nsw curriculum and learning innovation centre introducing the measurement aspect of the numeracy...

NSW Curriculum and Learning Innovation Centre

Introducing the Measurement aspect of the Numeracy continuum


What is measuring?

A measurement results from dividing a continuous quantity, such as length, area or volume, into identical units and determining the number of units.


Measuring a continuous quantity

• Increasing sophistication in measuring makes use of the structure of units.

• That is, rather than simply counting units, we use the structure of units to calculate accumulated distance, area or volume.

Measuring or counting

How tall?

When using a ruler, what answers would you expect?

Four or five, counting the spaces or the marks.

Measuring length is recording an accumulation of distance.


Measuring a continuous quantity

To measure a continuous quantity, such as length of a desk, ……

the length has to be partitioned into units that can be counted by ……

either repeating the unit along the length, or subdividing the length into units of a given size.


Why bother with informal units?

• When informal or non-standard units are used to measure a length, the units have to be either aligned along the length, or one unit has to be repeated and the endpoint of each length marked in some way.

• If students are not shown the relationship between the informal and formal measurement procedures, they may not understand the principle underlying the use of a ruler.

• Measuring areas and volumes with informal units assists students to understand the calculation formulae when these are taught, providing the principles underlying the informal and formal processes are understood.


Key understandings

• A key understanding of the measurement process is the repetition (or iteration) of units.

• Unit iteration involves knowledge of repeatedly placing identical tightly packing units so that there are no overlaps or gaps.

• For example, accurately aligning units along a length, constructing an array of units to measure the area of a rectangle, or packing a container to determine its volume.


Key understandings

• Another key understanding is that the units must be equal in size.

• As well as recognising the need for units to be equal in size, students need to recognise the relationship between number and units of measure.

• The understanding is that if they use a larger unit to measure they will need a smaller quantity of units.


Unit structure of length, area and volume

Direct alignment

Transitive comparison

Multiple units Indirect comparison

Iterates the unit

Composite unit

Repeated layers

Direct comparison of the size of two objects (alignment).

Direct comparison of the size of three or more objects (transitivity).Indirect comparison by copying the size of one of the objects.

Uses multiple units of the same size to measure an object (without gaps and overlaps).Chooses and uses a selection of the same size and type of units to measure an object (without gaps and overlaps).

States the qualitative relationship between the size and number of units (i.e. with bigger units you need fewer of them).Chooses and uses a selection of the same size and type of units to measure by indirect comparison.

Uses a single unit repeatedly (iterating) to measure or construct length.Can make a multi-unit ruler by iterating a single unit and quantifying accumulated distance.Identifies the quantitative relationship between length and number of units (i.e. if you halve the size of the units you will have twice as many units in the measure).

Creates the row-column structure of the iterated composite unit of area.

Uses the row-column structure to find the number of units to measure area.

Creates the row-column layer structure of the iterated layers when measuring volume.

Uses the row-column-layer structure to find the number of units to measure volume


Direct alignment

• Students initially determine length and area by direct comparison.

• Students directly compare two objects.

• With direct comparison, aligning baselines or edges is important.

Superimposing


Teaching activities

• Longer than or shorter than our string.Students cut a piece of string and move around the room to compare the length of the string to other objects in the room.

• Find a bigger area.Students are given a piece of paper and they have to find three areas that are bigger, smaller and about the same size.

• Packing them in.Students have a container each and pack blocks into the container to determine who can fit the most blocks into their container.


Transitive comparison

• Direct comparison of the size of three or more objects.

• Indirect comparison by copying the size of one of the objects


Teaching activities

• Straws in order.Students are given a number of straws of different lengths to place in order from longest to shortest.

• Find a smaller face.Students are given a small rectangle and find three other objects in the room with a smaller area than their shape by superimposing.

• How will I pack?Pack clear perspex containers and discuss which is better to fill them with to compare the volume.


Multiple units

• Chooses and uses a selection of the same size and type of units to measure an object.

• Students need to discover that the measurement has to be without gaps and overlaps for the measurement to be accurate.


Teaching activities

• Does the tallest person have the longest feet?Students work in groups to measure their foot lengths in units and record in order of length. They also record their heights in order and compare the results.

• Which is best?Pairs of students select a unit and use multiple copies of the unit to cover different shapes.

• How many will fit?Students are given a small number of blocks and asked to work out how many would be needed to fill a rectangular container. Students calculate the total by counting the number


Indirect Comparison

• States the qualitative relationship between the size and the number of units.

• Chooses and uses a selection of the same size and type of units to measure by indirect comparison.


Teaching activities

• Are they the same?Students have two identical lengths of streamers. They paste one onto butchers’ paper, then cut the other one up. Are they still the same length and how can you prove that?

• Make Me!Students make area of a specific size, eg Make a rectangle that has an area of 12 square units. Explain why the different rectangles cover the same area.

• Displacement Students discover which rock has the greatest volume by immersing rocks one-by-one in a container of water and measuring the amount of water displaced.


Iterates the unit

• When the unit is used repeatedly end to end, it is called iterating the unit.

• Learning how spatially organised units fit together, and how they may be counted systematically, is basic to

understanding the measurement

of length, area and volume.


Teaching activities

• Make a rulerStudents make their own rulers based on an informal unit. Students align the units end to end and mark the scale on the ruler.

• Which is bigger?Using a 10cm x 10cm tile, students compare the areas of two rectangles taped onto the floor.

• How many cubes?Students make simple rectangular constructions using a given number of cubes. Other students have to work out how many cubes were used. Discuss the strategies the students used to calculate the volume.


Composite unit

• Creates the row-column structure of the iterated composite unit of area.

• Uses the row-column structure to find the number of units to measure area.

• Calculating area by identifying rows or columns as composite units and adding, skip counting, or multiplying.


Composite unit

• The student is able to create the structure of a rectangular array.


Composite unit

• The student is able to visualise a column and row structure and has moved beyond simple counting of squares along a one-dimensional path. The rows and columns are conceptualised as composite units.


Teaching activities

• Hidden SquaresStudents are shown a square grid with a large rectangular square superimposed on it. Students have to work out how many small units are hidden and explain how they worked it out.

• Large and smallStudents are shown a rectangle covered with large squares and given some small squares (four to a large square). Students work out how many small squares would cover the rectangle and explain why.


Repeated layers

• Creates the row-column layer structure of the iterated layers when measuring volume.

• Uses the row-column-layer structure to find the number of units to measure volume


12

24

36

Repeated layers

• The student is able to create and use the structure of a repeated layers in determining the volume of a rectangular prism.


9 18 27 36

Calculating volume by identifying vertical layers and adding, skip counting, or multiplying the number of layers


Teaching activities

• Layer CountStudents make a rectangular prism with centimetre blocks. Students commence with a base which has 12 blocks. Students record the array pattern in the first layer. Two more layers are added and the total volume calculated. Students look at the number pattern of the layers to predict how many blocks will be needed for five layers.

• Side ViewsStudents build rectangular prisms from interlocking cubes. The students draw plans of 3 faces and then give the plans to another student who build the prism from the drawn plan.

nsw curriculum and learning innovation centre introducing the measurement aspect of the numeracy...

Documents

units of measure

informal units

structure of units

iteration of units

identical units

number of units

nonstandard units

array of units