These two fraction division tasks use the same context and ask ŇHow much in one group?Ó but require students to divide the fractions in the opposite order. Students struggle to understand which order one should divide in a fraction division context, and these two tasks give them an opportunity to think carefully about the meaning of fraction division.

Unit 4: Dividing Fractions
Lesson 4: How Many Groups? (Part 1)

This lesson and the next one extend the “how many groups?” interpretation of division to situations where the “group” can be fractional. This builds on the work in earlier grades on dividing whole numbers by unit fractions.

Students use pattern blocks to answer questions about how many times a fraction goes into another number (e.g., how many 2/3s are in 2?), and to represent multiplication and division equations involving fractions. In this lesson, they focus on situations where the quotient (the number of groups) is a whole number.

This lesson is the first in a group of six lessons that trace out a gradual progression of learning—from reasoning with specific quantities, to using a symbolic formula for division of fractions (MP8).

Unit 4: Dividing Fractions
Lesson 5: How Many Groups? (Part 2)

In this lesson, students continue to work with division situations involving questions like “how many groups?” or “how many of this in that?” Unlike in the previous lesson, they encounter situations where the quotient is not a whole number, and they must attend to the whole when representing the answer as a fraction (MP6). They represent the situations with multiplication equations (e.g., “? groups of 1/2 make 8” can be expressed as ? x 1/2 = 8) and division equations (8 / 1/2 = ?).

The purpose of this task is to help students see the connection between aÖb and ab in a particular concrete example. The relationship between the division problem 3Ö8 and the fraction 3/8 is actually very subtle.

Unit 4: Dividing Fractions
Lesson 8: How Much in Each Group? (Part 1)

Previously, students looked at division situations in which the number of groups (or the fraction of a group) was unknown. They interpreted division expressions as a way to find out that number (or fraction) of groups. In this lesson, students encounter situations where the number of groups is known but the size of each group is not. They interpret division expressions as a way to answer “how much in a group?” questions.

Students use the same tools—multiplication and division equations and tape diagrams—and the same structure of equal-sized groups to reason about “how much in a group?” questions (MP7). They also continue to relate their reasoning in quantitative contexts to their reasoning on abstract representations (MP2). Students find both whole-number and non-whole-number quotients, recognizing that, like the number of groups, the amount in one group can also be a whole number or a fraction.

Unit 4: Dividing Fractions
Lesson 9: How Much in Each Group? (Part 2)

This lesson allows students to practice finding the amount in one group, and to interpret, represent, and solve different kinds of division problems with less scaffolding. In one activity, students are not explicitly told whether the division situations involve finding the number of groups or finding the amount in each group. They decide on an interpretation that would enable them to solve a division problem. Students are also required to identify relevant information (from a video, a picture, or written statements) that would help them answer questions.

Because the tasks in this lesson are not scaffolded, students will need to make sense of the problems and persevere to solve them (MP1). As students move back and forth between the contexts and the abstract equations and diagrams that represent them, they reason abstractly and quantitatively (MP2).

These problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them.

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division.

This short video and interactive assessment activity is designed to teach fifth graders about identifying equivalent fractions (missing number).

This short video and interactive assessment activity is designed to teach fifth graders about inequalities with addition and subtraction of fractions.

Unit 4: Dividing Fractions
Lesson 3: Interpreting Division Situations

In an earlier lesson, students were reminded of the connection between multiplication and division. They revisited the idea of division as a way to find a missing factor, which can either be the number of groups, or the size of one group.

In this lesson, students interpret division situations in story problems that involve equal-size groups. They draw diagrams and write division and multiplication equations to make sense of the relationship between known and unknown quantities (MP2).

This Nrich problem challenges children to calculate with fractions and provides a good context in which to encourage learners to be curious about different methods of approach.

The purpose of this task is to present students with a situation where it is natural to add fraction with unlike denominators; it can be used for either assessment or instructional purposes.

The purpose of this task is to present students with a context where they need to explain why two simple fractions are equivalent.

This Nrich problem uses the context of sports training to offer opportunities for learners to explore division and/or multiplication. Pupils will be required to consider the relationships between multiplication, division and fractions, which will help reveal their level of understanding.

This is the first Nrich problem in a set of three linked activities. Egyptian Fractions and The Greedy Algorithm follow on.

It's often difficult to find interesting contexts to consolidate addition and subtraction of fractions. This problem offers that, whilst also requiring students to develop and analyze different strategies and explain their findings.

This Nrich activity is an unusual context in which pupils can consolidate recognizing, finding, naming and writing fractions. The rich environment also gives them the opportunity to identify, name and write equivalent fractions of a given fraction, represented visually as a chain. Furthermore, learners will be adding and subtracting fractions with the same denominators and denominators which are multiples of the same number.