After geometric series, this Nrich problem is one of the simplest infinite series with finite sum, all of whose terms are positive. This geometric demonstration of the result requires students to continue a pattern and to use several steps of reasoning to deduce that the sum is bounded by 2. Summing infinite geometric series also play an important role in the this proof, so this could be used to show an application of them in a larger proof. (It would be useful for students to be able to sum 12+14+18+⋯ before tackling this problem.)

In this activity, students will show fractions with tangrams and explain their thinking using SeeSaw.

This Nrich problem offers an excellent opportunity for students to practice converting fractions into decimals, while also investigating a wider question that connects their knowledge of prime factors and place value.

There are fascinating patterns to be found in recurring decimals. This Nrich problem explores the relationship between fraction and decimal representations. It's a great opportunity to practice converting fractions to decimals with and without a calculator.

This series of word problems requires students to determine which problems can be solved by multiplying 1/8_2/5.

This Nrich problem introduces an intriguing trick which provides a context for practicing manipulation of fractions. Watching the video, or perhaps trying the trick out for themselves, can engage students' curiosity, and lead to some intriguing mathematics to explore and explain.

Both of the questions in this task are solved by the division problem 12Ö3 but what happens to the ribbon is different in each case.

The goal of this task is to provide examples for comparing two fractions by finding a benchmark fraction which lies in between the two.

Unit 4: Dividing Fractions
Lesson 6: Using Diagrams to Find the Number of Groups

This is the second lesson in a series of three lessons exploring the “how many groups?” interpretation of division in situations involving fractions.
In the preceding lesson and in this one, the number of groups in each given situation is 1 or greater. In the next lesson, students find the number of groups that is less than 1 (“what fraction of a group?”).

Students have used different diagrams to represent multiplication and division. In this lesson, tape diagrams are spotlighted and used more explicitly. They are more abstract and more flexible than other representations students may have chosen for thinking about division problems that involve fractions. Because they use measurement along the length of the tape, tape diagrams are closer to the number line representation of fractions, and ultimately help students visualize division problems on the number line. (Students are not required to do that in this lesson, however.)

Students continue to make the journey from reasoning with concrete quantities to reasoning with abstract representations of fraction division (MP2).

Students will predict how many amusement parks are in their state. They will then analyze census data on the numbers of amusement parks in all 50 states in 2016. (Data in this activity do not include the District of Columbia or Puerto Rico.) Then students will write numbers as fractions and create a visual model of the data.

This task highlights a slightly different aspect of place value as it relates to decimal notation. More than simply being comfortable with decimal notation, the point is for students to be able to move fluidly between and among the different ways that a single value can be represented and to understand the relative size of the numbers in each place.

Unit 4: Dividing Fractions
Lesson 11: Using an Algorithm to Divide Fractions

In the previous lesson, students began to develop a general algorithm for dividing a fraction by a fraction. They complete that process in this lesson. Students calculate quotients using the steps they observed previously (i.e., to divide bya/b , we can multiply by a and divide by b), and compare them to quotients found by reasoning with a tape diagram. Through repeated reasoning, they notice that the two methods produce the same quotient and that the steps can be summed up as an algorithm: to divide by a/b, we multiply by b/a (MP8). As students use the algorithm to divide different numbers (whole numbers and fractions), they begin to see its flexibility and efficiency.

Unit 4: Dividing Fractions
Lesson 15: Volume of Prisms

In this lesson, students complete their understanding of why the method of multiplying the edge lengths works for finding the volume of a prism with fractional edge lengths, just as it did for prisms with whole-number edge lengths. They use this understanding to find the volume of rectangular prisms given the edge lengths, and to find unknown edge lengths given the volume and other edge lengths.

Problems about rectangles and triangles in the previous two lessons involved three quantities: length, width, and area; or base, height, and area. Problems in this lesson involve four quantities: length, width, height, and volume. So finding an unknown quantity might involve an extra step, for example, multiplying two known lengths first and then dividing the volume by this product, or dividing the volume twice, once by each known length.

In tackling problems with increasing complexity and less scaffolding, students must make sense of problems and persevere in solving them (MP1).

This Nrich activity offers free exploration which will help learners develop a deep understanding of halves and halving. The task gives a context in which to discuss the importance of the part-whole relationship of fractions so that children realise halves can be different sizes, depending on the whole.

Unit 4: Dividing Fractions
Lesson 7: What Fraction of a Group?

In the previous three lessons, students explored the “how many groups?” interpretation of division. Their explorations included situations where the number of groups was a whole number or a mixed number. In this lesson, they extend the work to include cases where the number of groups is a fraction less than 1, that is, situations in which the total amount is smaller than the size of 1 group. In such situations, the question becomes “what fraction of a group?”.

Students notice that they can use the same reasoning strategies as in situations with a whole number of groups, because the structure
(number of groups) x (size of groups) = (total amount)
is the same as before (MP7). They write multiplication equations of this form and for the corresponding division equations.

Throughout the lesson, students practice attending to details (in diagrams, descriptions, or equations) about how the given quantities relate to the size of 1 group.

When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder or a mixed number.

The purpose of this task is to help students understand and articulate the reasons for the steps in the usual algorithm for converting a mixed number into an equivalent fraction.