In this unit we explore the volume (in cubic units) of our skyscraper constructions and those of others in the class. We investigate the most efficient way to pack cuboids in a confined space and conclude with "water play" as we investigate the relationship between millilitres and cubic centimetres
use a formula to calculate the volume of cuboids by measuring the length of each of the three dimensions
investigate the relationship between millilitres and cubic centimetres
This unit leads to an application of the formula for the volume of cuboids, namely that the volume is found by multiplying the length by the width by the height.
Volume of a cuboid (rectangular prism) = l x w x h
In the application, different volumes are calculated by combining cuboids to make a variety of shapes. This reflects a common approach to finding volume (or area) by breaking up complicated shapes into simpler ones whose volume (or area) we can easily find.
The unit also leads to the discovery of the fact that 1000 centimetre cubes occupy the same space as one litre of water. They can then deduce that one cubic centimetre and 1 millilitre represent the same amount of space although it when you pour a millilitre of water onto a surface it is difficult to believe it has the same volume as a centimetre cube.
- Begin the unit by talking about the world’s tallest buildings. While the more recent tall buildings, like the Auckland Skytower, are in thin cylindrical-shaped tower form, some of the older structures are combinations of cuboid shapes. For example, the Sears Tower (see Copymaster), in Chicago, and the Rialto Tower, in Melbourne, are collections of cuboid in structure. Pictures of these buildings are easily found online.
- Tell the students that they are to make a scale model of a tall building by gluing together at least three different cardboard packets from those brought to school. Each building is then glued to a base to form a skyscraper, as seen in large cities throughout the world.
- Allow students time to develop their skyscrapers in groups of four or five. Bring the class back together and discuss what statistics could be displayed about each building. Suggestions might include height, width, length, and volume. Invite suggestions about how the volume, in cubic centimetres, might be worked out. Ideas might include building a cuboid of similar size and counting the cubes (successful but inefficient), making one layer of the building, counting the cubes in one layer and using equal additions to make up the height of the building and multiplying by the edge lengths. Highlight the efficiency of the edge length approach.
- Send the students away to label their structures, with stickers in appropriate places, giving the name of the building (eg. The Toa Tower), and the various edge lengths in centimetres. Note that for some packets these lengths may need to be rounded to the nearest centimetre. Monitor the students’ accuracy in measuring to the nearest centimetre.
- When each building is labelled, instruct the students to choose any five buildings from around the room and calculate the volume (office space) of each structure in cubic centimetres. After a time the original creator of each building can calculate its volume and display it using a sticker. Each student can then compare their solutions and discuss discrepancies with the builder.
- The class tries to identify the room’s biggest buildings by height and volume. This data could be displayed in a graph as an extension and a suitable scale (eg. 1cm = 10 m) established to relate the models to their actual size in real life. Note that the scale has interesting implications for volume in cubic metres. A scale of 1cm = 10m will mean that 1 cm3 = 1000 m3.
- The first exploratory problem is based on finding the most efficient way to pack a collection of cuboids (rectangular prisms) into a confined space. This has many real life contexts that you may wish to use as a story shell, including packing the car boot for a holiday or groceries at the supermarket. The context of the NASA first space trip to Mars may create more interest. We all know that space is at a premium on spacecraft and advanced technology is very tidy!
- Provide the students with a large number of multilink cubes, they will need their 30-cm ruler as well. Instruct them to build the following space packets from the cubes:
4 cm x 4 cm x 6 cm,
2 cm x 8 cm x 6 cm,
6 cm x 6 cm x 4 cm,
2 cm x 6 cm x 4 cm
Their task as the NASA engineer is to find a way of putting the parcels together in the most confined space possible. Allow the students time to attempt the problem. Stress (i) the need to record their solutions and (ii) that they should not join the packets together so that they can be unpacked if they haven’t been joined together correctly. Arranging the packets by manipulation is quite easy.
- Ask the students to work out the total number of cubes used. This creates an interesting crosscheck where the number of cubes in the combined cuboid can be checked against the sum of the cubes in the four packets. Ask the students how knowing the total volume of the combined cuboid could help in arranging the packages in a more difficult problem of this type. In this example the total volume is 48 cubes so that limits the dimensions of the cuboid (eg. 4 x 6 x 2, 3 x 4 x 4).
- Tell the students to use the cubes they have to make up a space packages problem for someone else to solve. The packages must fit together to form a cuboid and they must record how they fit together as a model answer.
- These package problems can be drawn on isometric dot paper if need be. Each problem should be shared with other students.
- The second session of the exploring time concerns Sunken Logs. Wood, like kauri, totara, and spruce that has aged in the bottom of swamps and lakes, is very sought after for making furniture and other items from waka to musical instruments. Tell the students that they are going to explore what happens to water level, as "logs" are submerged. Fill a one litre-measuring container with water to the 500-mL mark. Take a yellow cuisenaire rod (representing a log) and push it into the measuring container until it is just submerged. Ask the students to explain what they think happens as you do this (the water level rises though not appreciably as the rod is small).
- Ask: How we could work out the amount of water a yellow rod displaces (pushes out)?
They may suggest that you need to use a number of yellow rods to get a noticeable rise in water level. Make a cuboid using twenty yellow rods, wrap a rubber band around it to keep it together.
Ask: What is the volume of this cuboid in cubic centimetres?
Ask students to explain how this was worked out (e.g. 4 cm x 5 cm x 5 cm = 100 cm3). Submerge the cuboid into the measuring container with your fingertip and get the students to note the rise in water, which should be about 100 millilitres.
Ask: How much water one yellow rod displaces (about 5 mL since 100 ÷ 20 = 5). Suggest that this may just be coincidence.
- Give each group of students a set of light green, crimson, dark green, black, and brown cuisenaire rods, a rubber band and a measuring container. Tell them that they are to find out how much water each single rod displaces using the dunk a cuboid method. Stress that they must record their methods and results for reporting back purposes.
- After a suitable time of investigation bring the class together to share results. They should find that the water displaced by a rod, in millilitres, matches the volume of the rod in cubic centimetres. This connection needs to be recorded on a chart for the students to refer to.
- The third session of exploring relates to constructing a fish tank of certain dimensions so that it contains enough water for the fish to live in. Begin by stating that this session will be about extending what they found out yesterday about cubic centimetres and millilitres to include larger volumes of water. Show the students a large place value cube (10 cm x 10 cm x 10 cm).
Ask: If this was made of water, how much water would that be?
Various answers are possible including 1000 mL, 1000 cm3, and one litre. Highlight that these measures are equal.
- If possible get a cuboid shaped fish tank to show the class. Note that fish tanks are made by joining together rectangular pieces of plate glass. When the tank is complete, these pieces of glass are the faces of the cuboid.
Pose the following problem:
Jess has bought a pet fish. The people at the shop tell her that she will need a tank that can hold 8 litres of water. Jess decides to build her own tank. What dimensions might the tank have?
- Send the students away with cardboard, scissors, ruler and tape to make a model of the fish tank. As you roam around the room ask students to explain how they know their tank will hold exactly eight litres. The different tanks can be compared for size and suitability. An important idea is that tanks need to have a large water surface area to allow oxygen to dissolve. Students can be asked to find this area for their tanks. If constructed strongly enough the capacity of the tanks can be checked by lining them with a supermarket bag, pouring water in until they are full, removing the bag and measuring its contents.
The exploring aspect of the unit is divided into three sessions.
- Remind the students of the link between the units of capacity and volume using place value blocks.
Hold up the large cube and ask:
How much water would have the same volume as this?
(1000 mL or 1 litre) Ensure that the students justify their reasoning. Repeat the question with the flat, the long and the small cube (100 mL, 10 mL, and 1 mL, respectively)
- Pose this problem: The Just Juice Company wants a new carton that will hold exactly 330 mL of juice. Each side of the new carton must be an exact number of centimetres long (e.g. a side cannot be 4.75 cm long).
- One possible carton would be 330 cm x 1 cm x 1 cm. Ask the students to imagine what that would look like and how we know it would hold 330mL. Suggest that this carton would not be very practical and invite them to design other cartons which are more appropriate.
- The students can make their cartons from centimetre squared paper. Issues such as the ease of fit in a person’s hand should be considered.
- Allow the students time to make a number of possible cartons and bring the class together to share their ideas. Focus on their use of the cuboid volume formula (width x breadth x height) and the application of factors in finding workable dimensions. For example, if 10 cm is to be the length of one edge then 330 ÷ 10 = 33 gives the product of the other two edge lengths. Therefore 10 x 3 x 11 are the dimensions of one possible carton.
- The possible carton sizes could be entered in a table, in a systematic way, to check if all possible cartons have been found. Spreadsheet formulae could be used to make calculations easier:
Edge One (cm) Edge Two (cm) Edge Three (cm) Volume (cm3) 1 1 330 330 1 2 165 330 1 3 110 330 1 5 66 330 1 6 55 330 1 10 33 330 1 11 30 330 1 15 22 330 2 3 55 330 2 5 33 330 2 11 15 330 3 5 22 330 3 10 11 330 5 11 6 330
- Use students’ solutions as samples to assess their grasp of the unit objectives.