up to 4-digit dividends (roughly speaking, the number “being divided”). (5.NBT.B.6) For example, in , 2445 is the dividend and 15 is the divisor. The expectation is that students divide whole numbers using strategies based on place value, the properties of operations, and the relationship between multiplication and division. They illustrate and explain their work using equations, rectangular arrays, and/or area models. This builds from Grade 4, where students divide with 1-digit divisors and up to four-digit dividends. (4.NBT.B.6) Like Grade 4, students in Grade 5 decompose the number being divided into base-ten units and find the quotient for each unit, starting from the largest unit (e.g., 31010 = 3 hundreds and 1 ten divided by 10, leading to 3 tens and 1 one (31) as the quotient).⁴ Progressions for the Common Core State Standards in Mathematics (draft): Number and Operations in Base Ten, K-5, p. 18. Progressions for the Common Core State Standards in Mathematics (draft): Number and Operations in Base Ten, K-5, p. 18. Progressions for the Common Core State Standards in Mathematics (draft): Number and Operations in Base Ten, K-5, p. 18. Eventually, students relate this work to the standard algorithm. The lesson below introduces students to division with two-digit divisors by using a dividend and divisor that are multiples of ten. Since students are familiar with multiplying and dividing by 10, this is a comfortable place to begin thinking about division by two-digit numbers.

In this example, students decompose 420 into base-ten units (4 hundreds and 2 tens). They use place value disks (and the place value chart) to show the division of each base-ten unit by the divisor (10). In subsequent lessons, students use rounding (of both the dividend and the divisor) to estimate the partial quotients, relate their estimates to the standard algorithm, and then check their work with multiplication (and addition with respect to remainders). Note that while students might begin to relate multi-digit division to the standard algorithm, fluency with the standard algorithm for division is not expected until Grade 6.6.NS.B.6 Rushing to teach the algorithm, absent connection to conceptual understanding, may cause confusion for students because memorizing a procedure without understanding why it works does not support long term retention of the mathematics.So as much as you want to jump to significant practice with the standard algorithm, don’t! Spend time practicing solving division problems using place value strategies, the properties of operations, and the relationship between multiplication and division demonstrating how they connect to the standard algorithm adequately in Grade 5, so students can then apply it fluently in Grade 6. The following example shows use of the standard algorithm with connection to place-value understanding:

In the lesson above, multiplying two-digit numbers by one-digit numbers (e.g., 3 x 17 and 4 x 17) should be quick mental computations for students in Grade 5, because in Grades 3 and 4, students work extensively with the distributive property as a strategy to multiply. In this case, 3 x 17 = 3 x (10 + 7) = (3 x 10) + (3 x 7) = 30 + 21 = 51.

Operations with decimals

Students use place value understanding to perform all four operations with decimals to hundredths. (5.NBT.B.7) Students perform these operations by applying the same conceptual strategies they use with whole numbers and by using concrete models and drawings. Similarly, they compose and decompose decimals in the same way they do with whole numbers. Fluency with standard algorithms for operations with decimals is not expected until Grade 6. (6.NS.B.3)

Addition and subtraction with decimals

Many of us grew up learning that we must “line up” the decimal points when adding and subtracting decimals. In Grade 5, the Standards require us to make sure students understand why; students must be able to connect strategies to written methods and explain the reasoning used. (5.NBT.B.7) The following example shows how students use a place value chart to decompose and compose decimals in order to subtract. The chart strategy is connected to a written method that looks similar to the standard algorithm for subtraction.

The place value chart shows regrouping of a one into 10 tenths and then students subtract like units from like units, which aligns the decimal point. To further emphasize this point we might ask students to explain why the decimal points “line up” when subtracting in this example. Students should be able to explain that the decimal points line up because we need to subtract like units (i.e., tenths from tenths, and ones from ones), which is the same approach taken when we add and subtract whole numbers.

Multiplication and division with decimals

Methods for multiplying and dividing whole numbers can be applied to multiplying and dividing decimals. In the following example, students multiply a decimal by a whole number using a place value chart and then relate the place value strategy to an area model and written method.

In Grade 5, students should also multiply decimals by decimals. They can multiply decimals to the hundredths, however, since there is focus on decimals to the thousandths, (5.NBT.A.3) the products should not extend beyond the thousandths. Similarly with division, students should divide decimals to the hundredths, and the quotients should not extend beyond the thousandths place; students should divide decimals by whole numbers as well as by decimals. In the example below, students are asked to divide a decimal by a decimal.

In this example, students divide a decimal by a decimal divisor by writing the division problem as a fraction and then multiplying the fraction by 10/10 in order to yield an equivalent fraction with a whole number in the denominator. Students then relate the resulting fraction to the division algorithm. Note this kind of reasoning relies on important prerequisite understandings of fractions: understanding fractions as division, (5.NF.B.3) multiplication as scaling, (5.NF.B.5.B) and fraction equivalence. (4.NF.A.)1

Using the four operations in problem-solving

In Grade 5, students are expected to solve problems using the four operations. There are various problem types that students should engage with while using grade-appropriate numbers. These problem types have the unknown in various positions and can range from one-step to multi-step. Table 2 below comes from the Standards, and describes the common multiplication and division situations students should master in Grades 3-5. In Grade 5, students should have experience with these varying situations using multi-digit whole numbers as well as decimals and fractions. It will be helpful to spend some time digging into all of the different problem situations shown in the table as we will refer to them later.

Common multiplication and division situations (1)

(1) The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable.

(2) Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations.

(3) The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples.

CCSSM Table 2.

Table 1 in the Standards describes the various addition and subtraction problem types appropriate for elementary school.⁶ CCSSM Table 1. CCSSM Table 1. CCSSM Table 1. It is important that students engage with the different problem situations in Table 1 using multi-digit whole numbers, decimals, and fractions. Though these tables describe one-step problems, students should have lots of opportunity to engage with two-step and multistep problems using any combination of the four operations. In Grade 5, students should continue practice with representing problems with equations and using a letter to represent the unknown. The example below shows a two-step division problem leading to a decimal as the quotient.

This is an example of a two-step equal groups problem in which the number of groups (i.e., the number of hours) is unknown. First, the students have to determine how much money Ava has yet to earn ($609). Then students divide 609 by 14 to find the number of hours. Note, dividing 609 by 14 requires them to move beyond finding a remainder and to reason about the placement of the decimal point in order to divide 70 tenths by 14 to get 5 tenths.

Part 2: How do Number & Operations in Base Ten relate to other parts of Grade 5?

There are lots of connections among standards in Grade 5; if you think about the standards long enough, you’ll probably start to see these relationships everywhere.⁷ The idea that standards relate strongly to one another is known as coherence, and is a distinctive feature of the Common Core State Standards for Math. If you’re interested in exploring more of the connections between standards, you might want to check out the Student Achievement Partners Coherence Map, which illustrates them visually. The idea that standards relate strongly to one another is known as coherence, and is a distinctive feature of the Common Core State Standards for Math. If you’re interested in exploring more of the connections between standards, you might want to check out the Student Achievement Partners Coherence Map, which illustrates them visually. The idea that standards relate strongly to one another is known as coherence, and is a distinctive feature of the Common Core State Standards for Math. If you’re interested in exploring more of the connections between standards, you might want to check out the Student Achievement Partners Coherence Map, which illustrates them visually. In this section, we’ll talk about the connection between the standards in the Number & Operations in Base Ten (NBT) domain and the standard in the first cluster of the Measurement & Data (MD) domain. The standard in this MD cluster is a supporting standard and can be used to support work with the NBT standards.

5.MD.A | Convert like measurement units within a given measurement system.

This standard builds on work with conversions in Grade 4, where students convert from larger to smaller units. (4.MD.A.1) In Grade 5, students build on this understanding to convert measurements from a smaller unit to a larger unit. Students relate the idea that a digit in one place is 1/10 the size of the digit in the place to its left (5.NBT.A.1 )to the idea that smaller metric units are 1/10, 1/100, or 1/1000 the size of larger units. The following lesson gives an example of this relationship.

Understanding the relationship between the metric units and the place value chart helps students know whether they are moving from a larger unit to a smaller unit or the reverse. It also illustrates the multiplicative relationship between units.

Since this is a supporting cluster, students use conversions to solve multistep, real-world measurement problems in order to support addition, subtraction, multiplication, and division with whole numbers, decimals, and fractions. This builds from work in Grade 4 where students engage with measurement contexts involving whole numbers and simple fractions and decimals. (4.MD.A.2) The contexts for the multistep problems should involve measurement systems the students are familiar with: distance, intervals of time, liquid volumes, masses of objects, and money. The example below involves converting from a smaller unit to a larger unit.

In this task, students divide a 4-digit number by a two-digit number to convert hours to minutes (2011 ÷ 60). Alternatively, as shown in the solution, students can use multiplication to solve the problem.

Part 3: Where do the Number & Operations in Base Ten Standards come from, and where are they going?

The Number & Operations in Base Ten domain in Grade 5 builds from the NBT, NF, and OA standards in Grades 3 and 4. Knowing the lead-up to the Grade 5 content will help you leverage content from previous grades in your lessons; also, mathematical learning should always be explicitly connected to previous understandings. Moreover, if your students are behind, seeing where Grade 5 ideas come from will allow you to adapt your curriculum and lessons to make new ideas accessible. Let’s look at the progression of content to Grade 5; then we’ll examine some ways that you might use this information to meet the unique needs of your students. After that, we’ll see how the ideas of multiplication and division extend beyond Grade 5.

Podcast clip: Importance of Coherence with Andrew Chen and Peter Coe (start 9:34, end 26:19)

https://soundcloud.com/unboundedu/the-mathematics-standards-and-shifts/s-tqwCA

Where does work with place value and operations in Grade 5 come from?

Grades 3-4: Multiplication and division with whole numbers

The Standards lay out a purposeful progression for multiplication and division with whole numbers. Students begin formal study of multiplication and division in Grade 3,⁸ There are foundational skills and experiences presented in Grade 2. For example, students begin to work with equal groups and arrays and use repeated addition to find the total. (2.OA.C.4 )You can read more about foundations with multiplication and division in Part 3 of the Grade 3 Operations & Algebraic Thinking Content Guide. There are foundational skills and experiences presented in Grade 2. For example, students begin to work with equal groups and arrays and use repeated addition to find the total. (2.OA.C.4 )You can read more about foundations with multiplication and division in Part 3 of the Grade 3 Operations & Algebraic Thinking Content Guide. There are foundational skills and experiences presented in Grade 2. For example, students begin to work with equal groups and arrays and use repeated addition to find the total.2.OA.C.4 You can read more about foundations with multiplication and division in Part 3 of the Grade 3 Operations & Algebraic Thinking Content Guide.which includes:

• Developing meaning for these operations with work on equal groups, arrays, and area, along with representing problems with models, drawings, and equations. (3.OA.A.3, 3.MD.C.7)
• Using strategies for multiplication and division based on place value, properties of operations (commutative, associative, and distributive), and the relationship between multiplication and division (e.g., understanding that a division problem can be thought of as an unknown factor problem). (3.OA.B.5, 3.OA.B.6)

The example below illustrates two different solution methods for an equal groups problem.

As noted, students might use the distributive property or a tape diagram to represent the problem. The distributive property is introduced to develop fluency with single-digit facts and students continue to apply it with multi-digit numbers.

In Grade 4, students develop a new understanding of multiplication as a comparison. Multiplicative comparison involves comparing quantities multiplicatively instead of additively, which students have been working on since Grade 1. (1.OA.A.1)

In additive comparisons, the underlying question is: What amount would be added to one quantity in order to result in another? By contrast, in multiplicative comparisons, the underlying question is: What factor would be multiplied by one quantity in order to result in the other? The example below illustrates this distinction.

Students learn to make the distinction between multiplicative and additive comparisons by engaging with various types of multiplicative comparison word problems. (4.OA.A.2) The task below shows some problems involving multiplicative comparison.

Similarly, more experience with additive comparison problems (e.g., with multi-digit whole numbers (4.NBT.B.4)) is necessary to clarify the distinction between multiplicative and additive comparison. In Table 2 from the Standards (see Part 1), the compare problems highlight the different multiplicative comparison situations students should encounter.⁹ CCSSM Table 2. CCSSM Table 2. CCSSM Table 2.

Also in Grade 4, students multiply and divide with larger numbers. They continue to multiply and divide using strategies based on place value, the properties of operations, and the relationship between multiplication and division. (4.OA.A.2,4.NBT.B.5, 4.NBT.B.6) And they continue to use visuals like area models, connecting them to equations and written computation methods, which provides meaning for algorithms. The example below shows how a student might represent multi-digit multiplication using a model that supports a written method typical in Grade 4.

The example shows a two-step problem involving multiplicative comparison and addition. It shows the use of place value strategies to compute partial products when multiplying a two-digit number by a one-digit number.

Some students in Grade 4 may progress from written methods that display all calculations for multiplication and division (e.g., partial products) to more compact written methods. The example below illustrates how different strategies relate to the standard algorithm.

Grades 3-4: Addition and subtraction with whole numbers

In Grade 5, applying strategies to add and subtract decimals based on place value, properties of operations, and the relationship between addition and subtraction rests on students’ prior experience applying these strategies to addition and subtraction of whole numbers. Fluency expectations for addition and subtraction of whole numbers begin in Kindergarten and culminate in Grade 4 with students being able to fluently add and subtract multi-digit whole numbers using the standard algorithm. (K.OA.A.5, 1.OA.C.6, 2.OA.B.2, 3.NBT.A.2, 4.NBT.B.4) As with multiplication and division, developing fluency with addition and subtraction is rooted in conceptual understanding. Students represent problems with models, drawings, and equations. They also learn to use strategies for addition and subtraction based on place value, properties of operations (commutative and associative), and the relationship between addition and subtraction (e.g., understanding that a subtraction problem can be thought of as an unknown addend problem).1.OA.B.5, 1.OA.B.6 In Grade 3, students add and subtract within 1,000 using place value strategies and algorithms. The example below shows how a student might use place value to decompose and compose numbers when adding and subtracting (e.g., 43 is decomposed into 40 + 3).

With larger numbers, it becomes more efficient to use more compact written methods. The following example from Grade 4 illustrates how students might relate place value models to the standard algorithm for addition for multi-digit numbers. Notice the composing of a hundred from 10 tens and that it is recorded on the addition line.

Grades K-4: Application of the four operations

In Grade 4, students engage in solving multistep word problems using the four operations. (4.OA.A.3) This is a culminating standard for a sequence beginning in Kindergarten. (K.OA.A.2, 1.OA.A.1, 2.OA.A.1, 3.OA.D.8) More specifically, students in Grade 2 began working with two-step problems that were limited to addition and subtraction, and expanded to include multiplication and division in Grade 3. In Grade 4, the problems are multistep, generally with no more than three steps, and can include any combination of the four operations with whole numbers and integrate concepts from MD standards.

The following examples illustrate different kinds of multistep word problems in Grade 4. In the first example, students are expected to solve one problem that involves several steps in order to determine the total amount of money earned for both days.

This solution is modeled using different kinds of representations. The step involving multiplicative comparison is modeled with a tape diagram, and labeled equations are used to show the multiplication.

Though the prior example was a single, extended task, multistep problems can also comprise a series of questions developing from a particular context, as in the next example. Students have to pay close attention to the language to determine whether the relationship being asked about is additive or multiplicative. Given that there are so many questions, the size of the numbers is restricted.

Notice that in part a, students have to interpret the meaning of a remainder, which is a new understanding in Grade 4. (4.OA.A.3) It’s important to emphasize that we want students to be able to interpret the remainder, rather than simply compute quotients that happen to have remainders.

Grade 4: Generalizing place value

Students begin thinking about place value in Kindergarten when they observe that the teen numbers are composed of 10 ones and some more ones. (K.NBT.A.1) Students formalize their understanding of the tens place in first grade when they bundle 10 ones to make a ten and that 20 is 2 bundles of ten, or 2 tens and so on. (1.NBT.B.2) In Grade 2, they bundle 10 tens to make a hundred and that 200 is 2 bundles of a hundred or 2 hundreds and so on. (2.NBT.A.1) Understanding place value in K-2 always begins with concrete representation so students can conceptualize the relationship between ones, tens, and hundreds. Students continue to work with numbers within 1000 in Grade 3. (3.NBT.A.2) Students expand their understanding of whole numbers in Grade 4 to numbers greater than 1000. As in in K-2, we want students to understand the relationship between each place value. Specifically, we want them to understand the multiplicative relationship. (4.NBT.A.1) As in previous grades this understanding should be explored concretely and pictorially (i.e., with drawings and manipulatives) before moving on to abstract representations.

In the example below from Grade 4, students show their understanding of place value concepts through use of a place value chart to multiply numbers by 10. This work lays the foundation for further work with place value in Grade 5. (5.NBT.A.1)

Grade 4: Understanding decimal notation for fractions

Students begin their work with decimals in Grade 4 through their work with fractions. (4.NF.C.5, 4.NF.C.6, 4.NF.C.7) Number lines and tape diagrams can be useful tools for helping students relate decimal notation to fractions.

As students gain traction with understanding decimal notation for simple fractions, they begin to synthesize what they know about fractions and decimals. Students are then able to more fluently express decimal notation for fractions. The following example shows how students compare fractions to decimals after spending considerable time working on the skill in Grade 4.

This work, plus what students already know about place value concepts, are the foundation for them to perform operations with multi-digit whole numbers and with decimals to hundredths in Grade 5. (5.NBT.B.7)

Suggestions for students who are below grade level

If students come to Grade 5 without a solid grasp of the ideas named above (or haven’t encountered them at all), what can you do? It’s not practical (or even desirable) to reteach everything students should have learned in Grades 3-4; there’s plenty of new material in Grade 5, so the focus needs to be on grade-level standards. At the same time, there are strategic ways of wrapping up “unfinished learning” from prior grades. Here are a few suggestions for adapting your instruction to bridge the gaps with respect to preparedness for multiplication and division with whole numbers.

• If a significant number of students struggle to grasp whole number or decimal operations, consider a few days of instruction that start with simpler addition, subtraction, multiplication and division examples (4.NBT.B). In this lesson students can practice adding multi-digit whole numbers. If subtraction is the issue, this topic has lessons that will help students practice subtracting multi-digit whole numbers. This topic has a series of lessons that will help students who require additional practice using place value strategies to multiply multi-digit numbers. You can also check here for lessons that focus on division of multi-digit whole numbers. Consider explicitly relating whole number operations (via a warm-up activity or other example) to decimal operations.

• If a significant number of students don’t have an understanding of place value, you could plan a few lessons that address the foundations of the meaning of each place and how they are related by multiplication (4.NBT.A.1) prior to fifth grade level instruction. Topic A of this Module provides an opportunity for students to practice manipulating and representing the place value of whole numbers. If lessons seem too much, consider a warm-up activity each day that requires students to identify place value in numerals and explain the relationships between places.

• If a significant number of students don’t have conceptual understanding of decimals, consider a few days of instruction that relate decimals to fractions and visual models (4.NF.C). This module will help students make the connection between what they know about fractions and relate that to decimals.

Beyond Grade 5: What’s next with the four operations?

Grades 6-7: The number system

It’s helpful to understand how students will build on their understanding of the four operations in later grades. Understanding the progression of this content can help to solidify the focus in Grade 5. It helps us to define the limits of our instruction and to see why we focus so intently on a small number of concepts in Grade 5. At the same time, we realize that we want to set our students up for future success, and knowing the next steps in their journey can help us focus our lessons on the knowledge and skills that matter most.

Having strong conceptual understanding of the four operations is really important to success with middle school mathematics. In Grade 6, students are expected to fluently divide multi-digit whole numbers using the standard algorithm. (6.NS.B.2) And fluency is expected with all four operations with decimals using standard algorithm. (6.NS.B.3) In Grade 7, students are expected to solve real-world and mathematical problems using the four operations with rational numbers. (7.NS.A.3) The examples below illustrate work with decimals in Grade 6 and rational numbers in Grade 7.

If you’ve just finished this entire guide, congratulations! Hopefully it’s been informative, and you can return to it as a reference when planning lessons, creating units, or evaluating instructional materials. For more guides in this series, please visit our Enhance Instruction page. For more ideas of how you might use these guides in your daily practice, please visit our Frequently Asked Questions page. And if you’re interested in learning more about Number & Operations in Base Ten in Grade 5, don’t forget these resources:

Student Achievement Partners: Focus Grade 5

Draft K-5 Progression on Number and Operations in Base Ten

EngageNY: Grade 5 Module 1 Materials

Illustrative Mathematics Grade 5 Tasks

Endnotes

FAQs

1. What is a Content Guide?

In our work with high academic standards, we often hear educators ask, “What does standards-aligned instruction look like?” Our Content Guides aim to answer this question by providing an in-depth look at one or a few clusters of math standards at a time. The Content Guides are grade-level and content area-specific, and there are guides for each grade or course, from Kindergarten to Algebra II. If you want to learn more about teaching Ratios and Proportional Relationships in Grade 6, for example,our associated Content Guide will give you a comprehensive but accessible explanation about these standards, multiple Open Educational Resource (OER) examples that are aligned to the standards, and concrete suggestions to support the teaching of Grade 6 ratios and proportional reasoning.

Our goal in creating the Content Guides has been to provide busy teachers with a practical and easy-to-read resource on what the grade-level math standards are saying, along with examples of instructional materials that support conceptual understanding, problem-solving, and procedural skill and fluency for students.

It’s important to note that content guides are not meant to serve as a curriculum (or any kind of student-facing document), a guide or source material for test-preparation activities, or any kind of teacher evaluation tool.

2. What’s in a Content Guide?

Each Content Guide is focused on a specific group of standards. Most Content Guides follow the same three-part structure:

• Part 1 makes clear the student skills and understandings described by this group of standards. This section illustrates the standards using multiple student tasks from freely available online sources. Teachers can use or adapt these tasks for their students.

• Part 2 explains how this group of standards is connected to other standards in the same grade. We highlight how these connections have implications for planning and teaching, and how this within-grade coherence can increase access for students. Part 2 also includes multiple student tasks from freely available online sources.

• Part 3 traces selected progressions of learning leading to grade-level content discussed in the specific Content Guide. This discussion segues into a series of concrete and practical suggestions for how teachers can leverage the progressions to teach students who may not be prepared for grade-level mathematics. Finally, Part 3 traces the progression to content in higher grades.

3. How can I use the Content Guides?

Teachers who have read our Content Guides say they see benefits for all educators. Here are some suggestions for how different educators might use them.

Teachers can use the Mathematics Content Guides to:

• Increase or refresh their knowledge of the standards and the expectations for what students should know by the end of the year.
• Adapt lessons and units using appropriate pre-requisites to support students who are behind grade-level.
• Gain access to the best available OER for math to use for introducing and/or reinforcing concepts
• Ensure their curriculum and/or units:
  • Focus on the major work of the grade and the appropriate depth of each standard.
  • Target the appropriate aspects of rigor—procedural skill and fluency, modeling and application, and conceptual understanding described by the standards.
  • Help students make coherent connections within and across grades.
• Create or revise their lessons and questioning to focus on important concepts in the standards.

Instructional coaches and school leaders can use the Mathematics Content Guides to:

• Refresh or increase their knowledge of the standards and the expectations for what students should know by the end of the year.
• Develop and communicate consistent expectations for lesson planning and instruction aligned to the standards.
• Provide a reference when planning and/or discussing instruction with teachers.
• Gain insight into what instruction and student work should look like in order to meet the demands of the standards.
• Develop and design content and standards-driven professional development sessions/workshops.
• Foster content rich, standards-based discussions among staff and build staff knowledge.
• Develop and/or revise school improvement plans in order to support and incorporate content and practice-based teaching and learning.

4. Why the Content Guides?

The transition to higher standards has led teachers all over the country to make significant changes in their planning and instruction, but only one-third of teachers feel they are prepared to help their students pass the more rigorous standards-aligned assessments (Kane et.al., 2016). This is to be expected because the new high standards are a significant departure from prior standards. The standards require a deeper level of understanding of the math content they teach; a different progression of what students need to learn by which grade; as well as different pedagogy that emphasizes student conceptual understanding, problem solving and procedural fluency in equal intensity.

The support for teachers to bring high standards to their classrooms, however, has lagged behind. Research shows that teacher training in the U.S. is currently insufficient in preparing teachers to teach the demanding new standards (Center for Research in Mathematics and Science Education, 2010). And though some resources exist that “unpack” the standards, few, if any, explain and illustrate the standards. “Unpacking” the standards one by one can also result in a disjointed presentation that neglects the structure and coherence of the standards. In creating the Content Guides, we aimed to provide busy teachers with a practical, easy-to-read resource on their grade-specific standards and how to help all students learn them. There is ample empirical evidence that when teachers have both strong knowledge of the math content that they teach, and the pedagogical knowledge to help students master that content knowledge, their students learn more (Baumert et. al., 2010; Hill, Rowan and Ball, 2005; Rockoff et. al., 2008). With the Content Guides in hand, we hope that teachers will find more success in helping their students make progress toward college- and career-readiness.

5. What is the relationship between the Content Guides and the Progressions?

The Progressions documents describe the grade-to-grade development of understanding of mathematics. These were informed by research on children’s cognitive development as well as the logical structure of mathematics. The Progressions explain why standards are sequenced the way they are. The Content Guides often highlight key ideas from the Progressions, but do not add new standards or change the expectations of what students should know and be able to do; they aim to explain and illustrate a group of standards at a time using freely available online sources. While the OER tasks and lessons in the Content Guides are one way to meet the grade-level standards, they are not the only means for doing so.

6. How were the resources selected?

We selected sample tasks and lessons from freely available online sources such as EngageNY, Illustrative Mathematics and Student Achievement Partners to illustrate the Standards. These sources are chosen because they are fully aligned to the new high standards based on national review of K-12 curricula or are created by organizations led by the writers of the new high standards. In addition, because they are open educational resources (OER), they are freely accessible for all uses. All UnboundEd materials are also OER, as part of our commitment to make high-quality, highly aligned content available to all educators.

7. Why are the Content Guides only about a few standards? Where are the rest of the standards?

Each Content Guide addresses a subset of the standards for the grade. The standards addressed in the first set of Content Guides for each grade usually address high-priority content; these standards are also often a good choice for teaching at the beginning of the year. More information about the selection of standards can be found in the introduction to each Content Guide. Over time, we will develop additional Content Guides for each grade and update existing ones. We plan to have four Content Guides for each grade or course, from Kindergarten to Algebra II. The guides will be published in waves, with each wave consisting of one guide for each grade. We plan to release a second set of Content Guides for each grade by the end of the 2016-17 school year.

8. How do I stay informed about new Content Guides?

If you would like to receive updates on content and events from UnboundEd, including new Content Guides, please sign up for UnboundEd announcements here.

Understand the place value system. Perform operations with multi-digit whole numbers and with decimals to hundredths. Convert like measurement units within a given measurement system. Look for and make use of structure. Understand the place value system. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Read, write, and compare decimals to thousandths. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Use place value understanding to round decimals to any place. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Use decimal notation for fractions with denominators 10 or 100. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Use place value understanding to round whole numbers to the nearest 10 or 100. Use place value understanding to round multi-digit whole numbers to any place. Use place value understanding to round decimals to any place. Write and evaluate numerical expressions involving whole-number exponents. Perform operations with multi-digit whole numbers and with decimals to hundredths. Fluently multiply multi-digit whole numbers using the standard algorithm. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Fluently multiply multi-digit whole numbers using the standard algorithm. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Read, write, and compare decimals to thousandths. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1. Extend understanding of fraction equivalence and ordering. Convert like measurement units within a given measurement system. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Relate area to the operations of multiplication and addition. Apply properties of operations as strategies to multiply and divide. Understand division as an unknown-factor problem. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Fluently add and subtract multi-digit whole numbers using the standard algorithm. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Fluently add and subtract within 5. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Fluently add and subtract multi-digit whole numbers using the standard algorithm. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. Use decimal notation for fractions with denominators 10 or 100. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Use place value understanding and properties of operations to perform multi-digit arithmetic. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Understand decimal notation for fractions, and compare decimal fractions. Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Solve real-world and mathematical problems involving the four operations with rational numbers. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.