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# Volume and Capacity Units of Work

Volume is the measure of space taken up by a three-dimensional object. The space within a container is known as its capacity but as the thickness of many containers is negligible, it has become acceptable to refer to the space inside as volume too. (The terms volume and capacity are used interchangeably throughout the measurement strand of the NZ curriculum document although the glossary defines capacity as the interior volume of an object.)

Two different practical situations need to be experienced by students as they learn about volume. One relates to experiences involving "how much space does a three-dimensional object occupy?" which eventually leads to measures of volume derived from measuring the length of the object’s dimensions. The other set of experiences relates to measures of fluids.

### Level 1 Volume and Capacity

 Achievement Objectives Learning Outcomes Unit title GM1-1 push, pull, lift and handle objects in order to become aware of masscompares 2 masses by pushing and liftingpack materials and fill containerspour liquids from and into containers Tricky Bags compare the volume of two containers by packing or pouringorder the volume of three or more containers by packing or pouringrecognise that two matched amounts of liquid remain the same when one amount is poured into a container of a different shape. Three Bears use non-standard volume units (cups, spoons, bottles) to fill a container and count the number usedbuild with blocks and count the number of blocks usedcompare and order volumes of containers using non-standard volume units Spoonfuls, Cupfuls and Handfuls GM1-1NA1-2 use non standard units to measure the volume of a containeraccurately count a set of up to 20 objects Dino Cylinders develop an understanding of 100 and the quantity for which it standsunderstand the relationship between 100 and 10 Counting on Measurement

### Level 2 Volume and Capacity

 Achievement Objectives Learning Outcomes Unit title GM2-1 use non-standard volume units (cups, spoons, bowls) to fill a container and count the number usedrecognise the need for a standard unit of volumemeasure to the nearest litre and half litre by using litre containers to fill and count Popcorn accurately measure volume using standard kitchen measuring cups How Much Cereal? use objects of 1 litre volume to estimate the volume of other objectsdiscuss the need for having and using standard measures of volumemake sensible estimates about the volume of given objectscarry out conversions between basic standard measures of volume (millilitres to litres)explain the meaning of metric prefix terminology (e.g kilo) Making Benchmarks GM2-1GM2-2 estimate volume using litres and millilitresaccurately measure volume using litres and millilitres Party Volumes

### Level 3 Volume and Capacity

 Achievement Objectives Learning Outcomes Unit title GM3-1 recognise the need for a standard unit of volumemeasure volume using teaspoons and table spoonsconvert between units of volume: teaspoons, tablespoons, half and quarter cups Rainbow Jelly construct three-dimensional objects using cubic centimetres and state their volumeconstruct a model of a one cubic metre Boxing on recognise that objects have many measurable attributesidentify and measure attributes of common objects Oranges L3 GM3-1NA3-1 recognise the need for a standard unit of volumeestimate and measure to the nearest litrerelate the litre to familiar everyday containers Slosh, Dribble and Plop estimate the volume/mass of one marshmallow using appropriate standard units.design an investigation to find the average volume/mass of one marshmallowcarry out an investigation to find the average volume/mass of one marshmallow Marshmallows

### Level 4 Volume and Capacity

 Achievement Objectives Learning Outcomes Unit title GM4-1 recognise that objects have many measurable attributesidentify and accurately measure attributes of common objects Oranges L4 GM4-1GM4-2 use a formula to calculate the volume of cuboids by measuring the length of each of the three dimensionsinvestigate the relationship between millilitres and cubic centimetres Spaced Out plan a mathematical investigation in a group.take measurements and make calculations to complete an investigationInterpret the accuracy of the investigation Measurement investigations I

### Level 5 Volume and Capacity

 Achievement Objectives Learning Outcomes Unit title GM5-1 plan a mathematical investigation in a group.take measurements to make calculations to complete an investigationinterpret the accuracy of the investigation Measurement investigations II GM5-1NA5-1NA5-4 convert between millilitres and litresconvert ratios to fraction and percentage expressionscompare ratioscombine ratiossolve problems involving rates Ratios GM5-4GM5-9NA5-4 use scale factors to investigate areas being enlargeduse scale factors to investigate volumes being enlargedsolve real life context probelms involving scale factors Scale Factors

#### Stage One: Identifying the attribute

As with other measures, students require practical experience to begin to form the concept of an object taking up space. Students need lots of experiences of filling and emptying containers with sand and water. They need to have pouring experiences with containers of similar shape but different capacity and vice versa. They also need to fill containers with objects and build structures with blocks. The use of language such as it’s full, it’s empty, there’s no space left and it can hold more, focus attention on the attribute of volume. The awareness of the attribute of volume is extended as comparisons of volume are made at the next stage.

#### Stage Two: Comparing and ordering

It is important that students experience activities in which they compare and order attributes as these extend their understanding of the attribute and introduce them to informal measuring processes. Containers can sometimes be measured directly by placing one container inside another. Most comparisons however, need to be made indirectly by pouring from one container to another container to see which holds more.

If students realise that two matched amounts of liquid remain the same when one amount is poured into a container of a different shape, they are said to conserve volume.

Now A > B, B > C implied A > C is a transitive relation. These are extremely important in life and even more so in maths. In life we actually put too much store by them and wonder why, when the All Black beat the Springboks and the Springboks beat the Wallabies, that the All Blacks can’t beat the Wallabies.

#### Stage 3: Non-Standard units

When a comparison between two containers requires the student to find out how much more one container holds, then a unit of volume is required. Measuring the area of objects using non-standard or informal units is the third stage in the learning sequence. Beginning with non-standard, but familiar units such as eggcupfuls and cupfuls allows the students to focus on the process of repeatedly using a unit as a measuring device.

In addition to lots of filling activities using liquids, the students can pack containers with marbles and blocks. They can also build different objects with blocks.

From the earliest of these experiences, students should be encouraged to estimate. Initially these estimates may be no more than guesses, but estimating involves the students in developing a sense of the size of the unit. As everyday life involves estimating at least as frequently as finding exact measures, the skill of estimating is important.

At this stage students can also be introduced to the appropriateness of units of measure. For example, a cup is more appropriate than a spoon for measuring the volume of a bucket.

Although non-standard units reinforce most of the basic measuring principles, students need to realise that they are limited as a means of communication. This can be highlighted through activities that involve the students measuring the volume of an object using different sized cups.

#### Stage 4: Standard units

When students can measure areas effectively using non-standard units, they are ready to move to the use of standard units. The motivation for moving to this stage, often follows from experiences where the students have used different non-standard units for the same volume. This allows them to appreciate that consistency in the units used allows for easier and more accurate communication.

The usual sequence used in primary school is to introduce the litre as a measurement of volume before using cubic centimetres and cubic metres.

Student’s measurement experiences must enable them to:

1. develop an understanding of the size of a litre and 10 millilitres. (1 millilitre is too small to be appreciated);
2. estimate and measure using litres and millilitres;
3. develop an understanding of the size of a cubic metre and a cubic centimetre;
4. estimate and measure using cubic metres and cubic centimetres.

The standard units can be made meaningful by looking at the volumes of everyday objects. For example, the litre milk carton, the 2-litre ice-cream container and the 100-millilitre yoghurt pottle. Students should be able to use measuring jugs and to say what the measuring intervals on the scale represent.

Cubic centimetres are usually in abundant supply in classrooms and can be used to construct larger cubes and other shapes. A cubic metre can be built using metre rulers and compared with spaces such as that under the teacher’s desk.

#### Stage 5: Applying and Interpreting

When the students are able to measure efficiently and effectively using standard units, their learning experiences can be directed to situations that encourage them to "discover" measurement formulae and investigate the relationship between litres and cubic centimetres.

Students could find out that a container that holds 1000 centimetre cubes can also hold one litre of water. They can then deduce that one cubic centimetre and 1 millilitre represent the same amount of space, although when you pour a millilitre of water onto a surface it is difficult to believe it has the same volume as a centimetre cube.

The students can also use sets of centimetre cubes to construct rectangular prisms and then calculate the volume and "discover" the volume formula. Given the links between area and volume it is important that students first understand how to calculate the area of a surface so that they can see how to use this is used in calculating the volume of an object.

Volume of a cube = l x l x l.

Volume of a rectangular prism = l x w x h