Teaching measurement:

Early Stage 1 and Stage 1

Stage 2 and Stage 3

An introductory workshop

Understanding measurement Length

• When measuring length, area and volume, the units of measure are aligned spatially.

• Most adults can confidently estimate and measure length, distance or height and are familiar with the units used to measure length - millimetres, centimetres, metres and kilometres.

Understanding measurementArea

• However, fewer adults can confidently estimate area.

• Many learnt to calculate area by using the

L x B formula, without understanding what the formula does.

• Students often apply the area and perimeter formulae wrongly or invent their own formulae.

• The use of units cm2, m2 and hectares is frequently confused.

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

Understanding measurementVolume

• And even fewer adults are confident of how to estimate or calculate volume.

• Students or adults who don’t understand how the area formula works, will have difficulty in understanding how to calculate volume.

Understanding measurementVolume

• Volume can be modelled as repeated layers. An understanding of how one layer is constructed (as in area) is necessary for the confident use of the formula L X B X H.

• Sometimes the formula is applied with no understanding of what the units of measure will be.

12

24

36

Calculating volume by identifying horizontal layers and adding, skip counting, or multiplying.

9 18 27 36

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

Understanding measurement Capacity, Mass

• In measurement of capacity or mass, the spatial arrangement of the units is not important.

• Finding the capacity of a container by filling with liquid or sand or rice is different from packing with blocks, which must be arranged spatially.

• When finding the mass of a quantity by balancing with blocks or weights, the arrangement of the mass does not matter.

Understanding measurement Capacity, Mass

•The measurement framework generally refers to capacity as liquid measure and volume as cubic measure.

•Within the framework, mass is slightly different from the other 3 sub-strands:

– Level 1.3 is an additional sub-level, to introduce the concept of an equal-arm balance

– There is no sub-level 3.1, as Structure of repeated units does not apply to mass.

Teaching measurement:Early Stage 1 and Stage 1Stage 2 and Stage 3

The teaching resources contain -

the measurement framework

lesson ideas

lesson plans

for the measurement sub-strands of length, area, volume and capacity and mass.

Teaching measurement:Early Stage 1 and Stage 1Stage 2 and Stage 3

• The measurement framework provides the fine detail of how students’ concepts develop.

• The framework focuses on an understanding of quantities, terminology used to describe measurements, the use of units to measure and the pattern created by repeated, tessellating units.

The measurement framework Levels 1and 2

Level 1Identification of the attribute

1.1 Make direct comparisons of quantities1.2 Order two or more quantities by direct comparison

Level 2Informal measurement

2.1 Choose and use appropriate units for measuring a quantity

2.2 Compare and order quantities by using identical units

The measurement framework Levels 3 and 4

Level 3

Structure of repeated units3.1 Use one unit to work out how many will be needed

altogether when making indirect comparisons.3.2 Explain the relationship between unit size and the

number of units used to measure a quantity.

Level 4

Measure using conventional units4.1 Measure and record 1 m, 1 m2, 1 L, 1 kg.4.2 Measure and record in cm, cm2, cm3, grams.

The measurement framework Levels 5 and 6

Level 5Relationships between formal measurement units

5.1 Measure quantities using standard units5.2 Measure and calculate quantities using standard units.

Level 6Knowing and representing large units

6.1 Calculate large quantities in standard units. Interpret a simple scale

6.2 Convert units to calculate and compare quantities. Use a simple scale.

The measurement frameworkLevel 1

The measurement frameworkLevel 2

Level Length Area Volume andCapacity

Mass

Informal Measurement

2.1 Choose and useappropriate unitsfor measuringlength

Choose and useappropriate unitsfor measuring area.

Choose and useappropriate unitsfor measuringvolume andcapacity

Choose appropriateunits and use them tomeasure a mass

2.2 Compare and orderlengths by usingidentical units foreach length

Compare and orderareas by coveringeach area withidentical units

Compare and ordervolumes andcapacities by fillingor packing withidentical units

Compare and ordermasses usingidentical units foreach mass

The measurement frameworkLevel 3

Level Length Area Volume andCapacity

Mass

Structure of repeated unitsRelationshipbetween units

3.1 Use one unit towork out how manywill be neededaltogether whenmaking indirectcomparisons

Use one unit towork out how manywill be neededaltogether whenmaking indirectcomparisons

Use one unit orcomposite unit towork out how manywill be neededaltogether whenmaking indirectcomparisons

3.2 Explain therelationshipbetween unit sizeand the number ofunits used tomeasure length

Explain therelationshipbetween unit sizeand the number ofunits used tomeasure area

Explain therelationshipbetween unit sizeand the number ofunits required to fillor pack a container

Explain therelationship betweenunit size and thenumber of unitsrequired to balance amass

The measurement frameworkLevel 4

Level Length Area Volume and Capacity

Mass

Measure using conventional units

4.1 Measure and record 1 m

Measure 1 m2

Measure and record 1 L

Measure and record 1 kg

4.2 Measure and record in cm

Measure and record area in m2

or cm2 using the structure of repeated units

Measure and record volume in cm3 by packing

Measure quantities less than 1 kg, in grams

The measurement frameworkLevel 5

Level Length Area Volume and Capacity

Mass

Relationships between formal measurement units

5.1 Measure lengths and perimeters in m and cm

Measure and record area in square cm and square metres

Measure and record capacity in L and mL

Measure and record mass in kg and gm

5.2 Measure and calculate lengths and perimeters in m, cm and mm

Measure and calculate area in square cm and square m

Measure and calculate volume in cubic cm

Measure and calculate mass in kg and gm

The measurement frameworkLevel 6

Level Length Area Volume and Capacity

Mass

Knowing and representing large units

6.1 Calculate lengths, distances and perimeters in kilometers

Interpret a simple scale

Calculate area in square km and hectares

Interpret a simple scale

Calculate volume in cubic metres

Calculate mass in tonnes

6.2 Convert units of length to calculate and compare lengths, distances and perimeters

Use a simple scale

Convert units of area to calculate and compare areas

Use a scale

Convert units of volume and capacity to calculate and compare quantities

Convert units of mass to calculate and compare quantities

Teaching measurement:Early Stage 1 and Stage 1Stage 2 and Stage 3

• The knowledge and strategies are listed for each level, in each substrand.

• These are the skills, understandings, and mathematical terminology which students use in demonstrating achievement of that level.

Teaching measurementEarly Stage 1 and Stage 1

Level Knowledge and strategies

L1.1 Identification of the attribute Make direct comparisons of mass

1. use mass vocabulary such as light, heavy, hard to push or lift

2. identify materials that are light and heavy

3. identify objects that have about the same mass

4. heft or push objects and decide which is heavier or lighter

L1.2 Identification of the attribute

Compare and order objects by hefting

1. use comparative language, heavier than, lighter than, heaviest, lightest

2. predict which object would be heavier than, lighter than, about the same and explain reasoning

3. compare masses systematically and explain why a mass fits into a particular ordering

L1.3 Identification of the attribute

Compare masses using an equal-arm balance

1. know that an identical object on each side will balance

2. know that the side that goes down has the greater mass

3. demonstrate how to use an equal-arm balance to compare three or more objects and report the results

Teaching measurementEarly Stage 1 and Stage 1

Level Knowledge and strategies

L2.1 Informal measurement

Choose and use appropriate units for measuring length

1. align identical units end to end along a given line without overlapping or leaving gaps

2. state or record that the length is the number and type of units used

3. use approximate language to explain parts of units, e.g. about half a unit

4. measure a circumference using string or paper strips, without overlapping ends

L2.2 Informal measurement

Compare and order lengths by using identical units for each length

1. choose identical units to measure lengths

2. know that the longer line has more units

3. estimate the number of units and explain the estimation strategy

4. know that length is conserved if rearranged

Teaching measurementEarly Stage 1 and Stage 1

Level Knowledge and strategies

L4.1 Measure using conventional units

Measure 1 square metre

1. identify areas which are approximately 1 square metre

2. use an array structure to calculate how many tiles of a given size will be needed to cover an area of 1 square metre

L4.2 Measure using conventional units

Measure and record area in square metres or square centimetres using the structure of repeated units

1. use the square metre as a unit to measure area

2. use the square centimetre as a unit to measure

3. record area using the abbreviations m2 and cm2

Teaching measurementEarly Stage 1 and Stage 1

Level Knowledge and skills

5.1 Relationships between formal measurement units

Measure and record capacity in litres and millilitres

1. measure and record capacities in millilitres and litres

2. record capacities using decimal notation to three decimal places

3. convert between mL and litres, e.g. 1600 mL = 1 litre 600 millilitres

4. Use overflow or change in level as a measure of volume to the nearest 100 mL

5.2 Relationships between formal measurement units

Measure and calculate volume in cubic centimetres

1. estimate then calculate the volume of a rectangular prism in cubic centimetres, given the volume of one layer and the number of layers

2. explain the relationship between 1 cubic centimetre and 1 millilitre (using displacement)

Teaching measurement:Early Stage 1 and Stage 1Stage 2 and Stage 3

The introductory section of the book contains important information about:• Fundamental measurement ideas

– Identification of the attribute being measured– Knowledge of units is fundamental to the process of

measuring– The principle of conservation is fundamental to

understanding measurement– Knowledge of unit iteration is fundamental to the

process of measuring spatially organised quantities.

Teaching measurement:Early Stage 1 and Stage 1Stage 2 and Stage 3

The introductory section of the book contains important information about: • Components of teaching measurement

– Estimation– Recording– Questioning

• The measurement framework• Before you begin• Glossary

Group activity Read, summarise, share and discuss

• Work in your table group.• Each teacher chooses and reads an

introductory page from length, area, volume and capacity, or mass. Highlight the important points.

• Share your readings.

Discuss: – What issues have been raised?– What are the implications for teaching measurement,

K-6?

Lesson ideas

• Five lesson ideas have been listed for each sub-level.

• Each lesson idea includes:

syllabus references

materials

knowledge and strategies.

• The lesson ideas in each sub-level are in approximate order.

Lesson ideas (Area 3.1)

Lesson plans

• One lesson idea in each sub-level has been expanded to a full lesson plan.

• All of the lessons have the same basic structure of: introduction, followed by practical activity and then whole-class discussion.

• The suggested questions provide samples of open-ended questioning.

• Teachers are strongly encouraged to modify or adapt the lessons to suit their students.

Lesson plans

Where should I start?

• Refer to the lists of knowledge and strategies

• Check the references to syllabus Stages on lessons

• Try a lesson, observe students’ knowledge, and then plan further lessons. You may need to go forwards or backwards in the framework.

Where should I start?

• It is not necessary to teach all of the lessons in each sub-level.

• Teach a block of 4 or 5 lessons before moving on to another substrand.

• Remember the sequence of concept development: length, then area, then volume and capacity.

Teaching measurement:Early Stage 1 and Stage 1Stage 2 and Stage 3

An emphasis on:• developing an understanding of the attributes of

length, area, volume and capacity, mass• using mathematical terminology to discuss and

record measurements• using knowledge of the structure of repeated

units to measure• developing and practising measurement skills• using standard units confidently, including

converting units.

• When measuring, ensure the students have an understanding of what they are actually measuring.

• Do not make assumptions that they are thinking the same as you.

Son to father:“Dad, is a Ferrari a red car with a small horse?”

Father:“That’s right, but why do you ask?”

Son:“I think there’s one trying to pass us on the right...”