Students learn about and use the relationship between multiplication and division, place value understanding, and the properties of operations to multiply and divide whole numbers within 100. They also represent and solve two-step word problems using the four operations.
Unit Narrative
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 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 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 , 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: ).
Use properties of operations, place-value understanding, and the relationship between multiplication and division to divide within 100.
Section Narrative
In this section, students perform division in which the quotient or divisor is greater than 10. They apply what they know about place value, the two interpretations of division, and the relationship between multiplication and division to divide greater numbers.
The numbers in the division expressions encourage students to see the divisor as either the number of groups or the number in each group. For example, students may interpret to mean dividing 57 into 3 equal groups. However, given , students may make groups of 15 and see how many are needed to make 90. This flexibility helps students choose methods that are most efficient for them for any given problem.
Students also use the relationship between multiplication and division and place-value understanding to find quotients. For instance, to find the value of , students may reason as follows:
In both cases, students see that there are 3 groups of 26 in 78.
Represent and solve “how many groups?” and “how many in each group?” problems.
Section Narrative
In this section, students encounter situations involving the questions “how many in each group?” and “how many groups?” They make sense of division in terms of finding the answers to these questions.
The focus here is on interpreting descriptions, diagrams, and expressions that represent division situations. Students see that the same diagram or expression can represent different questions. For example, the expression can represent two different questions about 6 blocks being put into stacks of 2 or into 2 equal stacks.
Later, students generalize their observations about division situations and interpret division expressions without a context.
Use properties of operations to develop fluency with single-digit multiplication facts, and their related division facts.
Section Narrative
In this section, students explicitly relate division to the unknown factor in a multiplication equation. For example, the quotient in is the unknown factor in . Students use this insight and their knowledge of multiplication facts to identify division facts.
To develop fluency, students reason about patterns in a multiplication table and notice that multiplication is commutative. For instance, if they know the value of , they also know that of .
Students also reason about the product of two factors by decomposing one of the factors. For instance, to find the value of , they can decompose the 7 into 5 and 2 and find the value of . Visually, the product can be represented by the area of a 7-by-3 rectangle that has been decomposed into two rectangles that are 5 by 3 and 2 by 3.
This line of reasoning develops students' intuition for the distributive property of multiplication. (Note that students are not expected to know the names of the properties of operations.)
Use properties of operations and place-value understanding to develop strategies to multiply within 100 and to multiply one-digit numbers by a multiple of 10.
Section Narrative
In this section, students use various strategies based on place value and properties of operations to multiply greater numbers.
Students first multiply one-digit numbers and multiples of 10 and observe the associative property of multiplication. They interpret to mean 3 groups of 2 tens, which is 6 tens. This means can be evaluated by finding or .
These insights enable students to then multiply other one- and two-digit factors (not limited to multiples of 10) and find products within 100.
The representations used here (base-ten diagrams, gridded rectangles, and ungridded area diagrams) encourage students to also use their understanding of place value and to decompose two-digit factors into tens and ones as they multiply.
