Students enter division and multiplication with their experience in equipartitioning. From this conceptual development,

a.      students understand that fair sharing of evenly divisible collections produces equal-sized groups,

b.      the size of a fair share can be described as 1/nth of the whole collection or as a particular number of elements out of the total; and

c.       the whole collection is n times as large as one person’s share.

As the Structural Overview illustrates, this Learning Trajectory first highlights the models of division and multiplication (Grades 2-3) that support students’ understanding of these two concepts. It then explores the different problem types, the properties and the strategies that may be used (Grades 3-4) reaching to the examination of factors and multiples at Grades 4-6.

Recent research suggests that division may be taught first as fair sharing and then the context is reversed to introduce multiplication. In this trajectory, both ideas are moved forward gradually to yield the two inverse operations but, unlike in the past where multiplication is taught first and division is taught later, the operations are linked from the outset.

Adding to these, this Learning Trajectory also extends division and multiplication problems to involve multi-digit whole numbers and fractions and decimals.

CCSS-M Description

Descriptor

2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Evenness and oddness are early concepts of “number theory.” They link activities of counting with activities of splitting in critical ways. This becomes apparent as there are two related meanings of evenness and oddness. The first meaning of evenness is that a number is even if it can be equipartitioned into two parts “without remainder.” For instance, a class of students can be asked to line up in twos or form two columns. If all students have a partner, then the number of students in the class is even.

Similarly, an odd number is a number that cannot be equipartitioned into two parts. Doing so results in a remainder of one. In the previous example, if there were an odd number of students in the class, one student would not have a partner.

Students can also use the term “evenness” to refer to sharing between more than two people. Sharing a group of 12 coins among 3 people to produce 4 coins per person can be described as coming out “even,” with the student attending to the fact that the shares came out equal. “Even” in this context relates to a broader usage of “evenness” that focuses on the absence of a remainder after sharing and connects to the fact that the number of people sharing is a factor of the number of objects shared. Evenness in divisibility is thus related to naming a number as “even” versus “odd.” It is important for the teacher to guide students in a) distinguishing the two meanings of “even,” and b) making a deeper connection between the closely related meanings of “evenness” as “being divisible by two” and “evenness” as “being divisible by some other natural number” based on equipartitioning. This is an example of how an apparent misconception evolves into a partial conception with alternative meanings or distinctions.

The second meaning of an even number relates to skip-counting. As students learn skip-counting, they begin to recognize number patterns. Students identify the number sequence “2, 4, 6…” and associate it with the term “even numbers.” Even numbers are produced from skip counting by 2’s (the difference), starting from 2 (the initial term). Odd numbers are produced by skip counting by 2’s (the difference), starting from 1 (the initial term).

Students can use number sentences to justify the “evenness” of a number in two ways. They can write the number as the sum of two identical values, e.g., 12 = 6 + 6, or they can identify the number as a double of that identical value. Students can also discuss doubles and halves of numbers either before the formal introduction of multiplication or division, or the use of the notations. They should be supported in developing the ability to double and halve informally, as an underlying foundation for further development of multiplication and division.

3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

In the next three standards, students are introduced to multiplication and division. These introduce the three primary models for multiplication and division; and these should be introduced gradually with ample time to compare and contrast the models.

Research has shown that very young children in preschool are often able to equipartition (or fair share) collections, before, or at the same time, as they learn to count. In order to find the size of the fair share in equipartitioning problems, children typically use “partitive strategies” (or dealing with one-to-one correspondence to each person). This strategy evolves into division. Hence, a standard about division is listed as the first model for division and multiplication.

Although the standard only references division, research has increasingly shown the benefits of introducing division and multiplication simultaneously as inverse operations that link to different ways in which one asks questions about a given situation. For example, if a student is asked to determine the fair share when six cookies are shared among three people, a problem that results in two cookies per person, this is coded as the division statement: 6 ÷ 3 = 2. However, if a child is asked how many cookies were in the whole collection that results in three children each of whom has a fair share of 2 cookies, then this is coded as multiplication: 3 × 2 = 6.

Thus, the next three descriptors specify one operation of division and multiplication, but the operation is then immediately extended to include its inverse operation, multiplication or division.

To introduce division, building from equipartitioning in earlier grades, students can use and explain the fair sharing box by identifying the meaning of the number in each cell (in terms of the referents of coins and people) (see Bridging Standard 2.OA.B in the Equipartitioning LT).

Figure 1

With this new standard, students write division facts to correspond to the problem including, for example, sharing 12 coins among 3 people, they can write: 12 coins ÷ 3 people = 4 coins per person. They reverse the problem for multiplication as, 4 coins per person × 3 people = 12 coins.

This model of division and multiplication is called Referent Transforming because the referent starts either with “a number of coins” and is divided by “a number of people” resulting in “coins per person” or starts with the “number of coins per person” which is multiplied by the “number of people “ resulting in “the total number of coins.” The referent changes across the process of the operation. In the beginning, students always write in the units because units help them to keep track of their reasoning in number facts.

They also recognize other ways to code division in a fair sharing problem. If students are asked instead, “How many people can share 12 coins if each person is given 4 coins?” They code the problem as the division, 12 coins ÷ 4 coins per person = 3 people. This division statement is also referent transforming because the referent changes across the quantities in the problem.

When the students, asked to find the number of people sharing, try to find how many fair shares make up the whole collection, they typically use repeated subtraction (to reach 0) or repeated addition (to reach the total). This strategy is called a quotative strategy as contrasted to their previous problem’s partitive strategy. Quotative strategy is also called a “measurement” strategy because it requires a measuring the set of 12 coins four times using 3 coins as a measure.

Note to teachers: The distinction between partitive division and quotative strategies is due to the process of obtaining the quotient and not a property of the situation or even the question asked, because students may use either strategy or even other strategies.

Students also recognize the role of the numbers of a division and multiplication. In the previous division:

• 12 is the dividend (the total number of a collection being divided),
• 4 is the divisor (the number of people dividing the collection), and
• the outcome of 3 is the quotient.

In the multiplication,

• 3 is the multiplicand (the number being multiplied),
• 4 is the multiplier (the number multiplying the multiplicand), and
• the outcome of 12 is the product.
• the two numbers which are multiplied together are also called factors.

Note to teachers: The labels for the terms should not be mistaken for the goal of instruction, but rather learning them can help describe one’s strategies or clarify one’s referent.

Students also know from equipartitioning that the reassembly of the shares produces a whole collection that is three times as many as one fair share (see Standard 1.G.B of the Equipartitioning LT). They can now represent this relationship by connecting two vertical cells of the fair sharing box with an arrow pointing from the bottom left cell to the upper left cell and labeling it “x 3” (see Figure 2 below). Students understand that likewise, on the opposite side of the fair share box (the right side), the relationship between one person (the bottom right cell) and the number of people sharing (the upper right cell) is three times as many, and they label this as “x 3”.  Students also recognize that they could have drawn their arrows in the reverse, from top to bottom and labeled them on both sides as “÷ 3. They know that because the operation of multiplication or division is the same on both sides of the box, the quantities are shared fairly.

In a similar way, students recognize that the movement from the lower right cell to the lower left cell can be described multiplicatively as a shift from one person to four coins. The shift from the upper right cell to the upper left cell can be described multiplicatively as 3 people to 12 coins. In both cases, the operation can be described as multiplying by 4 coins per person. Again, the same operation ensures that the referent transforming change is the same in both rows.

Students can use a complete table to write division and multiplication problems. They also see a fundamental pattern: if they connect the cells by same-directional arrows to represent the operation of division or multiplication, the opposite side is connected by the identical operation. When asked to explain the reasons, they can explain the vertical multiplier or divider as “times as many” (this is related to a different model of multiplication in the next standard). They can explain the horizontal multiplier or explaining how the problem switches between people and coins or vice versa. The direction of the arrows and their values should be the same and is a way to check one’s answers for ensuring a fair sharing process.

Figure 2

Adding arrows to the fair share box to show that for each pair of relationships, both horizontal and vertical columns have identical multiplicative relations (and both rows likewise) is critical to understanding multiplicative (or division) structures. When students add the arrows into the fair share box to define explicitly the multiplicative (or division) relation, the box is called a division/multiplication box or D/M box. D/M boxes, like fair sharing boxes, always have a one in a cell, which can be organized so that it is always placed in the bottom right cell. This is because in the problems, there is always a hidden fourth quantity that is equal to one. In the problem, 12 coins shared among three people = 4 coins per person, the hidden one is the single person that goes with the four coins in the phrase “per person.” Later, the D/M box will transform into a “ratio box” where the fourth cell is no longer required to equal the value of one. One can say that fair share boxes and D/M boxes are precursors to ratio boxes in which one cell is equal to one.

Note to teachers: Having a variety of experiences with referent transforming in division and multiplication is critical in developing sound models of these two operations. Students are not required to learn this term, but only be able to describe the quantities in the problems in terms of units and how they change.

Referent transforming problems include two kinds of units: extensive (units that measure a quantity directly) and intensive (units that involve a “per” description). Consider the following problem: “At a garage, there is a stack of twenty brand new tires. Knowing that each new car needs four new tires installed, how many cars can receive their tires?” Students make a D/M box with headers of tires and cars. They know they have 20 tires but don’t know how many cars go with that. They know that 4 tires go with 1 car and solve the problem by reasoning: 20 tires ÷ 4 tires per car = 5 cars. The problem is referent transforming (from tires to cars) and involves two extensive quantities, 20 tires and 5 cars, and one intensive quantity 4 tires per car.

3.OA.G Compare the relative size of a collection to the size of a group that divides evenly into it as “times as many” and the relative size of the group to the collection as “times fewer.”

This Bridging Standard is added because comparing the relative size of the group to a collection helps students develop foundations for making multiplicative comparisons between quantities (see Standard 4.OA.1 later).

Students can compare the size of a collection to the size of a group and write related multiplication and division problems. For example, for the problem “How many times larger is a collection of 18 coins than a group of 3 coins?” students find the answer “6 times” by writing

18 coins ÷ 3 coins = 6 (or they solve the related multiplication problem,

6 × 3 coins = 18 coins). Another possible division problem is, “What is 18 coins 6 times as big as?” Students find the answer “3 coins” by writing 18 coins ÷ 6 = 3 coins (or 3 coins × 6 = 18 coins). Students may use quotative (or measurement) or partitive strategies in solving these problems.

This model for division and multiplication for this type of problems is called Referent Preserving because the problems start and end with coins as the referent being operated on, and both are extensive units. There is no intensive unit in this case; students recognize that in writing the number facts for both division statements and the associated multiplication problems, the units stay the same. They also notice that the 6 in “6 times as large (or big)” is written without a unit.  This can also be called “Scaling” as it describes the process of increasing or decreasing the size of a quantity by a value.

Students have a variety of ways of representing and solving these problems. A common one involves circling sets of three coins and counting how many sets are formed, using a “many-as-one” relationship.

Figure 3a

Another strategy is to use a tree diagram that shows the 18 coins over the three coins and shows that each of the three coins corresponds to six of the 18 coins, forming a “many to one” relationship.

Figure 3b

To understand “times as many,” students often benefit from starting with a group and then redrawing it to show one times as many. A second group is appended and the two groups as a whole represent two times as many. A third group is appended to make a group that is three times as large and called “three times as many”. While this outcome is equivalent in number to “counting” groups, bracketing and forming new whole groups is beneficial for students to understand the implications on the size of the whole group through multiplicative comparison instead of additive comparison (15 more). Students can also describe the relative size of the group to the whole collection as “6 times fewer” which is interpreted as an operation of dividing by 6.

If student describe the collection as “two, three, four, etc. times more than” a single person’s share, they are revealing a misconception by combining additive and multiplicative comparisons, when asked for a multiplicative comparison. This may lead to confusion unless students become attuned to a distinction between multiplicative comparisons and additive comparisons. Later, when students discuss percentage increase and decrease in Grade 6 (see Standard 6.RP.3.c of the Ratio and Proportion, and Percents LT), this idea will resurface and be taught in depth.

Note to teachers: From equipartitioning, students can also describe the share as 1/6th of the collection and will relate this to multiplication by 1/6th (see Standard 4.OA.1 later).

Note to teachers: It is not easy to create a D/M box for a referent preserving division/multiplication problem, because there is only one referent, “coins.” Figure 4a and 4b shows it is possible by having a second column labeled “groups.” A “groups” column is valuable in making explicit the relationship of how many coins are in one group and how many groups there are. However, referent preserving operations are more easily displayed in a single column as shown in Figure 5a with the arrow notation describing the “times as many” relationship between the upper row and the bottom one in order to preserve the size of the group. This can be referred to as a “Referent Preserving” D/M box. Figure 4b and 5b show the relationships using the operation of division.

Figure 4a                                              Figure 4b

Figure 5a                                              Figure 5b

The arrow represents “times as many”                    The arrow represents “times fewer”

Students also describe the multiplication and division with the referent preserving problems. Another way to represent referent preserving or scaling models of multiplication and division is to use the double number line. The double number line is useful in demonstrating that there is only one referent that is segmented into different sized units (units of one in the original and units of greater than one in the other number line).

Figure 6 The unit in the upper line is three times as large as that in the lower line

Within referent preserving problems, multiplication problems were posed in measurement and scaling contexts. In both of these cases, one of the quantities has units and the second defines how many times larger the product will be. Therefore, the product also has units. The multiplication is of the form Y Units × “Times as Large” Number = Z Units. The associated divisions are of the form Z Units ÷ “Times as Large” Number = Y Units and Z Units ÷ Y Units = “Times as Large” Number.

For example, “a whole ribbon measuring 20 inches in length is cut into 4-inch long strips. Compare the length of one strip to the length of the whole ribbon.” As a multiplicative comparison problem, the whole ribbon is 5 times as long as the strip. An analysis of the units in this problem anticipates scale factors (which are not introduced until 8th grade):

20 inches ÷ 4 inches = 5 (a scale factor).

As an equation, this problem can be written as:

20 ÷ 4 =

or

4 ×   = 20

3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

This standard discusses multiplication using simple groups as a model. An example of a simple groups problem is: “There are seven shelves of books. Each shelf has eight books. What is the total number of books?” The total number of books in this case is 7 × 8 = 56 books. An associated division problem would be, “There are 56 books. Each shelf has eight books. How many shelves are holding the books?” A second associated division problem is “There are 56 books placed on 7 shelves. Each shelf holds the same number of books. How many books per shelf are there?”

Groups and number of groups problems can be classified as either a referent transforming and referent preserving model of division and multiplication. As a referent transforming model, the associated D/M box looks like:

Figure 7

As a referent transforming model, the multiplication is written as 7 shelves × 8 books per shelf = 56 books; the referent changes from shelves to books. The associated divisions would be written as 56 books ÷ 7 shelves = 8 books per shelf or 56 books ÷ 8 books per shelf = 7 shelves.

As a referent preserving operation, the D/M table is shown below.

Figure 8

The shelves are viewed as a means to separate the books into groups of eight, producing seven shelves. When interpreting a shelf as a group, students write the multiplication and division as 7 groups (shelves) × 8 books = 56 books and 56 books ÷ 8 books = 7 groups (shelves). Both the multiplication and division statements can be thereby viewed as referent preserving because the unit “books” is the only referent being operated on.

Students also solve the associated division problem by preserving the referent. For instance, “There are 7 shelves holding 56 books. Each shelf holds the same number of books. How many books are there on each shelf?” They write the division statement as 56 books ÷ 7 groups = 8 books.

Note to teacher: There is a difference between two strategies students used to solve division problems depending on what question is asked or how the students approach the problem. Partitive strategies are often used to find the size of a group whereas quotative (measurement) strategies are often used to find the number of groups. For a further discussion of the two strategies, see Standard 3.OA.2 above.

The third model for multiplication is called referent composing and its inverse operation of division is introduced in Bridging Standards 3.OA.F, 3.OA.D, 3.OA.B and Standard 3.OA.3. First, some preliminary standards are presented to prepare students for it through the introduction of arrays.

3.OA.F Students reason with arrays using multiplicative relationships.

This Bridging Standard is added because the representation of the array is an important tool to transition to area representations. It also provides a conceptual basis for students’ understanding of commutative property of multiplication as m × n = n × m, in an m by n array; it is easy to show that n rows of m equals m columns of n as required in the later Standard 3.OA.5.

Students use arrays to represent problems that are best modeled with multiplication and division. There are many strategies they may use in finding the total (in addition to counting and skip counting from Standard 2.NBT.2 of the Counting LT and the earlier Standard 2.OA.4). Developing multiple strategies supports flexibility in solving multiplication and division problems. These strategies include, but are not limited to:

1)      If there is an even number of rows or column, students find the total for half of the array and then double to get the total,

2)      They may use strategies that represent the application of compensation to the multiplicative problem.

3)      For example, given the 6 × 8 array, students see that they can cut it in half in one direction and rearrange the halves to reconfigure the original array to be 12 × 4 or 24 × 2 or 48 × 1. This is akin to quantitative compensation as experienced in equipartitioning (see Bridging Standard 1.G.B of the Equipartitioning LT) and allows them to see how multiplication facts can be related. Students could be given the 6 × 8 array and asked: “Is there any way to rearrange the rows or columns (but not both) in this array to create different arrays?”

Figure 9

Students who learn to operate within multiplicative space, rather than by only shifting to interpret these problems as skip-counting or repeated addition, will strengthen their multiplication and division reasoning to become more integrated.

Students connect the area of a rectangle to the multiplication of length and width by constructing an array representation of the area using unit squares (see Standard 3.MD.7.a and 3.MD.5a of the Length, Area, and Volume LT).

3.OA.D Students learn to code composition of splits as multiplication and can state the associated division problem.

This Bridging Standard is added here because composition of splits supports students’ multiplicative thinking.

Based on the work with composition of splits, students can describe a variety of ways to equipartition (or fold) a rectangle into n parts. These include all the factors of n (although finding the factors of a number is included in Standard 4.OA.4 later). For example, to make a 12-split, the students can fold it vertically (or horizontally) into 12 parts, do a 2-split followed by a 6-split, or a 3-split followed by a 4-split. They avoid the additive misconception that a 6-split followed by a 6-split would result in a 12-split.

In this standard, students learn to code these compositions of splits as multiplication so that a 2-split followed by a 6-split is linked to 2 × 6 =12. The inverse operation of 12 ÷ 6 = 2 is understood by asking students: “A whole rectangle is folded into 12 equal parts. It is folded into 6 parts in the horizontal direction. How many parts is it folded into in the vertical direction?” Therefore, they could code this composition of splits as:

2 × 6 = 12

12 ÷ 6 = 2

Equipartitioning is used in this standard to support the development of multiplication whereas it was used to support division in the previous standard. It is important to stress to students that they are describing the number of parts rather than the size of the parts relative to the whole. Under either description, however, the whole rectangle is 12 times as large as one of the parts. Students can either say the part is 1/12th of the whole, or the whole consists of 12 parts; it depends on what unit is referenced.

They can also write 3 × 4 = 12 and 1 × 12 = 12 with their associated division problems.

3.OA.B Relate multiplication and division problems to rectangular area (e.g., 3 inches × 4 inches = 12 square inches) and Cartesian products (e.g., 3 pants × 2 shirts = 6 possible outfits).

This Bridging Standard is added to highlight the importance of area and Cartesian products in developing students’ modeling of division and multiplication as referent composing.  They are referent composing, because they create the need for new referents and new units (for example, area and square inches), from the referents for the original quantities (length and inches). Area is introduced in Standard 3.MD.7.a of the Length, Area, and Volume LT and this bridging standard links the two trajectories.

Students enjoy systematically figuring out combinations of two characteristics (e.g., the number of possible outfits that can be created from 3 pants and 2 shirts). In the beginning, many students list possible combinations unsystematically. They then begin to find ways to systematize their suggestions and aim to find all possible combinations. The representations below show possible approaches:

Systematic Listing

(P for pants and S for Shirts)

P1S1, P1S2, P1S3, P2, S1.P2S2, P2S3

S1P1, S1P2, S2P1, S2P2, S3P1, S3P2, S3P3

Array (2 dimensions)

Students use a table like the one in the figure above to represent all possible outcomes of combining two characteristics. For example, in the table above, the three different shirt colors are listed on the horizontal axis and the two different pant styles are listed on the vertical axis and students use the representation to see that there are 6 possible combinations of outfits that can be created. The 2-dimensional use of the table connects to students’ array and area models.

A discussion of whether order matters in naming possible combinations will lead to a distinction between combinations and permutations. Later in high school, students learn formally about permutations where order does matter.

3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Based on the previous standards we have established three general models. Below is a list of the kinds of problems associated to each model:

1.      Model 1: Referent Transforming

a.      Fair sharing builds on students’ earlier experience with equipartitioning.

Multiplication example: “To give candies fairly to all students, 5 candies are placed in each of 4 baskets. How many candies are there in total?”

Division example: “20 candies are to be placed in 4 baskets, with each basket having the same number of candies. How many candies per basket will there be?”

b.      Rate and cost problems always involve two quantities with different units, e.g., a number of miles and a number of hours.

Multiplication Example: “Teesha walked at 3 miles per hour for 4 hours. How far did she walk?”

Division example: “Julie paid 35 cents for 5 sticks of gum. What is the cost of one stick of gum?”

c.       Equal Groups problems use a group as a means of transforming the referent.

Multiplication example: “There are nine tables in a cafeteria. Each table has five chairs. What is the total number of chairs in the cafeteria?”

Division example: “Forty-five chairs are arranged in a cafeteria so that each table has five chairs. How many tables are there in the cafeteria?”

Students can build a D/M box for referent transforming models and fill in the missing value in any of the cells that is not equal to one. For rate problems involving unit rates, see Standards 6.RP.2 of the Ratio and Proportion, and Percents LT.

2.      Model 2: Referent Preserving

a.      The Unit Conversion builds on students’ previous experience with measurement.

Multiplication example: “12 inches make 1 foot. How many feet can you make from 36 inches?”

Division example: “36 inches is as long as 3 feet. How many inches is the same as 1 foot?”

b.      Scaling problems involve the multiplication and division of a quantity by a scale factor.

Multiplication Example: “Given a square with sides 3 units long and you enlarge the square by a scale factor of 5, what will be the length of a side of the new square created?

Division Example: “After enlarging a square by a scale factor of 5, the length of the sides is 15 units long, what will the length of a side of a square before enlarging?”

c.       Equal groups problems can be treated as referent preserving if the groups are viewed as segmenting the original referent.

Multiplication Example: “There are two groups of coins. Each group has seven coins. What is the total number of coins when they are all put together?”

Division Example: “There are 14 coins split among two groups. How may coins are there in each group?”

Students can build a D/M box for referent preserving problems but prefer to build 2 × 1 reduced box with the arrow to represent the underlying concept of “times as many”.

1. Model 3: Referent Composing

Area or combination (Cartesian product) models involve referent composing. The factors referents (and units) differ from those of the product: length and width (inches) vs. area (square inches) for area models; Cartesian components (shirts vs. pants) and its products (outfits) in combinations model. Array models (because they start and end with dots or elements) can be viewed as referent preserving but act as an important transitional tool to area models, due to their use of two dimensions (rows and columns transition to length and width).

A related division problem is:

a.      Array problems assist in learning commutativity.

Multiplication Example: “A rectangular candy box has 3 rows and 4 columns of candy. How many pieces of candy does the box hold?”

Division example: “A box holds 12 candies. There are 4 rows of candies in the box. How many columns are there in the box?”

b.      Area problems provide the first example of referent composing.

Multiplication Example: “A closet floor is covered with tiles whose dimensions are one foot by one foot. The closet measures six feet by eight feet. How many tiles cover the floor?”

Division Example: “The floor of a closet room is covered with 48 square tiles whose dimensions are one foot by one foot. One side of the closet floor is measured to be eight feet long. How long is the other side of the closet floor?”

c.       Cartesian products provide students opportunities to explore combinations

Multiplication Example: “Ruben has 3 different T-shirts and 4 different pairs of shorts. How many different outfits (with one T-shirt and one pair of shorts, each) can Ruben make?”

Division Example: “Ruben has twelve different outfits (with one T-shirt and one pair of shorts, each) to choose from. These outfits are created using 3 different T-shirts. How many different pairs of shorts does Ruben have?”

No D/M Box is used for referent composing; students can use rectangular displays or tree diagrams.

For each of the models above, students write equations using a symbol (e.g., a box, a blank, a letter) to represent the unknown number and are able to solve with the unknown in any position.

It will be important in later standards to see to what extent and how the three models support students in using fractional values for each of the factors. For example, Cartesian products and arrays only make sense in the context of using whole numbers while area models can be extended. Grouping and fair sharing models support the use of a fraction for the amount in a group but require a whole number for the number of groups (shares). Area, rate, and scaling models all support the use of fraction for both factors. Even though these extensions do not occur until later, recognizing their implications for later conceptual development is helpful.

Research shows that students are hampered by weak foundations in multiplication and division into and beyond middle grades. Too often this is simply attributed to them not having “adequately memorized their multiplication and division tables.” This explanation is in error. By introducing students to multiple models, they are provided scaffolding to revise and extend their ideas of multiplication and division and this includes their knowledge of models, and their understanding of the structure and interrelationships of number facts and properties.

3.OA.6 Understand division as an unknown-factor problem.

Students recognize that multiplication and division are inverse operations. For instance, when presented with a division problem with a missing value in the divisor, they can solve the problem by using a related multiplication fact.

For example, 32 ÷ = 8 can be solved by using the number fact

8 × 4 = 32 or 4 × 8 = 32. Students may associate pairs of multiplication and division facts based on the inverse relationship between multiplication and division. In connection with the Standard 3.OA.4 of LT Early Equations and Expressions, this is one of the approaches students use to solve multiplication or division equations.

This standard, linking multiplication and division, also represents a movement to generalization. Students begin to recognize situations in which one of the three models apply and know that they can write the associated problems with or without units. At this level, students also begin to lose their awareness of the difference between partitive and quotative or measurement division as they translate the problems into division and solve them with their knowledge of the numeric relationships. This use of generalization involves “curtailing” a mental process, and it should be accompanied by the ability to reconstruct that process for the purposes of explanation, justification or when extending the problem to new situations or new types.

4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

Building from 3.OA.G, students can compare the size of a collection and the size of a group in two ways. For example, for the problem “How many times as bigger is a collection of 18 coins than a group of 3 coins?” Students find the answer “6 times” by writing 18 coins ÷ 3 coins = 6 (or they solve the related multiplication problem, 6 × 3 coins = 18 coins). For the problem, “How many times fewer is a group of 3 coins than a collection 18 coins?”, students find the answer “6 times fewer” by writing 18 coins ÷ 3 coins = 6 (or 6 × 3 coins = 18 coins).

In this standard, students can also describe the size of the group in relation to the size of a collection as “1/6th as large.” This harkens back to Standard 3.NF.1 of the Equipartioning LT where they name the share of 3 coins as 1/6th as many as 18 coins in terms of the relative size.

Figure 10

Another way to see the same comparative relationship is by using the D/M box. The comparison made between 18 coins and 3 coins is the same as between 6 and 1.

Figure 11

Students notice there is also a multiplicative relationship between the number of coins and the number of people. This can be stated as “there are three times as many coins as there are people” or “there is one-third as many people as there are coins.” They recognize that the relationship holds for both rows. Therefore, there are three times as many coins as people. Students write this equation as p · 3 = c. Students also interpret 18 as 3 being made 6 times or 3, 6 times, i.e., 6 is the operator on the quantity of 3 coins.

There is a well-known problem that reveals students’ misconception about equation expressing multiplicative comparisons. The problem is stated, “There are six times as many students as professors. Write an equation to express this relationship.” Students of all ages tend to code this directly as 6S = P rather than correctly as 6P = S. The use of a table of values and a D/M box can assist in avoiding having students develop such a misconception by recognizing that the first sentence expresses a relationship not an equation.

Students can also compare two numbers multiplicatively using the number line to show “times as many.” For instance, in the problem, “Annie’s mother is three times as old as Annie. If Annie is 12 years old, how old is her mother?”, a comparison is made between Annie’s age and her mother’s age. This can be shown on the number line where the top loops represent Annie’s age and the bottom represents her mother’s age as follows:

Figure 12

In Equipartitioning, sharing a whole also involves relational naming as a form of multiplicative comparison. Students understand that if n people share 1 whole, each person’s share is 1/nth of the whole. They create a fair-share box for this case and put a fractional value as the size of the share.

Figure 13

Students compare a single whole shared amongst n people and a single person’s share in two ways.

4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

When given comparison problems, students make intuitive choices about whether to solve the problems using subtraction or division. For example, given the following data chart of students and circumference of heads:

When asked to compare the circumference of Margot’s head to Tyrone’s head, students are likely to respond that Margot’s head is 3 inches larger.

Another problem is given the following data chart:

When asked to compare the time for flight from DC to Tokyo to the length of the flight from DC to Dallas, students respond in two ways. While they may say it is 12 hours longer, most say that it is 5 times as long. Based on the previous Standard, they can also say the time flight from DC to Dallas is 1/5th times as long.”

Therefore, students learn from the beginning that there are two possibilities for comparing two quantities: additive and multiplicative comparison. The additive comparison refers to the difference between the two numbers (the value which added to the smaller number makes the larger number). In the example above, students say, “the flight to Tokyo is 12 hours longer than the flight to Dallas,” or “the flight to Dallas is 12 hours shorter.” The multiplicative comparison refers to how many times longer (or how much longer) the bigger number is than the smaller number (the value which multiplied to the smaller number makes the larger number). In the example above, students say, “the time for flying to Tokyo is 5 times as long as the time for flying to Dallas,” or “the time for flying to Dallas is 5 times shorter or 1/5th as long.”

3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56 (Distributive property.)

The commutative property of multiplication states that a × b = b × a.

However, students should explore this property using representations rather than being presented with it merely as a memorization exercise. For instance, students sometimes use either a physical array of objects (see Bridging Standard 3.OA.F above) or a rectangular area model to find out if 5 × 3 is the same as 3 × 5.

A possible area problem might be: “Rectangle A has a length of 5 inches and a width of 3 inches. Rectangle B has a length of 3 inches and a width of 5 inches. Which rectangle has the larger area and why?”

Figure 14

This problem challenges students to reason whether or not a × b = b × a by rotating the rectangle.

Students recognize that division is not commutative in the sense that ÷ b is not equal to b ÷ a. They do develop flexibility in relating division problems to a given multiplication problem. They know, for instance, 20 ÷ 4 = 5 implies that 20 ÷ 5 = 4, and this follows from knowing that division and multiplication are inverse operations and applying the commutative property of multiplication.

In context, consider how to use an array for the problem, “At an animal hospital, a vet counts 20 dog paws. How many dogs are there?”

Students may solve the problem as quotative: “How many groups of 4 paws are in 20 paws?” or they may build an array as shown below:

Figure 15

Using the array, they could view each row as four paws of a dog or each column as one specific paw of all the dogs (left front, right front, left rear, and right rear). The fact that the array can be looked at either way tends to convince students that both models of division are represented. Using a coded division problem, eventually they may just say: 20 ÷ 4 = 5 or 20 ÷ 5 = 4 and the distinction between the two different ways of thinking become curtailed.

The associative property of multiplication states that

(a × b) × c = a × (b × c).

However, students should explore this property, with an understanding that the grouping of factors does not change the result of the product, using representations rather than being presented with it merely as a memorization exercise.

Students solve problems related to associativity such as: “Suppose there are 6 bottles of soda in a six-pack and 4 six-packs in a case. How many bottles of soda are there in 3 cases? Show two different ways to solve the problem and justify their equivalence.” Students solve the problem

6 × 4 × 3 by either reasoning that there are 6 bottles per six-pack × 4 six-packs = 24 bottles of soda in a case and there are 3 cases, so there are 24 bottles per case × 3 cases = 72 bottles. They can also reason that there are 4 six-packs per case × 3 cases = 12 six-packs and 6 bottles per six-pack, so there are 6 bottles per six pack × 12 six-packs. Therefore, students recognize the associativity of multiplication as showing

(6 × 4) × 3 = 6 × (4 × 3).

Students may use a 3D array to code the total number of dots as

6 × 4 × 3 = (6 × 4) × 3 = 6 × (4 × 3) thereby anticipating later development of volume (see Standard 5.MD.5.a in the Length, Area, and Volume LT). How they view these number sentences in the representations below needs to be carefully discussed.

Figure 16

The distributive property states that a product of two numbers can be found by decomposing a number into two addends, multiplying each addend separately by the other number, and then adding their products together.

For example, 5 · 32 can be broken into 5 · (30 + 2) = (5 · 30) + (5 · 2). In either case the product equals 160.

Multiplication of 12’s provides a reason to introduce the distributive property. The distributive property is stated as a × (b + c) = a × b + a × c. To multiply 12’s, they treat 12 as 10 + 2, so that, for instance,

8 × 12 = 8 × (10 + 2) = 8 × 10 + 8 × 2 = 80 + 16 = 96. Representing 12 as one ten stick and two units can assist in understanding distributive property. If this represents a dozen eggs and students were asked to find the total of seven dozen eggs, they visualize seven ten sticks and fourteen units as the total. Composing fourteen units into a ten and four ones, the total becomes 84.

The distributive property becomes essential in understanding one-digit by two-digit multiplication (see Standard 4.NBT.5 later for multiplication of one digit by two-digit).

12

x     7

14

70

84

Students also use array and area models to justify the distributive property of multiplication. For example, they decompose a 12 by 7 rectangle lengthwise into two rectangles of 10 by 7 and 2 by 7 and explain that 12 × 7 = (10 + 2) × 7 = 10 × 7 + 2 × 7. Likewise, they decompose the rectangle widthwise into 12 by 3 and 12 by 4 and explain that 12 × 7 = 12 × (3 + 4) = 12 × 3 +12 × 4.

Figure 17

Through reasoning that the sum of the two smaller areas equals the original area, students recognize that the distributive property of multiplication can be applied to either factor within a multiplication problem.

Students develop an understanding of the distributive property where division is used instead of multiplication [(a + b) ÷ c = (a ÷ ) + (b ÷ c)] in solving partitive division problems such as: “Share 174 candies among 3 friends. How many candies does each friend get?” Some students solve this problem by breaking 174 candies into 150 candies and 24 candies. They reason that each person would get 50 candies from the 150 and 8 candies from the 24 for a total of 58 candies each. This can be written as 174 ÷ 3 = (150 + 24) ÷ 3 = (150 ÷ 3) + (24 ÷ 3) = 50 + 8 = 58. This anticipates the distributive property of multiplication involving 1/nth later after students learn to code 174 ÷ 3 as 1/3 × 174.

The Identity property for multiplication states that for any number a,

a × 1 = a.

The Zero property for multiplication states that for any number a,

a × 0 = 0.

Students learn that multiplication where either or both factors are 0 equals 0. They may apply this idea by thinking about the context of bottles of soda in a pack. For example, if all packs of soda are empty (e.g., there are 0 sodas in a pack), and there are 5 packs, then students code this as 0 sodas per pack × 5 packs = 0 sodas which they relate to five empty packs. They also recognize that if there are 0 packs of soda left, and there are 5 bottles of soda in a pack, they end up with 0 bottles and code this as 5 sodas per pack × 0 packs = 0 bottles.

Another way to introduce the Zero property is to ask students to figure out what 5 × 0 is by examining the pattern:

5 × 3 = 15

5 × 2 = 10

5 × 1 = 5

5 × 0 = ?

They may reason from a repeated subtraction model that the products decrease by 5 each time the second factor decreases by 1 so 5 × 0 = 5 - 5 = 0.

Note to teachers: 12 × 0 = 0 but 0 ÷ 0 is not 12. In fact, 0 ÷ 0 has many possibilities, and thus undefined, when one considers the fact that any number times 0 equals to 0. This discussion may take place because of the inverse relationship between division and multiplication but a formal discussion takes place at a later grade level (see Standard 5.NF.7.b).

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Developing fluency in multiplication and division facts should accompany students’ experiences with the different models, representations, and problem types. Students’ sole reliance on rote memorization of multiplication facts (skip counting or repeated addition) will limit their progress in multiplication and division problems and in subsequent related topics of mathematics in later grades. Using the ideas of factors and multiples, and interrelationships among multiplication and division facts are therefore crucial to build students’ fluency and automaticity with multiplication facts. The introduction of the terminology of multiplication and division (factors, product, dividend, divisor, quotient) should be used to facilitate communication and not memorized solely as an exercise in vocabulary building.

Students develop fluency for multiplication and division facts from previous experience with doubling, halving, skip counting, and partitioning and reassembling. They also use properties (e.g., commutative, associative, and distributive properties). In the end, it turns out that the only facts not covered by these strategies are perfect squares and products of primes (excluding 5). Note that students should learn division facts in tandem with multiplication facts.

Below, the multiplication table can be learned more easily by applying the above strategies. As a result, it is not memorized as much as it is built as a means of expressing interrelationships. Color coding is used to show those interrelationships.

1.      Students begin with multiplication/division by 2’s as doubling and halving and the identity property of multiplication (coded as yellow).

Figure 18

2.      Students multiply by 4’s as double-doubling, and by 8’s as double-double-doubling (coded in orange).

3.      They multiply by 10’s building on previous experience with counting by tens and place value. Students know the meaning of “10 times as much/many” and “10 times less,” explain how these two operations compose and decompose a place value unit into units of the next higher or lower place values, and come to recognize the pattern of appending or removing a zero when multiplying or dividing by 10 (coded in green).

4.      Multiplication by 5’s coordinates multiplying by 10 with division by 2. It can be reinforced with skip counting by 5’s. Likewise, division by 5’s is division by 10 with multiplication by 2 (coded in green).

5.      Multiplying/ dividing by 3 is learned by skip counting, but then multiplication by 6 is the double of each 3-fact, and multiplication by 9 is the triple of the 3 facts (by 3s, coded in white; by sixes, coded in olive; by nines, coded in tan, if they were not already coded by another color).

6.      Multiplication by 1’s emphasizes the idea of the multiplicative identity (a × 1 = a) (coded as blue).

7.      Students know the zero property of multiplication as (a) “0 × a = 0”, but also (b) as an extension of the following pattern 5 × 3 = 15, 5 × 2 = 10, 5 × 1 = 5, so 5 × 0 = 0 (coded as blue).

8.      Multiplication and division by 7 is the most difficult, but the only fact left is 7 × 7 (coded as grey).

9.      Multiplication by 11 exhibits a pattern such that the number multiplied by 11 results in that number in both the tens and ones places, up to 11 × 9 = 99.

Students notice patterns on the multiplication chart, such as, the products in each row and column increase by the same number (skip counting). Students also notice patterns concerning multiplication and division involving two odds, two evens and an odd and an even.

Students also identify sequences with differences of twos, fives, and tens, starting from any whole number and relate this to skip counting (see Bridging Standard 2.OA.A of the Early Equations and Expressions LT about arithmetic sequences). They develop flexibility with units and groupings by skip counting by 2’s, 5’s and 10’s starting from any whole number. For example, starting at 7 and counting by 10 would result in the sequence: 7, 17, 27, ... These experiences lead them to discover the patterns in the place value system (see Standards 2.NBT.2 of the Counting LT and 1.NBT.2.c of the Place Value and Decimals LT).

Students explain patterns by using properties of operations, such as:

·         Even numbers are always divisible by 2.

·         Multiples of even numbers (2, 4, 6, and 8) are always even numbers.

·         An odd number only results from multiplication of two odds numbers (see white squares above)

Note to teachers: The patterns identified in the tables are all arithmetic sequences. The main diagonal of the multiplication table consists of a pattern of perfect squares.

4.OA.4 Find all factor pairs for a whole number in the range 1 - 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 - 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1 - 100 is prime or composite.

Students distinguish the terms “factors” and “multiples” in multiplication.

A factor of a number is a number that divides evenly into that number with no remainder.

A multiple of a number is a product of that number with any whole number.

For example, 3 is a factor of 12 because 12 is evenly divided by 3, and 12 is a multiple of 3 because 3 × 4 = 12.

Students build fluency with factors and multiples when given challenges such as: Change from 24 to 30 using only multiplication and division. These types of problems are called Daisy Chain Problems.

Students a) divide 24 by 8, multiply the result by 10, b) divide 24 by 4, and multiply the result by 5, or c) multiply by 5, and divide the result by 4, etc. Eventually, students generalize that they can divide by 24 and multiply by 30, or vice versa. This type of exercise strengthens students’ sense of multiplicative reasoning, and anticipates multiplication by rational numbers. Students predict when a daisy chain will result in a number which is bigger or smaller than the original number based on the relative size of the multiplier and divisor.

These problems are effective because they encourage students to stay within multiplicative/divisional relations without switching into addition/subtraction. Students who gain proficiency in these activities become less dependent on repeated addition and subtraction to do multiplication and division which is cognitively more efficient and sophisticated.

Students explore simple rules of divisibility to recognize when a number is divisible by 2, 10, 5, 3, and 9. For example, a number such as 171 is divisible by 3 because the sum of its digits is 1+7+1 = 9, which is divisible by 3. Rules for divisibility by 6 and 4 can be derived as extensions, such as 84 is even and the sum of its digits is 12, which is divisible by 3. Therefore it is divisible by 6.

4.OA.A Find the prime factorization of a whole numbers less than or equal to 144

This Bridging Standard is added here because prime factorization of whole numbers helps students build a conceptual relationship between factors and multiples.

Students distinguish among special kinds of numbers. A prime number is a natural number greater than 1 whose factor pairs are only the number itself and one. A composite number is a natural number > 2 which has factors other than itself and one. (e.g., 6, 22, 100). A square number is a natural number, or 1, with one pair of factors that are identical (e.g., 1, 4, 9, 16, …), and which is visualized as the area of a square. They factor all the products produced by the values in the 12 × 12 multiplication table.

A prime factorization is the set of the prime factors of the number usually listed from smallest to largest. Prime factoring is often represented using factor trees or chains of division.

For example, the prime factorization of 24 is 2 × 2 × 2 × 3. The prime factorization for 36 is 2 × 2 × 3 × 3. The factor trees below show these prime factorizations.

Figure 19

Students use prime factoring to simplify fractions, by identifying “names for one” and recognizing that multiplication by one leaves the value of the number unchanged (see Standard 5.NB.5.b later). This understanding helps students in addition and subtraction of fractions in fifth grade (see Standard 5.NF.1 in the Fractions LT). It is a tool for finding the greatest common factor and the least common multiple in the next Standard (6. NS.4).

6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

Students are introduced to the greatest common factor (GCF) and the least common multiple (LCM), using contextual examples.

The GCF is the largest factor that divides into any set of numbers.

The LCM is the smallest multiple of a set of numbers.

For example, a student is asked to clap every fourth beat. Another student is asked to clap every sixth beat. The class is asked to observe and determine when the two students clap at the same time. The answer (12) is the least common multiple (LCM) of 4 and 6. Then, the class is asked how often a third student would have to clap every time each of the other two students claps (without clapping every single beat). The answer (2) is a common factor of 4 and 6. If this is the largest common factor, the number is the GCF, or Greatest Common Factor.

Venn diagrams are effective ways to display GCF and LCM. Each of the circles of the Venn diagram represents a number, and contains the prime factorization of that number. In the intersection of any two circles, the common factors are listed once, and the rest of the factors are placed in the non-intersecting portion of the circle. The product of the factors in the intersection gives the GCF. The product of all the factors on the diagram, known as the “union” of all the factors, gives the LCM.

Figure 20

Students can be asked to prove that the product of the LCM and GCF equals the product of the original two numbers. The Venn diagram is a powerful tool in proving this from elementary number theory.

LCM provides an efficient means to add and subtract fractions with unlike denominators (see Standard 5.NF.1 in the Fractions LT for an example of using Venn diagrams to find the common denominator).

4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Students’ ability in the multiplication of numbers with two or more digits is preceded by their place value understanding (see Standards 1.NBT.2.a, 2.NBT.1.a, 2.NBT.1.b, 3.NBT.3, and 4.NBT.1 of the Place Values and Decimals LT for the development of students’ understanding in place values. In particular, their understanding of “10” being the factor for composing higher place value units).

Students solve multiplication problems involving a single-digit number and larger numbers of up to four digits. When multiplying multi-digit numbers by single-digit numbers, students apply expanded notation and the distributive property. For example, 421 × 3 = (400 + 20 + 1) × 3 = (400 × 3) + (20 × 3) + (1 × 3) = 1200 + 60 + 3 = 1263. They demonstrate conceptual understanding by applying the distributive property to the operations and showing partial products, and by applying the standard algorithm, including regrouping (see Standards 3.NBT.3 of the Place Value and Decimals LT).

Students multiply a two-digit number by a two-digit number correctly, and demonstrate an understanding of how and why they line up the partial products. This is best illustrated by the using of an area model with sides subdivided according to the expanded notation for each factor, and the resulting regions calculated separately and then added by the distributive property.

For example, students rewrite 12 × 27 = 12 × (20 + 7) = 12 × 20 + 12 × 7 = (10 + 2) × 20 + (10 + 2) × 7 = (10 × 20 + 2 × 20) + (10 × 7 + 2 × 7). They recognize that the 12 × 7 corresponds to the first partial product and 12 × 20 corresponds to the second partial product, and that each partial product is a sum of two products by distributive property.

Figure 21

Solutions may include the placement of the zero in the second partial product to emphasize multiplication by a tens digit, or explain why the partial product is shifted one place to the left.

Common student misconceptions and errors in applying the standard algorithm for multiplication are shown below:

12                                                           12

x   27                                                      x   27

84                                                           84

24                                                         24?

108                                                         324

Figure 22

In the first example, students do not understand to place the partial product one space to the left. In the second example, students place it one space over but cannot explain that the missing value is a zero because multiplication is by tens. Further misconceptions occurred commonly when the multiplier has a zero, and the student can either create a row of zeroes or add a zero and move to the next place value multiplication.

26                                                           26

x   30                                                      x   30

00                                                        780

780

780

Figure 23

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Students know how to divide two- and three-digit numbers by one-digit numbers, with and without remainders. They apply the standard algorithm of long division, and multiply the outcomes of algorithm to check their answers. If there is a remainder, students show how to multiply the quotient by the divisor, and then add the remainder to the result to regenerate the original dividend. For example, students model the long division of 42 ÷ 3 using the base-ten blocks to find the quotient to be 14 with no remainders.

So 42 ÷ 3 gives 1 ten and four ones, that is, 14.

Figure 24

4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Problem solving situations can illustrate how the practices of mathematics add to the conceptual understanding. They can help students to use mental strategies, to round off and to assess the reasonableness of a result as shown in the following problem:

Students can solve the problem, “How many buses are needed to transport 135 people if each bus can transport 40 persons?”

They describe the unknown quantity as n and write the statement

40 • n + b = 135. By trial and error, they write 40 * 3 +15 = 135. Students also judge when and how to report the remainder 15, i.e. whether to round the quotient up or down or change the remainder to a fraction by placing it as a numerator and the divisor as a denominator. For example, if asked how many buses are needed to transport 135 people, and each bus can transport 40 people, the correct response, from an initial answer of 3 buses with a remainder of 15, is 4 buses. Other students may write the equation as 40 • nb = 135 and generate the solution of

40 • 4 – 25 = 135. Their interpretation is that one would need four buses and have 25 extra seats. Students compare the two solutions to see they generate the same answer with different reasoning.

Students also use mental computation to estimate a reasonable range of the solution. Using multiplication facts and properties, they calculated

3 • 40 to be 120 and 5 • 40 to be 200 and determine that the number of buses needed is greater than 3 and less than 5. Subsequently, they reason that the solution is 4 buses since it is the only possibility.

Students flexibly read and write multiplication using × and •, and they are careful to avoid using the × when they are using the variable x. They use both the / and ÷ to indicate division.

5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

Students solve multiplication problems involving multi-digit whole numbers. When multiplying three-digit numbers by two-digit numbers, students use standard form and explain why the partial products are shifted one place to the left. When one of the digits of the multiplier is 0, students explain why the next partial product shifts by two digits to the left. Students use calculators and estimate products to judge the reasonableness of their answers.

5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Students solve long division problems with and without remainders. Students experience practice extending the algorithm for division from single digit numbers to double digit numbers enough to learn how to estimate the results. They can interpret how to reverse the division problem to produce the related multiplication problem plus the remainder. However, long division is assumed to be carried out primarily with the use of the calculator or a related spreadsheet application. Therefore the emphasis needs to be on assessing the reasonableness of the results and estimating results using rounding.

It is important that students make the connection between computation and the context of the problem and the strategy being used. Students recognize multi-digit division problems in the context of all three models: referent transforming, referent preserving, and referent composing.

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm

Students use all of their previously learned division strategies to help them divide multi-digit numbers using the standard algorithm and select and use appropriate tools strategically. They fluently connect the computation used with the context of the problem and apply estimation strategies and assess for reasonableness.

Extending division and multiplication to include non-whole numbers (fractions and decimals) requires adjustments in student understanding of the three models. Extending early meanings to make sense of new situations is characteristic of mathematics.  In this case, the extensions result in the fact that multiplying two numbers does not always produce a larger number and dividing two numbers does not always result in a smaller number.  Knowing why and how this happens helps students learn how mathematics grows and avoid a common misconception.

It is important to recall that in a times b, the quantity a is made “b times” as large as a (see Standards 3.OA.G and 4.OA.1 earlier). The question students consider is “what does it mean to make something 1/d times as large or by c/d times as large?” Students confront the fact that at first 1/d times as many does not seem to make sense.  Then students extend the meaning of multiplication so that it does make sense, but in a different way. After understanding 1/d times as many, students extend the meaning to define c/d times as many. Likewise, the meaning for division needs to be extended to explain what it means to “divide by 1/d, and to “divide by c/d.”

It is also crucial that students are NOT taught only how to procedurally multiply and to divide fractions. Cute rhymes to teach division of fractions such as “Ours is not to reason why, just invert and multiply” may appear amusing, but are detrimental to the development of students’ mathematical reasoning.  Procedures and concepts complement each other and should be taught hand-in-hand. Previous multiplication and division provide considerable resources including: 1) simple primitive meanings of a × b as a b times (b is the multiplier of the multiplicand a) and a ÷ b as a divided into b equipartitions (b is the divisor of the dividend a), 2) established inverses of division and multiplication, 3) three models, and 4) properties to apply and extend.

To extend to multiplication and division by fractions, students recall that multiplication and division, as operators, start with one number and produce another one. This can be pictured as:

Figure 25

Likewise division can be viewed as:

Figure 26

To understand multiplication and division with fractions completely, students learn to understand multiplication and division by fractions when placed in the input position, the operator position or both. These cases are shown in the grid below which specifies the types of numbers used as either the input (A) or the operator (B) in the multiplication of A × B: whole numbers, unit fractions, or regular (non-unit) fractions.  The categories down the left side represent the input or A in A × B, and those along the top represent the operator or B. For example, the multiplication of A × B in the first cell is of the form Whole Number × Whole Number which results in a Whole Number such as 5 × 7 = 35. Then to be complete, for each case, students explore the three models: referent preserving, transforming and composing. Luckily, careful application of the commutative and the associate properties simplifies the number of cases to be considered to those marked in yellow.

A × B where A is the vertical column and B is the horizontal row.

The cells are named in relation to their row and then their column. So the first discussion is of (2, 1) or row 2, column 1 or (1/d × W).

4.NF.4.a Understand a fraction a/b as a multiple of 1/b.

One interpretation of this standard is that it represents student understanding of: 1) how to multiply a unit fraction as input by a whole number as operator and 2) how to multiply a whole number as input by a unit fraction as an operator.

Unit fraction as input quantity operated by a whole number operator.

Students explain the meaning of a fraction times a whole number by referring back to equipartitioning of a whole. Students share one cake (a whole) among three people and code the process as the division 1 (cake) ÷ 3 (people) = 1/3 (cake per person). Reversing this process (using reassembly of the whole) results in a meaning for: 1/3 × 3 = 1 or a unit fraction times a whole number. This is an example applying the referent preserving model by scaling.

Figure 27

Stating the same relationship using mathematical symbols, students recognize how the result of one whole is obtained from 1/3 × 3, that is, 1/3, 3 times, results in 3/3 which equals one. From this, it follows that to multiply,

Applying the same reasoning to a referent transforming problem, such as, “Bob paints 1/10  square meter on a wall in one minute. How much surface can he paint in 12 minutes?” Students make a D/M box with headers of surface (in square meter) and minutes. They know that painting 1/10 square meter goes with 1 minute. To find out what corresponds with 12 minutes, they know they must multiply :

1/10 square meter per minute × 12 minutes = 12/10 square meters.

Applying the same reasoning to referent composing, students reason with area. Students visualize a rectangular area of 1/3 × 5 square units dynamically as the product of “sweeping” a line segment of width 1/3 unit over a length of 5 units (see Standard 3.MD.5.a in the Length, Area, and Volume LT). They interpret the distance of sweeping (5 units) as an operator on the width quantity (1/3 unit). The area created by the sweep is thus 5/3 square units.

Figure 28

Whole number as input quantity operated by a unit fraction as an operator.

Students explain the meaning of a whole number times a fraction by referring back to equipartitioning of a collection. When sharing a collection of 24 coins among 6 pirates as shown in the D/M box below, 24 can be described as 6 times as large as 4 or 4 can be described as 6 times smaller than 24 (see Standard 4.OA.2 earlier). Students also name each pirate’s share in relation to the collection as 4/24 or 1/6 of the collection. Therefore students reason that 24 ÷ 6 = 4 and 1/6th of 24 = 4. They write 1/6 of 24 as a number sentence: 24 × 1/6. Students view this using an input-output machine with 1/6th as the operator. In this case, the operator is referent preserving by scaling.

Figure 29

Thus, for conceptual consistency, students interpret “multiplying by 1/b” to be the equivalent operation as “dividing by b”. As a result, students recognize that 24 ÷ 6 = 4 and 24 × 1/6= 4 are two ways to operate on 24 to produce 4.

Stating the same relationship using mathematical symbols, students write:

Just by exploring different ways of expressing the referent preserving relationships from equipartitioning, students can move flexibly vertically up or down the D/M box.  In the example below they can move from: (1) top to bottom using division by 6, (2) top to bottom using multiplication by 1/6, or (3) bottom to top using multiplication by 6. Although not yet thoroughly discussed, some students reason what the missing fourth arrow’s label is (4) bottom to top using division by 1/6.

Figure 30

Students can represent the equivalence of dividing by b and multiplying by 1/b using a number line as referent preserving, such as showing that 1/6 of 24 on the number line is equal to 4.

Figure 31

Applying the same reasoning to a referent transforming problem, such as, “It takes a car one hour to travel 30 miles on dirt roads. How far does it travel in one-third of an hour?”, students make a D/M box with headers of distance (miles) and time (hours) and fills in the bottom row with 30 miles and 1 hour. To find out what corresponds to 1/3 of an hour, they must multiply:

30 miles/hour × 1/3 hours = 10 miles.

Applying the same reasoning to a referent composing area problem, such as 5 × 1/3 they create a length of 1/3 from a unit square, and sweep it across a width of 5 to get 5/3 square units.

Figure 32

Commutative Property of multiplication involving unit fraction

Although the two problem types with a unit fraction as an input quantity and as a multiplier (operator) are understood by students separately, they use their referent composing model to show it applies for unit fractions as input or operator.

They extend this understanding to the multiplication of a unit fraction by a unit fraction. In an area model, students can explain why 1/3 × 5 and

5 × 1/3 both produce the same area of 5/3 dynamically using rotation or sweeping.

Figure 33

Students can now multiply unit fractions × whole numbers and whole numbers × unit fractions for the three models: referent transforming, referent preserving, and referent composing. They also generalize that the commutative property holds for multiplication involving unit fractions. Based on this idea, they understand that:

1)      a/b is a times as large as 1/b

2)       a/b is a multiple of 1/b

3)       a ÷ b = a × 1/b = 1/b × a = a/b.

In the case table, the following cells are now completed with depth and insight.

A × B where A is the vertical column and B is the horizontal row.

Students recognize that multiplication of a whole number by a unit fraction always produces a number that is smaller than the original input. Since multiplying by 1/n is the same as dividing by n where n is a natural number, it makes sense that it would be getting smaller. In terms of input-output machines, multiplication by a unit fraction as an operator is called a “shrinker” operation. Therefore, students know that one number multiplied by another does not automatically produce a larger number.

4.NF.A Understand a fraction 1/c × 1/d =

1/(c x d).

This bridging standard is placed here to ensure student understanding of the multiplication of two unit fractions. They do this for referent preserving (scaling), referent transforming and referent composing models.

Unit fractions as Input Quantity and Operator

The final activity with unit fractions is to multiply a unit fraction by another unit fraction.

Students can explain this from an equipartitioning context where they are sharing a fractional part fairly. For example, sharing ¼ of a cookie among 2 people would produce 1/8 of a cookie per person as shown in the D/M box below.

Figure 34

The vertical direction can be described as 1/4 ÷ 2= 1/8 or as 1/4 × 1/2 = 1/8. This is referent preserving.

Using mathematical symbolism, students write this as:

Note other inverse operations are also obtained, such as, 1/8 × 2 = 1/4 or 1/8 ÷ 1/2 = 1/4. The first was discussed previously (see Standard 4.NF.4.a earlier) and the second is discussed as a division of fractions standard (see 5.NF.7.a and 5.NF.7.c later).

On the number line, students see that 1/4 × 1/3 = 1/12 by first splitting the number line into fourths. They split each of the fourths into thirds and notice that this results in 1/12th. For example, suppose that a ribbon measures 1/4th yard and it is split into thirds. They also interpret multiplication of unit fractions as the two consecutive divisors operating on the quantity 1 in a daisy chain (see Standard 4.NF.4.a)

Figure 35

Applying the similar reasoning to a referent transforming problem, students explain a problem such as: if a water pipe takes 1/3 of an hour to fill a tank fully, how much time is needed to fill the tank to a-quarter full?

Applying similar reasoning to a referent composing problem, students can illustrate why 1/2 × 1/3 = 1/6. They draw a 1 × 1 unit square and mark it clearly as the referent unit. They equipartition it to create a width of 1/2 and sweep it across a length of 1/3 or they create a length of 1/3 and sweep it across a width of 1/2 and the result is 1/6. This model also convinces students that the commutative property can be applied when two unit fractions are multiplied or divided.

Figure 36

This example further convinces students that multiplication does not always produce a larger answer. In this case, one can view the problem as divided a whole by 2 and then dividing again by 3 so that the result of a smaller fraction than either of the factors comes as no surprise.

Note to teachers: the meaning of factor is extended to include unit fractions, for example, 1/2 × 6 = 6/2 = 3 implies that 1/2 and 6 are factors that multiply to 3. Therefore, if whole number factoring is intended, it should be specified as such.

The students can now solve the four cases with check marks and only need to extend their understanding to the case of “generalized non-unit fractions”.

A × B where A is the vertical column and B is the horizontal row.

4.NF.4.b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.

The previous standard establishes that a generalized non-unit fraction can be viewed as a product of a whole number and a unit fraction. That is a/b is equal to a × 1/b. Since students know that a × 1/b is also equal to

a ÷ b, then they can write a/b = a × 1/b = a ÷ b.  The implication of this for students is that they can 1) treat generalized non-unit fractions as a product of a whole number and a unit fraction, or 2) transform the generalized non-unit fraction into a division problem.

This standard now defines how to treat a/b as a quantity or operator within multiplication.  Since a/b itself can be viewed as entailing a multiplication or division problem, then to multiply with a/b involves a string of two multiplication problems or a multiplication added to a division problem. Since students have the associative property for multiplication, at first it may be easier to translate these problems into a string of multiplications.

Thus, students also know how to multiply a triplet of whole numbers using associative property (see Standard 3.OA.5 earlier) and now extend the application of the property to the multiplication c  × a × 1/b. Thus, students know that c × (a × 1/b ) = (c × a) × 1/b Since (c × a) = c a, then to multiply a whole number times a fraction, student know that

c × a/b = c × (a × 1/b )= ( c × a) × 1/b = ca/b

The third statement shows that “a multiple of a/b [is] a multiple of 1/b, and the last statement shows what that product equals.

4.NF.4.c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.

Students apply the earlier Standard 4.NF.4.b to word problems in order to multiply a fraction by a whole number. Recall that the three models of multiplication were:

Model 1: Referent Transforming: To demonstrate their understanding of how to multiply a generalized non-unit fraction by a whole number, students solve referent transforming problems related to associativity. For example, students solve problems such as: “Suppose each bottle of soda has 1/8 gallon and there are 6 bottles of soda in a case. How much soda is there in 5 cases? Show two different ways to solve the problem and justify their equivalence.” Students solve the problem 1/8 × 6 × 5 by either reasoning that there are 1/8 gallon per bottle × 6 bottles = 6/8 gallon of soda in a bottle and there are 5 cases, so there are 6/8 gallon per case × 5 cases = 30/8 (or 15/4) gallons. They can also reason that there are 6 bottles per case × 5 cases = 30 bottles and 1/8 gallon per bottle, so there are 1/8 gallon per bottle × 30 bottles. Therefore, students recognize the associativity of multiplication as showing (1/8 × 6) × 5 = 1/8 × (6 × 5). The total is 15/4 or 3 3/4 gallons.

Model 2: Referent Preserving: To demonstrate their understanding of how to multiply a generalized non-unit fraction by a whole number, students also solve referent preserving models of multiplication by extending the use of the fair sharing table of values using additional rows. For example, if students know that one small ice cream cake feeds three people, they know that each person receives 1/3 of a cake. Suppose then a problem asks them how much cake to get for a family of 5? From the previous standard, they can multiple both the 1 person and the 1/3 cake by 5 to get 5 persons and 5/3 cakes (or 1 2/3 cakes).  Now suppose the problem is extended again to say that four families show up, then how much cake is needed? This requires them to multiply the 5 persons and the 5/3 cakes by 4, to produce 20/3cakes (or 6 2/3 cakes) for 20 people. They recognize that they cannot purchase 6 2/3 cakes, so the answer to the problem, how many cakes should they purchase for 20 people is 7 ice cream cakes.  This problem demonstrates how to multiply a generalized non-unit fraction by a whole number.

Figure 37

Students use of the extended table of values also allows them a way to define multiply a whole number by a/b as the operator.

Suppose students know that 6 pan pizzas shared fairly can feed 3 people. Suppose that four people show up. Students first multiply both quantities by 1/3 to give 2 pizzas per person. Then the multiply this by 4 recognizing that 4 people will consume 8 pizzas.

Figure 38

Students recognize that they obtain the values in the third row by multiplying the values in the first row by 1/3 to obtain the values in the second row, and then by 4 to obtain the values in the third row. Thus students know that to go from serving 3 people to serving 4 people, they multiply by 4/3.

Students relate this problem to their work with “daisy chains” (see Standard 4.OA.4 and Bridging Standard 4.NF.A earlier). To go from 3 to 4, they would divide by three and multiply by 4. Based on this problem they know that dividing by 3 is the same as multiplying by 1/3. Therefore, to go from 3 to 4, one multiplies by 4/3.  Daisy chains can now be directly related to multiplying by a generalized non-unit fraction.

Learning to multiply 1/b × a × c can also be viewed on the number line as a model of referent preserving multiplication. In the figure below, starting with 1/3, the distance is quintupled to produce a value of 5/3. This quantity of 5/3 is then quadrupled to produce a value of 20/3. By comparing 1/3 to 20/3, students see the final result is 20 times the starting quantity (1/3).

Figure 39

Model 3: Referent Composing: As a model of referent composing multiplication of a fraction times a whole number, student can explain why (1/b × a) × c and 1/b × (a × c) both produce the same area. Consider the problem: A rectangular strip of paper measures 5/3 inches × 4 inches. What is the area of the paper? Starting by creating the width of 5/3 inches which is the same as 1/3 × 5 (either as sweeping a length 1/3 unit by 5 units), students interpret the number 4 in (1/3 × 5) × 4 as an operator on that width or the sweeping distance of 4 inches producing the area desired as 20/3. They also recognize that the result of extending the width of 5 units by 4 times is the same as increasing the area 1/3 × 5 by 4 times, and thus (1/3 × 5) × 4 = 1/3 × (5 × 4) = 20/3. Later students also use a volume model to illustrate the associative property of the multiplication of three numbers, including fractions (see Standard 5.MD.5.a in the Length, Area, and Volume LT).

Figure 40

Students understand how to multiply a whole number by a generalized non-unit fraction and a generalized non-unit fraction by a whole number across all three models of multiplication thus completing two more cells in the grid below. To complete their understanding of multiplication involving fractions, they learn to multiply a unit fraction by a generalized non-unit fraction and a generalized non-unit fraction by a generalized non-unit fraction, in Standard 5.NF.4.b later.  They also complete division of fractions in subsequent Standards 5.NF.7.a, 5.NF.7.b, 5.NF.7.c later.

A × B where A is the vertical column and B is the horizontal row.

5.NF.4.a Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

The meaning of this standard rests in the recognition that when students multiply a fraction by a whole number operator, they can divide the whole number q into b parts (q ÷ b) and take a of them (× a). In simpler language when multiplying a fraction by a whole number, the entire expression can be rewritten to involve without fractions and only involve a combination of multiplication and division, that is multiplying a times q and then dividing by b. While this standard does not complete student understanding of any new cells in the grid, it offers increased flexibility in interpreting fractional multiplication in relation to whole number multiplication and division.

Students can construct a symbolic and property-based explanation of the standard. They know that a/b = a × 1/b Therefore, they know that (a/b) × q = (a × 1/b) × q. Applying associativity for multiplication, they know that

(a × 1/b) × q = a × (1/b × q). Applying commutativity for multiplication, they know that a × (1/b × q) = a × (q × 1/b) and finally, they know that

a × (q × 1/b)= (a × q) × 1/b= a × (q ÷ b)= a × q ÷ b.

This insight can be captured by the D/M box shown below.

Figure 41

Students interpreting this D/M box should conclude that for any generalized non-unit fractional amount associated with one person, they could scale it up to q people by multiplying that fraction by q

Because the standard permits them to move flexibly from statements involving one multiplication and one division to multiplication by a generalized non-unit fraction, students can move directly between the first row and the third row of a fair-share box by curtailing the division and multiplication into one operation. Students already know that dividing by 3 is the same as multiplying by 1/3, and they have learned that 5/3 is 5 times as large as 1/3. Therefore, to divide by 3 and multiply by 5 is the same as multiplication by 5/3.

Figure 42

Figure 43

In fact, students can choose any two rows of a fair-share table and form a fair-share box and relate the rows by combining the division and multiplication in the table into a single rational operator.

Figure 44

Figure 45

For instance, to relate (9, 6) to (6, 4), a student can describe dividing by 3 as multiplication by 1/3, and then combine the two operators (1/3 and 2) resulting in multiplication by 2/3. Students check to see that this combination of operators changes 9 into 6 and 6 into 4. This anticipates the use of ratio boxes (D/M boxes without the requirement of having a value of “1” in a cell.

Students can also reverse the process and relate (6, 4) to (9, 6) to see that it requires division by 2 and multiplication by 3. Reasoning similarly to before, students can produce an operator of 3/2. Students can be asked to imagine what should happen if the two operators, 2/3 and 3/2, were combined to act sequentially (in either order) on a single row which anticipates an introduction to multiplicative inverses, such that the inverse of a/b is b/a.

Figure 46

Knowing a/b × c = a × (c ÷ b) also support students’ fluent use of strategies that simplify the denominator b and the multiplier c using their common factors. For example, student reason that 3/14 × 21 = 3/2 × 3 because 3/14 × 21 = 3 × (21 ÷ 14) = 3 × 21/14 = 3 × 3/2. This also anticipates similar strategies for multiplying two fractions.

5.NF.4.b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Students model the multiplication of two generalized non-unit fractions (e.g. 2/3 × 5/4) area models in this standard. To multiply 2/3 × 5/4 using the area model, students begin with a 1 × 1 unit square. They partition each of the sides to represent the denominators of the two fractions and create a rectangular region whose measure is the product of the two denominators, for example, a 1/12th unit with sides of 1/3rd and 1/4th. Students scale each of the sides to be the size the number of 1/bth units, i.e. scale by 2 in the 1/3rd direction to represent 2/3 , and 5 in the 1/4th direction to represent 5/4. The region bounded by the new area is comprised of 10 units of 1/12th. Students name the product, explain why it is the product of the numerators and the denominators, and identify the unit of 1 that the answer is defined in relation to. Students can also name the region by its equivalent fractional value in lowest common terms and explain their answer. In this example, that would be 5/6.

Figure 47

Students reason that 2/3 × 5/4 is the same as (1/3 × 2) × (1/4 × 5) as well as (2 × 5) × (1/3 × 1/4). Students can also write the multiplication as a daisy chain (2 ÷ 3) × (5 ÷ 4) and recognize that the numerators are the multipliers and the denominators are the divisors.

It is critical that students are able to identify the referent whole to which any particular fraction is applied. A fraction is always understood relative to an explicit or an implied whole, and students identify the whole with respect to their solution.

Based on their knowledge of the commutative property and associative property of multiplication involving unit fractions and whole numbers (see Standard 4.NF.4.a and 4.NF.4.b earlier), students are able to carry out all the types of multiplication shown in the grid. They complete the last three cells by learning to multiply unit fractions times generalized non-unit fractions and generalized non-unit fractions times generalized non-unit fractions in this standard which illustrates the model of referent composing. They can explain why they did not need common denominators for multiplication since in multiplication they compose new units.

A × B where A is the vertical column and B is the horizontal row.

5.NF.A Find the product of two fractions a/b × c/d by interpreting it as

a × 1/b × c × 1/d and applying the associative and commutative properties of multiplication flexibly to support multiple solution strategies.

This bridging standard was added to complete the interpretation of the cases in the grid to include referent preserving and referent transforming of the product of generalized non-unit fractions.

Students can use the D/M box, to show the product of 2/3 × 5/4. The problem could be stated, if a 2/3 cups of sugar goes with one cup of flour, and a person wants to increase the recipe to use 5/4 cups of flour, then how much sugar should she use? The student multiplies 2/3 by 5/4 as shown in the extended D/M table to get the result of 10/12 or 5/6. The student checks the reasonableness of the result by recognizing that they had increased the flour by 1/4 of the original amount. Since 2/3 = 4/6, to increase it by 1/4 of the original amount would mean add 1/6th more to get 5/6 cups of sugar. This reasoning is referent preserving.

Using referent transforming reasoning, they realize that for her sugar, she needs 2/3 as much flour. That is, the amount of sugar is 2/3 × the amount of flour. If the flour is 5/4, then 2/3 cups sugar/ 1 cup of flour × 5/4 flour= 10/12 or 5/6 cups of sugar.

Figure 48

To be comprehensive in the approaches, students can also multiply fractions on the number line. They begin with the line segment of 1 unit, shrink it by 3 times and then stretch it by 2 times to obtain 2/3. To multiply it with 5/4, they shrink it by 4 times and then stretch it by 5 times to obtain 10/12.

Figure 49

5.NF.5.b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence

a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

Because students can translate multiplication by a fraction into a combination of multiplication and division (see Standard 4.NF.4.c), they can predict whether a fractional operator will increase or decrease the size of the quantity it operates on.  If the operator is greater than one, then the numerator is greater than the denominator. When switched into a “daisy chain”, the numerator is the multiplier and the denominator is the divisor. If a number is multiplied by a larger number than it is divided, then the result or product is larger than the quantity operated on. Students predict for example that in multiplying 16 by 5/3, the product will be greater than 16.

Similarly, for the fraction as operator, if the denominator is greater than the numerator, then the fraction is less than one, and the quantity is divided by a larger number than it is multiplied, resulting in a smaller number. Students predict, for example, that in multiplying 2/3 by 3/8, the product will smaller than 2/3rds.

Students recognize that a/b = (n × a)/(n × b) by rewriting it as a daisy chain and reason that the combined operator, “n times larger followed by n times smaller,” does not change the input quantity a/b.

5.NF.5.a Interpret multiplication as scaling (resizing), by: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

Not only can students predict the outcome of using operators that are less than, equal to or greater than one on the input, but they can predict whether the overall effect of the combination of both factors in multiplication in reference to an original assumed referent unit of one. For instance, dividing 2/3 by 3/4 is less than one because each factor is less than one.

If both numbers are greater than one, it is also easy to predict. However, if one factor is greater than one and the other is less than one, prediction by students needs to take into consideration the full effects of the various scale factors.

Multiplying mixed numbers, multiplying mixed numbers and fractions less than one, and multiplying fractions less than one in different contexts can provide students such challenges. Students are already familiar with mixed numbers (see Standard 3.NF.2.b of the Fractions LT).

By reasoning about the scaling effect of multiplication, and whether the each quantities being multiplied is a proper fraction, an improper fraction, or a mixed numbers, students can predict the value of the product relative to the value of any of the quantities.

5.NF.6  Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Students solve equations and word problems by multiplying fractions and mixed numbers. This can either be a fraction by a mixed number (a/b × c d/e where c d/e  is a mixed number) or a mixed number by a mixed number (a b/c × d e/f where  a b/c  and  d e/f  are mixed numbers). Students are already familiar with mixed numbers (see Standard 3.NF.2.b of the Fractions LT) and they also use them in addition and subtraction.

There are three different types of real world problems involving multiplication of fractions and mixed numbers:

 Model 1: Referent Transforming Model 2: Referent Preserving Model 3: Arrays, Area, and Cartesian Products Fair Sharing (Partitive) Measurement (Quotative) Arrays Rate Scaling Area Equal Groups Cartesian Product

Model 1: Referent Transforming

a.         A fair-sharing multiplication problem is, “There are 4 1/2 barrels of soda and each barrel holds 3 2/3 liters of soda. How much soda is there in total?”  Students predict the final outcome is greater than one since both factors are greater than one.

b.         An example of rate problem involving multiplication might be: “Jessica walked at 4 1/2 miles per hour for 1 2/3 hours. How far did she walk?” Students predict the final outcome is greater than one since both factors are greater than one.

c.         An example of equal groups problem involving multiplication with transformation of units might be: “There are 5 1/2 bags of candies. Each bag holds 2/3 pound candies. What is the total amount of candies in all the bags?” Students predict the final outcome is greater than one because the problem, once coded as multiplication can be viewed as × 11 ÷ 2 × 2 ÷ 3. Thus overall, the original one is scaled up by 22 and down by 6, so it is greater than one.

Model 2: Referent Preserving

a.         The Measurement (Quotative) multiplication problem is“12 1/3 ribbons each measuring 3 1/2 inches long are taped together to create a single ribbon. How long is the ribbon created?” Students predict the final outcome is greater than one, since both factors are greater than one.

b.         An example of scale problem involving multiplication might be: “Given a square with sides 3 1/2 units long and you shrink the square by a scale factor of 1/5, what will be the length of a side of the new square created?” Students predict the final outcome is less than one because the problem, once coded as multiplication can be viewed as × 7 ÷ 2 × 1 ÷ 5. Thus overall, the original one is scaled up by 7 and down by 10 so the final outcome becomes less than one.

c.         An example of Equal Groups problem involving multiplication with preservation of units might be: “Lucy uses 1 1/5 cups of vinegar in her salad dressing recipe. How much vinegar would Lucy use to make 3 1/2 recipes?” Students predict the final outcome is greater than one, since both factors are greater than one.

Model 3: Arrays, Area, and Cartesian Products

An example of an area problem involving multiplication is: “A closet floor is covered with tiles whose dimensions are one foot by one foot. The closet measures 5 3/4 feet by 7 1/3 feet. How many tiles cover the floor?” Students predict the final outcome is greater than one, since both factors are greater than one.

Note to teachers: Before addressing the standards directly associated with division of fractions, it is important to note how extensively the topic of multiplication of fractions has been treated. It has been treated across a set of nine cases, based on the use of whole numbers, unit fractions and generalized non-unit fractions. It has been developed across all three models: referent transforming, referent preserving and referent composing. And, it has been developed by carefully linking multiplication and division as inverse operations and applying the properties of multiplication (commutativity, associativity, identity and even inverse). This treatment is in marked contrast to a teacher who tells his or her students that to multiply fractions, multiply the numerators and then multiply the denominators and simplify. Such instruction diminishes the foundation created by multiplication and division of fractions and weakens students’ ability to reason multiplicatively and to model problems with mathematics. It is hoped that this extensive treatment shows the importance of the topic and its full conceptualization.

5.NF.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

Similar to the multiplication grid specifying the types of numbers involved, there is an associated division grid for extending to fraction-related division specifying the types of numbers used in either as the input (A) or the operator (B) in the division of A ÷ B. Whole Number ÷ Whole Number could be, for example, 20 ÷ 7 = 20/7; a division problem associated with Unit Fraction ÷ Unit Fraction could be 1/2 ÷ 1/12 = 6.

A ÷ B where A is the vertical column and B is the horizontal row.

However, since students enter this set of topics with a number of resources, solving these types of problems can be viewed more simply. The students already have the three models for multiplication, the ability to change multiplication by fractions into strings of whole number multiplication and division, and a whole set of cases of multiplication with fractions, for each of which they could write an associated division problem as an inverse. In addition, students know much more about how to use their properties to simplify problems.

Therefore, in introducing division of fractions, students learn to work with only four distinct cases (1) dividing a unit fraction by a whole number, (2) dividing a whole number by a unit fraction, (3) dividing unit fractions by unit fractions, and then they can move directly to: (4) dividing generalized fractions by a generalized fraction.

For each case, students associate the division (of a quantity) with an equivalent multiplication based on inverse operations and work out the product based on their knowledge about the particular type of multiplication involving fractions and whole numbers (see Standards 4.NF.4.a, 4.NF.4.b, 5.NF.4.a, 5.NF.4.b, and Bridging Standard 4.NF.A for the multiplication types).

Whole number as divisors. For example, to define 1/9 ÷ 3, one can ask, what number times 3 equals 1/9. The answer is 1/27.  Since students know the answer, they must find ways to generalize the approach and justify their solutions.  In this case, students know that dividing by 3 is the same as multiplying by 1/3 and so 1/9 ÷ 3 = 1/9 × 1/3 = 1/27.

Students generate contextual problems for each type of division and 1) explain their answers in terms of meanings of the operations in a variety of contexts and models and 2) predict the size of the output relative to the input based on the type and value of number used.

To explore contexts, students work with the three most common representations, the D/M box, the number line representation and the area model.

To create a problem where a unit fraction is divided by a whole number, students create a fair sharing context associating a unit fraction with a number greater than one.  For example, if a chocolate torte is so rich that 1/3 of this luscious three-layer cake serves 4 people, then how much would serve one person. To solve this problem using the D/M box, students know to co-split the top rows by 4 to produce the problem 1/3 ÷ 4. The answer is generated by recalling that dividing by 4 is the same as multiplying by 1/4, hence the answer is 1/12. Because it is a co-split, the example is referent preserving.

Figure 50

A referent transforming division problem would ask if a slug travels 1/3 of a meter in 4 minutes, how far does it travel in one minute? The problem would be 1/3 meters divided by 4 minutes = 1/12 meters per minute.

Area Models illustrate length × width equal area. Therefore they also represent area divided by length (width) equals width (length). For the problem 1/3 ÷ 4, we are asking what width coupled with a length of 4 would result in an area of 1/3. The representation below shows the answer:

Figure 51

5.NF.7.b Interpret division of a whole number by a unit fraction, and compute such quotients.

To first reason about these problems, students write, for example,

30 ÷ 1/2 = C. The answer has to solve the multiplication problem

C × 1/2 = 30. This means also that C ÷ 2 = 30. The students know that

C = 60.  They also reason this by writing the division problem itself as a fraction:

From their work with fractions and in Standard 5.NF.5.b, they know that they can multiply this by 2/2 and preserve the value of the fraction. This results in the problem 60/1. Hence 30 ÷ ½ = 60.

5.NF.7.c Solve real world problems involving division of unit fractions using visual models and equations.

In the context of the D/M Box, this problem can represent the question, if there are 30 oranges in a half a case, how many oranges are in a full case?  To go from a half case to a whole case, they can divide the half case by a half, producing one. Likewise, they can divide 30 by ½ to get a result of 60 oranges.

Figure 52

Students check their reasoning that 30 ÷ 1/2 = 60 by reasoning that 60 × 1/2 = 30. They learn to re-represent the division by a unit fraction as a multiplication by the denominator of the fraction, based on the previous idea of n 1/n ths equals one (see Standard 3.NF.3.c of the Fractions LT).

A referent transforming rate problem would be: “Teesha drove 30 miles for 1/2 hour. What speed (miles per hour) did she drive at?” To find the distance travelled for an hour, they solve the problem:

distance/ time = rate, or 30 miles ÷ 1/2 hour. Students recognize that 2 1/2th-hours makes an hour, and the associated distance of 30 miles is multiplied by 2 correspondingly to produce 60 miles per hour

A number line type measurement problem would ask, “How many 1/2-inch pieces of cake can be cut from a 30-inch wide Christmas log cake?” The quotative strategy of imagining how many pieces could be produced gives the answer of 60 pieces. This is referent preserving.

An area problem for 6 ÷ 1/5 could be if the area of a piece of cloth is 6 square yards, and one side is 1/5 of a yard, how long is the piece of cloth. Each of the square yards would be cut into strips one fifth wide and one yard long. They would be lined up to produce a piece of cloth 30 feet long.  This is referent composing.

5.NF.B. Students know that dividing by zero is prohibited.

Students are also asked what the outcome of dividing a number by 0 is, e.g., “What is the result of 40 ÷ 0?” They may hold the misconception that the result is 0 or 40. To help them counter these misconceptions, students are asked to examine the trend of the result of dividing 40 by a decreasing unit fraction:

Students recognize that the while the unit-fraction divisors decreases, the result of the division increases instead of being zero or stay constant as 40. From the pattern, they may infer that when the unit fraction eventually decreases to zero, the result of 40 ÷ 0 would be something that cannot be represented by any written number and hence, undefined. Using the inverse relationship between multiplication and division, students explain that finding the result of 40 ÷ 0 is the same as finding an unknown number that multiplies with 0 to produce 40. Because the multiplication with zero is zero, students reason that this unknown number cannot be determined.

Student may also use the area model to interpret the result of 40 ÷ 0. As they draw a series of rectangles whose areas are 40 square units and widths decreasing, they recognize that the lengths are increasing. When the width of the rectangle becomes 0, the length for making an area of 40 square units becomes undefined.

They also re-represent the division 40 ÷ 0 as an unknown factor problem (see Standard 3.OA.6 earlier) “? × 0 = 40” and reason that 0 or any other number is not a solution to the unknown.

5.NF.C Interpret division of a unit fraction by a unit fraction, compute such quotients and solve related real world problems.

Students extend their repertoire to solving problems with division of a unit fraction by a unit fraction. Students know that division by a unit fraction is the same as multiplication by the denominator of the fraction. Therefore they reason that 1/a ÷ 1/b = (1/a) × b= b/a. They check this to see that b/a × 1/b = 1/a. Also they can justify their answers by writing the problem as;

Students understand how to use this information in solving problems such as:

“Cheryl needs 1/3 yard of fabric for making curtains. The fabric store only sells the fabric by the yard and 1/2-yard. If she buys 1/2 yard of the fabric, how much of the 1/2 yard does she need to use?”

Two approaches to this problem surface among students who can explain their reasoning visually and in context.

Approach One: Students approach this problem as a comparison problem. The question is 1/3 is how much of 1/2?  They visualize the lengths on a number line and seek to compare the two lengths in relation to each other.

Figure 53

Using a single piece of paper, students fold the strip into halves (light blue), unfold it and fold it again, this time into thirds (red lines). They notice if they fold again after folding it into thirds, they can mark sixths and sixths will measure both thirds and halves. Because two sixths measure a third and three sixths measure a half, they conclude that one third is 2/3 of a half. They can now write the problem as: 1/3 ¸ 1/2 = 2/3 so that Molly would use 2/3 of the fabric. Checking their answer, they use the inverse relationship between multiplication and division to write the related fact, 2/3 × 1/2 = 1/3. This related fact is read as 2/3 of 1/2 is 1/3.

Approach two: The covariation approach to pattern completion provides a second way to support successful solutions of these new problems (see the earlier Standard 5.NF.7.b).

They build a table for this relationship labeling the two columns for the fabric needed and the fabric bought.

 Fabric Needed Fabric Bought 1/3 1/2 ? 1

Students reason that if they want to know the comparison between 1/3 and 1/2, it would be easy to answer if the amount they had bought was 1 yard. Then the corresponding amount they needed would produce the relationship they were seeking. Using the idea of covariation, they know that what they do with one quantity should be done with the other. They choose to multiply the one half yard of fabric bought by two, they get the value one as if they had bought one yard. To co-vary the relationship, they double 1/3 to get 2/3. So they conclude the relationship between 1/3 and 1/2 is the same as the relationship between 2/3 and 1.

In this sense, they know that division to two unit fractions is the same as finding the relationship between their quotient and one. This is a restatement of the standard a ¸ b = (a/b) / 1. In the case of unit fractions, they have shown that 1/p ¸ 1/q = (q/p) /1. They note that the numerator of the equivalent fraction is q/p, which is the multiplicative inverse of p/q.

Experience of comparing two unit fractions multiplicatively help students generalize and be able to justify that: 1) dividing a unit fraction with a smaller denominator by another unit fraction with a larger denominator results in a fraction less than 1; and 2) dividing a unit fraction with a larger denominator by a unit fraction with a smaller denominator will always result in an answer greater than 1.

6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

When faced with the problem of dividing two generalized, non-unit fractions, students can use the variety of strategies they have developed to simplify the problem. They write it as a complex fraction and then multiply the numerator and the denominator by the inverse of the denominator and simplify.

While this procedure produces an effective way to solve the problems, they also can solve the problems in different contexts using a variety of models.

The D/M box below can be used to explore division of fractions by fractions.

In the following D/M box below, students work on the division 1/4 ÷ 2/3.

Figure 54

When dividing 1/4 by 2/3, students apply the inverse relationship between division and multiplication to reason that dividing by 2/3 is the same as multiplying by 3/2. They recognize that finding the outcome of dividing 1/4 by 2/3 is equivalent to determine the outcome such that 2/3 of it is 1/4. This implies that 1/3 of the outcome equals to 1/2 of 1/4 and thus the outcome is 3 times as large as 1/2 of 1/4, i.e., 1/4 × 1/2 × 3. Therefore, 1/4 ÷ 2/3 = 1/4 × 3/2.

There are three types of dividing fractions by fraction: i) Dividing two "normal" fractions, ii) Dividing a mixed number by a fraction, and iii) Dividing two mixed numbers.

As stated during Standard 3.OA.3 earlier, division problems are classified in three models depending on their nature:

 Model 1: Referent Transforming Model 2: Referent Preserving Model 3: Referent Composing Fair Sharing (Partitive) Measurement (Quotative) Arrays Rate Scaling Area Equal Groups Cartesian Product

Note that Arrays and Cartesian products are not viable models for interpreting division of fractions or mixed numbers in this standard.

1. Referent Transforming

a.         A fair-sharing example might be: “Ally made 4/5 of a pound of trail mix to be shared among her friends. She places them into bags of each having 2/25 pounds for each person. How much trail mix will there be in each bag?” (Solution: 4/5 pounds ÷ 2/25 pounds per person = 4/5 × 25/2 people = 100/10 people = 10 people)

b.         An example of rate problems involving partitive division might be: “A long distance runner covers 2 3/4 miles in 2/5 hour. How many miles can the runner cover in 1 hour?” (Solution: 2 3/4 miles ÷ 2/5 = 9/4 ÷ 2/5 = 9/4 × 5/2 = 45/8 miles) (For rate problems involving unit rates, see Standard 6.RP.2 of the Ratio, Proportion, and Percents LT.)

c.         An example of Equal Groups problem involving division with transformation of units might be: “If a box of cereal holds 28 1/2 ounces, then how many servings will there be if each serving is 31/2 ounces?” (Solution: 28 1/2 ÷ 3 1/2 = (28 1/2 × 2) ÷ (3 1/2 × 2) = 8 1/7)

2. Referent Preserving

a.         A measurement example might be, “A whole ribbon measuring 3/4 foot in length is cut into 3/8 -foot long strips. How many strips can be made from the whole ribbon?” (Solution: 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2) The whole ribbon is twice as long as the strip (An analysis of the units in this problem anticipates scale factors at 8th grade: inches ÷ inches = a scale factor).

b.         Scaling problems involve the division of a quantity with a scale factor. An example of a scaling problem involving division might be: “After shrinking a square by a scale factor of 1/4, the length of the square is 12/15 feet long, what will the length of a side of a square before enlarging?” (Solution: 12/15 ÷ 1/4 = 12/60 = 1/5 feet long)

c.         An example of Equal Groups problem involving division with preservation of unit units might be: “A bottle of medicine contains 8 2/3 oz. You can have 12 1/2 doses from this bottle. How many ounces are in each dose?” (Solution: 8 2/3 ÷12 1/2 = 26/3 ÷ 25/2= 26/3 × 2/25 = 52/75 ounces in each dose)

3. Referent Composing

An example of an area problem involving division is: “Mr. Jones plans to build a rectangular garden outside of his house that will cover 2/3 of a square mile. One dimension of the garden area will be determined by a fence that is 3/4 of a mile long. What is the other dimension of the garden area?” (Solution: 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9)

A common way to approach division by fractions is by using the “invert and multiply rule”. However, this procedure is often taught without context.

Multiplication and division of decimals are unpacked in the Place Value and Decimals LT (see Standards 4.NF.6, 5.NBT.1, 5.NBT.2, 5.NBT.3.a, and 5.NBT.7).

Selected References for M & D LT

Behr, M., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity: A conceptual basis common to additive and multiplicative structures. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 123-180). Albany, NY: SUNY Press.

Charles, K., & Nason, R. (2000). Young Children's Partitioning Strategies. Educational Studies in Mathematics, 43(2), 191-221.

Clark, F. B., & Kamii, C. (1996). Identiﬁcation of multiplicative thinking in children in grades 1–5. Journal for Research in Mathematics Education, 27, 41–51.

Cobb, P., Confrey, J., diSessa, A. A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.

Confrey, J. (2008). A synthesis of the research on rational number reasoning: A learning progressions approach to synthesis. Paper presented at the Eleventh International Congress of Mathematics Instruction, Monterrey, Mexico.

Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291-330). Albany: State University of New York Press.

Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 231-265). Mahwah, NJ: Lawrence Erlbaum Associates.

Confrey, J., Maloney, A. P., & Nguyen, K. H. (Eds.). (in progress). Learning over time: Learning trajectories in mathematics education: Information Age.

Confrey, J., Maloney, A. P., Nguyen, K. H., Mojica, G., & Myers, M. (2009). Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. Paper presented at the 33rd Conference of the International Group for the Psychology of Mathematics Education, Thessaloniki, Greece.

Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3-17.

Greer, B. (1994). Extending the meaning of multiplication and division. In G. Harel & J. Confrey (Eds.). The development of multiplicative reasoning in the learning of mathematics (pp. 61–85). Albany: State University of New York Press.

Greer, B. (1988). Nonconservation of multiplication and division: Analysis of a symptom. Journal of Mathematical Behavior 7 (3): pp. 281-298.

Harel, G., Behr, M., Post, T., & Lesh, R. (1994). The impact of number type on the solution of multiplication and division problems: Further considerations. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 365-388). Albany, NY: SUNY Press.

Harel, G., Behr, M., Post, T. & Lesh, R. (1991). Variables Affecting Proportionality: Understanding of Physical Principles, Formation of Quantitative Relations, and Multiplicative Invariance. In F. Furinghetti (Ed.) Proceedings of PME XV Conference (pp. 125-133). Assisi, Italy: PME.

Harel, G. and Confrey, J. (eds): 1994, The Development of Multiplicative Reasoning in the Learning of Mathematics, Albany, New York: SUNY Press.

Lamon, S. J. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89-121). Albany, NY: SUNY Press.

Nesher, P. (1988). Multiplicative School Word Problems: Theoretical Approaches and Empirical Findings. In J. Hiebert & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades. Mahwah: NJ: Lawrence Erlbaum Association.

Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford: Wiley

Piaget, J. (1970). Genetic epistemology. New York, NY: W.W. Norton & Company, Inc.

Piaget, J., Grize, J., Szeminska, A., & Bangh, V. (1977). Epistemology and psychology of functions (F. Castelanos & V. Anderson, Trans.). Dordrecht, The Netherlands: Reidel

Piaget, J., Inhelder. B.. and Szeminska, A. (1960). The child’s conception of geometry. London: Routledge & Keagan Paul.

Pothier, Y., & Sawada, D. (1983). Partitioning: The Emergence of Rational Number Ideas in Young Children. Journal for Research in Mathematics Education, 14(4), 307-3317.

Squire, S., & Bryant, P. . (2002). The influence of sharing on children's initial concept of division. Journal of Experimental Child Psychology, 81(1), 1-43.

Squire, S., Davies, C., & Bryant, P. (2004). Does the cue help? Children’s understanding of multiplicative concepts in different problem contexts. British Journal of Educational Psychology, 74, 515–32.

Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht, the Netherlands: Kluwer Academic Publishers.

Thomson, P. W., & Saldanha, L. A. (2003). Fractions and Multiplicative Reasoning. In J. Kilpatrick, G. Martin & D. Schifter (Eds.), Research Companion to the Principles and Standards for School Mathematics (pp. 95-113). Reston, VA: National Council of Teachers of Mathematics.

Toluk, Z., & Middleton, J. A. . (2004). The development of children's understanding of the quotient: A teaching experiment. International Journal for Mathematics Teaching and Learning.

Van de Walle J. A., Karp K. S. and Bay-Williams J. M. (2013) Developing strategies for multiplication and division computation. In Elementary and Middle school mathematics: Teaching developmentally: The professional development edition for mathematics coaches and other teacher leaders. New Jersey: Pearson, pp. 236-257.

Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 141-161). New York, NY: Academic Press.

Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 141-161). Mahwah, NJ: Lawrence Erlbaum Association.