## Scope and Sequence

### 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:

1. Factors and Multiples
2. Fraction Equivalence and Comparison
3. Extending Operations to Fractions
4. From Hundredths to Hundred-thousands
5. Multiplicative Comparison and Measurement
6. Multiplying and Dividing Multi-digit Numbers
7. Angles and Angle Measurement
8. Properties of Two-dimensional Shapes
9. 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
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.

PLC: Lesson 1, Activity 3, ¿Qué áreas pueden construir?

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

Standards Alignments
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.

PLC: Lesson 6, Activity 2, Casilleros dudosos

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
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.

PLC: Lesson 4, Activity 3, Fracciones en rectas numéricas

#### Section B: Fracciones equivalentes

Standards Alignments
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.

PLC: Lesson 8, Activity 2, Rectas numéricas útiles

#### Section C: Comparación de fracciones

Standards Alignments
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.

PLC: Lesson 12, Activity 2, La mayor de todas

Estimated Days: 16 - 17