# Measuring a Solid

Many students never really understand volume or surface area, although they can memorize the formulas and even apply them on tests. These resources have been selected with an eye to helping students enter into the concepts of volume and surface area through practical problems, hands-on experiences, and applets they can manipulate to actually see how these measurements are affected by change in a figure’s dimensions. Please add your ideas on how to teach these concepts in the comments section.

Keeping Cool: When Should You Buy Block Ice or Crushed Ice?
Which would melt faster: a large block of ice or the same block cut into three cubes? The prime consideration is surface area. A complete solution demonstrates how to calculate the surface area of the cubes as well as the large block of ice. Related problems involve finding surface area and volume for irregular shapes and examining the relationship between surface area and volume in various situations.

How High? Geometry (Grades 6-8)
Using an excellent online simulation, students pour a liquid from one container to a container of the same shape, but of a larger size. Students choose from four shapes: rectangular prism, cylinder, cone, and pyramid. The smaller version of the selected shape is shown partially filled with liquid; the base dimensions of both containers are given. Using this information, students use a slider to predict how high the liquid will rise when poured into the larger container. On “pouring” the liquid, students can compare their prediction with the results. Multiple problems are available for each of the shapes.

Popcorn: If You Like Popcorn, Which One Would You Buy?
Students are directed to use popcorn to compare the volumes of tall and short cylinders formed with 8-by-11-inch sheets of paper. A simple but visual and motivating way of comparing volume to height in cylinders! The solution offered explains clearly all the math underlying the problem.

Surface Area and Volume
With this applet students explore both rectangular and triangular prisms. They can set the dimensions (width, depth, and height), observing how each change in dimension affects the shape of the prism as well as its volume and surface area. This is a quick way to collect data for a discussion of the relationship between surface area and volume or have students practice computing these measurements.

Pyramid Applet
This applet allows students to set the width, height and length of a pyramid. They then see the initial cutout (the net) and watch it fold into the pyramid specified. For better viewing, the pyramid can be rotated. At this point, the surface area and the volume are shown. No activities accompany the applet, except for the challenge to try to minimize the surface area while maximizing the volume.

Three Dimensional Box Applet: Working with Volume
With this applet, students create boxes online; for each box, its dimensions, surface area, and volume are displayed onscreen.  Since various sizes of boxes can be created, data can be quickly collected and the relationship between volume and surface area explored.  A visual and “hands-on” experience!

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# Close Encounters with Ratios

Understanding ratio and proportion, one of NCTM’s Focal Points for grade 7, presents a real challenge for all levels of middle school. Here are classroom-friendly ways to explore the topic from several angles. Each involves visuals or hands-on activities that bring students into close contact with the abstract concept of ratio. Let other teachers hear your ideas on teaching this topic! Post a comment below.

Constant Dimensions
In this carefully developed lesson, students measure the length and width of a rectangle using standard units of measure as well as nonstandard units such as pennies, beads, and paper clips. When students mark their results on a length-versus-width graph, they find that the ratio of length to width of a rectangle is constant, in spite of the units. For many middle school students, not only is the discovery surprising but also opens up the whole meaning of ratio.

Discovering the Value of Pi
Students measure the diameter and circumference of several circles, using a handy applet, record their data, and reach conclusions about the ratio of circumference to diameter. A genuine guided exploration!

Math-Kitecture
Math-Kitecture is about using architecture to do math (and vice versa). Activities engage students in doing real-life architecture while learning estimation, measuring skills, proportion, and ratios. In Floor Plan Your Classroom, for example, exact directions are set out and illustrated on how to make a copy to scale of your classroom.

What’s My Ratio?
What would happen to a picture in the pocket of someone who is shrunk or enlarged? This question hooks students into a study of similar figures. As they compare the measurements of corresponding parts of pictures that have been either decreased or increased in size, they can investigate concepts of similarity, constant ratio, and proportionality.

Figure and Ratio of Area
A page shows two side-by-side grids, each with a blue rectangle inside. Students can change the height and width of these blue rectangles and then see how their ratios compare — not only of height and width but also, most importantly, of area. The exercise becomes most impressive visually when a tulip is placed inside the rectangles. As the rectangles’ dimensions are changed, the tulips grow tall and widen or shrink and flatten. An excellent visual experience!

Capture-Recapture: How Many Fish in the Pond?
To estimate the number of fish in a pond, scientists tag a number of them and return them to the pond. The next day, they catch fish from the pond and count the number of tagged fish recaptured. From this, they can set up a proportion to make their estimation. Hints on getting started are given, if needed, and the solution explains the setup of the proportion.

Size and Scale
This is a challenging and thorough activity on the physics of size and scale. The final product is a scale model of the Earth-moon system, but the main objective is understanding the relative sizes of bodies in our solar system and the problem of making a scale model of the entire solar system. The site contains a complete lesson plan, including motivating questions for discussion and extension problems.

Scaling Away
For this one-period lesson, students bring to class either a cylinder or a rectangular prism, and their knowledge of how to find surface area and volume. They apply a scale factor to these dimensions and investigate how the scaled-up model has changed from the original. Activity sheets and overheads are included, as well as a complete step-by-step procedure and questions for class discussion.

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# Hands-On Measuring

Students need active learning experiences to understand measurement concepts and develop important skills. These resources provide opportunities for students to problem solve with hands-on and virtual measurements in real-world and online environments.

The Global Sun Temperature Project
Bigger than hands-on, this is an annual real-world, international and interdisciplinary research project for students. Classes gather local data, post data online, and use the aggregated data to see how average daily temperatures and hours of sunlight relate to distance from the equator.

It Takes Ten
Students use metric units to estimate and measure weight, length, and volume, and to determine area.

Open-Ended Math Problems: Get Ready, Get Set
Select a month and scroll down to find open-ended measurement problems at three levels of difficulty. Students build mathematics understanding and see how mathematics is used in everyday life.

Pentagon Puzzles
This measurement lesson is one of 37 hands-on projects focused on mathematics. See http://www.math.nmsu.edu/~breakingaway/lessons.html for more lessons.

Popcorn Math
Here is a volume estimating activity for students to do on their own or with others.

Surface Area and Volume
Examine prisms from multiple views, adjust dimensions, rotate prisms, and see how dimension changes impact volume and surface area. Students can also calculate volume and surface area.

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# 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

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# Density

Density is a property of materials included in the National Science Education Standards Physical Science Content Standard B. It is a property by which mixtures can be separated but has much more profound applications outside the classroom such as rock formation, severe weather and living systems. But none of these concepts are fully comprehensible without a fundamental conceptual understanding of density. This is an abstract concept combining the concepts of mass and volume. The resources here provide examples designed to help you facilitate student acquisition of a conceptual understanding of density.

Funny Water
This page from the National Science Education Standards starts the story of a middle school teacher’s inquiry lesson on density.

Density Balloon
What happens to the density of a balloon as it is heated and cooled? Here students use a hair dryer to heat a helium-filled Mylar balloon, causing it to rise, and let it cool, causing it to drop. The activity includes a description, a list of science process skills and complex reasoning strategies being used. Also provided are content topics, a list of necessary supplies, instructions, and presentation techniques. The content of the activity is explained, and assessment suggestions are provided.

Liquid Rainbow
In this density activity, students determine the relative densities of five liquids and layer them atop one another in a straw.

Atmospheric Properties: Convection
This page describes convection currents in terms of density differences.

Potato Float
How can a potato wedge be made to hover in the center of a glass of liquid? Students investigate density using potato wedges and water and sugar water solutions.

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