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, students 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. Students find as many pairs of whole-number side lengths as they can given a rectangle with a specific area. 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.

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 1 partitioned into \(b\) equal parts. Students used unit fractions to build non-unit fractions, including fractions greater than 1, and represented 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.

In this unit, students deepen their understanding of how fractions can be composed and decomposed, and they 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 \(\frac{45}{100}\).

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. In this unit, students 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 decimals.

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 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. Students 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.

In this unit, students make sense of multiplication as a way to compare quantities. They use this understanding to solve problems about measurement.

In earlier grades, students related two quantities and made an additive comparison, where the key question was “How many more?” Here they make a multiplicative comparison, in which the underlying question is “How many times as many?” For example, if Mai has 3 cubes and Tyler has 18 cubes, we can say that Tyler has 6 times as many cubes as Mai does.

Initially, students reason, using concrete manipulatives and discrete images. Later, they reason more abstractly, using tape diagrams and equations. Comparative language, such as “_____ times as many (or much) as ____” is emphasized, offering students opportunities to attend to precision as they communicate mathematically (MP6).

Write a multiplication equation to compare
the pages read by Elena and Clare.
Use a symbol to represent the unknown.

Next, students use the idea and language of multiplicative relationships to learn about various units of length, mass, capacity, and time, and to convert from larger units to smaller units, within the same system of measurement. For example, they describe 1 kilometer as 1,000 times as long as a meter. Students then use their new knowledge to solve measurement problems.

Elena’s disc went 3 times as far as Clare’s did.
Andre’s disc went 4 times as far as Tyler’s did.

How far did Elena and Tyler throw the disc?

student	distance
Han	17 yards
Lin	\(51\frac{1}{2}\) feet
Clare	\(21 \frac{1}{3}\) feet
Andre	22 yards 2 feet
Elena
Tyler

In this unit, students extend their knowledge of multiplication and division to find products and quotients of multi-digit numbers.

In IM Grade 3, students learned that they could find the value of a product by decomposing one factor into smaller parts, finding partial products, and then combining them. To support this reasoning, they used base-ten diagrams (decomposing two-digit factors into tens and ones) and area diagrams (decomposing one side length into smaller numbers). In this unit, students use those understandings to multiply up to four digits by single-digit numbers, and to multiply a pair of two-digit numbers.

Students begin by generating geometric and numerical patterns that follow a given rule. Students describe features of the patterns that are not explicit in the rule and use ideas and language related to multiplication and multiplicative relationships (such as factors, multiples, double) to explain what they notice. As they generate and analyze patterns, they deepen their understanding of properties of operations.

Next, students reason about products of multi-digit numbers. They transition from using base-ten diagrams to using algorithms to record partial products.

Students learn that they can multiply the factors by place value, one digit at a time, and then organize the partial products vertically. Here are two ways to show partial products for \(3,\!419 \times 8\).

Later in the unit, students divide dividends up to four-digit by single-digit divisors. Students see that it helps to decompose a dividend into smaller numbers and find partial quotients, just as it helps to decompose factors and find partial products.

They also recognize that sometimes it is most productive to decompose a dividend by place value. For instance, to find \(465 \div 5\), we can divide each 400, 60, and 5 by 5.

Students encounter various ways to record the division process, including an algorithm that records partial quotients in a vertical arrangement.

At the end of the unit, students apply their expanded knowledge of operations to solve multi-step problems about measurement in various contexts—calendar days, distance, and population.

\(\begin{align} 400\div 5&= 80\\ 60\div 5 &= 12\\ 5 \div 5 &= \phantom{0}1 \\ \overline {\hspace{5mm}465 \div 5} &\overline{\hspace{1mm}=  93 \phantom{000}} \end{align}\)

In this unit, students deepen and refine their understanding of geometric figures and measurement.

In earlier grades, students learned about two-dimensional shapes and their attributes, which they described informally early on but with increasing precision over time. Here, students formalize their intuitive knowledge about geometric features and draw them. They identify and define some building blocks of geometry (points, lines, rays, and line segments), and develop concepts and language to more precisely describe and reason about other geometric figures.

Jada says figure A shows an angle,
but figure B does not. Do you agree?

Students analyze cases where lines intersect and where they don’t (for example, parallel lines). They learn that an angle is a figure composed of two rays that share the same starting point.

Later, students compare the sizes of angles and consider ways to quantify the comparison. They learn that angles can be measured in terms of the amount of turn one ray makes relative to another ray that shares the same vertex.

Students learn that a 1-degree angle is \(\frac{1}{360}\) of a full turn or full circle and can be used to measure angles. They use a protractor to measure angles in whole-number degrees.

Students also learn that angles are additive. When an angle is composed of multiple non-overlapping parts, the measure of the whole is the sum of the angle measures of the parts. These insights enable students to classify angles (as acute, obtuse, right, or straight) and to solve problems about unknown angle measurements in concrete and abstract contexts.

How many degrees is each marked angle on the clock? Show your reasoning.

In this unit, students deepen their understanding of the attributes and measurement of two-dimensional figures.

Prior to this unit, students learned about some building blocks of geometry—points, lines, rays, segments, and angles. Students identified parallel and intersecting lines, measured angles, and classified angles based on their measurement. In this unit, they apply those insights to describe and reason about characteristics of shapes.

In the first half of the unit, students analyze and categorize two-dimensional shapes—triangles and quadrilaterals—by their attributes. They classify two-dimensional shapes based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Students also learn about symmetry. They identify line-symmetric figures and draw lines of symmetry.

Quadrilaterals N, U, and Z are parallelograms.

Quadrilaterals AA, EE, and JJ are rhombuses.

Write 4–5 statements about the sides and angles of the quadrilaterals in each set.
Each statement must be true for all the shapes in the set.

The second half of the unit gives students opportunities to apply their understanding of geometric attributes to solve problems about measurements (side lengths, perimeters, and angles).

Included in this unit are three optional lessons that offer opportunities for students to strengthen and extend their understanding of symmetry and other attributes of two-dimensional figures.

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 learn about comparing fractions, adding and subtracting fractions, and multiplying fractions and whole numbers. In Section B, they strengthen their ability to add and subtract multi-digit numbers fluently, using the standard algorithm. They also multiply and divide numbers by reasoning about place value and practice doing so strategically.
 

Here are the times of the runners for two teams.
Which team won the relay race?

runner	Diego’s team, time (seconds)	Jada’s team, time (seconds)
1	\(10\frac{25}{100}\)	\(11\frac{9}{10}\)
2	\(11\frac{40}{100}\)	\(9\frac{8}{10}\)
3	\(9\frac{7}{10}\)	\(9\frac{84}{100}\)
4	\(10\frac{5}{100}\)	\(10\frac{60}{100}\)

In Section C, students practice making sense of situations and solving problems that involve reasoning with multiplication and division, including multiplicative comparison and interpreting remainders. In the final section, 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 (Estimation Exploration, Number Talk, and Which Three Go Together?).

The sections in this unit stand alone and are not required to be completed in order. Within a section, lessons also can be completed selectively, 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.