Scope and Sequence

(with Spanish)

Narrative

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

Unit 1: Encontremos volúmenes

Unit Learning Goals
  • Students find the volume of right rectangular prisms and solid figures composed of two right rectangular prisms.

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.

Square prism.
Solid Figure.

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.

Rectangular prism. 4 cubes by 2 cubes by 3 cubes.
Prism.
Rectangular prism.

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.


Section A: Cubos unitarios y volumen

Standards Alignments
Addressing 5.MD.C.3, 5.MD.C.3.b, 5.MD.C.4, 5.MD.C.5.a, 5.OA.A.2
Section Learning Goals
  • Describe volume as the space taken up by a solid object.
  • Measure the volume of a rectangular prism by finding the number of unit cubes needed to fill it.
  • Use the layered structure in a rectangular prism to find 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.

Unfilled cube.

PLC: Lesson 4, Activity 2, Capas de prismas rectangulares


Section B: Expresiones para encontrar volumen

Standards Alignments
Addressing 5.MD.C.4, 5.MD.C.5.a, 5.MD.C.5.b, 5.OA.A.1, 5.OA.A.2
Section Learning Goals
  • Describe the calculations from the previous section as $\text{length} \times \text{width} \times \text{height}$ or $\text{area of the base} \times \text{height}$.
  • Find volume using $\text{length} \times \text{width} \times \text{height}$ or $\text{area of the base} \times \text{height}$.

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.


PLC: Lesson 5, Activity 3, Un prisma que crece


Section C: Volumen de figuras sólidas

Standards Alignments
Addressing 5.MD.C, 5.MD.C.5, 5.MD.C.5.c, 5.OA.A.1, 5.OA.A.2
Section Learning Goals
  • Find the volume of a figure composed of rectangular prisms.

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.

6-sided rectangular prism.

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.


PLC: Lesson 10, Activity 3, Encontremos el volumen de diferentes maneras


Estimated Days: 11 - 12

Unit 2: Fracciones como cocientes y multiplicación de fracciones

Unit Learning Goals
  • Students develop an understanding of fractions as the division of the numerator by the denominator, that is $a \div b = \frac{a}{b}$, and solve problems that involve the multiplication of a whole number and a fraction, including fractions greater than 1.

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.


Section A: Fracciones como cocientes

Standards Alignments
Addressing 5.NF.B.3
Section Learning Goals
  • Represent and explain the relationship between division and fractions.
  • Solve problems involving division of whole numbers leading to answers that are fractions.

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  \(1\frac{1}{2}\).


PLC: Lesson 3, Activity 3, Interpretemos expresiones


Section B: Fracciones de números enteros

Standards Alignments
Addressing 5.NF.B, 5.NF.B.3, 5.NF.B.4, 5.NF.B.4.a, 5.OA.A.2
Section Learning Goals
  • Connect division to multiplication of a whole number by a non-unit fraction.
  • Connect division to multiplication of a whole number by a unit fraction.
  • Explore the relationship between multiplication and division.

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:

\(4 \times \frac{2}{3}\)

\(\frac{2}{3} \times 4\)

\(4 \times (2 \div 3)\)

\(2 \times (4 \div 3)\)

4 Diagrams of equal length. 3 parts. 2 parts shaded. Total length, 1.

PLC: Lesson 7, Activity 2, ¿Qué distancia corrieron?


Section C: Área y lados de longitud fraccionaria

Standards Alignments
Addressing 5.NF.B, 5.NF.B.3, 5.NF.B.4, 5.NF.B.4.a, 5.NF.B.4.b, 5.OA.A, 5.OA.A.1
Section Learning Goals
  • Find the area of a rectangle when one side length is a whole number and the other side length is a fraction or mixed number.
  • Represent and solve problems involving the multiplication of a whole number by a fraction or mixed number.
  • Write, interpret and evaluate numerical expressions that represent multiplication of a whole number by a fraction or mixed number.

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.

Area 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 \((2 \times 3) + \left (2 \times \frac{2}{5}\right)\).


PLC: Lesson 11, Activity 2, Mayor que uno


Estimated Days: 15 - 17

Unit 3: Multipliquemos y dividamos fracciones


Unit 4: Concluyamos multiplicación y división con números de varios dígitos


Unit 5: Patrones entre valores posicionales y operaciones con decimales


Unit 6: Más operaciones con decimales y fracciones


Unit 7: Figuras en el plano de coordenadas


Unit 8: Conectemos todo