The big ideas in grade 5 include: developing fluency with addition and subtraction of fractions, developing understanding of multiplication and division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions), extending division to two-digit divisors, developing understanding of operations with decimals to hundredths, developing fluency with whole number and decimal operations, and developing understanding of volume.

The mathematical work for grade 5 is broken into 8 units:

1. Finding Volume
2. Fractions as Quotients and Fraction Multiplication
3. Multiplying and Dividing Fractions
4. Wrapping Up Multiplication and Division with Multi-digit Numbers
5. Place Value Patterns and Decimal Operations
6. More Decimal and Fraction Operations
7. Shapes on the Coordinate Plane
8. Putting it All Together

This unit introduces students to the concept of volume by building on their understanding of area and multiplication.

In grade 3, students learned that the area of a two-dimensional figure is the number of square units that cover it without gaps or overlaps. They first found areas by counting squares and began to intuit that area is additive. Later, they recognized the area of a rectangle as a product of its side lengths and found the area of more-complex figures composed of rectangles.

Here, students learn that the volume of a solid figure is the number of unit cubes that fill it without gaps or overlaps. First, they measure volume by counting unit cubes and observe its additive nature. They also learn that different solid figures can have the same volume.

Next, they shift their focus to right rectangular prisms: building them using unit cubes, analyzing their structure, and finding their volume. They write numerical expressions to represent their reasoning strategies and work with increasingly abstract representations of prisms.

Later, students generalize that the volume of a rectangular prism can be found by multiplying its side measurements (\(\text{length} \times \text{width} \times \text{height}\)), or by multiplying the area of the base and its height (\(\text{area of the base} \times \text{height}\)). As they analyze, write, and evaluate different expressions that represent the volume of the same prism, students revisit familiar properties of operations from earlier grades.

Later in the unit, students apply these understandings to find the volume of solid figures composed of two non-overlapping rectangular prisms and solve real-world problems involving such figures. In doing so, they also progress from using cubes to using standard units to measure volume.

In this section, students make sense of volume as a measurement of three-dimensional figures by building objects with unit cubes and counting the cubes. They experiment with different figures made from the same number of cubes and see them as having the same volume.

Students then build right rectangular prisms and analyze images of prisms constructed of unit cubes. To find the volume of these solids, students look at their structure and relate the number of horizontal and vertical layers to the total number of cubes (MP7). They engage with the commutative and associative properties of multiplication as they reason about the volume of rectangular prisms that are oriented in different ways.

In this section, students continue to work with right rectangular prisms and to relate side measurements to volume. They observe that multiplying the number of layers of cubes in a prism by the number of cubes in one layer gives its volume. They also see that the number of cubes in one layer is in essence the area of a rectangle. 

Students then generalize the volume of a right rectangular prism as the product of its side lengths, \(\text{length} \times \text{width} \times \text{height}\) and as the product of the area of its base and its height, \(\text {base area} \times \text {height}\). 

To promote flexible use of measurements and sense making in finding volume, students connect these mathematical terms to numerical expressions that represent volume, rather than relying on algebraic formulas. This work reinforces the associative property of multiplication and highlights that the volume of a rectangular prism can be represented with equivalent multiplication expressions.

In this section, students apply their understanding of volume to solve real-world and mathematical problems. They encounter solid figures that are composed of two or more right rectangular prisms, which reinforces their understanding of the additive nature of volume.

Students also work with side lengths that are larger than those in earlier sections, prompting them to activate multiplication strategies from earlier grades. The work reminds students that they can decompose multi-digit factors by place value to find their product, paving the way toward the standard algorithm for multiplication in a later unit.

In this unit, students learn to interpret a fraction as a quotient and extend their understanding of multiplication of a whole number and a fraction.

In grade 3, students made sense of multiplication and division of whole numbers in terms of equal-size groups. In grade 4, they used multiplication to represent equal-size groups with a fractional amount in each group and to express comparison.

For instance, \(4 \times \frac{1}{3}\) can represent “4 groups of \(\frac{1}{3}\)” or “4 times as much as \(\frac{1}{3}\).”

The amount in both situations can be represented by the shaded parts of a diagram like this:

Here, students learn that a fraction like \(\frac{4}{3}\) can also represent:

• a division situation, where 4 objects are being shared by 3 people, or \(4 \div 3\)
• a fraction of a group, in this case, \(\frac{1}{3}\) of a group of 4 objects, or \(\frac{1}{3} \times 4\)

Students also interpret the product of a whole number and a fraction in terms of the side lengths of a rectangle. The expression \(6 \times 1\) represents the area of a rectangle that is 6 units by 1 unit. In the same way, \(6 \times \frac{2}{3}\) represents one that is 6 units by \(\frac{2}{3}\) unit.

The commutative and associative properties become evident as students connect different expressions to the same diagram. The distributive property comes into play as students multiply a whole number and a fraction written as a mixed number, for instance: \(2 \times 3\frac{2}{5} = (2 \times 3) + (2 \times \frac{2}{5})\). 

Throughout this unit, it is assumed that the sharing is always equal sharing, whether explicitly stated or not. For example, in the situation above, 4 objects are being shared equally by 3 people.

In this section, students learn to see a fraction as a quotient, a result of dividing the numerator by the denominator. They solve a sequence of problems about situations that involve sharing a whole number of objects equally. Through repeated reasoning, they notice regularity in the result of division (MP8) and generalize that \(\frac{a}{b} = a \div b\).

For example, 3 objects being shared equally by 2 people can be represented by the expression \(3 \div 2\) and by a diagram. Each person’s share can be shown by the shaded parts in a diagram such as:

or

Each person would get half of the 3 objects, or 3 groups of \(\frac{1}{2}\) an object. The value of this expression is \(\frac{3}{2}\) or  \(1\frac{1}{2}\).

In grade 4, students saw that a non-unit fraction can be expressed as a product of a whole number and a unit fraction, or a whole number and a non-unit fraction with the same denominator. For instance, \(\frac{8}{3}\) can be expressed as \(8 \times \frac{1}{3}\), as \(4 \times \frac{2}{3}\), or as \(2 \times \frac{4}{3}\). In the previous section, students interpreted a fraction like \(\frac{8}{3}\) as a quotient: \(8 \div 3\).

This section allows students to connect these two interpretations of \(\frac{8}{3}\) and relate \(8 \times \frac{1}{3}\) and \(8 \div 3\).

Students use diagrams and contexts to make sense of division situations that result in a fractional quotient. As they interpret and write expressions that represent the quantities, students observe the commutative property of multiplication. For example, they interpret \(8 \times \frac{1}{3}\) and \(\frac{1}{3} \times 8\) as 8 groups of a third and a third of 8, respectively, and recognize that both are equal to \(\frac{8}{3}\).

These understandings then help students make sense of other multiplication and division expressions that can be represented by the same diagram and have the same value:

In this section, students learn that they can reason about the area of a rectangle with a fractional side length the same way they had with rectangles with whole-number side lengths: using diagrams and multiplication.

To find the area of such rectangles, students work through a progression of fractional side lengths: a unit fraction (\(\frac{1}{3}\)), a non-unit fraction (\(\frac{2}{3}\)), a fraction greater than 1 (\(\frac{5}{3}\)), and a mixed number (\(1\frac{2}{3}\)). They write and interpret multiplication expressions, such as \(6 \times \frac{1}{3}\) and \(6 \times \frac{5}{3}\), that represent the area of such rectangles. Students use shaded diagrams and their understanding of fractions to reason about the value of the expressions.

Along the way, the associative property of multiplication becomes evident. For instance, students see that the expressions \(6 \times \frac{2}{5}\), \(6 \times 2 \times \frac{1}{5}\), and \(12 \times \frac{1}{5}\) can all describe the area of the shaded region in this diagram.

The distributive property is illustrated as students reason about the area of a rectangle where the side lengths are a whole number and a mixed number. To find \(2 \times 3 \frac{2}{5} \), for example, students may decompose the rectangle by grouping the whole-number units and the fractional units and multiply them separately before combining them, resulting in an expression such as \((2 \times 3) + \left (2 \times \frac{2}{5}\right)\).

In this unit, students find the product of two fractions, divide a whole number by a unit fraction, and divide a unit fraction by a whole number.

Previously, students made sense of multiplication of a whole number and a fraction in terms of the side lengths and area of a rectangle. Here, they make sense of multiplication of two fractions the same way. Students interpret area diagrams with two unit fractions for their side lengths, then a unit fraction and a non-unit fraction, and then two non-unit fractions.

Through repeated reasoning, students notice regularity in the value of the product (MP8). They generalize that it can be found by multiplying the numerators and multiplying the denominators of the factors:
 

\(\displaystyle{\frac{a}{b} \times \frac{c}{d}  = \frac{a\times c}{b\times d}}\)

For example, \(\frac{2}{4} \times \frac{3}{5}\) is \(\frac{2 \ \times \ 3}{4 \ \times \ 5}\) ​​​because there are \(4 \times 5\) equal parts in the whole square and \(2 \times 3\) parts are shaded.

​​​​​​

Next, students make sense of division situations and expressions that involve a whole number and a unit fraction. They recall that division can be understood in terms of finding the number of equal-size groups or finding the size of each group.

For instance, students interpret \(\frac{1}{3} \div 4\) to mean finding the size of one part if \(\frac{1}{3}\) is split into 4 equal parts, and \(4 \div \frac{1}{3}\) to mean finding how many \(\frac{1}{3}\)s are in 4.

Students consider how changing the dividend or the divisor changes the value of the quotients and look for patterns (MP8). They use tape diagrams to represent and reason about division situations and expressions.

Later in the unit, students apply what they learned to solve problems. The relationship between multiplication and division is reinforced when they notice that both operations can be used to solve the same problem.

In this section, students reason about multiplication of two fractions. They begin by considering situations that involve finding a fraction of a fraction. They represent the situations by drawing diagrams that make sense to them.

For example, “A pan of macaroni and cheese is \(\frac {1}{3}\) full. Kiran eats \(\frac {1}{4}\) of the macaroni and cheese. How much of the whole pan did Kiran eat?”

By partitioning the first third of a pan into fourths and doing the same with the other two thirds, students can see that Kiran ate \(\frac{1}{12}\) of the whole pan.

Students connect the product of two fractions to the area of a rectangle with fractional side lengths. When multiplying unit fractions, students see the denominator as the number of equal parts in the unit square, structured as an array. So partitioning one side of a rectangle into fourths and the other into thirds create a 4-by-3 array. Each part in the array is \(\frac{1}{12}\) of 1 whole.

 The area of a rectangle that is \(\frac{1}{4}\) by \(\frac{1}{3}\) is thus \(\frac{1}{12}\), or \(\frac{1}{4} \times \frac{1}{3} = \frac{1}{4 \ \times \ 3} = \frac{1}{12}\). Students generalize this as:

 \(\displaystyle {\frac{1}{b} \times \frac{1}{d} = \frac{1}{b \ \times \ d}}\) 

They extend this insight to find the product of non-unit fractions, including fractions greater than 1.

For example, the value of \(\frac{3}{4} \times \frac{7}{5}\) is \(\frac{3 \ \times \ 7}{4 \ \times \ 5}\) because \(3 \times 7\) parts are shaded and there are \(4 \times 5\) equal parts in 1 whole.

\(\displaystyle{\frac{a}{b} \times \frac{c}{d} = \frac{a \ \times \ c}{b \ \times \ d}}\)

In grade 3, students learned that division can be understood in terms of equal-size groups and can be interpreted in two ways. For example, \(8 \div 4\) can mean finding the size of each group if 8 is put into 4 equal groups, or finding how many groups of 4 are in 8.

In this section, students extend this idea to divide a unit fraction by a whole number and divide a whole number by a unit fraction. They interpret \(\frac{1}{2} \div 5\) to mean finding the size of one part if \(\frac{1}{2}\) is split into 5 equal parts, and \(5 \div \frac{1}{2}\) as a way of finding how many \(\frac{1}{2}\)s are in 5.

\(\frac{1}{2} \div 5\)

\(5 \div \frac{1}{2}\)

To build this understanding, students reason about situations, diagrams, and expressions that represent division. They look for patterns and assess the reasonableness of the quotients they find.

Students may notice that to find \(5 \div \frac{1}{2}\), they can multiply 5 by 2 because there are 2 halves in each of the 5 wholes. It is not essential, however, that students generalize division of fractions at this point, as they will do so in grade 6.

In this section, students solve problems involving multiplication and division of fractions. As they reason about situations and interpret tape diagrams, they see that the same situation or diagram can be expressed with multiplication or division.

For example, if \(\frac{1}{2}\) gallon of lemonade is shared equally by 5 friends, each friend gets \(\frac{1}{2} \div 5\) gallon of lemonade. This also means that each friend gets \(\frac{1}{5}\) of the \(\frac{1}{2}\) gallon, which can be expressed by \(\frac{1}{5} \times \frac{1}{2}\).

Students interpret situations and diagrams in terms of one or both operations, depending on what makes sense in the given context. In this diagram, the shaded part represents both \(\frac{1}{2} \div 5\) and \(\frac{1}{5} \times \frac{1}{2}\).

In this unit, students multiply multi-digit whole numbers using the standard algorithm and begin working toward end-of-grade expectation for fluency. They also find whole-number quotients with up to four-digit dividends and two-digit divisors.

In grade 4, students used strategies based on place value and properties of operations to multiply a one-digit whole number and a whole number of up to four digits, and to multiply a pair of two-digit numbers. They decomposed the factors by place value, and used diagrams and algorithms using partial products to record their reasoning. 

Here, students build on those strategies to make sense of the standard algorithm for multiplication. They recognize that it is also based on place value but records the partial products in a condensed way.
 

Han and Elena used different algorithms to find the value of \(3 \times 318\).

Han

Elena

Explain to your partner what Han and Elena did. What does the 2 represent in Elena's algorithm?

In grade 4, students also found whole-number quotients using place-value strategies and the relationship between multiplication and division. They decomposed dividends in various ways and found partial quotients. The numbers they encountered then were limited to four-digit dividends and one-digit divisors. In this unit, they extend that work to include two-digit divisors.

As they build their facility with multi-digit multiplication and division, students solve problems about area and volume and reinforce their understanding of these concepts.

This section introduces the standard algorithm for multiplication, extending students’ earlier work on multiplication. In grade 4, students used diagrams and partial-products algorithms to find the product of a one-digit number and a number up to four digits, and the product of 2 two-digit numbers. They attended to the role of place value along the way.

Students revisit these strategies and representations here, but work with factors with more digits than encountered in grade 4. They make connections between the partial products in diagrams and previous algorithms to the numbers in the standard algorithm. They also learn the notation for recording new place-value units that result from finding partial products.

When using the standard algorithm to multiply a two-digit number and a three-digit number, students account for the place value of the digits being multiplied, as they had done before.

For example, the 3 in 23 represents 3 ones, so \(3 \times 123\) is 369.

The 2 in 23, however, represents 2 tens, so the partial product is \(2 \times 10 \times 123\) or 2,460, instead of \(2 \times 123\) or 246.

The partial products 369 and 2,460 can be seen in a diagram as well.

Once students have practiced recording products this way, they learn to multiply factors that require composing new units, such as \(264 \times 38\).

In grade 4, students found whole-number quotients and remainders with up to four-digit dividends and one-digit divisors using strategies based on place value and partial quotients. In grade 5, they extend this work to include quotients involving two-digit divisors.

Students begin with an exploration that relates division of large numbers to a real-world context. They use strategies based on place value and the relationship between multiplication and division to estimate how the world’s longest noodle could be shared. Then, they analyze and use different ways to decompose a dividend.

For instance, here are two ways to divide 448 by 16:
 

\(\displaystyle{\begin{align} 160\div 16&= 10\\ 160\div 16 &= 10\\ 80 \div 16 &= \phantom{0}5 \\48 \div 16 &= \phantom{0}3  \\\overline {\hspace{5mm}448 \div 16} &\overline{\hspace{1mm}=  28 \phantom{000}}\end{align}}\)

Students see that some decompositions may be more helpful than others for finding whole-number quotients. They use this insight to make sense of algorithms using partial quotients that are more complex.  

Note that use of the standard algorithm for division is not an expectation in grade 5, but students can begin to develop the conceptual understanding needed to do so. The algorithms using partial quotients seen here are based on place value, which will allow students to make sense of the logic of the standard algorithm they’ll learn in grade 6.

The final section invites students to use multiplication and division of whole numbers to estimate large quantities and solve real-world and mathematical problems.

Students encounter area and volume problems in the context of geography—the area of states—and everyday consumption—the volume of milk consumed, the area of plastic waste in the Pacific Ocean, and the volume of recyclable plastic shipped abroad for processing.

New Mexico is about 596 km long and 552 km wide. Which is larger, the garbage patch or New Mexico?

The section ends with an additional opportunity for mathematical modeling. Students estimate and calculate the weight of food waste produced in the United States per year, using an average per-person amount. They also estimate and reflect on the amount of their own food waste.

In this unit, students expand their knowledge of decimals to read, write, compare, and round decimals to the thousandths. They also extend their understanding of place value and numbers in base ten by performing operations on decimals to the hundredth.

In grade 4, students wrote fractions with denominators of 10 and 100 as decimals. They recognized that the notations 0.1 and \(\frac{1}{10}\) express the same amount and are both called “one tenth.” They used hundredths grids and number lines to represent and compare tenths and hundredths.

Here, students likewise rely on diagrams and their understanding of fractions to make sense of decimals to the thousandths. They see that “one thousandth” refers to the size of one part if a hundredth is partitioned into 10 equal parts, and that its decimal form is 0.001. Diagrams help students visualize the magnitude of each decimal place and compare decimals.

Locate 0.001 on each number line.
 

Students then apply their understanding of decimals and of whole-number operations to add, subtract, multiply, and divide decimal numbers to the hundredths, using strategies based on place value and the properties of operations.

They see that the reasoning strategies and algorithms they used to operate on whole numbers are also applicable to decimals. For example, addition and subtraction can be done by attending to the place value of the digits in the numbers, and multiplication and division can still be understood in terms of equal-size groups.

In grade 6, students will build on the work here to reach the expectation to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

In this section, students reason about decimals to the thousandths place. They begin by representing decimals on gridded area diagrams, where the large square has a value of 1, and each small square within represents \(\frac{1}{100}\). Students learn that if they partition each small square into tenths, each of those parts represents a thousandth of the large square.

The diagram highlights the relationships between place values. For instance, each thousandth is \(\frac{1}{10}\) of a hundredth and each hundredth is 10 thousandths.

It also helps to illustrate the structure of the number in its expanded form. In this case, the shaded region includes 3 tenths, 6 hundredths, and 8 thousandths, which can be written as \((3 \times 0.1) + (6 \times 0.01) + (8 \times 0.001)\).

This awareness helps to prepare students for multiplication of a decimal by a whole number later in the unit. 

Students then move on to using number lines to represent decimals and to compare, order, and round them. This number line shows that \(92.415 < 92.451\) because 92.415 is further to the left. It also shows that 92.451 rounded to the nearest hundredth is 92.45 and rounded to the nearest tenth it is 92.5.

In this section, students add and subtract decimals to the hundredths. They begin by adding and subtracting in ways that make sense to them, which prompts them to relate the operations to those on whole numbers. It also allows the teacher to take note of the strategies and algorithms they choose, including the standard algorithm and those that use expanded form.

Adding and subtracting decimals using the standard algorithm brings up a new question in terms of how the digits should be aligned. To highlight this consideration, students analyze a common error as shown here.

Before using the standard algorithm, students use place-value reasoning to decide whether sums and differences are reasonable and to ensure that the digits in the numbers are aligned correctly. As they take care to align tenths with tenths and hundredths with hundredths, students practice attending to precision (MP6).

In this section, students learn to multiply decimals. They continue to think in terms of place value, make connections with whole-number operations, and use diagrams to support their reasoning.

Students begin by multiplying a whole number and a decimal. To find \(2 \times 0.43\), for instance, they shade 43 hundredths in each of two large squares, and see that the 86 shaded pieces or 86 hundredths, which is 0.86.

Diagrams also help students relate products of decimals and products of whole numbers. This diagram shows 2 groups of 43 shaded pieces where each piece is 0.01. The combined shaded region therefore represents \((2 \times 43) \times 0.01\).

Likewise, \(15 \times 0.26\) can be viewed as 15 groups of 26 hundredths or \(15 \times 26\) hundredths. Because \(15 \times 26\) is 390, the value of \(15 \times 0.26\) is 390 hundredths or 3.90.

Next, students reason about the product of two decimals. Diagrams are helpful here as well.

For example, \(1.5 \times 0.4\) can be represented by the area of a rectangle with side lengths of 1.5 and 0.4.

Students can see that the result is 60 hundredths or 0.60 because there are \(15 \times 4\)  or 60 shaded pieces and each represents a hundredth.

They also recognize that they can decompose the shaded region and find \(1 \times 0.4\) (the shaded area in the first large square) and \(0.5 \times 0.4\) (the shaded area in the second large square) and add these partial products: \((1 \times 0.4) + (0.5 \times 0.4)\).

In this section, students use the relationship between multiplication and division and the idea of equal groups to make sense of division of decimals, just as they had done with whole numbers and fractions.

Students learned previously that the expression \(8 \div 2\) can mean finding how many are in one group if 8 is put into 2 equal groups, or it can mean finding how many groups of 2 are in 8.

Here, students interpret \(8 \div 0.1\) to mean finding how many groups of 1 tenths are in 8. There are 10 tenths in 1, so there must be 80 tenths in 8, so \(8 \div 0.1 = 80\). This understanding provides a foundation for students to divide a whole number by any amount of tenths or hundredths.

For instance, to find the value of \(2 \div 0.2\), we can see how many groups of 2 tenths are in 2.

There are 5 groups of 2 tenths in 1, so there must be \(2 \times 5\) or 10 groups of 2 tenths in 2, as shown in the diagram.
 

\(2 \div 0.2 = 10\)

When dividing a decimal by a whole number, the other interpretation of division may be more intuitive.

For example, \(0.2 \div 5\) can mean putting 0.2 into 5 equal groups and finding the size of each group. The diagram shows 4 hundredths in each group, so \(0.2 \div 5 = 0.04\).

Later in the unit, students use equivalent expressions to find quotients. For example, they reason that \(6 \div 0.4\) is equivalent to \(60 \div 4\) because both the dividend and divisor are multiplied by 10. If the value of \(60 \div 4\) is 15, then the value of \(6 \div 0.4\) is also 15.

In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4.

In grade 4, students learned the value of each digit in a whole number is 10 times the value of the same digit in a place to its right. Here, they extend that insight to include decimals to the thousandths. Students recognize that the value of each digit in a place (including decimal places) is \(\frac {1}{10}\) the value of the same digit in the place to its left.

Next, students turn their attention to fractions. In earlier grades, students made sense of equivalent fractions, added and subtracted fractions with the same denominator, and added tenths and hundredths. In this unit, they add and subtract fractions with different denominators. They see that the key is to find a common denominator and analyze different techniques for doing so.

Students then solve problems that involve measurement data (in halves, fourths, and eighths) that are displayed on line plots.

In the final section, students reason about the size of a product of fractions and that of the factors. This work builds on the multiplicative comparison work in grade 4, in which students compared a whole number as “_____ times as many (or as much) as” another whole number. Here, students reason about products of a whole number and a fraction without finding the value of each product. They use diagrams and expressions to support their reasoning.

Write \(<\), \(>\), or \(=\) in each blank to make true statements.

In this section, students extend their understanding of place value and apply it to perform conversions between different, mostly metric, units.

Students begin by observing that the value of the digit in each place is 10 times the value of the same digit in the place to its right and \(\frac{1}{10}\) the value of the same digit in the place to its left. They see that this applies not only to whole-number places but also to decimal places. Students then learn to use exponential notation for powers of 10 and use this notation to represent very large numbers such as 1 million or 1 billion.

Next, students reason about measurement conversions in metric and customary units. Conversion in metric units further highlights place-value relationships in numbers in base ten. For example, this table shows some distances in centimeters, meters, and kilometers.

Students notice that multiplying or dividing by a power of 10 shifts the position of the digits in a decimal number to the right or left. 

As they perform conversions from a larger unit to a smaller unit and the other way around, students apply what they learned about performing operations on whole numbers and decimals.

In this section, students learn to add and subtract fractions (including mixed numbers) with unlike denominators and apply this learning to solve problems. 

Students begin to add and subtract fractions using strategies and diagrams that make sense to them, relying what they know about adding and subtracting fractions with like denominators and with equivalent fractions. They then consider ways to write equivalent fractions so that the fractions in an expression have the same denominator. Later, they analyze and then use numerical strategies for finding common denominators, such as multiplying the denominators and finding a common multiple.

At the end of the section, students create line plots to display measurement data in fractional units (halves, fourths, and eighths), interpret the data on line plots, and use all four fraction operations to solve problems involving fractional measurements.

Do all of Mai’s apricots together weigh more or less than a pound?

In this section, students build on their understanding of multiplication to include the concept of scaling. They interpret multiplication expressions as a quantity that is resized or scaled by a factor. This idea builds on the multiplicative comparison work students did with whole numbers in grade 4.

To develop an understanding of this concept, students compare the value of multiplication expressions without performing the multiplication. Early in the section, the expressions are such that one factor is the same and the other one is different.

Students use visual representations to help them compare products. For instance, the following diagrams can represent \(\frac{2}{7} \times 3\) and \(\frac{2}{7} \times 5\).

 

 

Students also reason about products with one unknown factor, which prompts them to make the comparisons based on the size of the other factor.

In this unit, students learn about the coordinate grid, deepen their knowledge of two-dimensional shapes, and use the coordinate grid to study relationships of pairs of numbers in various situations.

Here, students learn about grids that are numbered in two directions. They see that the structure of a coordinate grid allows us to precisely communicate the location of points and shapes.

Students also continue to study two-dimensional shapes and their attributes. In grade 3, they classified triangles and quadrilaterals by the presence of right angles and sides of equal length. In grade 4, they learned about angles and parallel and perpendicular lines, which allowed them to further distinguish shapes. In this unit, students use these insights to make sense of the hierarchy of shapes.
 

Later in the unit, students analyze and generate numerical patterns based on pairs of rules and graph pairs of numbers on the coordinate grid. They also interpret points on the coordinate grid in terms of situations, plot points to better understand the relationship between two sets of numbers, and use the coordinate grid to solve problems.

This section introduces students to the coordinate grid. 

Students begin by drawing rectangles based only on verbal descriptions. They first do so without a grid, then on an unmarked grid, and finally on a coordinate grid. Along the way, they recognize that numbered grid lines allow them to locate points and communicate the features of shapes precisely. 

Students then learn to use the numbers on the horizontal axis and vertical axis to describe the position of points and plot them on the coordinate grid. They learn that pairs of numbers such as \((1,4)\), called coordinates, describe the numbers of units a point is from the axes and the point \((0,0)\), which is called the origin. 

For example, \((7,0)\) is 7 units to the right of \((0,0)\) and is on the horizontal axis. The point \( (1,4)\) is 1 unit to the right of \( (0,0)\) and 4 units up.

In other words, the first number tells us its horizontal position, and the second number tells us its vertical position.

Students then practice plotting points given their coordinates and identifying the coordinates of points on the grid.

In this section, students classify quadrilaterals and triangles into different categories and study the relationships between the categories.

They begin by sorting a large set of quadrilaterals in a way that makes sense to them, using attributes such as angle measures (especially right angles) and pairs of parallel sides. Then, they focus on relating the attributes of trapezoids, rectangles, parallelograms, squares, and rhombuses.

Students explore two ways of defining trapezoids. One way is to say a parallelogram is a trapezoid, and the other is to say that a parallelogram is not a trapezoid. In this course, the former (inclusive) definition is used.

Students then study the relationship between squares and rhombuses, and between rectangles and parallelograms. They build these shapes with toothpicks, and see that a square is a special kind of rhombus and a rectangle is a special kind of parallelogram.

As they learn more about the relationships between quadrilateral categories, students use a Venn diagram to highlight their understanding.

In this section, students apply the concepts of this unit as they analyze numerical relationships between two quantities in different contexts.

Students begin by examining patterns in numbers generated by following a pair of rules. They record the patterns in a table and interpret the relationships between the pairs of numbers. Students learn that they can form ordered pairs using corresponding terms from each pattern and these pairs can be graphed on the coordinate grid, which allows them to better understand the behavior of the patterns.
 

Rule 1: Start at 0. Keep adding 10.
Rule 2: Start at 0. Keep adding 40.

Use the rules to complete the table.

	A	B	C	D	E	F
rule 1
rule 2

Plot the numbers in the table on the coordinate grid. Label the points.

Next, students use the coordinate grid to explore the relationship of pairs of values in different situations. For instance, they look at the numbers of heads and tails that result from flipping a coin a certain number of times, the number of coins and the value of coins, and the length and width of rectangles with a fixed perimeter or a fixed area.

In this unit, students revisit major work and fluency goals of the grade, applying their learning from the year.

In section A, students deepen their understanding of the standard algorithm for multiplication and practice using it to find the value of products. They also revisit algorithms that use partial quotients to divide whole numbers. In Section B, students solve real-world problems about volume and have opportunities to model with mathematics.

The base of the Great Pyramid of Egypt is a square.
One side length of the base is 230 meters.
The pyramid is 140 meters tall.

If the pyramid was shaped like a rectangular prism,
what would be the volume of the prism?

Section C focuses on operation with decimals and fractions. In the final section, students review major work of the grade as they create activities in the format of the warm-ups routines they have encountered throughout the year (Notice and Wonder, Estimation Exploration, Number Talk, True or False, and Which One Doesn’t Belong?).

The sections in this unit are standalone sections, not required to be completed in order. Within a section, lessons can also be completed selectively and without completing prior lessons. The goal is to offer ample opportunities for students to integrate the knowledge they have gained and to practice skills related to the expected fluencies of the grade.

In this section, students reinforce their understanding of the standard algorithm for multiplication and practice using it. They use estimation to determine the reasonableness of their answers, recognize and explain place-value patterns when multiplying multi-digit numbers, and learn how to use the algorithm when one or more of the factors has several zeros.
 

Here is how Kiran found the value of \(650 \times 27\).
Is the answer reasonable?

Find the value of each product.​​​​​​
 

What's the relationship between \(260 \times 35\) and \(2,\!600 \times 35\)?

Students also practice dividing multi-digit whole numbers using an algorithm involving partial quotients, which they learned in unit 4.

In this section, students revisit the meaning of volume and apply their understanding to solve problems. In each lesson, students estimate and calculate the volume of rectangular prisms in various contexts. The numbers used in this section are larger than the numbers students used in the opening unit, when they first learned how to calculate the volume of rectangular prisms.
 

A company packages 126 sugar cubes in each box.
The box is a rectangular prism.
What are some possible ways they could pack the cubes?

The side lengths of the box are about \(1\frac{7}{8}\) inches by \(3\frac{3}{4}\) inches by \(4\frac{3}{8}\) inches.
What can we say about how the sugar cubes are packed?

The work here prompts students to make reasonable estimates, consider appropriate sizes of units in a given context, and take unit conversion into account in solving problems about volume.

Use the picture of the wagon to make an estimate of the length, width, and height of the wagon bed.

Then, improve your estimate for the volume of the wagon.

In this section, students strengthen their understanding of operations with fractions and decimals by playing a variety of games. Each lesson is structured as a game day.

Students begin with games that involve adding and subtracting fractions and in which the goal is to find the largest or the smallest sums or differences.
 

Round 1

\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}} \,- \, \frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, =\)

• Spin the spinner.
• Write the number in an empty box. Keep the chosen box hidden from view.
• Once a number is written down, it cannot be changed.
• Continue spinning and writing numbers until all 4 boxes have been filled.
• Find the difference. The person with the greatest difference wins the round.

Next, students practice adding and subtracting decimals. The games here likewise prompt students to meet certain goals, such as finding the largest decimal or reaching 1, 0.1, or 0.01.

At the end of the section, students play a game that involves multiplying fractions. All the games about fractions invite students to consider the meaning of the numerator and the denominator and to make strategic choices about the numbers they use in those positions.

Throughout the course, students have engaged in warm-up routines such as Notice and Wonder, Exploration Estimation, True or False, Number Talk, and Which One Doesn’t Belong? This section enables them to apply the mathematics they have learned to design warm-ups that incorporate these routines.

Each lesson is devoted to a particular routine. Students begin by completing at least two partially created tasks, each with more missing parts to complete than the previous one. They practice anticipating responses that others might give to the prompts they pose.
 

Choose 3 shapes from the set of cards.
Draw a fourth shape to complete the Which One Doesn’t Belong?
For each shape, discuss one reason why each shape does not belong.

Which one doesn’t belong?

A
 

B
 

C
 

D