Students solve word problems that require dividing and multiplying with fractions and mixed numbers.Key ConceptsStudents apply their knowledge about multiplying and dividing fractions to solve word problems. This includes applying the general methods for dividing fractions learned in previous lessons:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.Goals and Learning ObjectivesApply knowledge of fraction multiplication and division to solve word problems.

Students explore whether multiplying by a number always results in a greater number. Students explore whether dividing by a number always results in a smaller number.Key ConceptsIn early grades, students learn that multiplication represents the total when several equal groups are combined. For this reason, some students think that multiplying always “makes things bigger.” In this lesson, students will investigate the case where a number is multiplied by a factor less than 1.Students are introduced to division in early grades in the context of dividing a group into smaller, equal groups. In whole number situations like these, the quotient is smaller than the starting number. For this reason, some students think that dividing always “makes things smaller.” In this lesson, students will investigate the case where a number is divided by a divisor less than 1.Goals and Learning ObjectivesDetermine when multiplying a number by a factor gives a result greater than the number and when it gives a result less than the number.Determine when dividing a number by a divisor gives a result greater than the number and when it gives a result less than the number.

Students critique and improve their work on the Self Check.Key ConceptsNo new concepts are introduced in this lesson. To solve the problems in the Self Check, students use fraction division and operations with decimals.Goals and Learning ObjectivesUse knowledge of fraction division and decimal operations to solve problems.

Students explore methods of dividing a fraction by a fraction.Key ConceptsStudents extend what they learned in Lesson 4 to divide a fraction by any fraction. Students are presented with two general methods for dividing fractions:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.These two methods will work for all cases, including cases in which one or both of the numbers in the division is a fraction or whole number.Goals and Learning ObjectivesUse models and other methods to divide fractions by fractions.

Students explore methods for dividing a whole number by a fraction.Key ConceptsIn earlier grades, students learned to think of a whole number division problem, such as 8 ÷ 4, in terms of two types of equal groups.Divisor as the Number of Groups Divide 8 into 4 equal groups and find the size of each group.Divisor as the Group Size Divide 8 into groups of 4 and find the number of groups.To divide a fraction by a whole number in Lesson 2, students used the first interpretation. For example, to find 89 ÷ 4, they divided 8 ninths into 4 equal groups and found that there were 2 ninths in each group.To divide a whole number by a fraction, the second interpretation is most helpful. For example, to find 3 ÷ 34, we find the number of groups of 3 fourths in 3 wholes. The diagram in the Opening shows that there are 4 groups, so 3 ÷ 34 = 4.Just as with whole number division, the quotient when a whole number is divided by a fraction is not always a whole number. Below is a model for 2 ÷ 35. The model shows that there are 3 groups of 3 fifths in 2 wholes plus 13 of another group (13 of a group of 3 fifths is 1 fifth). Therefore, 2 ÷ 35 = 313. Notice that once we have divided the 2 wholes into fifths, we are finding the number of groups of 3 fifths in 10 fifths. This is simply 10 ÷ 3.These models can help explain that the “multiply by the reciprocal” method of dividing a whole number by a fraction works. To find 2 ÷ 35, we can multiply 2 by 5 to find the total number of fifths in 2 and then divide the result (10) by 3 to find the number of groups of 3 of these fifths in 2. So, 2÷35=2×53=2×53.ELL: Encourage students to verbalize their explanations. To help students gain confidence and increase their understanding, allow those that share the same language of origin to speak in small groups using their prefered language.Goals and Learning ObjectivesUse models and other methods to divide a whole number by a fraction.

This supplemental resource provides problems and activities related to Numerical and Algebraic Operations & Analytical Thinking in Middle School Mathematics.

Open Educational Resources (OER) offer opportunities for increasing equity and access to high-quality K–12 education. Many state education agencies now have offices devoted to identifying and using OERs and other digital resources in their states. To help states, districts, teachers, and other users determine the degree of alignment of OERs to the Common Core State Standards, and to determine aspects of quality of OERs, Achieve has developed eight rubrics in collaboration with leaders from the OER community.

This task builds on a fifth grade fraction multiplication task, Ň5.NF Running to School, Variation 1.Ó This task uses the identical context, but asks the corresponding ŇNumber of Groups UnknownÓ division problem. See Ň6.NS Running to School, Variation 3Ó for the ŇGroup Size UnknownÓ version.

It is much easier to visualize division of fraction problems with contexts where the quantities involved are continuous. It makes sense to talk about a fraction of an hour. The context suggests a linear diagram, so this is a good opportunity for students to draw a number line or a double number line to solve the problem.

In this activity, students work with measurement and shapes using live web maps and images of the Earth at various scales using ArcGIS Online. Time required: 2 class periods (100 minutes total).

This task could be used in instructional activities designed to build understandings of fraction division. With teacher guidance, it could be used to develop knowledge of the common denominator approach and the underlying rationale.

In this task, students see a bunch of candies poured on the table. How many candies should each of four people receive if you know how many there are in two thirds of the whole?