## Scope and Sequence

### (with Spanish)

### Narrative

The big ideas in grade 4 include: developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

The mathematical work for grade 4 is partitioned into 9 units:

- Factors and Multiples
- Fraction Equivalence and Comparison
- Extending Operations to Fractions
- From Hundredths to Hundred-thousands
- Multiplicative Comparison and Measurement
- Multiplying and Dividing Multi-digit Numbers
- Angles and Angle Measurement
- Properties of Two-dimensional Shapes
- Putting it All Together

### Unit 1: Factores y múltiplos

**Unit Learning Goals**

- Students apply understanding of multiplication and area to work with factors and multiples.

In this unit, students extend their knowledge of multiplication, division, and the area of a rectangle to deepen their understanding of factors and to learn about multiples.

In grade 3, students learned that they can multiply the two side lengths of a rectangle to find its area, and divide the area by one side length to find the other side length.

To represent these ideas, they used area diagrams, wrote expressions and equations, and learned the terms “factors” and “products.”

In this unit, students return to the concept of area to make sense of factors and multiples of numbers. Given a rectangle with a particular area, students find as many pairs of whole-number side lengths as they can. They make sense of those side lengths as factor pairs of the whole-number area, and the area as a multiple of each side length.

Students also learn that a number can be classified as prime or composite based on the number of factor pairs it has.

Throughout the unit, students encounter various contexts related to school, gatherings, and celebrations. They are intended to invite conversations about students’ lives and experiences. Consider them as opportunities to learn about students as individuals, to foster a positive learning community, and to shape each lesson based on insights about students.

#### Section A: Comprendamos factores y múltiplos

**Standards Alignments**

Addressing | 4.OA.B.4, 4.OA.C.5 |

**Section Learning Goals**

- Determine if a number is prime or composite.
- Explain what it means to be a factor or a multiple of a whole number.
- Relate the side lengths and area of a rectangle to factors and multiples

In this section, students revisit the ideas of area and factors from grade 3 and encounter the idea of multiples. They begin by building rectangles given specific side lengths and identifying possible areas when only one side length is known. Students use tiles and diagrams to build their understanding before learning new terminology.

Next, students build rectangles given a certain area. They see that the side lengths of the rectangles represent the factor pairs of the given area value. Students also observe the commutative property of multiplication when they see that rectangles with the same pair of side lengths have the same area, regardless of their orientation.

*Build 5 different rectangles with the given width. Record the area of each rectangle in the table.*

\(\hspace{1in}\)area of rectangle\(\hspace{1in}\) | |||||
---|---|---|---|---|---|

2 tiles wide | |||||

3 tiles wide | |||||

4 tiles wide |

Students discover that for some whole-number values of area, only one rectangle can be built, and for other values, more than one rectangle is possible. Likewise, some numbers have only one factor pair (the number itself and 1) and other numbers have more than one factor pair. Students learn that we call the former “prime numbers” and the latter “composite numbers.”

The section closes with an optional game day, which is an opportunity to see students' fluency with multiplication and division within 100.

#### Section B: Encontremos parejas de factores y múltiplos

**Standards Alignments**

Addressing | 4.OA.A.3, 4.OA.B, 4.OA.B.4 |

**Section Learning Goals**

- Apply multiplication fluency within 100 and the relationship between multiplication and division to find factor pairs and multiples.

In this section, students apply and deepen their understanding of the ideas of factors and multiples as they play games and solve problems in context. The activities prompt students to look for patterns in factors, multiples, and prime and composite numbers, and use them to make predictions and generalize their observations.

*Twenty students are playing a game with 20 lockers in a row.
The first student starts with the first locker and opens all the lockers.
The second student starts at the second locker and shuts every other locker.
The third student stops at every third locker and opens it if it is closed or closes it if it is open.*

*Which locker numbers does the third student touch?
How many students touch locker 17?*

In the last lesson, students have a chance to use the ideas from this unit to create geometric art.

Estimated Days: 6 - 8

### Unit 2: Equivalencia y comparación de fracciones

**Unit Learning Goals**

- Students generate and reason about equivalent fractions and compare and order fractions with the following denominators: 2, 3, 4, 5, 6, 8, 10, 12, and 100.

In this unit, students extend their prior understanding of equivalent fractions and comparison of fractions.

In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction \(\frac{1}{b}\) results from a whole partitioned into \(b\) equal parts. They used unit fractions to build non-unit fractions, including fractions greater than 1, and represent them on fraction strips and tape diagrams. The denominators of these fractions were limited to 2, 3, 4, 6, and 8. Students also worked with fractions on a number line, establishing the idea of fractions as numbers and equivalent fractions as the same point on the number line.

Here, students follow a similar progression of representations. They use fraction strips, tape diagrams, and number lines to make sense of the size of fractions, generate equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Students generalize that a fraction \(\frac{a}{b}\) is equivalent to fraction \(\frac{(n \times a)}{(n \times b)}\) because each unit fraction is being broken into \(n\) times as many equal parts, making the size of the part \(n\) times as small \(\frac{1}{(n \times b)}\) and the number of parts in the whole \(n\) times as many \((n \times a)\). For example, we can see \(\frac{3}{5}\) is equivalent to \(\frac{6}{10}\) because when each fifth is partitioned into 2 parts, there are \(2 \times 3\) or 6 shaded parts, twice as many as before, and the size of each part is half as small, \(\frac{1}{(2 \times 5)}\) or \(\frac{1}{10}\).

As the unit progresses, students use equivalent fractions and benchmarks such as \(\frac{1}{2}\) and 1 to reason about the relative location of fractions on a number line, and to compare and order fractions.

#### Section A: Tamaño y ubicación de fracciones

**Standards Alignments**

Addressing | 4.NF.A.1, 4.NF.A.2 |

**Section Learning Goals**

- Make sense of fractions with denominators 2, 3, 4, 5, 6, 8, 10, and 12 through physical representations and diagrams.
- Reason about the location of fractions on the number line.

In this section, students revisit ideas and representations of fractions from grade 3, working with denominators that now include 5, 10, and 12. They use physical fraction strips, diagrams of fraction strips, tape diagrams, and number lines to make sense of the size of fractions and fractional relationships.

Students reason about the relationship between fractions where one denominator is a multiple of the other denominator (such as \(\frac{1}{5}\) and \(\frac{1}{10}\), or \(\frac{1}{6} \) and \(\frac{1}{12}\)). They consider different ways to represent these relationships. Students also compare fractions to benchmarks such as \(\frac{1}{2}\) and 1.

The work in this section prepares students to reason about equivalence and comparison of fractions in the subsequent lessons.

#### Section B: Fracciones equivalentes

**Standards Alignments**

Addressing | 4.NF.A.1 |

**Section Learning Goals**

- Generate equivalent fractions with the following denominators: 2, 3, 4, 5, 6, 8, 10, 12, and 100.
- Use visual representations to reason about fraction equivalence, including using benchmarks such as $\frac{1}{2}$ and 1.

In this section, students develop their ability to reason about and generate equivalent fractions. They begin by using number lines as a tool for finding equivalent fractions and verifying equivalence of two fractions.

Through repeated reasoning, students notice regularity in the visual representations and begin to make sense of a numerical way to determine equivalence and generate equivalent fractions (MP8). They generalize that fraction \(\frac{a}{b}\) is equivalent to fraction \(\frac{n \times a}{n \times b} \).

Note that students do not need to describe this generalization in algebraic notation. Given their understanding of the size of fractions and relationship between fractions, however, they should be able to explain it with fractions that have denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

As they identify and generate equivalent fractions numerically, students apply their knowledge of factors and multiples from an earlier unit.

#### Section C: Comparación de fracciones

**Standards Alignments**

Addressing | 4.NF.A, 4.NF.A.1, 4.NF.A.2 |

**Section Learning Goals**

- Use visual representations or a numerical process to reason about fraction comparison.

By the time they reach this section, students have an expanded set of understandings and strategies for reasoning about the size of fractions. Here, they further develop these skills and work to compare fractions with different numerators and different denominators.

To make comparisons, students may use visual representations, equivalent fractions, and their understanding of the size of fractions (for instance, relative to benchmarks such as \(\frac{1}{2}\) and 1). They may rely on the meaning of the numerator and denominator, and choose a way to compare based on the numbers at hand. Students record the results of comparisons with symbols \(<\), \(=\), or \(>\).

At the end of the section, students learn to write equivalent fractions with a particular denominator as a way to compare any fractions, another opportunity to apply the idea of factors and multiples. Having a numerical strategy notwithstanding, students are still encouraged to use flexible methods to reason about the relative size of fractions.

Estimated Days: 16 - 17

### Unit 3: Extendamos las operaciones a las fracciones

**Unit Learning Goals**

- Students learn that a fraction $\frac{a}{b}$ is a product of a whole number $a$ and a unit fraction $\frac{1}{b}$, or $\frac{a}{b} = a \times \frac{1}{b}$, and that $n \times \frac{a}{b} = \frac{(n \space \times \space a)}{b}$. Students learn to add and subtract fractions with like denominators, and to add and subtract tenths and hundredths.

In this unit, students deepen their understanding of how fractions can be composed and decomposed, and learn about operations on fractions.

In grade 3, students partitioned a whole into equal parts and identified one of the parts as a unit fraction. They learned that non-unit fractions and whole numbers are composed of unit fractions. They used visual fraction models, including tape diagrams and number lines, to represent and compare fractions. In a previous unit, students extended that work and reasoned about fraction equivalence.

Here, students multiply fractions by whole numbers, add and subtract fractions with the same denominator, and add tenths and hundredths. They rely on familiar concepts and representations to do so. For instance, students had represented multiplication on a tape diagram, with equal-size groups and a whole number in each group. Here, they use a tape diagram that shows a fraction in each group.

\(3 \times \frac{1}{5} = \frac{3}{5}\)

In earlier grades, students used number lines to represent addition and subtraction of whole numbers. Here, they use number lines to represent the decomposition of fractions into sums, and to reason about addition and subtraction of fractions with the same denominator, including mixed numbers.

\( \frac{4}{3} + \frac{6}{3} = \frac{10}{3}\)

\(\frac{11}{6} – \frac{7}{6} = \frac{4}{6}\)

Students then apply these skills in the context of measurement and data. They analyze line plots showing fractional lengths and find sums and differences to answer questions about the data.

Lastly, students use fraction equivalence to find sums of tenths and hundredths. For instance, to find \(\frac{3}{10} + \frac{15}{100}\), they reason that \(\frac{3}{10}\) is equivalent to \(\frac{30}{100}\), so the sum is \(\frac{30}{100} + \frac{15}{100}\), which is \(\frac{45}{100}\).

#### Section A: Grupos iguales de fracciones

**Standards Alignments**

Addressing | 4.NF.B.4, 4.NF.B.4.a, 4.NF.B.4.b, 4.NF.B.4.c |

**Section Learning Goals**

- Recognize that $n \times \frac{a}{b} = \frac{(n \space \times \space a)}{b}$.
- Represent and explain that a fraction $\frac{a}{b}$ is a multiple of $\frac{1}{b}$, namely $a \times \frac{1}{b}$.
- Represent and solve problems involving multiplication of a fraction by a whole number.

In this section, students extend their earlier understanding of multiplication as equal groups of whole numbers of objects to now include equal groups of fractional pieces.

*How many do you see? How do you see them?*

Students begin by reasoning about groups containing unit fractions. For instance, they interpret the 5 plates with half an orange each as \(5 \times \frac{1}{2}\), which is \(\frac{5}{2}\). Later, they also reason about groups of non-unit fractions and write expressions to represent the quantities. For instance, 5 groups of \(\frac{3}{4}\) can be expressed as \(5 \times \frac{3}{4}\) or \(\frac{15}{4}\).

Later, students reason with diagrams and equations. Through repeated reasoning, they see regularity in the product of a whole number and a fraction (MP8). The numerator in the resulting fraction is the product of the whole number and the numerator of the fractional factor, and the denominator is the same as in the fractional factor.

\(4 \times \frac{2}{3} = \frac{8}{3}\)

These diagrams also help students see that some fractions can be represented by more than one multiplication expression. Students can reason that \(\frac{8}{3}\) is \(8 \times \frac{1}{3}\), which is also equivalent to \(4 \times 2 \times \frac{1}{3}\) and \(2 \times 4 \times \frac{1}{3}\), and is therefore equivalent to \(4 \times\frac{2}{3}\) and \(2 \times\frac{4}{3}\), respectively.

By circling the diagram in various ways, students can visualize the different combinations of groups, understand their equivalence, and observe the associative property of multiplication. In doing this work, students practice looking for and making use of structure (MP7).

Students then solve problems that involve fraction multiplication, using diagrams and equations to show their reasoning. These diagrams will also be useful in later grades, when students make sense of fractions as quotients.

#### Section B: Sumemos y restemos fracciones

**Standards Alignments**

Addressing | 4.MD.B.4, 4.NF.B.3, 4.NF.B.3.a, 4.NF.B.3.b, 4.NF.B.3.c, 4.NF.B.3.d, 4.NF.B.4.c |

**Section Learning Goals**

- Create and analyze line plots that display measurement data in fractions of a unit ($\frac18, \frac14, \frac12$).
- Represent and solve problems that involve the addition and subtraction of fractions and mixed numbers, including measurements presented in line plots.
- Use various strategies to add and subtract fractions and mixed numbers with like denominators.

In this section, students learn to add and subtract fractions by decomposing them into sums of smaller fractions, writing equivalent fractions, and using number lines to support their reasoning.

Students begin by thinking about a fraction as a sum of unit fractions with the same denominator and then as a sum of other smaller fractions. They represent different ways to decompose a fraction by drawing “jumps” on number lines and writing different equations.

\(\frac{13}{10} = \frac{10}{10} + \frac{3}{10}\)

\(\frac{13}{10} = \frac{5}{10} + \frac{8}{10}\)

Working with number lines helps students see that a fraction greater than 1 can be decomposed into a whole number and a fraction, and then be expressed as a mixed number. This can in turn help us add and subtract fractions with the same denominator. For example, to find the value of \(3 - \frac{2}{5}\), it helps to first decompose the 3 into \(2 + \frac{5}{5}\), and then subtract \(\frac{2}{5}\) from the \(\frac{5}{5}\).

Later in the section, students organize fractional length measurements (\(\frac{1}{2}\), \(\frac{1}{4}\), and \(\frac{1}{8}\) inch) on line plots. They apply their ability to interpret line plots and to add and subtract fractions to solve problems about measurement data.

*What is the difference between the largest and smallest shoe lengths?
Explain or show your reasoning.*

#### Section C: Sumemos décimos y centésimos

**Standards Alignments**

Addressing | 4.NF.A.1, 4.NF.A.2, 4.NF.B.3.c, 4.NF.B.3.d, 4.NF.B.4, 4.NF.B.4.c, 4.NF.C.5 |

**Section Learning Goals**

- Reason about equivalence to add tenths and hundredths.
- Reason about equivalence to solve problems involving addition and subtraction of fractions and mixed numbers.

In this section, students apply their understanding of fraction equivalence to add tenths and hundredths.

In the previous unit, students learned that \(\frac{1}{10} = \frac{10}{100}\). They use this reasoning to add tenths and hundredths by generating equivalent fractions. They also apply what they learned in the previous section to strategically use decomposition and the associative and commutative properties to add three or more tenths and hundredths, including mixed numbers.

This section ends with an optional lesson that allows students to apply what they have learned about multiplication, addition, and subtraction of fractions and mixed numbers to solve a design problem.

Estimated Days: 18 - 20

### Unit 4: De centésimas a cienmilésimas

**Unit Learning Goals**

- Students read, write and compare numbers in decimal notation. They also extend place value understanding for multi-digit whole numbers and add and subtract within 1,000,000.

In this unit, students learn to express both small and large numbers in base ten, extending their understanding to include numbers from hundredths to hundred-thousands.

In previous units, students compared, added, subtracted, and wrote equivalent fractions for tenths and hundredths. Here, they take a closer look at the relationship between tenths and hundredths and learn to express them in decimal notation. Students analyze and represent fractions on square grids of 100 where the entire grid represents 1. They reason about the size of tenths and hundredths written as decimals, locate decimals on a number line, and compare and order them.

Students then explore large numbers. They begin by using base-ten blocks and diagrams to build, read, write, and represent whole numbers beyond 1,000. Students see that ten-thousands are related to thousands in the same way that thousands are related to hundreds, and hundreds are to tens, and tens are to ones.

As they make sense of this structure (MP7), students see that the value of the digit in a place represents ten times the value of the same digit in the place to its right.

Students then reason about the size of multi-digit numbers and locate them on number lines. To do so, they need to consider the value of the digits. They also compare, round, and order numbers through 1,000,000. They also use place-value reasoning to add and subtract numbers within 1,000,000 using the standard algorithm.

Throughout the unit, students relate these concepts to real-world contexts and use what they have learned to determine the reasonableness of their responses.

#### Section A: Decimales con décimas y centésimas

**Standards Alignments**

Addressing | 4.NF.C, 4.NF.C.5, 4.NF.C.6, 4.NF.C.7 |

**Section Learning Goals**

- Represent, compare, and order decimals to the hundredths by reasoning about their size.
- Write tenths and hundredths in decimal notation.

Previously, students learned that there are 10 hundredths in 1 tenth and explored tenths and hundredths in fraction notation. In this section, they learn to represent and reason about tenths and fractions in decimal notation.

Students relate \(\frac{1}{10}\) to the notation 0.1 and \(\frac{1}{100}\) to 0.01. They learn to read 0.1 as “one tenth” and 0.01 as “one hundredth,” the same way these numbers are called when written in fraction notation. To see the connections between the fraction notation, decimal notation, and the word name, students reason with unit squares (representing 1) divided into hundredths.

The squares in this section are shaded from left to right, to reflect the digits in a decimal. For example, the number 1.33 is represented by shading a full square that represents 1, 3 columns in the next large square, and 3 small squares in the adjacent column.

The structure of the unit square grid helps to illustrate the equivalence of \(\frac{10}{100}\) and \(\frac{1}{10}\). It also allows students to see that 0.10 is equivalent to 0.1, and to generalize it to other equivalent tenths and hundredths, for instance \(0.20 = 0.2\) and \(0.5 = 0.50\).

In these materials, decimals less than 1 are expressed with a leading zero. Consider explaining to students the zero is sometimes omitted and this doesn’t impact the value of the decimal.

Later in the section, students use benchmarks such as 0.5 and the relationship between tenths and hundredths to locate and label decimals on a number line. They compare and order decimals based on size and write comparison statements using the symbols <, >, and =.

#### Section B: Relaciones entre valores posicionales hasta 1,000,000

**Standards Alignments**

Addressing | 4.NBT.A.1, 4.NBT.A.2, 4.NBT.B.4 |

**Section Learning Goals**

- Read, represent, and describe the relative magnitude of multi-digit whole numbers up to 1 million.
- Recognize that in a multi-digit whole number, the value of a digit in one place represents ten times what it represents in the place to its right.

In this section, students make sense of whole numbers up to the hundred-thousands place, learn to read and write them, and deepen their understanding of place value.

Students begin by using base-ten blocks and diagrams to represent and reason about multi-digit numbers. They quickly see the limits of using base-ten blocks to represent large numbers when the smallest cube represents 1. For example, this collection represents 1,325. If the smallest block has a value of 10 or ten times as much, however, the same collection would represent 13,250. The reasoning here prepares them to think about place-value relationships.

As students analyze and draw base-ten diagrams and write multi-digit numbers in expanded form, they observe structure and begin to understand the value of the digit in each position (MP7). They see the “ten times” relationship between the value of a digit in one place and that of the same digit in a place to its right. For example, \(300,\!000 = 10 \times 30,\!000\), so the 3 in 347,000 has a value ten times that of the 3 in 34,700.

Students also see this “ten times” relationship as they locate numbers on a number line. If the endpoints of a number line are each ten times those on another number line, points that are in the same position on the two number lines are related by a factor of 10 as well.

Students use these observations of structure to compare, order, and round numbers in the next section.

#### Section C: Comparemos, ordenemos y redondeemos

**Standards Alignments**

Addressing | 4.NBT.A.2, 4.NBT.A.3 |

**Section Learning Goals**

- Compare, order, and round multi-digit whole numbers within 1,000,000.

In grade 3, students compared, ordered, and rounded numbers within 1,000. In this section, they extend that work to include numbers within 1,000,000.

Students begin by placing multi-digit numbers on a number line with increasing levels of precision and then making comparisons. In comparing numbers, including those that are missing digits in some places, they make use of structure to determine the size of numbers and the significance of the value of the digits (MP7).

*Is it possible to fill in the two blanks with the same digit to make:*

\(\boxed{4} \ \boxed{\phantom{0}} \ , \boxed{3} \ \boxed{0} \ \boxed{0}\) less than \(\boxed{3} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\) ?

\(\boxed{\phantom{0}} \ \boxed{4} \ , \boxed{3} \ \boxed{0} \ \boxed{0}\) less than \(\boxed{\phantom{0}} \ \boxed{3} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)?

Previously, students rounded numbers to the nearest multiple of 10 or 100. Here, they round numbers within 1,000,000 to the nearest multiples of 10, 100, 1,000, 10,000, and 100,000. When a number is exactly halfway between two consecutive multiples of 1,000, 10,000, or 100,000, they round up, following the convention used in grade 3 when rounding to the nearest multiple of 10 or 100.

Students apply their understanding of place value and rounding to solve contextual problems. They also engage in aspects of mathematical modeling as they consider the implications of rounding large numbers in different situations (MP4).

#### Section D: Sumemos y restemos

**Standards Alignments**

Addressing | 4.NBT.A, 4.NBT.A.2, 4.NBT.B.4, 4.NF.B.3.c |

**Section Learning Goals**

- Add and subtract multi-digit whole numbers using the standard algorithm.

In grade 3, students used various representations and strategies to add and subtract within 1,000, including strategies that rely on place value. In this section, they build on those strategies while also learning about the standard algorithm for addition and subtraction. They begin working toward the end-of-grade expectation of fluency with addition and subtraction within 1,000,000.

As in earlier grades, students attend to the relationship between addition and subtraction, and find sums and differences by composing and decomposing numbers. They compare an algorithm that uses expanded form and the standard algorithm, and observe the role of place value in both algorithms.

Students start by finding sums that do not require composing a unit in any given place and progress towards those that require composing a unit multiple times.

Likewise, they start by subtracting numbers that don’t require decomposing a unit and move towards differences that require multiple decompositions. Students practice adding and subtracting numbers both in and out of context.

Estimated Days: 22 - 23