Volume and capacity by direct comparison and use of informal units

Here are two containers (A and B).

• Which container holds the most water? (Let the student predict.)

Check which container holds more using this water.

Signs of fluency and understanding

• Uses visual comparison to correctly decide which container holds more. Pours water from one container to another to verify which container holds more.

What to notice if they don’t solve the problem fluently

• Gets confused by the height of the containers, may think that taller containers hold more water, and is unsure of how to compare the capacity of the containers by pouring from one to the other, which may indicate that the student would benefit from water play to learn how to compare the capacity of containers.

Supporting activity

• Comparing capacities

Here are three more containers (C, D, and E, including A and B).

• Put the containers in order from smallest to largest based on how much water they hold.

Let the student order the containers.

• Use this water to check if the order you have is correct. Explain what you are doing.

Signs of fluency and understanding

• Orders the five containers by visual comparison with the indication that both cross-section and height are considered when comparing two containers. Organises the containers logically by comparing them two at a time.
• Tests containers by pouring from one to the other in sequence. Ideally, the student begins with the container that they believe is the largest. Explains what each pair comparison shows.

What to notice if they don’t solve the problem fluently

• Attends to height only, rather than both cross-sectional area and height, when ordering the containers. This may indicate that the student needs more experience directly comparing the capacities of containers with varying heights and cross-sectional areas.
• Unable to order five containers logically by comparing them two at a time may indicate that the student needs support organising the results of paired comparison of objects into a sequence when more than two objects are involved. This is also likely to apply when comparing lengths, areas, and weights.

Supporting activity

• Ordering capacities

Here are two packets. (Long and narrow and short and broad, for example, toothpaste and baking powder)

• Which box do you think holds the most rice? Explain how you know.

Use this rice to check which packet holds the most. (Provide the students with a container of rice.)

Signs of fluency and understanding

• Compares the boxes by volume, correctly attending to visual appearance. The student may align the boxes by height or compare base areas. Pours from one box into the other to check if the comparison is correct. Explains that there is some rice left over when they pour from the largest box to the smallest, or that there is not enough rice if they pour from the smallest to the largest.

What to notice if they don’t solve the problem fluently

• Attends to one dimension, such as height, when comparing the boxes. Incorrectly thinks the taller or longer box holds more. This may indicate that the student needs experience comparing boxes of different heights and widths by pouring from one to the other.
• Unable to check the comparison by pouring rice from one box to the other, or is unsure what the pouring shows, particularly if the result is inconsistent with their previous view. The student may benefit from pouring from each box into containers with identical cross-sections.

Supporting activity

• Comparing boxes by volume

Here are three more boxes (as well as those used for question 3).

• Put the boxes in order from smallest to largest based on how much rice they hold.
• Let the student order the boxes.

Use this rice to check if the order you have is correct. Explain what you are doing.

Signs of fluency and understanding

• Orders the five boxes by visual comparison with the indication that all three dimensions are considered when comparing two boxes. Organises the boxes logically by comparing them two at a time.
• Tests volumes of boxes by pouring from one to the other in sequence. Ideally, the student begins with the box that they believe is the largest. Explains what each pair comparison shows.

What to notice if they don’t solve the problem fluently

• Attends to one dimension only, such as height, rather than both cross-sectional area and height, when ordering the boxes. This may indicate that the student needs more experience directly comparing the volumes of boxes with varying heights and cross-sectional areas.
• Unable to order five boxes logically by comparing them two at a time may indicate that the student needs support organising the results of paired comparisons of objects into a sequence when more than two objects are involved. This is also likely to apply when comparing lengths, areas, and weights.

Supporting activity

• Ordering boxes by volume

• How many cubes will it take to completely fill this box?
• Explain how you worked the number out.

Choose one of the boxes used for question 4.

Signs of fluency and understanding

• Makes a tower of cubes and works out how many towers will fit into the box and may use repeated addition or multiplication to find the total number of cubes.
• Makes a layer by creating a rectangle that matches one face of the box and works out how many layers fit into the box. Uses skip counting, repeated addition, or multiplication (ideally) to calculate the volume.

What to notice if they don’t solve the problem fluently

• Fills the box with cubes, tips them out, and counts the number of cubes, which may indicate that the student needs experience structuring units of volume in three-dimensional arrays.

Supporting activity

• Measuring volumes using cubic units