# Geoboard Geometry

Sometimes geoboards are left on the shelf because we don’t know what to do with them. They can be powerful tools for students to study, length, area and perimeter. (But remember to be careful with the perimeter part because the length of one unit is only measured on the horizontal or vertical, not the diagonal.) Geoboards can help students experience area so that they can develop area formulas for themselves.

Geoboards in the Classroom
This unit deals with the length and area of two-dimensional geometric figures using the geoboard as a pedagogical device. Five lesson plans are provided.

The Online Geoboard
An applet simulates the use of an actual geoboard without the usual limitations of working with rubber bands. Most materials designed for real geoboards may be used with this online version.

Rectangle: Area, Perimeter, Length, and Width
This applet features an interactive grid for forming rectangles. The student can form a rectangle and then examine the relationships among perimeter, area, and the dimensions of the rectangle as the rectangle dimensions are varied.

Investigating the Concept of Triangle and the Properties of Polygons: Making Triangles
These activities use interactive geoboards to help students identify simple geometric shapes, describe their properties, and develop spatial sense.

National Library of Virtual Manipulatives: Geometry (Grades 6—8)
This site has a number of virtual manipulatives related to the NCTM geometry standards.

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
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 12/07/2011.

# 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 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
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.

# Linear Measurement

Sure middle schoolers know how to measure length! But if students need a little more experience with linear measurement, here are challenging resources that put linear measurement into a practical context, or can be used for review before introducing a new measurement topic.

Big Tree: Have You Ever Seen a Tree Big Enough to Drive a Car Through?
In this thought-provoking, discussion-generating challenge, students use common sense, given information, and their calculated tree diameters to answer the question. Teachers should investigate the links at the bottom of the web page. All activities on this site are printable.

Constant Dimensions
Students measure length and width of a rectangle using both standard and nonstandard units of measure. After creating a length vs. width graph, students observe an interesting and important fact — the ratio of length to width in a rectangle is constant.

Inclined Plane
Here is a terrific multiday lesson for teachers interested in having students apply their linear measurement skills and engage in mathematics discussions. Students use length and height data from hands-on experiments to draw conclusions. Useful teacher information is included.

Perimeter Explorer
Using this applet, students determine perimeters for irregular shapes on a grid. The applet is part of a complete lesson reinforcing students’ concept of perimeter and skills for finding perimeters. Shape area can be varied and a table comparing perimeter and area can be generated.

Reaching New Heights
Measuring, collecting and interpreting data, using variables—this complete lesson has it all! Students measure height and arm span, create a scatterplot, and draw conclusions about the correlation. This lesson is an excellent way to build the foundation for the study of functions. Teacher support and information about supporting research is included.

Rectangle: Area, Perimeter, Length, and Width
Using this simulation, students can instantaneously see the interplay among perimeter, area, and the rectangle dimensions. The rectangle size or shape is changed by dragging a point.

# 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