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 IM Grade 3, students made sense of multiplication and division of whole numbers in terms of equal-size groups. In IM 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:

Students learn that a fraction like \(\frac{4}{3}\) can also represent:

• A division situation, where 4 objects are being shared equally 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 a rectangle 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 is used 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.

In this unit, students find the product of two fractions, divide a whole number by a unit fraction, and divide a unit fraction by a whole number.

Previously, students made sense of multiplication of a whole number and a fraction in terms of the side lengths and area of a rectangle. In this unit, students make sense of multiplication of two fractions the same way. Students interpret area diagrams with two unit fractions for their side lengths, a unit fraction and a non-unit fraction, and then two non-unit fractions.

Through repeated reasoning, students notice regularity in the value of the product (MP8). They generalize that it can be found by multiplying the numerators and multiplying the denominators of the factors:
 

\(\displaystyle{\frac{a}{b} \times \frac{c}{d}  = \frac{a\times c}{b\times d}}\)

For example, \(\frac{2}{4} \times \frac{3}{5}\) is \(\frac{2 \ \times \ 3}{4 \ \times \ 5}\) ​​​because there are \(4 \times 5\) equal parts in the whole square and \(2 \times 3\) parts are shaded.

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Next, students make sense of division situations and expressions that involve a whole number and a unit fraction. They recall that division can be understood in terms of finding the number of equal-size groups or finding the size of each group.

For instance, students interpret \(\frac{1}{3} \div 4\) to mean finding the size of one part if \(\frac{1}{3}\) is split into 4 equal parts, and \(4 \div \frac{1}{3}\) to mean finding how many \(\frac{1}{3}\)s are in 4.

Students consider how changing the dividend or the divisor changes the value of the quotients and look for patterns (MP8). They use tape diagrams to represent and reason about division situations and expressions.

Later in the unit, students apply what they learned to solve problems. The relationship between multiplication and division is reinforced when they notice that both operations can be used to solve the same problem.

In this unit, students multiply multi-digit whole numbers, using the standard algorithm, and begin working toward end-of-grade expectations for fluency. They also find whole-number quotients, with up to four-digit dividends and two-digit divisors.

In IM Grade 4, students used strategies based on place value and the properties of operations to multiply a whole number of up to four digits by a one-digit whole number, and to multiply a pair of two-digit numbers. They decomposed the factors by place value, and used diagrams and algorithms using partial products to record their reasoning. 

Here, students build on those strategies to make sense of the standard algorithm for multiplication. They recognize that it also is based on place value but records the partial products in a condensed way.
 

Han and Elena used different algorithms to find the value of \(3 \times 318\).

Explain to your partner what Han and Elena did. What does the 2 represent in Elena's algorithm?

In grade 4, students also found whole-number quotients, using place-value strategies and the relationship between multiplication and division. They decomposed dividends in various ways and found partial quotients. The numbers they encountered then were limited to four-digit dividends and one-digit divisors. In this unit, they extend that work to include two-digit divisors.

As they build their facility with multi-digit multiplication and division, students solve problems about area and volume and reinforce their understanding of these concepts.

In this unit, students expand their knowledge of decimals to read, write, compare, and round decimals to the thousandths. They also extend their understanding of place value and numbers in base ten by performing operations on decimals to the hundredth.

In IM Grade 4, students wrote fractions with denominators of 10 and 100 as decimals. They recognized that the notations 0.1 and \(\frac{1}{10}\) express the same amount and are both called “one tenth.” Students used hundredths grids and number lines to represent and compare tenths and hundredths.

Students rely on diagrams and their understanding of fractions to make sense of decimals to the thousandths. They see that “one thousandth” refers to the size of one part if a hundredth is partitioned into 10 equal parts, and that its decimal form is 0.001. Diagrams help students visualize the magnitude of each decimal place and compare decimals.

Locate 0.001 on each number line.
 

Students then apply their understanding of decimals and of whole-number operations to add, subtract, multiply, and divide decimal numbers to the hundredths, using strategies based on place value and the properties of operations.

Students see that the reasoning strategies and algorithms they used to operate on whole numbers are also applicable to decimals. For example, addition and subtraction can be done by attending to the place value of the digits in the numbers, and multiplication and division can still be understood in terms of equal-size groups.

In IM Grade 6, students will build on the work to reach the expectation to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4.

Next, students turn their attention to fractions. In earlier grades, students made sense of equivalent fractions, added and subtracted fractions with the same denominator, and added tenths and hundredths. In this unit, they add and subtract fractions with different denominators. They see that the key is to find a common denominator and analyze different techniques for doing so.

Students then solve problems that involve measurement data (in halves, fourths, and eighths) that are displayed on line plots.

In the final section, students reason about the size of a product of fractions and the sizes of the factors. This work builds on the multiplicative comparison work in grade 4, in which students compared a whole number as “_____ times as many (or as much) as” another whole number. Here, students reason about products of a whole number and a fraction, without finding the value of each product. They use diagrams and expressions to support their reasoning.

Write \(>\), \(<\), or \(=\) in each blank to make true statements.

In this unit, students learn about the coordinate grid, deepen their knowledge of two-dimensional shapes, and use the coordinate grid to study relationships of pairs of numbers in various situations.

Students learn about grids that are numbered in two directions. They see that the structure of a coordinate grid allows us to precisely communicate the location of points and shapes.

Students also continue to study two-dimensional shapes and their attributes. In IMG Grade 3, they classified triangles and quadrilaterals by the presence of right angles and sides of equal length. In IM Grade 4, they learned about angles and parallel and perpendicular lines, which allowed students to further distinguish shapes. In this unit, students use these insights to make sense of the hierarchy of shapes.
 

Later in the unit, students analyze and generate numerical patterns based on pairs of rules and graph pairs of numbers on the coordinate grid. They also interpret points on the coordinate grid in terms of situations, plot points to better understand the relationship between two sets of numbers, and use the coordinate grid to solve problems.

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

In Section A, students deepen their understanding of the standard algorithm for multiplication and practice using it to find the value of products. They also revisit algorithms that use partial quotients to divide whole numbers. In Section B, students solve real-world problems about volume and have opportunities to model with mathematics.

The base of the Great Pyramid of Giza is a square.
Each side of the base is 230 meters long.
The pyramid is now 137 meters tall.

If the pyramid was shaped like a rectangular prism,
what would be the volume of the prism?

Section C focuses on operations with decimals and fractions. 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 (Notice and Wonder, Estimation Exploration, Number Talk, True or False? and Which Three Go Together?).

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