Unit 4: Dividing Fractions
Lesson 12: Fractional Lengths

This is the first of four lessons in which students use multiplication and division of fractions to solve geometric problems. In this lesson, they encounter problems involving fractional lengths. They use their understanding of the two interpretations of division—“how many groups?” and “how much in each group?”—to solve problems that involve multiplicative comparison (MP7).

In these geometry-themed lessons, students work with a wider range of fractions and mixed numbers, which gives them opportunities to choose their methods and tools for problem solving.

Unit 4: Dividing Fractions
Lesson 14: Fractional Lengths in Triangles and Prisms

In this transitional lesson, students conclude their work with area and begin to explore volume of rectangular prisms. First, they extend their work on area to include triangles, using division to find the length of a base or a height in a triangle when the area is known. Second, they undertake a key activity for extending their understanding of how to find the volume of a prism.

In previous grades, students learned that the volume of a prism with whole-number edge lengths is the product of the edge lengths. Now they consider the volume of a prism with dimensions 1 1/2 inch by 2 inches by 2 1/2 inches. They picture it as being packed with cubes whose edge length is 1/2 inch, making it a prism that is 3 cubes by 4 cubes by 5 cubes, for a total of 60 cubes, because 3 x 4 x 5 = 60. At the same time, they see that each of these 1/2-inch cubes has a volume of 1/8 cubic inches, because we can fit 8 of them into a unit cube. They conclude that the volume of the prism is 60 x 1/8 = 7 1/2 cubic inches.

In the next lesson, by repeating this reasoning and generalizing (MP8), students see that the volume of a rectangular prism with fractional edge lengths can also be found by multiplying its edge lengths directly (e.g., (1 1/2) x 2 x (2 1/2) = 7 1/2).

This short video and interactive assessment activity is designed to teach fifth graders about the conversion of fractions to decimals.

This Nrich problem provides an opportunity to find equivalent fractions and carry out some simple additions and subtractions of fractions in a context that may challenge and motivate students.

This Nrich activity gives the pupils opportunities to use and develop their visualizing skills in conjunction with the knowledge of fractions. It's quite a contrast to just dealing with fractions numerically.

This simple Nrich game is designed to help children become more familiar with common coins in the UK, and to find fractional quantities of amounts of money. The fractions involved are slightly more challenging than those found in Fraction Card Game. Higher-order thinking is required in order to play strategically.

This Nrich game offers students an opportunity to practice calculating fractions and percentages of amounts.

This Nrich problem gives practice in calculating with fractions in a challenging setting. It also requires the use of factors and multiples. While doing the problem learners will need to express a smaller whole number as a fraction of a larger one and find equivalent fractions. This activity will require some estimating and trial and improvement, combined with working systematically.

This short video and interactive assessment activity is designed to teach fifth graders about fractions with common denominators - word problems.

This Nrich problem gives practice in calculating with fractions and encourages different ways of representing and recording.

In Module 8, the final module of the year, students extend their understanding of part–whole relationships through the lens of geometry.  As students compose and decompose shapes, they begin to develop an understanding of unit fractions as equal parts of a whole.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This 20-day module gives students their first opportunity to explore decimal numbers via their relationship to decimal fractions, expressing a given quantity in both fraction and decimal forms.  Utilizing the understanding of fractions developed throughout Module 5, students apply the same reasoning to decimal numbers, building a solid foundation for Grade 5 work with decimal operations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

In Module 3, students' understanding of addition and subtraction of fractions extends from earlier work with fraction equivalence and decimals. This module marks a significant shift away from the elementary grades' centrality of base ten units to the study and use of the full set of fractional units from Grade 5 forward, especially as applied to algebra.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

Grade 5’s Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions.  Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication.  The module proceeds to fraction by fraction multiplication in both fraction and decimal forms.  An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa.  Students are introduced to the work of division with fractions and decimal fractions.  Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers.  Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients.  Throughout the module students are asked to reason about these important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving real-world, multistep problems which include all fraction operations supported by the use of tape diagrams.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This Nrich problem follows on from Keep It Simple and Egyptian Fractions
These three problems together offer students an opportunity to engage with some mathematical ideas in depth and not just with the rather mechanical process of adding and subtracting fractions.

This problem in particular requires students to compare fractions and may deepen their understanding of their relative sizes.

This task involves fraction multiplication that can be solved with pictures or number lines.

This Nrich problem provides a fraction-based challenge for students who already possess a good understanding of fraction addition and subtraction, and it leads to algebraic manipulation of that same process.