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# Length Units of Work

Length is a one-dimensional concept related to the geometric concepts of direction and line. Put more simply length is usually thought of as the extent of line. Length measures need to be investigated in the many practical situations in which they occur, for example, as length, width, depth, height, thickness or closeness of objects. Most initial experiences relate to straight lines, but distances along curves or around plane shapes are also relevant. See the links in the "Learning Sequence" column for more detail on the steps in the sequence.

Learning Sequence

Curriculum Achievement Objectives

Specific Learning Outcomes
The students will be able to:

Units of Work

Identifying the attribute

Level 1 Measurement AO1

• Compare lengths from the same starting point
• Use materials to make a long or short construction
• Use materials to compare large and small areas

Worms and more

Comparing and ordering

Level 1 Measurement AO1

• compare the length of two objects
• select objects which are about the same length as a given object
• order three or more objects by length

Time capsule

Non-standard units

Level 1 Measurement AO1

• compare a group of objects by length
• measure length with non-standard units
• use measuring language to compare length, width, and height

Teddy Bears and Friends

Taller, Wider, Longer

Standard units

Link to Learning Objects Level 2

Link to Learning Objects Level 3

Level 2 Measurement AO1

• demonstrate a personal benchmark for 1 metre, 1/2 metre
• identify and use external benchmarks to carry out practical measuring tasks
• discuss the need for having and using standard measures of length
• make sensible estimates about the lengths of given objects

Making Benchmarks

Level 2 Measurement AO1

• recognise the need for a standard unit of length
• recognise a metre length
• estimate and measure to the nearest metre

Pirate Plays

Level 2 Measurement AO1

• carry out practical measuring tasks using appropriate metric units.
• make measurement estimates using appropriate metric units
• pose measurement questions

Make a measurement trail

Level 2 Measurement AO1

• recognise the need for a standard unit of length
• recognise a centimetre length
• estimate and measure to the nearest centimetre

 Level 2 Measurement AO1 Level 2 Measurement AO2
• estimate using metres amd centimetres
• measure to the nearest metre and centimetre

Paper Planes L2

 Level 2 Measurement AO1 Level 2 Measurement AO2
• find objects that they estimate to be a 1cm, 10cm, 50cm and one metre long
• measure lengths of approximately one metre to the nearest cm

Scavenger Hunt

Level 3 Measurement AO1

• make estimates of lengths between approximately 50cm and 1.5m
• measure lengths in metres and centimetres
• convert between metres and centimetres

Stepping Out

Level 3 Measurement AO1

• develop a concept of a km
• measure a distance of 1km and the time taken to cover it

How far is a km?

 Level 3 Measurement AO1 Level 3 Statistical investigations AO1
• measure accurately using metres, centimetres and millimetres
• record data in tables
• use scatter plot graphs to identify relationships between variables

Paper Planes L3

 Level 3 Measurement AO1 Level 3 Number stategies and knowledge AO1
• make appropriate measurements accurately using standard units
• perform addition calculations to find perimeters of objects

Perplexing Perimeters

 Level 3 Measurement AO1 Level 3 Statistical investigations AO1
• measure accurately using centimetres and millimetres
• organise and record data, in tables and graphs
• interpret trends and identify number relationships
• apply mathematical knowledge to practical problem solving

Giant Mystery

Reasoned Measurement

Link to Learning Objects Level 4

Level 3 Measurement AO1

• make estimates of lengths and areas
• measure using a variety of techniques to check their estimates

How can you measure this?

 Level 4 Measurement AO1 Level 4 Measurement AO4
• describe a method to measure the length of circular objects
• measure length using metres, centimeters and millimeters
• calculate the circumference of a circle from a measurement of diameter

Applying and Interpreting

Link to Learning Objects Level 5

 Level 4 Measurement AO2 Level 4 Transformation AO1
• follow instructions, in diagram form, to construct two-dimensional mathematical shapes, e.g. triangles, quadrilaterals, pentagons and hexagons
• enlarge and reduce two-dimensional mathematical shapes by a given scale factor
• identify invariant properties when enlarging and reducing two-dimensional mathematical shapes
• convert between mm and cm measurement.
• multiply whole numberss by a decimal

Team Puzzles

 Level 4 Measurement AO2 Level 4 Measurement AO4 Level 4 Number strategies and knowledge AO1
• calculate speed from measured distance and time
• solve multiplication problems using doubling and halving strategies
• use known multiplication facts to solve multiplication problems

How fast is fast?

 Level 5 Measurement AO1 Level 5 Measurement AO4
• draw a plan to scale, of an object based on a rectangle and two semi-circles
• understand the relation of length on the plan to actual length
• find lengths and areas use these for costing purposes
• be able to link speed, distance and time (given two find the third)
• estimate distances and times
• calculate practical percentages

Round the Track

 Level 5 Transformation AO2 Level 5 Measurement AO3 Level 5 Shape AO1
• apply Pythagoras' theorem
• use their knowledge of the sum of interior angles of a polygon
• construct angles based on halving and combining 90° and other straightforward angles
• apply knowledge of length and area

How High?

 Level 5 Transformation AO2 Level 5 Measurement AO4 Level 5 Number strategies and knowledge AO4
• use scale factors to investigate areas being enlarged
• use scale factors to investigate volumes being enlarged
• solve real life context problems involving scale factorsmeasure the lengths of the sides of sets of similar right angled riangles and find the ratio of sides

Scale Factors for Areas and Volumes

 Level 5 Measurement AO4
• measure the circumference of circles
• state the relationship between the circumference and diameter of a circle

Around we go

### Stage One: Identifying the attribute

Early length experiences must develop an awareness of what length is, and of the range of words that can be used to discuss length. Young students usually begin by describing the size of objects as big and small. They gradually learn to discriminate in what way an object is big or small and use more specific terms. The use of words such as long, short, wide, close, near, far, deep, shallow, high, low and close, focus attention on the attribute of length. The awareness of the attribute of length is extended as comparisons of lengths are made at the next stage.

### Stage Two: Comparing and ordering

Comparison activities are a measurement process in their own right, in that adults often measure in real life without using units. Students will advance from using direct comparison to indirect comparison as a more efficient way to obtain the information that is required. For example, using your own height to give a rough check of the space for a fridge or whether a car will fit into a particular parking space.

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. Early comparison activities should involve two similar objects that differ only on the attribute of length. These objects should initially be compared using the same starting position.

Young students are easily deceived by their eyes into thinking that the length of an object can be altered when the object is moved.

The student may see that these two lines are the same length when their starting points are together. However, they may think that the second line is longer if its starting point is moved to the right as in the diagram. If the student thinks that both are the same length, they are said to conserve the property of length, because they are able to reason that neither length has been added to or reduced by the movement.

When students can compare two lengths they should be given the opportunity to order three or more lengths. The process of ordering three or more lengths is not a simple extension as it involves thinking that if A is longer than B and B is longer than C then A is longer than C.

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 Three: Non-Standard units

From the earliest of these experiences, students should be encouraged to estimate. Initially these estimations 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 estimates at least as frequently as exact measures, the skill of estimation is important.

At this stage students can also be introduced to the appropriateness of units of measure. For example, a pace is more appropriate that a hand span? for measuring the length of a room. In addition to using their own bodies, there are many other readily available objects that can be used, for example, linked cubes, rods, pencils, paper clips, straws and sticks.

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 a single object using non-standard units, for example, hand spans.

### Stage Four: Standard units

Students’ measurement experiences must enable them to:

1. develop an understanding of the size of the standard unit;
2. estimate and measure using the unit.

The usual sequence used in primary school is to introduce the centimetre first, then the metre, followed later by the kilometre and the millimetre.

The centimetre is often introduced first because it is small enough to measure common objects. The size of the centimetre unit can be established by constructing it, for example by cutting 1-centimetre pieces of paper or straws. Most primary classrooms also have a supply of 1-cm cubes that can be used to measure objects. An appreciation of the size of the unit can be built up through lots of experience in measuring everyday objects. The students should be encouraged to develop their own reference for a centimetre, for example, a fingertip.

As the students become familiar with the size of the centimetre they should be given many opportunities to estimate before measuring. After using centimetre units to measure objects the students can be introduced to the centimetre ruler. It is a good idea to let the students develop their own ruler to begin with. For example, some classrooms have linked cubes which can be joined to form 10 cm rulers. Alternatively pieces of drinking straw could be threaded together.

The correct use of a ruler to measure objects requires specific instruction. The correct alignment of the zero on the ruler with one end of the object needs to be clarified.

Metres and millimetres are established using a similar sequence of experiences: first construct the unit and then use it to measure appropriate objects.

The kilometre may be established by marking out the distance on the school playground and having the students walk the distance. (This may require the students making several laps of the school grounds.)

### Stage Five: Reasoned measurement

When students become proficient with the use of standard units they will come to realise that there are often more efficient ways to obtain the required measurement information. For example, to check the height of a bookshelf to see if a book will fit on it, it is not necessary to use a ruler, handspans as an indirect comparison or a direct comparison using the book itself is more reasonable. Using logic to choose the best way to carry out a measurement task is an important skill to learn.

Reasoned measurement also includes realising how accurate a measurement needs to be. For example, there is no need to measure the width of the classroom to the nearest millimeter, but the thickness of a book in meters isn't much use either.

### Applying and interpreting

When the students are able to measure efficiently and effectively using standard units, their leaning experiences can be directed to situations that encourage them to "discover" measurement formulae?. For example, they can discover the perimeters of common polygons and circles. The perimeter formula for squares and rectangles can be readily found using squared paper.

Perimeter of square = 4 x length. Perimeter of rectangle = 2 x (length + width). (Note that the perimeter formulae for squares and rectangles are exactly the same!)

Rolling circular objects such as wheels or using string to measure the circumference and then comparing this with the length of the diameter and radius, leads to the discovery of the formula that links these measures.

Circumference of a circle = 2 x Pi x radius.

When students can measure lengths 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 length. They can then appreciate that consistency in the units used would allow for the easier and more accurate communication of length measures.

Measuring the length of objects using non-standard or informal units is the third stage in the learning sequence. Beginning with non-standard but familiar units allows the students to focus on the process of repeatedly using a unit as a measuring device. Parts of the body provide interesting units for introductory use. Students should therefore be given lots of opportunities to measure a wide range of objects using their own body parts as appropriate – finger width, hand span), foot, pace, arm span. (Later on parts of the body provide the opportunity to introduce discussion of the history of measurement, for example, foot from human feet lengths, the yard from pace lengths.

Comparing the lengths of objects is the second stage in developing an understanding of length. This is first done by physically placing the objects side by side and making a comparison, for example, the red pen is longer than the yellow one.

Early length experiences must develop an awareness of what length is, and of the range of words that can be used to discuss length. Young students usually begin by describing the size of objects as big and small. They gradually learn to discriminate in what way an object is big or small and use more specific terms. The use of words such as long, short, wide, close, near, far, deep, shallow, high, low and close, focus attention on the attribute of length. The awareness of the attribute of length is extended as comparisons of lengths are made at the next stage.

Early length experiences must develop an awareness of what length is, and of the range of words that can be used to discuss length. Young students usually begin by describing the size of objects as big and small. They gradually learn to discriminate in what way an object is big or small and use more specific terms. The use of words such as long, short, wide, close, near, far, deep, shallow, high, low and close, focus attention on the attribute of length. The awareness of the attribute of length is extended as comparisons of lengths are made at the next stage.

Early length experiences must develop an awareness of what length is, and of the range of words that can be used to discuss length. Young students usually begin by describing the size of objects as big and small. They gradually learn to discriminate in what way an object is big or small and use more specific terms. The use of words such as long, short, wide, close, near, far, deep, shallow, high, low and close, focus attention on the attribute of length. The awareness of the attribute of length is extended as comparisons of lengths are made at the next stage.