Fractions: Multiplying and Dividing

Is there anything more difficult to explain at 5th and 6th grade level than the rules for multiplying and dividing fractions? These resources offer support in explaining the concepts that underlie the rules. Visual, interactive models are provided where possible. You will also find opportunities for your students to practice their skills in this area of arithmetic. If you have other approaches to teaching this topic, please share! Just use the comment box below!

Multiplication of Fractions
Visualize and practice multiplying fractions using an area representation. With the “Show Me” option selected, the virtual manipulative is used to graphically demonstrate, explore, and practice multiplying fractions. A rectangular grid, representing a whole, shows the areas of two fractions to be multiplied, one fraction in red on the left and another in blue at the bottom. The area of the overlapping region, shown in purple, represents the product of their multiplication. The “Test Me” option provides problems to be solved using the same graphical representation.

Multiplying Fractions
This tutorial site offers instruction as well as practice in multiplication of fractions. The fractions are modeled with either circles or lines (rectangular areas). The visual display matched with the numerical makes an effective demonstration.

Divide and Conquer
This lesson is based on the idea that middle school students can better understand the procedure for dividing fractions if they analyze division through a sequence of problems. Students start with division of whole numbers, followed by division of a whole number by a unit fraction, division of a whole number by a non-unit fraction, and finally division of a fraction by a fraction. Activity sheets and guiding questions are included.

Dividing Fractions
In this activity, students divide fractions using area models. They can adjust the numerators and denominators of the divisor and dividend and see how the area model and calculation change. Full access to ExploreLearning is available through an annual subscription, but you can apply for a month’s free access in order to test out the applets.

Fraction Bars
This applet offers a classroom-adaptable idea of how to explain division of fractions. Adjustable colored bars are used to illustrate arithmetic operations with fractions on the number line. The initial seeding shows a division problem, dividing 7/5 by ½.

We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/08/2011.

Fractions: Adding and Subtracting

As teachers, we all want our students to understand those concepts underpinning addition and subtraction of fractions— the concept of equivalent fractions, in particular. These resources offer teaching ideas on both the concepts and the skills, including demonstrations of addition and subtraction as well as problems for practice. I looked hard for material that would visually or interactively engage middle school students. If you have ideas to share on how you teach these operations, please use the comments area to put them online.

Visual Fractions
An exceptional tutorial on fractions, including step-by-step, illustrated explanations of addition and subtraction. Both circle and line models help students visualize the operations with like and unlike denominators. Interactive problems allow students to use these visual models as they figure the numerical answers.

Adding Fractions
Students must do the usual exercise of finding equivalent fractions with common denominators, but here the fractions are represented visually as portions of a square. Once the computer checks that the fractions are correct, students can drag the representations into a third box and enter the sum of the fractions. This is a real learning experience!

The Fractionator
Created by math teacher Jeff LeMieux, the site offers online and offline tools to help students understand fractions. The online tools use unit squares to model two fractions to be added (or subtracted) and then create equivalent fraction models; with this visual aid, students complete the operation. They can request a new problem for each exercise or enter the two new fractions themselves. Also provided are links to printable materials.

Soccer Shootout
Practice time! Students can practice the addition and subtraction of fractions at levels of difficulty ranging from Easy to Super Brain. Students play against the computer and are provided with a full solution when a wrong answer is entered.

Classic Middle-Grades Problems for the Classroom
The king finds a bowl of mangoes and eats 1/6 of them; the queen eats 1/5 of the remaining mangoes; the prince eats 1/4, etc., until only 3 are left. How many were in the bowl to begin with? A complete lesson plan is given, including activity sheets. A thought-provoking problem to cap this work on fractions!

We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/10/2011.

Let’s Talk Teaching: Games in Math Class

In my years of teaching grades 6 through 8, I generally used games only for reviewing before a test. What I didn’t realize was how effective games can be for teaching the content. Each of the games below has a learning objective; each could be embedded in a lesson plan for middle school math. And, as you know, games focus students’ attention as few other teaching strategies can. Use our comment box below to share with other teachers the games you use in class!

Polygon Capture
This excellent lesson uses a game to stimulate conversation about the properties of polygons. A player draws two cards, one about the sides of a polygon, such as “All sides are equal,” and one about the angles, such as “Two angles are acute.” The player then captures all the polygons on the table that fit both of the properties. Provided here are handouts of the game cards, the polygons, and the rules of the game.

Maze Game
This online activity allows the player to practice their point plotting skills by having them move a robot through a mine field to a target location.  Great for learning to visualize coordinates on the Cartesian plane!

The Factor Game
In this two-player game, one person circles a number from 1 to 30 on a game board. The second person circles (in a different color) all the proper factors of that number. When no numbers remain with uncircled factors, the person with the largest total wins. A lesson plan outlines how to help students analyze the best first move in the game, which leads to class discussion of primes and squares as well as abundant and deficient numbers.

Data Picking
In this interactive game, students first create a table using data they collect from the onscreen characters. They then select a scatter plot, a histogram, a line graph, or a pie chart that best represents the data. The amount of data increases and the type of data representation changes according to which of three levels of difficulty is selected.

Fraction Track
Working in two-player competition or individually students practice finding equivalent fractions and ways of combining fractions as they move their pieces across the board. Both sites use applets, but the basic game play can be set up using only paper game boards and chips.

We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/10/2011.

Making Math Visual

The abstract concepts of mathematics, usually expressed through symbols and un-common vocabulary, can frustrate the visual learners in your middle school classroom. Here is where the computer can become a powerful teaching tool. Such commonplace but abstract concepts as fractional equivalence and the “size” of large numbers can be made visual through technology. Students can interact with virtual manipulatives to change algebraic variables on a balance scale, or rotate a 12-sided solid to see its regularity and symmetry.

Below are a few examples of what I mean. If you have found other sites that make math visual for your students, please use our comment box below to share them with other teachers!

The MegaPenny Project
This site shows arrangements of large quantities of U.S. pennies. It begins with only 16 pennies, which measure one inch when stacked and one foot when laid in a row. The visuals build to a thousand pennies and in progressive steps to a million and even a quintillion pennies! All pages have tables at the bottom listing the value of the pennies on the page, size of the pile, weight, and area covered (if laid flat). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

The Pythagorean Theorem
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” Five interactive, visual exercises require students to delve deeper into the mystery; each exercise is a hint that motivates and entices. The tutorial ends with information on Pythagoras and problems that rely on the theorem for their solutions.

Fraction Sorter
Using this applet, the student represents two to four fractions by dividing and shading areas of squares or circles and then ordering the fractions from smallest to largest on a number line. The applet even checks if a fraction is correctly modeled and keeps score. A visual support to understanding the magnitude of fractions!

Algebra Balance Scales — Negatives
This virtual balance scale offers students an experimental way to learn about solving linear equations. Blue blocks represent positives and red balloons represent negatives. The student solves an equation by adding or removing the blocks and balloons, while a record of the steps taken, written in algebraic terms, is shown on the screen.

Geometric Solids
This tool allows learners to investigate various geometric solids and their properties. They can manipulate and color each shape to explore the number of faces, edges, and vertices, and to answer the following question: For any polyhedron, what is the relationship between the number of faces, vertices, and edges?

Transmorgrapher 2
Another way to “explain” geometric transformations! Using this applet, students explore the world of translations, reflections, and rotations in the Cartesian coordinate system by transforming polygons on the plane.

Cynthia Lanius’ Fractal Unit
This unit developed for middle school students begins with a discussion of why we study fractals at all. Lessons then provide step-by-step explanations of how to make fractals, first by hand and then using Java applets—an excellent strategy! But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike.

We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/10/2011.

Action with Fractions!

Really understanding what fractions are, how they fit on the number line, and how to operate with them – add, subtract, multiply, divide – is central to learning decimals and percentages. According to the NCTM Principles and Standards, students in the middle grades should be expected to acquire a deeper understanding of fractions, decimals, and percents and an increased flexibility in using them to solve problems. Yet students may reach even the higher grades of middle school without a firm grasp of fractions. Perhaps a change in strategy could help? Let’s offer compelling visuals and hands-on manipulation of those fractured numbers!

COUNTDOWN: Number and Operations – Fractions
COUNTDOWN is an interactive television math program broadcast on cable television in Chicago. This web site contains the archives of those broadcasts. The 4-7 minute math movies consist of direct instruction and are reinforced with literature, manipulatives, activities and related computer instruction. Topics include logic, perimeter, area, probability, graphing, congruence, integers and much more. Movies are organized according to content standards established by the National Council of Teachers of Mathematics. Many of the movies have associated worksheets that can be printed.

Visual Fractions
A thorough tutorial on the topic—from identifying and comparing fractions to operations with them! Students work through interactive exercises and games. A complete step-by-step, illustrated explanation of each subtopic is included. Don’t miss the game of Finding Grampy, where students practice finding a mixed number in lowest terms as they look for Grampy on the number line.

Fraction Sorter
Using this online manipulative, students represent two to four fractions—such as 7/13, 2/7, 8/9, and 2/3—by dividing and shading areas of squares or circles. They then order the fractions from smallest to largest on a number line. The visual representation here is powerful.

National Library of Virtual Manipulatives
Under the middle school offerings, you will find the activity Adding Fractions. Students must do the usual exercise of finding equivalent fractions with common denominators, but here the fractions are represented visually as portions of a square. Once the computer checks that the fractions are correct, the students can drag the representations into a third box and enter the sum of the fractions. This is a learning experience! There are other activities on fractions as well, all worth checking out.

The Maths File Game Show
This BBC site offers a game that requires players to match fractions with equivalent decimals or percentages, Go to Saloon Snap. When a question is answered correctly, the player can move his or her piece across the board. The goal is to be the first to create a path across a 10-by-10 grid. It’s a real math exercise and it’s motivating!

We Want Your Feedback
We want and need your ideas, suggestions, and observations. What
would you like to know more about? What questions have your
students asked? We invite you to share with us and other readers
by posting your comments. Please check back often for our newest
posts or download the RSS feed for this blog. Let us know what
you think and tell us how we can serve you better. We appreciate
your feedback on all of our Middle School Portal 2
publications. You can also email us at msp@msteacher.org. Post
updated 11/18/2011.