In this unit, students encounter the concept of area, relate the area of a rectangle to multiplication, and solve problems involving area.

In grade 2, students explored attributes of shapes, such as number of sides, number of vertices, and lengths of sides. They measured and compared lengths (including side lengths of shapes). 

In this unit, students make sense of another attribute of shapes: a measure of how much space a shape covers. They begin informally, by comparing two shapes and deciding which one covers more space. Later, they compare more precisely by tiling shapes with pattern blocks and square tiles. Students learn that the area of a flat figure is the number of square units that cover it without gaps or overlaps. 

Students then focus on the area of a rectangle. They notice that a rectangle tiled with squares forms an array, with the rows and the columns as equal-size groups. This observation allows them to connect the area of a rectangle to multiplication—as a product of the number of rows and the number of squares per row.

To transition from counting to multiplying side lengths, students reason about area, using increasingly more abstract representations. They begin with tiled or gridded rectangles, move to partially gridded rectangles or those with marked sides, and end with rectangles labeled with their side lengths. 

\(6\times 3=18\)

Students also learn some standard units of area—square inch, square centimeter, square foot, and square meter—and solve real-world problems involving the areas of rectangles.

Later in the unit, students find the area and the unknown side lengths of figures composed of non-overlapping rectangles. This work includes cases with two non-overlapping rectangles that share one side, which lays the groundwork for understanding the distributive property of multiplication in a later unit.

In this unit, students work toward the goal of fluently adding and subtracting within 1,000. They use mental math strategies developed in grade 2, and learn algorithms based on place value.

In grade 2, students added and subtracted within 1,000, using strategies based on place value, properties of operations, and the relationship between addition and subtraction. When students combine hundreds, tens, and ones, they use place-value understanding. When they decompose numbers to add or subtract, they rely on the commutative and associative properties. When students count up to subtract, they use the relationship between addition and subtraction.

To move toward fluency, students learn a few different algorithms that work with any numbers and are generalizable to greater numbers and decimals. Students work with a variety of algorithms, starting with those that show expanded form, and moving toward algorithms that are more streamlined and closer to the standard algorithm.

Students explore various algorithms but are not required to use a specific one. They should, however, move from the strategy-based work of grade 2 to algorithm-based work, to set the stage for using the standard algorithm in grade 4. If students begin the unit with knowledge of the standard algorithm, it is still important for them to make sense of the place-value basis of the algorithm.

Understanding of place value also comes into play as students round numbers to the nearest multiples of 10 and of 100. Students do not need to know a formal definition of “multiples” until grade 4. At this point, it is enough to recognize that a multiple of 10 is a number called out when counting by 10, or the total in a whole-number of tens (such as 8 tens). Likewise, a multiple of 100 is a number called out when counting by 100, or the total in a whole-number of hundreds (such as 6 hundreds). Students use rounding to estimate answers to two-step problems and to determine if answers are reasonable.

This unit introduces students to the concept of division and its relationship to multiplication.

Previously, students learned that multiplication can be understood in terms of equal-size groups. The expression \(5 \times 2\) can represent the total number of objects when there are 5 groups of 2 objects, or when there are 2 groups of 5 objects.

Here, students make sense of division also in terms of equal-size groups. For instance, the expression \(30 \div 5\) can represent putting 30 objects into 5 equal groups, or putting 30 objects into groups of 5. Students see that, in general, dividing can mean finding the size of each group, or finding the number of equal groups.

30 objects put into 5 equal groups

 

30 objects put into groups of 5

 

Students use the relationship between multiplication and division to develop fluency with single-digit multiplication and division facts. They continue to reason about products of two numbers in terms of the area of rectangles whose side lengths represent the factors, decomposing side lengths and applying properties of operations along the way.

As they multiply numbers greater than 10, students see that it is helpful to decompose the two-digit factor into tens and ones and distribute the multiplication. For instance, to find the value of \(26 \times 3\), they can decompose the 26 into 20 and 6, and then multiply each by 3. 

Toward the end of the unit, students solve two-step problems that involve all four operations. In some situations, students work with expressions that use parentheses to indicate which operation is completed first (for example: \(276 + (45 \div 5) = {?}\)).

In this unit, students make sense of fractions as numbers, using various diagrams to represent and reason about fractions, compare their sizes, and relate them to whole numbers. The denominators of the fractions explored here are limited to 2, 3, 4, 6, and 8.

In grade 2, students partitioned circles and rectangles into equal parts and used the language “halves,” “thirds,” and “fourths.” Students begin this unit in a similar way, by reasoning about the sizes of shaded parts in shapes. Next, they create fraction strips by folding strips of paper into equal parts, and later represent the strips as tape diagrams.

Using fraction strips and tape diagrams to represent fractions prepare students to think about fractions more abstractly as lengths and locations on the number line. This work builds on students’ prior experience with representing whole numbers on the number line.

In each representation, students take care to identify 1 whole. This helps them reason about the size of the parts and whether a fraction is less than or greater than 1. (Fractions greater than 1 are not treated as special cases.)

Students then use these representations to learn about equivalent fractions and to compare fractions.

They see that fractions are equivalent if they are the same size or at the same location on the number line, and that some fractions are the same size as whole numbers.

\(3 = \frac{12}{4}\)

Later in the unit, students compare fractions with the same denominator and those with the same numerator. They recognize that as the numerator gets larger, more parts are counted, and as the denominator gets larger, the size of each part that makes up the whole gets smaller.

In this unit, students measure length, weight, liquid volume, and time. They begin with a study of length measurement, building on their recent work with fractions.

In grade 2, students measured lengths using informal and formal units to the nearest whole number. They also plotted such length data on line plots. Here, students explore length measurements in halves and fourths of an inch. They use a ruler to collect measurements and then display the data on line plots, learning about mixed numbers and revisiting equivalent fractions along the way.

Kiran says that the worm is \(4\frac{2}{4}\) inches long.
Jada says that the worm is \(4\frac{1}{2}\) inches long.
Use the ruler to explain how both of their measurements are correct.

Next, students learn about standard units for measuring weight (kilograms and grams) and liquid volume (liters). To build a sense of the weight of 1 gram or 1 kilogram, students hold common objects, such as paper clips and bottles of water.

To gain familiarity with liters, students measure the volume of a container by filling it with water by the liter and estimate the volume of everyday containers, such as pots, tubs, and buckets. They then use the scale on measurement tools to measure and represent the volume of liquids.

From there, students move on to measure time. In grade 2, they told and wrote time to the nearest 5 minutes. Now, they tell time to the minute, using the relationship between the hour hand and the minute hand to make sense of times such as 3:57 p.m.

In the final section of the unit, students make sense of and solve problems related to all three measurements. The work here allows students to continue to develop their fluency with addition and subtraction within 1,000 and understanding of properties of operations. It also prompts them to use the relationship between multiplication and division to solve problems.

In this unit, students reason about attributes of two-dimensional shapes and learn about perimeter.

Students learn to describe, compare, and sort two-dimensional shapes in earlier grades. In this unit, students continue to develop language that is increasingly more precise to describe and categorize shapes. Students learn to classify broader categories of shapes (quadrilaterals and triangles) into more specific subcategories based on their attributes. For instance, they study examples and non-examples of rhombuses, rectangles, and squares, to recognize their specific attributes.

These are rectangles.

These are not rectangles.

Students also expand their knowledge about attributes that can be measured.

Previously, they learned the meaning of area and found the area of rectangles and figures composed of rectangles. In this unit, students learn the meaning of perimeter and find the perimeter of shapes. They consider geometric attributes of shapes (such as opposite sides having the same length) that can help them find perimeter.

Find the perimeter of this rectangle.

As the lessons progress, they consider situations that involve perimeter, and then those that involve both perimeter and area. These lessons aim to distinguish the two attributes (which are commonly confused) and reinforce that perimeter measures length or distance (in length units) and area measures the amount of space covered by a shape (in square units).

At the end of the unit, students solve problems in a variety of contexts. They apply what they learn about geometric attributes of shapes, perimeter, and area, to design a park, and a West African wax print pattern. They then solve problems within the context of their design.

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

In Section A, students reinforce what they learned about fractions, their sizes, and their locations on the number line. In Section B, students deepen their understanding of perimeter, area, and scaled graphs by solving problems about measurement and data. Two of the lessons invite students to design a tiny house that meets certain conditions and to calculate the cost for furnishing it.

Section C enables students to work toward multiplication and division fluency goals through games. In Section D, students review major work of the grade as they create activities in the format of the Warm-up routines they have encountered throughout the year (Notice and Wonder, Estimation Exploration, Number Talk, and How Many Do You See?).

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

The concepts and skills strengthened in this unit prepare students for major work in grade 4: comparing, adding, and subtracting fractions, multiplying and dividing within 1,000, and using the standard algorithm to add and subtract multi-digit numbers within 1 million.

The sections in this unit are standalone sections, with no requirement to be completed in order. Within each section, many lessons also can be completed independently of those preceding them. 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.