Fracciones como cocientes y multiplicación de fracciones
Unit Goals
Students develop an understanding of fractions as the division of the numerator by the denominator, that is , and solve problems that involve the multiplication of a whole number and a fraction, including fractions greater than 1.
In this unit, students learn to interpret a fraction as a quotient and extend their understanding of multiplication of a whole number and a fraction.
In IM Grade 3, students made sense of multiplication and division of whole numbers in terms of equal-size groups. In IM Grade 4, they used multiplication to represent equal-size groups with a fractional amount in each group and to express comparison.
For instance, can represent “4 groups of ” or “4 times as much as .”
The amount in both situations can be represented by the shaded parts of a diagram like this:
Students learn that a fraction like can also represent:
A division situation, where 4 objects are being shared equally by 3 people, or .
A fraction of a group, in this case, of a group of 4 objects, or .
Students also interpret the product of a whole number and a fraction in terms of the side lengths of a rectangle. The expression represents the area of a rectangle that is 6 units by 1 unit. In the same way, represents a rectangle that is 6 units by unit.
The commutative and associative properties become evident as students connect different expressions to the same diagram. The distributive property is used as students multiply a whole number and a fraction written as a mixed number, for instance: .
Throughout this unit, it is assumed that the sharing is always equal sharing, whether explicitly stated or not. For example, in the situation above, 4 objects are being shared equally by 3 people.
Represent and explain the relationship between division and fractions.
Solve problems involving division of whole numbers leading to answers that are fractions.
Section Narrative
In this section, students learn to see a fraction as a quotient, a result of dividing the numerator by the denominator. They solve a sequence of problems about situations that involve sharing a whole number of objects equally. Through repeated reasoning, they notice regularity in the result of division (MP8) and generalize that .
For example, 3 objects being shared equally by 2 people can be represented by the expression and by a diagram. Each person’s share can be shown by the shaded parts in a diagram such as:
or
Each person would get half of the 3 objects, or 3 groups of an object. The value of this expression is or .
Connect division to multiplication of a whole number by a non-unit fraction.
Connect division to multiplication of a whole number by a unit fraction.
Explore the relationship between multiplication and division.
Section Narrative
In IM Grade 4, students saw that a non-unit fraction can be expressed as a product of a whole number and a unit fraction, or a whole number and a non-unit fraction with the same denominator. For instance, can be expressed as , as , or as . In the previous section, students interpreted a fraction like as a quotient: .
This section allows students to connect these two interpretations of and relate and .
Students use diagrams and contexts to make sense of division situations that result in a fractional quotient. As they interpret and write expressions that represent the quantities, students observe the commutative property of multiplication. For example, they interpret and as 8 groups of a third and a third of 8, and recognize that both are equal to .
These understandings then help students make sense of other multiplication and division expressions that can be represented by the same diagram and have the same value:
Find the area of a rectangle when one side length is a whole number and the other side length is a fraction or mixed number.
Represent and solve problems involving the multiplication of a whole number by a fraction or mixed number.
Write, interpret, and evaluate numerical expressions that represent multiplication of a whole number by a fraction or mixed number.
Section Narrative
In this section, students learn that they can reason about the area of a rectangle with a fractional side length the same way they had with rectangles with whole-number side lengths: using diagrams and multiplication.
To find the area of such rectangles, students work through a progression of fractional side lengths: a unit fraction (), a non-unit fraction (), a fraction greater than 1 (), and a mixed number (). They write and interpret multiplication expressions, such as and , to represent the area of such rectangles. Students use shaded diagrams and their understanding of fractions to reason about the value of the expressions.
Along the way, the associative property of multiplication becomes evident. For instance, students see that the expressions , , and can all describe the area of the shaded region in this diagram.
The distributive property is illustrated as students reason about the area of a rectangle where the side lengths are a whole number and a mixed number. To find , for example, students may decompose the rectangle by grouping the whole-number units and the fractional units. Then students multiply the whole-number units and the fractional units separately before combining them, resulting in an expression such as .
