# Triangles Online

How much you want your middle school students to learn about triangles depends on many factors you take into account as you plan. If lesson ideas that are “hands-on,” actually or virtually, enter into that planning, you may find this wide range of resources useful. Please share your own teaching ideas with colleagues by commenting on this post!

Discovering the Area Formula for Triangles
In this lesson, students develop the area formula for a triangle. Students find the area of rectangles and squares, and compare them to the areas of triangles derived from the original shape. Student handouts are included here.

With this virtual manipulative, students arrange sides and angles to construct congruent triangles. They drag line segments and angles to form triangles and flip the triangles as needed to show congruence. Options include constructing triangles given three sides (SSS), two sides and the included angle (SAS), and two angles and an included side (ASA). But the option that will motivate most discussion is constructing two triangles given two sides and a nonincluded angle (SSA). The question in this case is: Can you find two triangles that are not congruent?

Transformations—Reflections
Here students can manipulate one of six geometric figures on one side of a line of symmetry and observe the effect on its image on the other side. A triangle may be selected and then translated and rotated. The line of symmetry can be moved as well, even rotated, giving more hands-on experience with reflection as students observe the effect on the image of the triangle.

The Pythagorean Theorem
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” The site’s author, Jacobo Bulaevsky, speaks directly to students, encouraging them throughout five interactive exercises to delve deeper into the mystery. Within each exercise he gives hints that will motivate and entice your students.

Do you find that your middle school students still become confused with decimals? Not unusual! The resources here offer support in explaining the concepts underpinning addition and subtraction of decimals, in particular, place value. They also include demonstrations of addition and subtraction and opportunities for practice. The goal for students, as stated in NCTM’s Focal Points, is to develop fluency with computing, solving problems, and making reasonable estimates. I hope these resources will visually and interactively engage your students, helping them toward that goal.

Builder Ted
This interactive game deals with place value in decimals, necessary to understanding addition and subtraction. In the game scenario, students help Builder Ted by placing numbered bricks on a ladder in numerical order. At the first level of difficulty, all numbers are positive, but the two higher levels include negative numbers as well. If a number is placed incorrectly, all the bricks immediately fall and the player begins again.

Place Value
The user can type in any number, such as 3601.076, or let the computer choose a number. As the student passes the mouse over each digit in the number, its place value is shown. Also, how to say the number is given, plus a short exercise asking the student to identify the digit in, say, the thousandth position.

Fractions, Decimals, and Percentages
Especially appropriate for tutoring or even review, the site introduces decimals (reading, writing, and comparing) and then offers examples and practice in operating with decimals.

Base Block Decimals
With this virtual manipulative, students can explore the meaning of place value and grouping as they add and subtract decimals. The base blocks can represent negative as well as positive numbers with one to four decimal places. Students exchange and group the blocks as needed to solve the problems created by either the student or the computer. All material is available in Spanish and French as well as English, including instructions for using the manipulative, information about bases and place value, and suggested questions for classroom use.

Decimals
This site offers bare-bones explanations of decimal topics and interactive practice. In the long list of topics are adding and subtracting decimals as well as adding and subtracting money. The computer sets the problem and gives immediate feedback to the student’s response. The bottom of each lesson page contains timed exercises.

# 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!

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# Speaking in Math Terms

—————————–“Words! Words! Words!” sang Eliza Doolittle in the musical My Fair Lady. “I’m so sick of words!” But students have to master the vocabulary of mathematics. This is especially difficult for those learning a new language even as they struggle with new mathematics vocabulary. The first two resources deal with mathematical terms themselves, while the other three offer classroom activities in Spanish and English.

A Maths Dictionary for Kids
This animated, interactive mathematics dictionary for kids explains over 500 common mathematical terms in simple language. Each term is illustrated and, often, accompanied by an interactive applet that makes visual and immediate the definition of the term.

Terms and Formulas from Beginning Algebra to Calculus
This interactive mathematics dictionary offers many terms and formulas appropriate for older middle school students. The illustrations, diagrams, and applets help define the terms every bit as much as the text does.

This is the Spanish equivalent of the National Library of Virtual Manipulatives. Together they offer interactive, web-based virtual manipulatives, mostly in the form of Java applets, for mathematics instruction. On any page, users can easily switch between English and Spanish by using a drop-down menu.

Multicultural Math Fair
Here are ten activities used successfully in a math fair at Frisbie Middle School in Rialto, California. Each activity is available in English and Spanish. Links connect the teacher to information on setting up a fair and locating software.

Who Wants Pizza?/Quien Quiere Pizza?
A fun way to learn about fractions in English and in Spanish. Four activities, each followed by practice exercises, lead students to explore the basic definition of fractions and addition with fractions.

# Area and Volume

Here are online resources with virtual manipulatives that can help make area and volume real for students. Be sure to check out the sites these resources are from — the sites contain many other interesting and useful mathematics learning resources.

Area Explorer
With this simulation, the student finds the areas for irregular shapes on a grid. Answers are checked and a table displays the perimeters and areas. The instructor page contains exploration questions to use to investigate the relationship between area and perimeter.

How High?
This virtual manipulative simulates pouring a liquid from one container to another container with different dimensions and the same or different shape. Students determine the volume of the liquid in the first container and predict the height of the liquid in the second. The container can be a cylinder, tank, or cone.

Neighborhood Math
Two of this site’s printable lessons, Math at the Mall and Math in the Park or City, feature hands-on activities where students use area or volume to explore their actual neighborhood.

Patios: Does Bigger Perimeter Mean Bigger Area?
This activity challenges students to think about the relationship between perimeter and area. Students must use a little ingenuity to find the dimensions of the tiles used to build two patios with the same area, but different shapes.

Scaling Away
In this hands-on lesson, students find the dimensions of a rectangular prism or cylinder and create a larger scale model of the same shape. After calculating surface areas and volumes, students draw conclusions about the relationship between surface area and volume.

Three Dimensional Box Applet: Working With Volume
Students create boxes by using their mouse to indicate how much of each corner should be cut from a grid. The dimensions of the box and its volume and surface are generated by the applet