Ordering boxes by volume

Note: Volume is the measure of space taken up by a three-dimensional object. The space within a container is known as its capacity. As the thickness of many containers is negligible, it has become acceptable to refer to the space inside as volume too. You might frame the purpose for finding capacity within a context that is relevant to your students' interests, cultural backgrounds, and to learning from other curriculum areas. The māori kupu for volume is rōrahi.

1. Show students the set of five packets labelled A-E. Ensure the packets are not presented in order of volume.
   Highlight packets A and B.
   Which packet holds the most rice, that is, has the greatest volume? How do you know?
   Look for students to attend to all three dimensions (height, width, depth) when comparing the packets by volume. They might suggest reorientating the packets so at least one dimension is similar, and say things like:
   A is taller than B but not as wide.
   B has more area in its base than A but is not as tall.
    
2. Check and confirm the comparative volumes by pouring rice from one packet to another.
    
3. Create a large continuum, with opposite ends labelled "smallest volume" and "largest volume". 

   
    

4. According to the order students give you, place packets A and B in appropriate positions.
    
5. Introduce packet C.
   Where should this packet go on the line? Explain how you know.
   Students may be willing to accept that C has less volume than either A or B. Confirm the relationship by pouring rice from the smallest of packets A and B into packet C. If the volume comparison is not obvious, discuss with students how they might predict which packet has greatest volume. 

• Invite students to suggest which pairs of packets can be compared to establish the correct order. For example:
  If we know that packet C has more volume than B but less volume than A, where does it go on the continuum?
• Confirm the volume of order of A, B and C by pouring rice from one packet to another.
• Place packet C on the continuum.
   

6. Ask students to draw a picture of the continuum, including the positions of containers A, C, and E. You might model this for students, or provide a template for them to use. Emphasise that the drawing should only include the key details (e.g. the relative size of each container B - C - A).
    
7. Remove packets C, B and A from the line. Take packet B and ask students to compare it, by volume, with packet D.
   Which packet has more volume, packet B or packet D?
   Think about where packets B and D should go on the line. 
    
8. Together, confirm the order of all five containers. This might be done as a class or paired discussion, or by having students construct their own representation of volume order. This might make use of language (e.g. I know the order of these packets because...), symbols (e.g. < and >), and/or drawings and physical comparisons. 
   Look for students to:

• Decide on which comparisons are critical, and which are redundant.
• Use their model and/or provided diagrams to organise and apply information about paired comparisons.
   

9. Letting students work collaboratively in appropriate groupings, provide time for them to carry out a similar investigation with the second set of packets labelled F-J. If needed, you could work with a small group of students, either initially or throughout the whole task, to support their participation and understanding. Look for students to: 

• Use all three dimensions of the packets in making predictions about volume.
• Record comparisons accurately and logically, with the effective use of materials, symbols, and/or mathematical language.
• Use transitive reasoning to minimise the number of pairs comparisons needed to complete the full ordering.
• Recognise that packets with the same volume will be represented with the symbol = and will occupy the same point on a continuum.
   

10. Gather together to compare the diagrams for both sets of packets.
    Can you predict the order we will put all ten packets in?
    Are there some pairs we need to check? Which packets do we need to compare? Why?
    Work together to order all ten packets on the continuum.

Next steps 

1. Provide problems in which students compare the volumes of different shaped containers with cuboid shaped packets. For example, compare the volume of a shampoo bottle with that of a milo packet.
   Check the predictions by pouring rice from one container to another.
   How do you compare the volume of very different three-dimensional containers?
   What do you need to consider?
    
2. Give students a packet and ask them to create a cuboid packet from paper and tape that has different dimensions but the same volume.
   This task could be extended to making a cylinder or cone shaped packet with the same volume.