Ordering capacities

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.

1. Show students the set of five containers, labelled A-E, presented in random order. Highlight containers A and E.
   Which bottle holds the most water? How do you know?
   Discuss students' methods for measuring and comparing. Suggestions might include physically aligning the containers to see which one is tallest, or putting the cross-sections together to see which one is widest. 
    
2. Create a large continuum, with opposite ends labelled "smallest capacity" and "largest capacity". 
   
    
3. As a class, compare the capacities of containers A and E by pouring from one container to the other. Ensure the students understand the logic of pouring as a means of comparison. Pouring from the smaller container into the larger results in the larger container being not completely full. Pouring from the larger into the smaller results in the smaller container being full and some water remaining in the larger container. Place containers A and E in appropriate positions on the continuum.
    
4. Introduce container C.
   Where should this container go on the line? Explain how you know?
   Students may be willing to accept that C has less volume than either A or E. Confirm the relationship by pouring in two pairings: AC and EC.
   If we know that this container holds less water than both A and E, where should it go on the continuum?
   Ask a student to place container C where they think it should go.
    
5. 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).
    
6. Remove containers C, E, and A from the line. Show students container D and compare it by capacity with container E.
   Which container holds more, D or E?
   Confirm students' comparisons by pouring. 
   Think about where container D should go on the line. (To the left or right of E depending on the result)
   Should we compare container D with container A? (Yes. That will help place D to the left of A if it holds less.)
   If D holds less than A ask: Should we compare container D with container C A? (Yes. Either container could have more capacity)
   Once the order is decided ask students to add to their drawing to show the placement of container D.
    
7. Finish ordering by considering container B. Carry out paired comparisons until the place of B is decided. 
   Carry out a similar investigation with the second set of containers. Label the containers with further letters to ease recording load. Let the students collaboratively establish the order. Look for the following:
   • Do students predict the capacity of each pair of containers using both height and cross-sections (horizontal and vertical)?
   • Do students record the pair comparisons using a continuum?
   • Do students carry out all the necessary pair comparisons to sequence the containers?
   • Do students record equal capacity as the same location on the continuum?
   • Do students show evidence of transitive thinking? For example, if A>B and C>A then C>B.
      
8. After the students have attempted the ordering, gather together to discuss the points above. To check students' understanding of the process, give them an example in which the order of containers is obviously wrong. Invite students to fix the order.
   What’s the smallest number of pairs we need to check to get the order correct?

Next steps

1. Pose problems where the containers have many distracting features, such as colour, shape, and height. See if students can ignore the distractions of other attributes and establish a reliable order by capacity.
2. Order containers by capacity using a third-party container rather than by direct comparison. For example, a clear jar might be used to record the order. Fill each container with water. One by one, pour the contents into the clear jar and mark the level of water (either with a pen, rubber band, or sticky note). After each pour, empty the jar. Ask students to use this system of comparison to develop their own continuum - either by drawing or using physical materials. This could be done in small groups.