Multiplying Fractions and Mixed Numbers

Abstract

This lesson is designed to reinforce skills associated with multiplying fractions and mixed numbers and allow students to visualize the effects of multiplying by a fraction or mixed number.

Objectives

Upon completion of this lesson, students will:

  • have practiced multiplying fractions and/or mixed numbers.
  • have explored the effects of multiplying fractions and mixed numbers.
  • have practiced predicting the effects of multiplying a number by a fraction or mixed number.

Standards Addressed

Grade 6

  • Numeration

    • The student demonstrates conceptual understanding of fractions (proper or mixed numbers), decimals, percents (whole number), or integers.
    • The student demonstrates conceptual understanding of fractions, mixed numbers, or percents.

Grade 7

  • Numeration

    • The student demonstrates conceptual understanding of rational numbers (fractions, decimals, percents, or integers).
    • The student demonstrates conceptual understanding of positive fractions, decimals, or percents.

Grade 8

  • Numeration

    • The student demonstrates conceptual understanding of real numbers.
    • The student demonstrates conceptual understanding of rational numbers (fractions, decimals, or percents including integers).

Grade 9

  • Estimation and Computation

    • The student accurately solves problems (including real-world situations).

  • Numeration

    • The student demonstrates conceptual understanding of real numbers.

Grade 10

  • Estimation and Computation

    • The student accurately solves problems (including real-world situations).

  • Numeration

    • The student demonstrates conceptual understanding of real numbers.

Third Grade

  • Number and Operations-Fractions

    • Develop understanding of fractions as numbers.

Fifth Grade

  • Number and Operations-Fractions

    • Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Sixth Grade

  • The Number System

    • Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Seventh Grade

  • The Number System

    • Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Grades 6-8

  • Numbers and Operations

    • Compute fluently and make reasonable estimates
    • Understand meanings of operations and how they relate to one another

Grade 5

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with non-negative rational numbers.

Grade 6

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

6th Grade

  • Numbers and Operations

    • The student will demonstrate through the mathematical processes an understanding of the concepts of whole-number percentages, integers, and ratio and rate; the addition and subtraction of fractions; accurate, efficient, and generalizable methods of multiplying and dividing fractions and decimals; and the use of exponential notation to represent whole numbers.

Textbooks Aligned

Grade Six

  • Bits and Pieces I

    • Investigation Two: Comparing Fractions
    • Investigation Three: Cooking with Fractions

  • Bits and Pieces II

    • Investigation Six: Computing with Decimals

6th

  • Module 3 - Mind Games

    • Section 6: Decimal Multiplication
      • Reason for Alignment: The Multiplying Decimals lesson calls on the students use of number sense with decimals, which ties in with Target Game and Estimating Decimal Products in the textbook. This lesson should reinforce and extend the concepts of decimal multiplication. The Sequencer activity in the lesson is useful here, with students entering "0" as the add-on. In this way, the multiplication can be repeated many times and patterns observed.

7th

  • Module 4 - The Art of Motion

    • Section 1: Multiplication and Division of Fractions
      • Reason for Alignment: The Multiplying Decimals and Fractions lesson contains a discussion about these skills. It strengthens the students' concepts as well as skills with the worksheets that accompany the lesson.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • multiply whole numbers.
    • recall some knowledge regarding multiplication of mixed numbers and fractions.
    • ability to interpret a line graph (helpful, but not necessary).
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Students will need:

Key Terms

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Can someone tell me the difference between a fraction and a mixed number?
      • Emphasize that the word "fraction", here, is used to refer to proper fractions only.
    • If I multiply the number 1 by a fraction, will the result be larger or smaller than 1? OR If I multiply 1 by a mixed number, will the result be larger or smaller than 1?
    • Entertain a discussion on fractions and mixed numbers including how to multiply these types of numbers.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we will be talking about multiplying fractions and mixed numbers.
    • We are going to use the computers to look at the effect of multiplying a number by a fraction or mixed number, but please do not turn your computers on or go to this page until I ask you to. I want to show you a little about this program first.

  3. Teacher Input

    Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use the computer applets on the Project Interactivate site.

    • Open your browser to The Sequencer in order to demonstrate this activity to the students.
    • Show the class that there is a box where they may enter a starting number, in other words, the number that will be multiplied by a fraction or mixed number.
    • Show the class the box where they will enter the multiplier, or the number that our starting number will be multiplied by.
    • Explain to the class that they should enter a "0" into the add-on box as this lesson is about multiplying fractions, not adding them. Show students where to enter the zero.

  4. Guided Practice

    • After answering all questions that the students might have regarding the use of The Sequencer, pass out the Multiplying Fractions Worksheet.
    • Walk the students through the worksheet, having all of the students in class use the same numbers, for example: 3 for the whole number and 3/10 for the fraction. For each question, ask two different students what they think the answer is. Ask the students to settle any disputes on what the answers are.

  5. Independent Practice

    • Allow students to work independently or in groups to complete the worksheet and circulate the room to offer help where necessary.
    • When the students are finished with the Fraction worksheet, pass out the Multiplying Mixed Numbers Worksheet and have them work through it independently.

  6. Closure

    Allow students to describe what steps are needed and how they differ when numbers are multiplied by a fraction or a mixed number.

Alternate Outline

This lesson can be rearranged in several ways.

  • You may choose to allow students to work in cooperative groups to make predictions about the effects of multiplying two particular numbers together and then check them as a class.
  • If only one computer is available, students can graph the "input" and "output" by hand to discover the pattern. Students who need additional help can use The Sequencer to help them visualize the pattern.