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.

Rate

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Understand quantity as a number used with a unit of measurement.
Solve problems involving quantities such as distances, intervals of time, liquid volumes, masses of objects, and money, and with the units of measurement for these quantities.
Understand that a ratio is a comparison of two quantities.
Write ratios for problem situations.
Make and interpret tables, graphs, and diagrams.
Write and solve equations to represent problem situations.

Lesson Flow

In this unit, students will explore the concept of rate in a variety of contexts: beats per minute, unit prices, fuel efficiency of a car, population density, speed, and conversion factors. Students will write and refine their own definition for rate and then use it to recognize rates in different situations. Students will learn that every rate is paired with an inverse rate that is a measure of the same relationship. Students will figure out the logic of how units are used with rates. Then students will represent quantitative relationships involving rates, using tables, graphs, double number lines, and formulas, and they will see how to create one such representation when given another.

In this lesson, students focus on the units used with rates. Students are given calculations without units and must determine the correct units to use.Key ConceptsWhen dividing quantity A by quantity B to find a rate, the unit of the quotient is expressed in the form A per B.When multiplying a B quantity by an A per B rate, you get an A quantity.Some rates, while mathematically correct, are physically impossible in the real world.Goals and Learning ObjectivesUnderstand the units that result from rate calculations.

Working With Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Compare and order positive and negative numbers and place them on a number line.
Understand the concepts of opposites absolute value.

Lesson Flow

The unit begins with students using a balloon model to informally explore adding and subtracting integers. With the model, adding or removing heat represents adding or subtracting positive integers, and adding or removing weight represents adding or subtracting negative integers.

Students then move from the balloon model to a number line model for adding and subtracting integers, eventually extending the addition and subtraction rules from integers to all rational numbers. Number lines and multiplication patterns are used to find products of rational numbers. The relationship between multiplication and division is used to understand how to divide rational numbers. Properties of addition are briefly reviewed, then used to prove rules for addition, subtraction, multiplication, and division.

This unit includes problems with real-world contexts, formative assessment lessons, and Gallery problems.

Students critique and improve their work on the Self Check. They then extend their knowledge with additional problems.Students solve problems that require them to apply their knowledge of multiplying and dividing positive and negative numbers. Students will then take a quiz.Key ConceptsTo solve the problems in the Self Check, students must apply their knowledge of multiplication and division of positive and negative numbers learned throughout the unit.Goals and Learning ObjectivesUse knowledge of multiplication and division of positive and negative numbers to solve problems.

Students solve fraction multiplication and division problems. To see the answers, learners need to roll their mouse over the colored area.

This lesson teaches students how to use colored blocks to model word problems. There are several sets of problems each for addition / subtraction, multiplication / division, and ratio word problems.

Download these free worksheets to sharpen your word problem skills. Sheets focus on addition, multiplication, subtraction, division, and other math topics.

Learn about the mathematics of fencing, including the ratio of touches to time remaining that an athlete needs to mentally calculate in order to stay in the match, in this video from Center for Asian American Media. In the accompanying classroom activity, students reflect on the mathematics involved and analyze the ratios and calculations involved in this numerical relationship. This resource is part of the Math at the Core: Middle School Collection. [3:26]

Math variety of concepts, this colorful website is sure to give your students the chance show what they know. Concepts covered include rounding, division with decimals, order of operations, ratio, percent, multiplication, and division of fractions, simplifying fractions, and more. Not an interactive site, but great practice activities.

Download these worksheets to sharpen your operations skills. Sheets focus on addition, subtraction, multiplication, division, mixed operations, input/output tables, the order of operations, and operations in other base number systems.

This short video and interactive assessment activity is designed to teach third graders about finding the number from the figure patterns.

Practice dividing multi-digit whole numbers. These problems use remainders. Students receive immediate feedback and have the opportunity to try questions repeatedly, watch a video or receive hints.

Khan Academy learning modules include a Community space where users can ask questions and seek help from community members. Educators should consult with their Technology administrators to determine the use of Khan Academy learning modules in their classroom. Please review materials from external sites before sharing with students.

This short video and interactive assessment activity is designed to teach fourth graders about multiples - word problems.

This short video and interactive assessment activity is designed to teach third graders about multiples - word problems.

Demonstrates how to make reasonable estimates using multiplication and division. [2:10]

Khan Academy learning modules include a Community space where users can ask questions and seek help from community members. Educators should consult with their Technology administrators to determine the use of Khan Academy learning modules in their classroom. Please review materials from external sites before sharing with students.

This short video and interactive assessment activity is designed to teach fourth graders about shortening multiplication expressions.

This short video and interactive assessment activity is designed to teach fourth graders about multiplying 2-digit by 2-digit numbers in columns.