Hands-on Teaching: Area and Volume
Geometry make more sense when kids can hold math in their hands.
- Grades: 3–5
Easy, Fun Math Manipulatives
Tools to teach area and volume are all around you.
- Cheez-It square crackers
- Graham crackers
- Colorful tiles from a tile store
- Square, laminated photos of students
- Styrofoam peanuts
- Dried beans
- Colored water
- Sugar cubes
Review the concept of volume by asking students to bring in clean, empty food containers from home, such as cans, jugs, and cartons. Explain that these containers are labeled with different ways we measure volume (e.g., gallons, ounces, liters, cups). Start a word wall that displays both the terms and the items, using tacks and glue to hold up containers. Kids can peel labels off cans and cut out catalog items and magazine ads. Soon they’ll be fluent in the language of volume.
Distribute rulers, and have students use them to help draw squares and rectangles of different sizes from graph paper, construction paper, or magazines. After they’ve cut out the shapes, challenge kids to create pictures or designs by gluing the shapes onto cardboard, making sure they don’t overlap. Ask them to find the area of each shape and then the total area of the design. Have students display their works by hanging them on a bulletin board or shelf in order from least to greatest area. Make a class mural by asking students to arrange their designs on one sheet of butcher paper and then calculate the total area of all the shapes.
Ask students if they can find the volume of a shoe box using Unifix cubes. Do they need to fill the box to find the answer? Line the bottom of the box with cubes. How many cover the bottom? What is the area of the base of the box? Now ask how many layers they will need to fill the box. Let students experiment in small groups. Some will want to fill the box entirely with cubes to find the volume, while others may realize they only need to find the height of the box in cubes and multiply it by the area of the base. When reflecting on the strategies they used to find the volume, students should discover that the “shortcut” to finding volume is to multiply length and width and height.
Tall and Short Containers
Use two identical pieces of construction paper to make cylinders—one tall and skinny, the other short and stout. Tape each cylinder together by lining up the seams so they do not overlap. Ask students which cylinder holds more. Place the tall, skinny cylinder inside the short, stout cylinder. Fill the tall cylinder to the top with dry ingredients such as rice, popcorn or Styrofoam peanuts. Lift the tall cylinder, letting the dry ingredients fill the short cylinder. Children will see that the short cylinder still has room for more, and thus has a larger capacity.
Give each student 20 square units and ask them to make any shape or design they’d like to with their squares. Lead students on a tour to see the shapes made by others in the class. Once they’ve returned to their seats, ask what all of the designs have in common (they’re all made of squares, and they all have an area of 20 square units). Next, have the students find the perimeter of their shapes by counting the units along the outside edges. Whose shape has the smallest perimeter? The largest? They’ll discover that perimeter does not depend on area.
Have students work in pairs with a partition between them. Give them each a pile of square tiles and challenge them to make a figure with a specified area of square units. Once they’ve constructed their figures, let them lift the partition and compare. Do two figures with the same area have the same perimeter or shape?
Give students sets of tangram puzzle pieces and have them find the area of each piece. (For the rhombus, they will need to divide it into two triangles and one square, find the area of each, and then find the total.) Have students write the area on each puzzle piece. Now ask them to put the pieces together to form one large square and find the area of the square. If they add the areas of the seven smaller pieces, their answer should be equal to (or close to) the total area of the large square.
Line up five or six containers of different sizes and shapes and have students figure out how they compare in volume. This can be done with water on a warm day, or with dry ingredients indoors. Have groups of students share the strategies they used.
Solid containers aren’t the only ones with capacity. Containers that change shape, such as balloons, sponges, and even our lungs, also have capacity. Bring in a selection of sponges that have different sizes, shapes, and density. Have students predict which sponges will hold the most water, arranging the sponges in order from least to greatest capacity according to their predictions. Place the sponges in a tub of water for 10 minutes, then let students squeeze the water into individual graduated cylinders to get an estimate of how much water each sponge held. Older students can measure the volume of each sponge (length times width times height for rectangular and square prism shapes) before and after it sits in the water, as if it were an empty container. How does the calculated volume relate to the amount of water a sponge can hold?
Finally, for a digital twist, introduce Setting the Stage with Geometry, a new math program designed to help students build basic skills for measuring 2D and 3D shapes. It includes extension activities and worksheets.